Simulated annealing
Quoted from the Wikipedia page : Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Simulated annealing interprets slow cooling as a slow decrease in the probability of temporarily accepting worse solutions as it explores the solution space.
Pseudo code from Wikipedia
Notations : T : temperature. Decreases to 0. s : a system state E(s) : Energy at s. The function we want to minimize ∆E : variation of E, from state s to state s_next P(∆E , T) : Probability to move from s to s_next. if ( ∆E < 0 ) P = 1 else P = exp ( - ∆E / T) . Decreases as T → 0 Pseudo-code: Let s = s0 -- initial state For k = 0 through kmax (exclusive): T ← temperature(k , kmax) Pick a random neighbour state , s_next ← neighbour(s) ∆E ← E(s) - E(s_next) If P(∆E , T) ≥ random(0, 1), move to the new state: s ← s_next Output: the final state s
Problem statement
We want to apply SA to the travelling salesman problem. There are 100 cities, numbered 0 to 99, located on a plane, at integer coordinates i,j : 0 <= i,j < 10 . The city at (i,j) has number 10*i + j. The cities are all connected : the graph is complete : you can go from one city to any other city in one step.
The salesman wants to start from city 0, visit all cities, each one time, and go back to city 0. The travel cost between two cities is the euclidian distance between there cities. The total travel cost is the total path length.
A path s is a sequence (0 a b ...z 0) where (a b ..z) is a permutation of the numbers (1 2 .. 99). The path length = E(s) is the sum d(0,a) + d(a,b) + ... + d(z,0) , where d(u,v) is the distance between two cities. Naturally, we want to minimize E(s).
Definition : The neighbours of a city are the closest cities at distance 1 horizontally/vertically, or √2 diagonally. A corner city (0,9,90,99) has 3 neighbours. A center city has 8 neighbours.
Distances between cities d ( 0, 7) → 7 d ( 0, 99) → 12.7279 d ( 23, 78) → 7.0711 d ( 33, 44) → 1.4142 // sqrt(2)
Task
Apply SA to the travelling salesman problem, using the following set of parameters/functions :
- kT = 1 (Multiplication by kT is a placeholder, representing computing temperature as a function of 1-k/kmax):
- temperature (k, kmax) = kT * (1 - k/kmax)
- neighbour (s) : Pick a random city u > 0 . Pick a random neighbour city v > 0 of u , among u's 8 (max) neighbours on the grid. Swap u and v in s . This gives the new state s_next.
- kmax = 1000_000
- s0 = a random permutation
For k = 0 to kmax by step kmax/10 , display k, T, E(s). Display the final state s_final, and E(s_final).
You will see that the Energy may grow to a local optimum, before decreasing to a global optimum.
Illustrated example Temperature charts
Numerical example
kT = 1 E(s0) = 529.9158 k: 0 T: 1 Es: 529.9158 k: 100000 T: 0.9 Es: 201.1726 k: 200000 T: 0.8 Es: 178.1723 k: 300000 T: 0.7 Es: 154.7069 k: 400000 T: 0.6 Es: 158.1412 <== local optimum k: 500000 T: 0.5 Es: 133.856 k: 600000 T: 0.4 Es: 129.5684 k: 700000 T: 0.3 Es: 112.6919 k: 800000 T: 0.2 Es: 105.799 k: 900000 T: 0.1 Es: 102.8284 k: 1000000 T: 0 Es: 102.2426 E(s_final) = 102.2426 Path s_final = ( 0 10 11 21 31 20 30 40 50 60 70 80 90 91 81 71 73 83 84 74 64 54 55 65 75 76 66 67 77 78 68 58 48 47 57 56 46 36 37 27 26 16 15 5 6 7 17 18 8 9 19 29 28 38 39 49 59 69 79 89 99 98 88 87 97 96 86 85 95 94 93 92 82 72 62 61 51 41 42 52 63 53 43 32 22 12 13 23 33 34 44 45 35 25 24 14 4 3 2 1 0)
Extra credit
Tune the parameters kT, kmax, or use different temperature() and/or neighbour() functions to demonstrate a quicker convergence, or a better optimum.
Ada
This implementation is adapted from the C, which was adapted from the Scheme. It uses fixed-point numbers for no better reason than to demonstrate that Ada has fixed-point numbers support built in.
----------------------------------------------------------------------
--
-- The Rosetta Code simulated annealing task in Ada.
--
-- This implementation demonstrates that Ada has fixed-point numbers
-- support built in. Otherwise there is no particular reason I used
-- fixed-point instead of floating-point numbers.
--
-- (Actually, for the square root and exponential, I cheat and use the
-- floating-point functions.)
--
----------------------------------------------------------------------
with Ada.Numerics.Discrete_Random;
with Ada.Numerics.Elementary_Functions;
with Ada.Text_IO; use Ada.Text_IO;
procedure simanneal
is
Bigint : constant := 1_000_000_000;
Bigfpt : constant := 1_000_000_000.0;
-- Fixed point numbers.
type Fixed_Point is delta 0.000_01 range 0.0 .. Bigfpt;
-- Integers.
subtype Zero_or_One is Integer range 0 .. 1;
subtype Coordinate is Integer range 0 .. 9;
subtype City_Location is Integer range 0 .. 99;
subtype Nonzero_City_Location is City_Location range 1 .. 99;
subtype Path_Index is City_Location;
subtype Nonzero_Path_Index is Nonzero_City_Location;
-- Arrays.
type Path_Vector is array (Path_Index) of City_Location;
type Neighborhood_Array is array (1 .. 8) of City_Location;
-- Random numbers.
subtype Random_Number is Integer range 0 .. Bigint - 1;
package Random_Numbers is new Ada.Numerics.Discrete_Random
(Random_Number);
use Random_Numbers;
gen : Generator;
function Randnum
return Fixed_Point
is
begin
return (Fixed_Point (Random (gen)) / Fixed_Point (Bigfpt));
end Randnum;
function Random_Natural
(imin : Natural;
imax : Natural)
return Natural
is
begin
-- There may be a tiny bias in the result, due to imax-imin+1 not
-- being a divisor of Bigint. The algorithm should work, anyway.
return imin + (Random (gen) rem (imax - imin + 1));
end Random_Natural;
function Random_City_Location
(minloc : City_Location;
maxloc : City_Location)
return City_Location
is
begin
return City_Location (Random_Natural (minloc, maxloc));
end Random_City_Location;
function Random_Path_Index
(imin : Path_Index;
imax : Path_Index)
return Path_Index
is
begin
return Random_City_Location (imin, imax);
end Random_Path_Index;
package Natural_IO is new Ada.Text_IO.Integer_IO (Natural);
package City_Location_IO is new Ada.Text_IO.Integer_IO
(City_Location);
package Fixed_Point_IO is new Ada.Text_IO.Fixed_IO (Fixed_Point);
function sqrt
(x : Fixed_Point)
return Fixed_Point
is
begin
-- Cheat by using the floating-point routine. It is an exercise
-- for the reader to write a true fixed-point function.
return
Fixed_Point (Ada.Numerics.Elementary_Functions.Sqrt (Float (x)));
end sqrt;
function expneg
(x : Fixed_Point)
return Fixed_Point
is
begin
-- Cheat by using the floating-point routine. It is an exercise
-- for the reader to write a true fixed-point function.
return
Fixed_Point (Ada.Numerics.Elementary_Functions.Exp (-Float (x)));
end expneg;
function i_Coord
(loc : City_Location)
return Coordinate
is
begin
return loc / 10;
end i_Coord;
function j_Coord
(loc : City_Location)
return Coordinate
is
begin
return loc rem 10;
end j_Coord;
function Location
(i : Coordinate;
j : Coordinate)
return City_Location
is
begin
return (10 * i) + j;
end Location;
function distance
(loc1 : City_Location;
loc2 : City_Location)
return Fixed_Point
is
i1, j1 : Coordinate;
i2, j2 : Coordinate;
di, dj : Coordinate;
begin
i1 := i_Coord (loc1);
j1 := j_Coord (loc1);
i2 := i_Coord (loc2);
j2 := j_Coord (loc2);
di := (if i1 < i2 then i2 - i1 else i1 - i2);
dj := (if j1 < j2 then j2 - j1 else j1 - j2);
return sqrt (Fixed_Point ((di * di) + (dj * dj)));
end distance;
procedure Randomize_Path_Vector
(path : out Path_Vector)
is
j : Nonzero_Path_Index;
xi, xj : Nonzero_City_Location;
begin
for i in 0 .. 99 loop
path (i) := i;
end loop;
-- Do a Fisher-Yates shuffle of elements 1 .. 99.
for i in 1 .. 98 loop
j := Random_Path_Index (i + 1, 99);
xi := path (i);
xj := path (j);
path (i) := xj;
path (j) := xi;
end loop;
end Randomize_Path_Vector;
function Path_Length
(path : Path_Vector)
return Fixed_Point
is
len : Fixed_Point;
begin
len := distance (path (0), path (99));
for i in 0 .. 98 loop
len := len + distance (path (i), path (i + 1));
end loop;
return len;
end Path_Length;
-- Switch the index of s to switch which s is current and which is
-- the trial vector.
s : array (0 .. 1) of Path_Vector;
Current : Zero_or_One;
function Trial
return Zero_or_One
is
begin
return 1 - Current;
end Trial;
procedure Accept_Trial
is
begin
Current := 1 - Current;
end Accept_Trial;
procedure Find_Neighbors
(loc : City_Location;
neighbors : out Neighborhood_Array;
num_neighbors : out Integer)
is
i, j : Coordinate;
c1, c2, c3, c4, c5, c6, c7, c8 : City_Location := 0;
procedure Add_Neighbor
(neighbor : City_Location)
is
begin
if neighbor /= 0 then
num_neighbors := num_neighbors + 1;
neighbors (num_neighbors) := neighbor;
end if;
end Add_Neighbor;
begin
i := i_Coord (loc);
j := j_Coord (loc);
if i < 9 then
c1 := Location (i + 1, j);
if j < 9 then
c2 := Location (i + 1, j + 1);
end if;
if 0 < j then
c3 := Location (i + 1, j - 1);
end if;
end if;
if 0 < i then
c4 := Location (i - 1, j);
if j < 9 then
c5 := Location (i - 1, j + 1);
end if;
if 0 < j then
c6 := Location (i - 1, j - 1);
end if;
end if;
if j < 9 then
c7 := Location (i, j + 1);
end if;
if 0 < j then
c8 := Location (i, j - 1);
end if;
num_neighbors := 0;
Add_Neighbor (c1);
Add_Neighbor (c2);
Add_Neighbor (c3);
Add_Neighbor (c4);
Add_Neighbor (c5);
Add_Neighbor (c6);
Add_Neighbor (c7);
Add_Neighbor (c8);
end Find_Neighbors;
procedure Make_Neighbor_Path
is
u, v : City_Location;
neighbors : Neighborhood_Array;
num_neighbors : Integer;
j, iu, iv : Path_Index;
begin
for i in 0 .. 99 loop
s (Trial) := s (Current);
end loop;
u := Random_City_Location (1, 99);
Find_Neighbors (u, neighbors, num_neighbors);
v := neighbors (Random_Natural (1, num_neighbors));
j := 0;
iu := 0;
iv := 0;
while iu = 0 or iv = 0 loop
if s (Trial) (j + 1) = u then
iu := j + 1;
elsif s (Trial) (j + 1) = v then
iv := j + 1;
end if;
j := j + 1;
end loop;
s (Trial) (iu) := v;
s (Trial) (iv) := u;
end Make_Neighbor_Path;
function Temperature
(kT : Fixed_Point;
kmax : Natural;
k : Natural)
return Fixed_Point
is
begin
return
kT * (Fixed_Point (1) - (Fixed_Point (k) / Fixed_Point (kmax)));
end Temperature;
function Probability
(delta_E : Fixed_Point;
T : Fixed_Point)
return Fixed_Point
is
prob : Fixed_Point;
begin
if T = Fixed_Point (0.0) then
prob := Fixed_Point (0.0);
else
prob := expneg (delta_E / T);
end if;
return prob;
end Probability;
procedure Show
(k : Natural;
T : Fixed_Point;
E : Fixed_Point)
is
begin
Put (" ");
Natural_IO.Put (k, Width => 7);
Put (" ");
Fixed_Point_IO.Put (T, Fore => 5, Aft => 1);
Put (" ");
Fixed_Point_IO.Put (E, Fore => 7, Aft => 2);
Put_Line ("");
end Show;
procedure Display_Path
(path : Path_Vector)
is
begin
for i in 0 .. 99 loop
City_Location_IO.Put (path (i), Width => 2);
Put (" ->");
if i rem 8 = 7 then
Put_Line ("");
else
Put (" ");
end if;
end loop;
City_Location_IO.Put (path (0), Width => 2);
end Display_Path;
procedure Simulate_Annealing
(kT : Fixed_Point;
kmax : Natural)
is
kshow : Natural := kmax / 10;
E : Fixed_Point;
E_trial : Fixed_Point;
T : Fixed_Point;
P : Fixed_Point;
begin
E := Path_Length (s (Current));
for k in 0 .. kmax loop
T := Temperature (kT, kmax, k);
if k rem kshow = 0 then
Show (k, T, E);
end if;
Make_Neighbor_Path;
E_trial := Path_Length (s (Trial));
if E_trial <= E then
Accept_Trial;
E := E_trial;
else
P := Probability (E_trial - E, T);
if P = Fixed_Point (1) or else Randnum <= P then
Accept_Trial;
E := E_trial;
end if;
end if;
end loop;
end Simulate_Annealing;
kT : constant := Fixed_Point (1.0);
kmax : constant := 1_000_000;
begin
Reset (gen);
Current := 0;
Randomize_Path_Vector (s (Current));
Put_Line ("");
Put (" kT:");
Put_Line (Fixed_Point'Image (kT));
Put (" kmax:");
Put_Line (Natural'Image (kmax));
Put_Line ("");
Put_Line (" k T E(s)");
Simulate_Annealing (kT, kmax);
Put_Line ("");
Put_Line ("Final path:");
Display_Path (s (Current));
Put_Line ("");
Put_Line ("");
Put ("Final E(s): ");
Fixed_Point_IO.Put (Path_Length (s (Current)), Fore => 3, Aft => 2);
Put_Line ("");
Put_Line ("");
end simanneal;
----------------------------------------------------------------------
- Output:
An example run. In the following, you could use gnatmake instead of gprbuild.
$ gprbuild -q simanneal.adb && ./simanneal kT: 1.00000 kmax: 1000000 k T E(s) 0 1.0 525.23 100000 0.9 189.97 200000 0.8 180.33 300000 0.7 153.31 400000 0.6 156.18 500000 0.5 136.17 600000 0.4 119.56 700000 0.3 110.51 800000 0.2 106.21 900000 0.1 102.89 1000000 0.0 102.89 Final path: 0 -> 10 -> 11 -> 21 -> 20 -> 30 -> 31 -> 32 -> 22 -> 23 -> 33 -> 43 -> 42 -> 52 -> 51 -> 41 -> 40 -> 50 -> 60 -> 70 -> 80 -> 90 -> 91 -> 92 -> 93 -> 84 -> 94 -> 95 -> 85 -> 86 -> 96 -> 97 -> 98 -> 99 -> 89 -> 88 -> 87 -> 77 -> 67 -> 57 -> 58 -> 68 -> 78 -> 79 -> 69 -> 59 -> 49 -> 39 -> 29 -> 19 -> 9 -> 8 -> 7 -> 6 -> 25 -> 24 -> 34 -> 35 -> 44 -> 54 -> 53 -> 63 -> 62 -> 61 -> 71 -> 81 -> 72 -> 82 -> 83 -> 73 -> 74 -> 64 -> 65 -> 75 -> 76 -> 66 -> 56 -> 55 -> 45 -> 46 -> 47 -> 48 -> 38 -> 37 -> 36 -> 26 -> 27 -> 28 -> 18 -> 17 -> 16 -> 15 -> 5 -> 4 -> 14 -> 3 -> 13 -> 12 -> 2 -> 1 -> 0 Final E(s): 102.89
C
For your platform you might have to modify parts, such as the call to getentropy(3).
You can easily change the kind of floating point used. I apologize for false precision in printouts using the default single precision floating point.
Some might notice the calculations of random integers are done in a way that may introduce a bias, which is miniscule as long as the integer is much smaller than 2 to the 31st power. I mention this now so no one will complain about it later.
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <string.h>
#include <unistd.h>
#define VECSZ 100
#define STATESZ 64
typedef float floating_pt;
#define EXP expf
#define SQRT sqrtf
static floating_pt
randnum (void)
{
return (floating_pt)
((double) (random () & 2147483647) / 2147483648.0);
}
static void
shuffle (uint8_t vec[], size_t i, size_t n)
{
/* A Fisher-Yates shuffle of n elements of vec, starting at index
i. */
for (size_t j = 0; j != n; j += 1)
{
size_t k = i + j + (random () % (n - j));
uint8_t xi = vec[i];
uint8_t xk = vec[k];
vec[i] = xk;
vec[k] = xi;
}
}
static void
init_s (uint8_t vec[VECSZ])
{
for (uint8_t j = 0; j != VECSZ; j += 1)
vec[j] = j;
shuffle (vec, 1, VECSZ - 1);
}
static inline void
add_neighbor (uint8_t neigh[8],
unsigned int *neigh_size,
uint8_t neighbor)
{
if (neighbor != 0)
{
neigh[*neigh_size] = neighbor;
*neigh_size += 1;
}
}
static void
neighborhood (uint8_t neigh[8],
unsigned int *neigh_size,
uint8_t city)
{
/* Find all non-zero neighbor cities. */
const uint8_t i = city / 10;
const uint8_t j = city % 10;
uint8_t c0 = 0;
uint8_t c1 = 0;
uint8_t c2 = 0;
uint8_t c3 = 0;
uint8_t c4 = 0;
uint8_t c5 = 0;
uint8_t c6 = 0;
uint8_t c7 = 0;
if (i < 9)
{
c0 = (10 * (i + 1)) + j;
if (j < 9)
c1 = (10 * (i + 1)) + (j + 1);
if (0 < j)
c2 = (10 * (i + 1)) + (j - 1);
}
if (0 < i)
{
c3 = (10 * (i - 1)) + j;
if (j < 9)
c4 = (10 * (i - 1)) + (j + 1);
if (0 < j)
c5 = (10 * (i - 1)) + (j - 1);
}
if (j < 9)
c6 = (10 * i) + (j + 1);
if (0 < j)
c7 = (10 * i) + (j - 1);
*neigh_size = 0;
add_neighbor (neigh, neigh_size, c0);
add_neighbor (neigh, neigh_size, c1);
add_neighbor (neigh, neigh_size, c2);
add_neighbor (neigh, neigh_size, c3);
add_neighbor (neigh, neigh_size, c4);
add_neighbor (neigh, neigh_size, c5);
add_neighbor (neigh, neigh_size, c6);
add_neighbor (neigh, neigh_size, c7);
}
static floating_pt
distance (uint8_t m, uint8_t n)
{
const uint8_t im = m / 10;
const uint8_t jm = m % 10;
const uint8_t in = n / 10;
const uint8_t jn = n % 10;
const int di = (int) im - (int) in;
const int dj = (int) jm - (int) jn;
return SQRT ((di * di) + (dj * dj));
}
static floating_pt
path_length (uint8_t vec[VECSZ])
{
floating_pt len = distance (vec[0], vec[VECSZ - 1]);
for (size_t j = 0; j != VECSZ - 1; j += 1)
len += distance (vec[j], vec[j + 1]);
return len;
}
static void
swap_s_elements (uint8_t vec[], uint8_t u, uint8_t v)
{
size_t j = 1;
size_t iu = 0;
size_t iv = 0;
while (iu == 0 || iv == 0)
{
if (vec[j] == u)
iu = j;
else if (vec[j] == v)
iv = j;
j += 1;
}
vec[iu] = v;
vec[iv] = u;
}
static void
update_s (uint8_t vec[])
{
const uint8_t u = 1 + (random () % (VECSZ - 1));
uint8_t neighbors[8];
unsigned int num_neighbors;
neighborhood (neighbors, &num_neighbors, u);
const uint8_t v = neighbors[random () % num_neighbors];
swap_s_elements (vec, u, v);
}
static inline void
copy_s (uint8_t dst[VECSZ], uint8_t src[VECSZ])
{
memcpy (dst, src, VECSZ * (sizeof src[0]));
}
static void
trial_s (uint8_t trial[VECSZ], uint8_t vec[VECSZ])
{
copy_s (trial, vec);
update_s (trial);
}
static floating_pt
temperature (floating_pt kT, unsigned int kmax, unsigned int k)
{
return kT * (1 - ((floating_pt) k / (floating_pt) kmax));
}
static floating_pt
probability (floating_pt delta_E, floating_pt T)
{
floating_pt prob;
if (delta_E < 0)
prob = 1;
else if (T == 0)
prob = 0;
else
prob = EXP (-(delta_E / T));
return prob;
}
static void
show (unsigned int k, floating_pt T, floating_pt E)
{
printf (" %7u %7.1f %13.5f\n", k, (double) T, (double) E);
}
static void
simulate_annealing (floating_pt kT,
unsigned int kmax,
uint8_t s[VECSZ])
{
uint8_t trial[VECSZ];
unsigned int kshow = kmax / 10;
floating_pt E = path_length (s);
for (unsigned int k = 0; k != kmax + 1; k += 1)
{
const floating_pt T = temperature (kT, kmax, k);
if (k % kshow == 0)
show (k, T, E);
trial_s (trial, s);
const floating_pt E_trial = path_length (trial);
const floating_pt delta_E = E_trial - E;
const floating_pt P = probability (delta_E, T);
if (P == 1 || randnum () <= P)
{
copy_s (s, trial);
E = E_trial;
}
}
}
static void
display_path (uint8_t vec[VECSZ])
{
for (size_t i = 0; i != VECSZ; i += 1)
{
const uint8_t x = vec[i];
printf ("%2u ->", (unsigned int) x);
if ((i % 8) == 7)
printf ("\n");
else
printf (" ");
}
printf ("%2u\n", vec[0]);
}
int
main (void)
{
char state[STATESZ];
uint32_t seed[1];
int status = getentropy (&seed[0], sizeof seed[0]);
if (status < 0)
seed[0] = 1;
initstate (seed[0], state, STATESZ);
floating_pt kT = 1.0;
unsigned int kmax = 1000000;
uint8_t s[VECSZ];
init_s (s);
printf ("\n");
printf (" kT: %f\n", (double) kT);
printf (" kmax: %u\n", kmax);
printf ("\n");
printf (" k T E(s)\n");
printf (" -----------------------------\n");
simulate_annealing (kT, kmax, s);
printf ("\n");
display_path (s);
printf ("\n");
printf ("Final E(s): %.5f\n", (double) path_length (s));
printf ("\n");
return 0;
}
- Output:
An example run:
$ cc -Ofast -march=native simanneal.c -lm && ./a.out kT: 1.000000 kmax: 1000000 k T E(s) ----------------------------- 0 1.0 383.25223 100000 0.9 195.81190 200000 0.8 186.58963 300000 0.7 152.46564 400000 0.6 143.59039 500000 0.5 130.91815 600000 0.4 126.53572 700000 0.3 112.85691 800000 0.2 103.72134 900000 0.1 103.07108 1000000 0.0 102.24265 0 -> 10 -> 20 -> 21 -> 31 -> 30 -> 40 -> 50 -> 60 -> 61 -> 62 -> 72 -> 82 -> 81 -> 71 -> 70 -> 80 -> 90 -> 91 -> 92 -> 93 -> 94 -> 84 -> 83 -> 73 -> 63 -> 64 -> 65 -> 66 -> 76 -> 86 -> 87 -> 77 -> 67 -> 68 -> 58 -> 57 -> 56 -> 55 -> 45 -> 35 -> 26 -> 36 -> 46 -> 47 -> 48 -> 38 -> 37 -> 27 -> 28 -> 29 -> 39 -> 49 -> 59 -> 69 -> 79 -> 78 -> 88 -> 89 -> 99 -> 98 -> 97 -> 96 -> 95 -> 85 -> 75 -> 74 -> 54 -> 53 -> 52 -> 51 -> 41 -> 42 -> 43 -> 44 -> 34 -> 33 -> 32 -> 22 -> 23 -> 14 -> 4 -> 5 -> 6 -> 7 -> 8 -> 9 -> 19 -> 18 -> 17 -> 16 -> 15 -> 25 -> 24 -> 13 -> 3 -> 2 -> 12 -> 11 -> 1 -> 0 Final E(s): 102.24265
C++
Compiler: MSVC (19.27.29111 for x64)
#include<array>
#include<utility>
#include<cmath>
#include<random>
#include<iostream>
using coord = std::pair<int,int>;
constexpr size_t numCities = 100;
// CityID with member functions to get position
struct CityID{
int v{-1};
CityID() = default;
constexpr explicit CityID(int i) noexcept : v(i){}
constexpr explicit CityID(coord ij) : v(ij.first * 10 + ij.second) {
if(ij.first < 0 || ij.first > 9 || ij.second < 0 || ij.second > 9){
throw std::logic_error("Cannot construct CityID from invalid coordinates!");
}
}
constexpr coord get_pos() const noexcept { return {v/10,v%10}; }
};
bool operator==(CityID const& lhs, CityID const& rhs) {return lhs.v == rhs.v;}
// Function for distance between two cities
double dist(coord city1, coord city2){
double diffx = city1.first - city2.first;
double diffy = city1.second - city2.second;
return std::sqrt(std::pow(diffx, 2) + std::pow(diffy,2));
}
// Function for total distance travelled
template<size_t N>
double dist(std::array<CityID,N> cities){
double sum = 0;
for(auto it = cities.begin(); it < cities.end() - 1; ++it){
sum += dist(it->get_pos(),(it+1)->get_pos());
}
sum += dist((cities.end()-1)->get_pos(), cities.begin()->get_pos());
return sum;
}
// 8 nearest cities, id cannot be at the border and has to have 8 valid neighbors
constexpr std::array<CityID,8> get_nearest(CityID id){
auto const ij = id.get_pos();
auto const i = ij.first;
auto const j = ij.second;
return {
CityID({i-1,j-1}),
CityID({i ,j-1}),
CityID({i+1,j-1}),
CityID({i-1,j }),
CityID({i+1,j }),
CityID({i-1,j+1}),
CityID({i ,j+1}),
CityID({i+1,j+1}),
};
}
// Function for formating of results
constexpr int get_num_digits(int num){
int digits = 1;
while(num /= 10){
++digits;
}
return digits;
}
// Function for shuffeling of initial state
template<typename It, typename RandomEngine>
void shuffle(It first, It last, RandomEngine& rand_eng){
for(auto i=(last-first)-1; i>0; --i){
std::uniform_int_distribution<int> dist(0,i);
std::swap(first[i], first[dist(rand_eng)]);
}
}
class SA{
int kT{1};
int kmax{1'000'000};
std::array<CityID,numCities> s;
std::default_random_engine rand_engine{0};
// Temperature
double temperature(int k) const { return kT * (1.0 - static_cast<double>(k) / kmax); }
// Probabilty of acceptance between 0.0 and 1.0
double P(double dE, double T){
if(dE < 0){
return 1;
}
else{
return std::exp(-dE/T);
}
}
// Permutation of state through swapping of cities in travel path
std::array<CityID,numCities> next_permut(std::array<CityID,numCities> cities){
std::uniform_int_distribution<> disx(1,8);
std::uniform_int_distribution<> disy(1,8);
auto randCity = CityID({disx(rand_engine),disy(rand_engine)}); // Select city which is not at the border, since all neighbors are valid under this condition and all permutations are still possible
auto neighbors = get_nearest(randCity); // Get list of nearest neighbors
std::uniform_int_distribution<> selector(0,neighbors.size()-1); // [0,7]
const auto [i,j] = randCity.get_pos();
auto randNeighbor = neighbors[selector(rand_engine)]; // Since randCity is not at the border, all 8 neighbors are valid
auto cityit1 = std::find(cities.begin(),cities.end(),randCity); // Find selected city in state
auto cityit2 = std::find(cities.begin(), cities.end(),randNeighbor);// Find selected neighbor in state
std::swap(*cityit1, *cityit2); // Swap city and neighbor
return cities;
}
// Logging function for progress output
void log_progress(int k, double T, double E) const {
auto nk = get_num_digits(kmax);
auto nt = get_num_digits(kT);
std::printf("k: %*i | T: %*.3f | E(s): %*.4f\n", nk, k, nt, T, 3, E);
}
public:
// Initialize state with integers from 0 to 99
SA() {
int i = 0;
for(auto it = s.begin(); it != s.end(); ++it){
*it = CityID(i);
++i;
}
shuffle(s.begin(),s.end(),rand_engine);
}
// Logging function for final path
void log_path(){
for(size_t idx = 0; idx < s.size(); ++idx){
std::printf("%*i -> ", 2, s[idx].v);
if((idx + 1)%20 == 0){
std::printf("\n");
}
}
std::printf("%*i", 2, s[0].v);
}
// Core simulated annealing algorithm
std::array<CityID,numCities> run(){
std::cout << "kT == " << kT << "\n" << "kmax == " << kmax << "\n" << "E(s0) == " << dist(s) << "\n";
for(int k = 0; k < kmax; ++k){
auto T = temperature(k);
auto const E1 = dist(s);
auto s_next{next_permut(s)};
auto const E2 = dist(s_next);
auto const dE = E2 - E1; // lower is better
std::uniform_real_distribution dist(0.0, 1.0);
auto E = E1;
if(P(dE,T) >= dist(rand_engine)){
s = s_next;
E = E2;
}
if(k%100000 == 0){
log_progress(k,T,E1);
}
}
log_progress(kmax,0.0,dist(s));
std::cout << "\nFinal path: \n";
log_path();
return s;
}
};
int main(){
SA sa{};
auto result = sa.run(); // Run simulated annealing and log progress and result
std::cin.get();
return 0;
}
- Output:
kT == 1 kmax == 1000000 E(s0) == 529.423 k: 0 | T: 1.000 | E(s): 529.4231 k: 100000 | T: 0.900 | E(s): 197.5111 k: 200000 | T: 0.800 | E(s): 183.7467 k: 300000 | T: 0.700 | E(s): 165.8442 k: 400000 | T: 0.600 | E(s): 143.8588 k: 500000 | T: 0.500 | E(s): 133.9247 k: 600000 | T: 0.400 | E(s): 125.9499 k: 700000 | T: 0.300 | E(s): 115.8657 k: 800000 | T: 0.200 | E(s): 107.8635 k: 900000 | T: 0.100 | E(s): 102.4853 k: 1000000 | T: 0.000 | E(s): 102.4853 Final path: 71 -> 61 -> 51 -> 50 -> 60 -> 70 -> 80 -> 90 -> 91 -> 92 -> 82 -> 83 -> 93 -> 94 -> 84 -> 85 -> 95 -> 96 -> 86 -> 76 -> 75 -> 74 -> 64 -> 65 -> 55 -> 45 -> 44 -> 54 -> 53 -> 43 -> 33 -> 34 -> 35 -> 26 -> 16 -> 6 -> 7 -> 17 -> 27 -> 37 -> 47 -> 57 -> 58 -> 48 -> 38 -> 28 -> 18 -> 8 -> 9 -> 19 -> 29 -> 39 -> 49 -> 59 -> 69 -> 68 -> 78 -> 79 -> 88 -> 89 -> 99 -> 98 -> 97 -> 87 -> 77 -> 67 -> 66 -> 56 -> 46 -> 36 -> 25 -> 24 -> 14 -> 15 -> 5 -> 4 -> 3 -> 2 -> 11 -> 21 -> 31 -> 41 -> 40 -> 30 -> 20 -> 10 -> 0 -> 1 -> 12 -> 13 -> 23 -> 22 -> 32 -> 42 -> 52 -> 62 -> 63 -> 73 -> 72 -> 81 -> 71
EchoLisp
(lib 'math)
;; distances
(define (d ci cj)
(distance (% ci 10) (quotient ci 10) (% cj 10) (quotient cj 10)))
(define _dists
(build-vector 10000 (lambda (ij) (d (quotient ij 100) (% ij 100)))))
(define-syntax-rule (dist ci cj)
[_dists (+ ci (* 100 cj))])
;; E(s) = length(path)
(define (Es path)
(define lpath (vector->list path))
(for/sum ((ci lpath) (cj (rest lpath))) (dist ci cj)))
;; temperature() function
(define (T k kmax kT)
(* kT (- 1 (// k kmax))))
#|
;; alternative temperature()
;; must be decreasing with k increasing and → 0
(define (T k kmax kT)
(* kT (- 1 (sin (* PI/2 (// k kmax))))))
|#
;; ∆E = Es_new - Es_old > 0
;; probability to move if ∆E > 0, → 0 when T → 0 (frozen state)
(define (P ∆E k kmax kT)
(exp (// (- ∆E ) (T k kmax kT))))
;; ∆E from path ( .. a u b .. c v d ..) to (.. a v b ... c u d ..)
;; ∆E before swapping (u,v)
;; Quicker than Es(s_next) - Es(s)
(define (dE s u v)
;;old
(define a (dist [s (1- u)] [s u]))
(define b (dist [s (1+ u)] [s u]))
(define c (dist [s (1- v)] [s v]))
(define d (dist [s (1+ v)] [s v]))
;; new
(define na (dist [s (1- u)] [s v]))
(define nb (dist [s (1+ u)] [s v]))
(define nc (dist [s (1- v)] [s u]))
(define nd (dist [s (1+ v)] [s u]))
(cond
((= v (1+ u)) (- (+ na nd) (+ a d)))
((= u (1+ v)) (- (+ nc nb) (+ c b)))
(else (- (+ na nb nc nd) (+ a b c d)))))
;; all 8 neighbours
(define dirs #(1 -1 10 -10 9 11 -11 -9))
(define (sa kmax (kT 10))
(define s (list->vector (cons 0 (append (shuffle (range 1 100)) 0))))
(printf "E(s0) %d" (Es s)) ;; random starter
(define Emin (Es s)) ;; E0
(for ((k kmax))
(when (zero? (% k (/ kmax 10)))
(printf "k: %10d T: %8.4d Es: %8.4d" k (T k kmax kT) (Es s))
)
(define u (1+ (random 99))) ;; city index 1 99
(define cv (+ [s u] [dirs (random 8)])) ;; city number
#:continue (or (> cv 99) (<= cv 0))
#:continue (> (dist [s u] cv) 5) ;; check true neighbour (eg 0 9)
(define v (vector-index cv s 1)) ;; city index
(define ∆e (dE s u v))
(when (or
(< ∆e 0) ;; always move if negative
(>= (P ∆e k kmax kT) (random)))
(vector-swap! s u v)
(+= Emin ∆e))
;; (assert (= (round Emin) (round (Es s))))
) ;; for
(printf "k: %10d T: %8.4d Es: %8.4d" kmax (T (1- kmax) kmax kT) (Es s))
(s-plot s 0)
(printf "E(s_final) %d" Emin)
(writeln 'Path s))
- Output:
(sa 1000000 1) E(s0) 501.0909 k: 0 T: 1 Es: 501.0909 k: 100000 T: 0.9 Es: 167.3632 k: 200000 T: 0.8 Es: 160.7791 k: 300000 T: 0.7 Es: 166.8746 k: 400000 T: 0.6 Es: 142.579 k: 500000 T: 0.5 Es: 131.0657 k: 600000 T: 0.4 Es: 116.9214 k: 700000 T: 0.3 Es: 110.8569 k: 800000 T: 0.2 Es: 103.3137 k: 900000 T: 0.1 Es: 102.4853 k: 1000000 T: 0 Es: 102.4853 E(s_final) 102.4853 Path #( 0 10 20 30 40 50 60 70 71 61 62 53 63 64 54 44 45 55 65 74 84 83 73 72 82 81 80 90 91 92 93 94 95 85 75 76 86 96 97 98 99 88 89 79 69 59 49 48 47 57 58 68 78 87 77 67 66 56 46 36 35 25 24 34 33 32 43 42 52 51 41 31 21 11 12 22 23 13 14 15 16 17 26 27 37 38 39 29 28 18 19 9 8 7 6 5 4 3 2 1 0)
Fortran
module simanneal_support
implicit none
!
! The following two integer kinds are meant to be treated as
! synonyms.
!
! selected_int_kind (2) = integers in the range of at least -100 to
! +100.
!
integer, parameter :: city_location_kind = selected_int_kind (2)
integer, parameter :: path_index_kind = city_location_kind
!
! selected_int_kind (1) = integers in the range of at least -10 to
! +10.
!
integer, parameter :: coordinate_kind = selected_int_kind(1)
!
! selected_real_kind (6) = floating point with at least 6 decimal
! digits of precision.
!
integer, parameter :: float_kind = selected_real_kind (6)
!
! Shorthand notations.
!
integer, parameter :: clk = city_location_kind
integer, parameter :: pik = path_index_kind
integer, parameter :: cok = coordinate_kind
integer, parameter :: flk = float_kind
type path_vector
integer(kind = clk) :: elem(0:99)
end type path_vector
contains
function random_integer (imin, imax) result (n)
integer, intent(in) :: imin, imax
integer :: n
real(kind = flk) :: randnum
call random_number (randnum)
n = imin + floor ((imax - imin + 1) * randnum)
end function random_integer
function i_coord (loc) result (i)
integer(kind = clk), intent(in) :: loc
integer(kind = cok) :: i
i = loc / 10_clk
end function i_coord
function j_coord (loc) result (j)
integer(kind = clk), intent(in) :: loc
integer(kind = cok) :: j
j = mod (loc, 10_clk)
end function j_coord
function location (i, j) result (loc)
integer(kind = cok), intent(in) :: i, j
integer(kind = clk) :: loc
loc = (10_clk * i) + j
end function location
subroutine randomize_path_vector (path)
type(path_vector), intent(out) :: path
integer(kind = pik) :: i, j
integer(kind = clk) :: xi, xj
do i = 0_pik, 99_pik
path%elem(i) = i
end do
! Do a Fisher-Yates shuffle of elements 1 .. 99.
do i = 1_pik, 98_pik
j = int (random_integer (i + 1, 99), kind = pik)
xi = path%elem(i)
xj = path%elem(j)
path%elem(i) = xj
path%elem(j) = xi
end do
end subroutine randomize_path_vector
function distance (loc1, loc2) result (dist)
integer(kind = clk), intent(in) :: loc1, loc2
real(kind = flk) :: dist
integer(kind = cok) :: i1, j1
integer(kind = cok) :: i2, j2
integer :: di, dj
i1 = i_coord (loc1)
j1 = j_coord (loc1)
i2 = i_coord (loc2)
j2 = j_coord (loc2)
di = i1 - i2
dj = j1 - j2
dist = sqrt (real ((di * di) + (dj * dj), kind = flk))
end function distance
function path_length (path) result (len)
type(path_vector), intent(in) :: path
real(kind = flk) :: len
integer(kind = pik) :: i
len = distance (path%elem(0_pik), path%elem(99_pik))
do i = 0_pik, 98_pik
len = len + distance (path%elem(i), path%elem(i + 1_pik))
end do
end function path_length
subroutine find_neighbors (loc, neighbors, num_neighbors)
integer(kind = clk), intent(in) :: loc
integer(kind = clk), intent(out) :: neighbors(1:8)
integer, intent(out) :: num_neighbors
integer(kind = cok) :: i, j
integer(kind = clk) :: c1, c2, c3, c4, c5, c6, c7, c8
c1 = 0_clk
c2 = 0_clk
c3 = 0_clk
c4 = 0_clk
c5 = 0_clk
c6 = 0_clk
c7 = 0_clk
c8 = 0_clk
i = i_coord (loc)
j = j_coord (loc)
if (i < 9_cok) then
c1 = location (i + 1_cok, j)
if (j < 9_cok) then
c2 = location (i + 1_cok, j + 1_cok)
end if
if (0_cok < j) then
c3 = location (i + 1_cok, j - 1_cok)
end if
end if
if (0_cok < i) then
c4 = location (i - 1_cok, j)
if (j < 9_cok) then
c5 = location (i - 1_cok, j + 1_cok)
end if
if (0_cok < j) then
c6 = location (i - 1_cok, j - 1_cok)
end if
end if
if (j < 9_cok) then
c7 = location (i, j + 1_cok)
end if
if (0_cok < j) then
c8 = location (i, j - 1_cok)
end if
num_neighbors = 0
call add_neighbor (c1)
call add_neighbor (c2)
call add_neighbor (c3)
call add_neighbor (c4)
call add_neighbor (c5)
call add_neighbor (c6)
call add_neighbor (c7)
call add_neighbor (c8)
contains
subroutine add_neighbor (neighbor)
integer(kind = clk), intent(in) :: neighbor
if (neighbor /= 0_clk) then
num_neighbors = num_neighbors + 1
neighbors(num_neighbors) = neighbor
end if
end subroutine add_neighbor
end subroutine find_neighbors
function make_neighbor_path (path) result (neighbor_path)
type(path_vector), intent(in) :: path
type(path_vector) :: neighbor_path
integer(kind = clk) :: u, v
integer(kind = clk) :: neighbors(1:8)
integer :: num_neighbors
integer(kind = pik) :: j, iu, iv
neighbor_path = path
u = int (random_integer (1, 99), kind = clk)
call find_neighbors (u, neighbors, num_neighbors)
v = neighbors (random_integer (1, num_neighbors))
j = 0_pik
iu = 0_pik
iv = 0_pik
do while (iu == 0_pik .or. iv == 0_pik)
if (neighbor_path%elem(j + 1) == u) then
iu = j + 1
else if (neighbor_path%elem(j + 1) == v) then
iv = j + 1
end if
j = j + 1
end do
neighbor_path%elem(iu) = v
neighbor_path%elem(iv) = u
end function make_neighbor_path
function temperature (kT, kmax, k) result (temp)
real(kind = flk), intent(in) :: kT
integer, intent(in) :: kmax, k
real(kind = flk) :: temp
real(kind = flk) :: kf, kmaxf
kf = real (k, kind = flk)
kmaxf = real (kmax, kind = flk)
temp = kT * (1.0_flk - (kf / kmaxf))
end function temperature
function probability (delta_E, T) result (prob)
real(kind = flk), intent(in) :: delta_E, T
real(kind = flk) :: prob
if (T == 0.0_flk) then
prob = 0.0_flk
else
prob = exp (-(delta_E / T))
end if
end function probability
subroutine show (k, T, E)
integer, intent(in) :: k
real(kind = flk), intent(in) :: T, E
write (*, 10) k, T, E
10 format (1X, I7, 1X, F7.1, 1X, F10.2)
end subroutine show
subroutine display_path (path)
type(path_vector), intent(in) :: path
integer(kind = pik) :: i
999 format ()
100 format (' ->')
110 format (' ')
120 format (I2)
do i = 0_pik, 99_pik
write (*, 120, advance = 'no') path%elem(i)
write (*, 100, advance = 'no')
if (mod (i, 8_pik) == 7_pik) then
write (*, 999, advance = 'yes')
else
write (*, 110, advance = 'no')
end if
end do
write (*, 120, advance = 'no') path%elem(0_pik)
end subroutine display_path
subroutine simulate_annealing (kT, kmax, initial_path, final_path)
real(kind = flk), intent(in) :: kT
integer, intent(in) :: kmax
type(path_vector), intent(in) :: initial_path
type(path_vector), intent(inout) :: final_path
integer :: kshow
integer :: k
real(kind = flk) :: E, E_trial, T
type(path_vector) :: path, trial
real(kind = flk) :: randnum
kshow = kmax / 10
path = initial_path
E = path_length (path)
do k = 0, kmax
T = temperature (kT, kmax, k)
if (mod (k, kshow) == 0) call show (k, T, E)
trial = make_neighbor_path (path)
E_trial = path_length (trial)
if (E_trial <= E) then
path = trial
E = E_trial
else
call random_number (randnum)
if (randnum <= probability (E_trial - E, T)) then
path = trial
E = E_trial
end if
end if
end do
final_path = path
end subroutine simulate_annealing
end module simanneal_support
program simanneal
use, non_intrinsic :: simanneal_support
implicit none
real(kind = flk), parameter :: kT = 1.0_flk
integer, parameter :: kmax = 1000000
type(path_vector) :: initial_path
type(path_vector) :: final_path
call random_seed
call randomize_path_vector (initial_path)
10 format ()
20 format (' kT: ', F0.2)
30 format (' kmax: ', I0)
40 format (' k T E(s)')
50 format (' --------------------------')
60 format ('Final E(s): ', F0.2)
write (*, 10)
write (*, 20) kT
write (*, 30) kmax
write (*, 10)
write (*, 40)
write (*, 50)
call simulate_annealing (kT, kmax, initial_path, final_path)
write (*, 10)
call display_path (final_path)
write (*, 10)
write (*, 10)
write (*, 60) path_length (final_path)
write (*, 10)
end program simanneal
- Output:
$ gfortran -std=f2018 -Ofast simanneal.f90 && ./a.out kT: 1.00 kmax: 1000000 k T E(s) -------------------------- 0 1.0 517.11 100000 0.9 198.12 200000 0.8 169.43 300000 0.7 164.66 400000 0.6 149.10 500000 0.5 138.38 600000 0.4 119.24 700000 0.3 113.69 800000 0.2 105.80 900000 0.1 101.66 1000000 0.0 101.66 0 -> 10 -> 11 -> 21 -> 31 -> 20 -> 30 -> 40 -> 41 -> 51 -> 50 -> 60 -> 70 -> 71 -> 61 -> 62 -> 72 -> 82 -> 81 -> 80 -> 90 -> 91 -> 92 -> 93 -> 83 -> 73 -> 74 -> 84 -> 94 -> 95 -> 96 -> 97 -> 98 -> 99 -> 89 -> 88 -> 79 -> 69 -> 59 -> 58 -> 48 -> 49 -> 39 -> 38 -> 28 -> 29 -> 19 -> 9 -> 8 -> 18 -> 17 -> 7 -> 6 -> 16 -> 15 -> 5 -> 4 -> 14 -> 24 -> 25 -> 26 -> 27 -> 37 -> 36 -> 35 -> 45 -> 46 -> 47 -> 57 -> 67 -> 68 -> 78 -> 77 -> 87 -> 86 -> 85 -> 75 -> 76 -> 66 -> 56 -> 55 -> 65 -> 64 -> 63 -> 54 -> 53 -> 52 -> 42 -> 43 -> 44 -> 34 -> 33 -> 32 -> 22 -> 23 -> 12 -> 13 -> 3 -> 2 -> 1 -> 0 Final E(s): 101.66
Go
package main
import (
"fmt"
"math"
"math/rand"
"time"
)
var (
dists = calcDists()
dirs = [8]int{1, -1, 10, -10, 9, 11, -11, -9} // all 8 neighbors
)
// distances
func calcDists() []float64 {
dists := make([]float64, 10000)
for i := 0; i < 10000; i++ {
ab, cd := math.Floor(float64(i)/100), float64(i%100)
a, b := math.Floor(ab/10), float64(int(ab)%10)
c, d := math.Floor(cd/10), float64(int(cd)%10)
dists[i] = math.Hypot(a-c, b-d)
}
return dists
}
// index into lookup table of float64s
func dist(ci, cj int) float64 {
return dists[cj*100+ci]
}
// energy at s, to be minimized
func Es(path []int) float64 {
d := 0.0
for i := 0; i < len(path)-1; i++ {
d += dist(path[i], path[i+1])
}
return d
}
// temperature function, decreases to 0
func T(k, kmax, kT int) float64 {
return (1 - float64(k)/float64(kmax)) * float64(kT)
}
// variation of E, from state s to state s_next
func dE(s []int, u, v int) float64 {
su, sv := s[u], s[v]
// old
a, b, c, d := dist(s[u-1], su), dist(s[u+1], su), dist(s[v-1], sv), dist(s[v+1], sv)
// new
na, nb, nc, nd := dist(s[u-1], sv), dist(s[u+1], sv), dist(s[v-1], su), dist(s[v+1], su)
if v == u+1 {
return (na + nd) - (a + d)
} else if u == v+1 {
return (nc + nb) - (c + b)
} else {
return (na + nb + nc + nd) - (a + b + c + d)
}
}
// probability to move from s to s_next
func P(deltaE float64, k, kmax, kT int) float64 {
return math.Exp(-deltaE / T(k, kmax, kT))
}
func sa(kmax, kT int) {
rand.Seed(time.Now().UnixNano())
temp := make([]int, 99)
for i := 0; i < 99; i++ {
temp[i] = i + 1
}
rand.Shuffle(len(temp), func(i, j int) {
temp[i], temp[j] = temp[j], temp[i]
})
s := make([]int, 101) // all 0 by default
copy(s[1:], temp) // random path from 0 to 0
fmt.Println("kT =", kT)
fmt.Printf("E(s0) %f\n\n", Es(s)) // random starter
Emin := Es(s) // E0
for k := 0; k <= kmax; k++ {
if k%(kmax/10) == 0 {
fmt.Printf("k:%10d T: %8.4f Es: %8.4f\n", k, T(k, kmax, kT), Es(s))
}
u := 1 + rand.Intn(99) // city index 1 to 99
cv := s[u] + dirs[rand.Intn(8)] // city number
if cv <= 0 || cv >= 100 { // bogus city
continue
}
if dist(s[u], cv) > 5 { // check true neighbor (eg 0 9)
continue
}
v := s[cv] // city index
deltae := dE(s, u, v)
if deltae < 0 || // always move if negative
P(deltae, k, kmax, kT) >= rand.Float64() {
s[u], s[v] = s[v], s[u]
Emin += deltae
}
}
fmt.Printf("\nE(s_final) %f\n", Emin)
fmt.Println("Path:")
// output final state
for i := 0; i < len(s); i++ {
if i > 0 && i%10 == 0 {
fmt.Println()
}
fmt.Printf("%4d", s[i])
}
fmt.Println()
}
func main() {
sa(1e6, 1)
}
- Output:
Sample run:
kT = 1 E(s0) 520.932463 k: 0 T: 1.0000 Es: 520.9325 k: 100000 T: 0.9000 Es: 185.1279 k: 200000 T: 0.8000 Es: 167.7657 k: 300000 T: 0.7000 Es: 158.6923 k: 400000 T: 0.6000 Es: 151.6564 k: 500000 T: 0.5000 Es: 139.9185 k: 600000 T: 0.4000 Es: 132.9964 k: 700000 T: 0.3000 Es: 121.8962 k: 800000 T: 0.2000 Es: 120.0445 k: 900000 T: 0.1000 Es: 116.8476 k: 1000000 T: 0.0000 Es: 116.5565 E(s_final) 116.556509 Path: 0 11 21 31 41 51 52 61 62 72 82 73 74 64 44 45 55 54 63 53 42 32 43 33 35 34 24 23 22 13 12 2 3 4 14 25 26 7 6 16 15 5 17 27 36 46 56 66 65 75 77 78 68 69 59 49 39 38 37 28 29 19 9 8 18 47 48 58 57 67 76 86 85 95 96 97 87 88 79 89 99 98 84 94 83 93 92 91 90 80 81 71 70 60 50 40 30 20 10 1 0
Icon
- Output:
An example run:
$ icont -s -u simanneal-in-Icon.icn && ./simanneal-in-Icon kT: 1.0 kmax: 1000000 k T E(s) -------------------------- 0 1.0 511.67 100000 0.9 206.16 200000 0.8 186.68 300000 0.7 165.92 400000 0.6 158.49 500000 0.5 141.76 600000 0.4 122.53 700000 0.3 119.47 800000 0.2 107.56 900000 0.1 102.89 1000000 0.0 102.24 0 -> 10 -> 20 -> 30 -> 31 -> 41 -> 40 -> 50 -> 60 -> 70 -> 71 -> 72 -> 62 -> 61 -> 51 -> 52 -> 53 -> 63 -> 54 -> 44 -> 45 -> 35 -> 34 -> 24 -> 25 -> 26 -> 27 -> 17 -> 7 -> 8 -> 9 -> 19 -> 29 -> 39 -> 49 -> 59 -> 69 -> 79 -> 89 -> 99 -> 98 -> 97 -> 96 -> 86 -> 76 -> 75 -> 84 -> 85 -> 95 -> 94 -> 93 -> 92 -> 91 -> 90 -> 80 -> 81 -> 82 -> 83 -> 73 -> 74 -> 64 -> 55 -> 65 -> 66 -> 56 -> 46 -> 36 -> 37 -> 47 -> 57 -> 67 -> 77 -> 87 -> 88 -> 78 -> 68 -> 58 -> 48 -> 38 -> 28 -> 18 -> 16 -> 6 -> 5 -> 15 -> 14 -> 4 -> 3 -> 2 -> 12 -> 13 -> 23 -> 33 -> 43 -> 42 -> 32 -> 22 -> 21 -> 11 -> 1 -> 0 Final E(s): 102.24
J
Implementation:
dist=: +/&.:*:@:-"1/~10 10#:i.100
satsp=:4 :0
kT=. 1
pathcost=. [: +/ 2 {&y@<\ 0 , ] , 0:
neighbors=. 0 (0}"1) y e. 1 2{/:~~.,y
s=. (?~#y)-.0
d=. pathcost s
step=. x%10
for_k. i.x+1 do.
T=. kT*1-k%x
u=. ({~ ?@#)s
v=. ({~ ?@#)I.u{neighbors
sk=. (<s i.u,v) C. s
dk=. pathcost sk
dE=. dk-d
if. (^-dE%T) >?0 do.
s=.sk
d=.dk
end.
if. 0=step|k do.
echo k,T,d
end.
end.
0,s,0
)
Notes:
E(s_final) gets displayed on the kmax progress line.
We do not do anything special for negative deltaE because the exponential will be greater than 1 for that case and that will always be greater than our random number from the range 0..1.
Also, while we leave connection distances (and, thus, number of cities) as a parameter, some other aspects of this problem made more sense when included in the implementation:
We leave city 0 out of our data structure, since it can't appear in the middle of our path. But we bring it back in when computing path distance.
Neighbors are any city which have one of the two closest non-zero distances from the current city (and specifically excluding city 0, since that is anchored as our start and end city).
Sample run:
1e6 satsp dist
0 1 538.409
100000 0.9 174.525
200000 0.8 165.541
300000 0.7 173.348
400000 0.6 168.188
500000 0.5 134.983
600000 0.4 121.585
700000 0.3 111.443
800000 0.2 101.657
900000 0.1 101.657
1e6 0 101.657
0 1 2 3 4 13 23 24 34 44 43 33 32 31 41 42 52 51 61 62 53 54 64 65 55 45 35 25 15 14 5 6 7 17 16 26 27 37 36 46 47 48 38 28 18 8 9 19 29 39 49 59 69 79 78 68 58 57 56 66 67 77 76 75 85 86 87 88 89 99 98 97 96 95 94 84 74 73 63 72 82 83 93 92 91 90 80 81 71 70 60 50 40 30 20 21 22 12 11 10 0
Julia
Module:
module TravelingSalesman
using Random, Printf
# Eₛ: length(path)
Eₛ(distances, path) = sum(distances[ci, cj] for (ci, cj) in zip(path, Iterators.drop(path, 1)))
# T: temperature
T(k, kmax, kT) = kT * (1 - k / kmax)
# Alternative temperature:
#T(k, kmax, kT) = kT * (1 - sin(π / 2 * k / kmax))
# ΔE = Eₛ_new - Eₛ_old > 0
# Prob. to move if ΔE > 0, → 0 when T → 0 (fronzen state)
P(ΔE, k, kmax, kT) = exp(-ΔE / T(k, kmax, kT))
# ∆E from path ( .. a u b .. c v d ..) to (.. a v b ... c u d ..)
# ∆E before swapping (u,v)
# Quicker than Eₛ(s_next) - Eₛ(path)
function dE(distances, path, u, v)
a = distances[path[u - 1], path[u]]
b = distances[path[u + 1], path[u]]
c = distances[path[v - 1], path[v]]
d = distances[path[v + 1], path[v]]
na = distances[path[u - 1], path[v]]
nb = distances[path[u + 1], path[v]]
nc = distances[path[v - 1], path[u]]
nd = distances[path[v + 1], path[u]]
if v == u + 1
return (na + nd) - (a + d)
elseif u == v + 1
return (nc + nb) - (c + b)
else
return (na + nb + nc + nd) - (a + b + c + d)
end
end
const dirs = [1, -1, 10, -10, 9, 11, -11, -9]
function _prettypath(path)
r = IOBuffer()
for g in Iterators.partition(path, 10)
println(r, join(lpad.(g, 3), ", "))
end
return String(take!(r))
end
function findpath(distances, kmax, kT)
n = size(distances, 1)
path = vcat(1, shuffle(2:n), 1)
Emin = Eₛ(distances, path)
@printf("\n# Entropy(s₀) = %10.2f\n", Emin)
println("# Random path: \n", _prettypath(path))
for k in Base.OneTo(kmax)
if iszero(k % (kmax ÷ 10))
@printf("k: %10d | T: %8.4f | Eₛ: %8.4f\n", k, T(k, kmax, kT), Eₛ(distances, path))
end
u = rand(2:n)
v = path[u] + rand(dirs)
v ∈ 2:n || continue
δE = dE(distances, path, u, v)
if δE < 0 || P(δE, k, kmax, kT) ≥ rand()
path[u], path[v] = path[v], path[u]
Emin += δE
end
end
@printf("k: %10d | T: %8.4f | Eₛ: %8.4f\n", kmax, T(kmax, kmax, kT), Eₛ(distances, path))
println("\n# Found path:\n", _prettypath(path))
return path
end
end # module TravelingSalesman
Main:
distance(a, b) = sqrt(sum((a .- b) .^ 2))
const _citydist = collect(distance((ci % 10, ci ÷ 10), (cj % 10, cj ÷ 10)) for ci in 1:100, cj in 1:100)
TravelingSalesman.findpath(_citydist, 1_000_000, 1)
- Output:
# Entropy(s₀) = 521.86 # Random path: 1, 2, 11, 80, 78, 73, 68, 19, 43, 69 86, 79, 66, 67, 77, 96, 26, 62, 60, 98 71, 3, 59, 37, 18, 40, 34, 92, 97, 6 84, 94, 29, 63, 36, 50, 87, 45, 83, 90 76, 28, 15, 38, 91, 58, 47, 44, 85, 17 25, 33, 31, 99, 27, 74, 53, 95, 16, 13 42, 88, 8, 4, 7, 64, 54, 9, 14, 41 5, 81, 65, 23, 75, 100, 89, 51, 20, 48 82, 12, 21, 55, 24, 70, 49, 10, 35, 72 52, 22, 61, 32, 46, 57, 30, 93, 39, 56 1 k: 100000 | T: 0.9000 | Eₛ: 184.4448 k: 200000 | T: 0.8000 | Eₛ: 175.3662 k: 300000 | T: 0.7000 | Eₛ: 169.0505 k: 400000 | T: 0.6000 | Eₛ: 160.8328 k: 500000 | T: 0.5000 | Eₛ: 147.1973 k: 600000 | T: 0.4000 | Eₛ: 132.9186 k: 700000 | T: 0.3000 | Eₛ: 126.9931 k: 800000 | T: 0.2000 | Eₛ: 122.0656 k: 900000 | T: 0.1000 | Eₛ: 119.7924 k: 1000000 | T: 0.0000 | Eₛ: 119.7924 k: 1000000 | T: 0.0000 | Eₛ: 119.7924 # Found path: 1, 2, 12, 13, 3, 4, 6, 7, 8, 9 19, 18, 17, 5, 14, 15, 16, 27, 28, 29 39, 38, 26, 25, 24, 23, 22, 10, 21, 20 30, 31, 32, 33, 34, 35, 36, 37, 49, 48 47, 46, 45, 44, 43, 42, 41, 40, 50, 51 52, 53, 54, 55, 56, 57, 58, 59, 69, 68 67, 65, 64, 63, 62, 61, 71, 60, 70, 80 81, 82, 72, 73, 74, 66, 78, 79, 89, 99 98, 97, 96, 95, 94, 85, 86, 87, 88, 77 76, 75, 84, 83, 93, 92, 91, 100, 90, 11 1
Nim
import math, random, sequtils, strformat
const
kT = 1
kMax = 1_000_000
proc randomNeighbor(x: int): int =
case x
of 0:
sample([1, 10, 11])
of 9:
sample([8, 18, 19])
of 90:
sample([80, 81, 91])
of 99:
sample([88, 89, 98])
elif x > 0 and x < 9: # top ceiling
sample [x-1, x+1, x+9, x+10, x+11]
elif x > 90 and x < 99: # bottom floor
sample [x-11, x-10, x-9, x-1, x+1]
elif x mod 10 == 0: # left wall
sample([x-10, x-9, x+1, x+10, x+11])
elif (x+1) mod 10 == 0: # right wall
sample([x-11, x-10, x-1, x+9, x+10])
else: # center
sample([x-11, x-10, x-9, x-1, x+1, x+9, x+10, x+11])
proc neighbor(s: seq[int]): seq[int] =
result = s
var city = sample(s)
var cityNeighbor = city.randomNeighbor
while cityNeighbor == 0 or city == 0:
city = sample(s)
cityNeighbor = city.randomNeighbor
result[s.find city].swap result[s.find cityNeighbor]
func distNeighbor(a, b: int): float =
template divmod(a: int): (int, int) = (a div 10, a mod 10)
let
(diva, moda) = a.divmod
(divb, modb) = b.divmod
hypot((diva-divb).float, (moda-modb).float)
func temperature(k, kmax: float): float =
kT * (1 - (k / kmax))
func pdelta(eDelta, temp: float): float =
if eDelta < 0: 1.0
else: exp(-eDelta / temp)
func energy(path: seq[int]): float =
var sum = 0.distNeighbor path[0]
for i in 1 ..< path.len:
sum += path[i-1].distNeighbor(path[i])
sum + path[^1].distNeighbor 0
proc main =
randomize()
var
s = block:
var x = toSeq(0..99)
template shuffler: int = rand(1 .. x.high)
for i in 1 .. x.high:
x[i].swap x[shuffler()]
x
echo fmt"E(s0): {energy s:6.4f}"
for k in 0 .. kMax:
var
temp = temperature(float k, float kMax)
lastenergy = energy s
newneighbor = s.neighbor
newenergy = newneighbor.energy
if k mod (kMax div 10) == 0:
echo fmt"k: {k:7} T: {temp:6.2f} Es: {lastenergy:6.4f}"
var deltaEnergy = newenergy - lastenergy
if pDelta(deltaEnergy, temp) >= rand(1.0):
s = newneighbor
s.add 0
echo fmt"E(sFinal): {energy s:6.4f}"
echo fmt"path: {s}"
main()
Compile and run:
nim c -r -d:release --opt:speed travel_sa.nim
- Output:
Sample run:
E(s0): 505.1591 k: 0 T: 1.00 Es: 505.1591 k: 100000 T: 0.90 Es: 196.5216 k: 200000 T: 0.80 Es: 165.6735 k: 300000 T: 0.70 Es: 159.3411 k: 400000 T: 0.60 Es: 144.8330 k: 500000 T: 0.50 Es: 131.7888 k: 600000 T: 0.40 Es: 127.6914 k: 700000 T: 0.30 Es: 113.9280 k: 800000 T: 0.20 Es: 104.7279 k: 900000 T: 0.10 Es: 103.3137 k: 1000000 T: 0.00 Es: 103.3137 E(sFinal): 103.3137 path: @[0, 10, 11, 22, 21, 20, 30, 31, 41, 40, 50, 51, 61, 60, 70, 71, 81, 80, 90, 91, 92, 93, 82, 83, 73, 72, 62, 63, 53, 52, 42, 32, 33, 23, 13, 14, 24, 34, 35, 25, 15, 16, 26, 36, 47, 48, 38, 39, 49, 59, 58, 57, 68, 69, 79, 89, 99, 98, 97, 96, 95, 94, 84, 74, 75, 85, 86, 87, 88, 78, 77, 67, 76, 66, 65, 64, 54, 43, 44, 45, 55, 56, 46, 37, 27, 28, 29, 19, 9, 8, 18, 17, 7, 6, 5, 4, 3, 2, 12, 1, 0]
Perl
use utf8;
use strict;
use warnings;
use List::Util qw(shuffle sum);
# simulation setup
my $cities = 100; # number of cities
my $kmax = 1e6; # iterations to run
my $kT = 1; # initial 'temperature'
die 'City count must be a perfect square.' if sqrt($cities) != int sqrt($cities);
# locations of (up to) 8 neighbors, with grid size derived from number of cities
my $gs = sqrt $cities;
my @neighbors = (1, -1, $gs, -$gs, $gs-1, $gs+1, -($gs+1), -($gs-1));
# matrix of distances between cities
my @D;
for my $j (0 .. $cities**2 - 1) {
my ($ab, $cd) = (int($j/$cities), int($j%$cities));
my ($a, $b, $c, $d) = (int($ab/$gs), int($ab%$gs), int($cd/$gs), int($cd%$gs));
$D[$ab][$cd] = sqrt(($a - $c)**2 + ($b - $d)**2);
}
# temperature function, decreases to 0
sub T {
my($k, $kmax, $kT) = @_;
(1 - $k/$kmax) * $kT
}
# probability to move from s to s_next
sub P {
my($ΔE, $k, $kmax, $kT) = @_;
exp -$ΔE / T($k, $kmax, $kT)
}
# energy at s, to be minimized
sub Es {
my(@path) = @_;
sum map { $D[ $path[$_] ] [ $path[$_+1] ] } 0 .. @path-2
}
# variation of E, from state s to state s_next
sub delta_E {
my($u, $v, @s) = @_;
my ($a, $b, $c, $d) = ($D[$s[$u-1]][$s[$u]], $D[$s[$u+1]][$s[$u]], $D[$s[$v-1]][$s[$v]], $D[$s[$v+1]][$s[$v]]);
my ($na, $nb, $nc, $nd) = ($D[$s[$u-1]][$s[$v]], $D[$s[$u+1]][$s[$v]], $D[$s[$v-1]][$s[$u]], $D[$s[$v+1]][$s[$u]]);
if ($v == $u+1) { return ($na + $nd) - ($a + $d) }
elsif ($u == $v+1) { return ($nc + $nb) - ($c + $b) }
else { return ($na + $nb + $nc + $nd) - ($a + $b + $c + $d) }
}
# E(s0), initial state
my @s = 0; map { push @s, $_ } shuffle 1..$cities-1; push @s, 0;
my $E_min_global = my $E_min = Es(@s);
for my $k (0 .. $kmax-1) {
printf "k:%8u T:%4.1f Es: %3.1f\n" , $k, T($k, $kmax, $kT), Es(@s)
if $k % ($kmax/10) == 0;
# valid candidate cities (exist, adjacent)
my $u = 1 + int rand $cities-1;
my $cv = $neighbors[int rand 8] + $s[$u];
next if $cv <= 0 or $cv >= $cities or $D[$s[$u]][$cv] > sqrt(2);
my $v = $s[$cv];
my $ΔE = delta_E($u, $v, @s);
if ($ΔE < 0 or P($ΔE, $k, $kmax, $kT) >= rand) { # always move if negative
($s[$u], $s[$v]) = ($s[$v], $s[$u]);
$E_min += $ΔE;
$E_min_global = $E_min if $E_min < $E_min_global;
}
}
printf "\nE(s_final): %.1f\n", $E_min_global;
for my $l (0..4) {
printf "@{['%4d' x 20]}\n", @s[$l*20 .. ($l+1)*20 - 1];
}
printf " 0\n";
- Output:
k: 0 T: 1.0 Es: 519.2 k: 100000 T: 0.9 Es: 188.2 k: 200000 T: 0.8 Es: 178.5 k: 300000 T: 0.7 Es: 162.3 k: 400000 T: 0.6 Es: 157.0 k: 500000 T: 0.5 Es: 148.9 k: 600000 T: 0.4 Es: 128.7 k: 700000 T: 0.3 Es: 129.5 k: 800000 T: 0.2 Es: 119.8 k: 900000 T: 0.1 Es: 119.5 E(s_final): 119.1 0 12 23 24 35 36 26 27 16 7 8 9 19 29 28 18 17 6 14 13 22 32 33 34 25 15 5 4 3 2 1 11 20 21 31 30 40 51 50 60 61 62 53 43 44 54 56 57 48 49 39 38 37 46 45 55 65 64 63 74 84 83 82 81 80 90 91 92 93 94 95 85 66 47 58 59 69 89 88 87 77 67 68 78 79 99 98 97 96 86 76 75 73 72 70 71 52 42 41 10 0
Phix
with javascript_semantics function hypot(atom a,b) return sqrt(a*a+b*b) end function function calc_dists() sequence dists = repeat(0,10000) for abcd=1 to 10000 do integer {ab,cd} = {floor(abcd/100),mod(abcd,100)}, {a,b,c,d} = {floor(ab/10),mod(ab,10), floor(cd/10),mod(cd,10)} dists[abcd] = hypot(a-c,b-d) end for return dists end function constant dists = calc_dists() function dist(integer ci,cj) return dists[cj*100+ci] end function function Es(sequence path) atom d = 0 for i=1 to length(path)-1 do d += dist(path[i],path[i+1]) end for return d end function -- temperature() function function T(integer k, kmax, kT) return (1-k/kmax)*kT end function -- deltaE = Es_new - Es_old > 0 -- probability to move if deltaE > 0, -->0 when T --> 0 (frozen state) function P(atom deltaE, integer k, kmax, kT) return exp(-deltaE/T(k,kmax,kT)) end function -- deltaE from path ( .. a u b .. c v d ..) to (.. a v b ... c u d ..) function dE(sequence s, integer u,v) -- (note that u,v are 0-based, but 1..99 here) -- integer sum1 = s[u-1], su = s[u], sup1 = s[u+1], -- svm1 = s[v-1], sv = s[v], svp1 = s[v+1] integer sum1 = s[u], su = s[u+1], sup1 = s[u+2], svm1 = s[v], sv = s[v+1], svp1 = s[v+2] -- old atom {a,b,c,d}:={dist(sum1,su), dist(su,sup1), dist(svm1,sv), dist(sv,svp1)}, -- new {na,nb,nc,nd}:={dist(sum1,sv), dist(sv,sup1), dist(svm1,su), dist(su,svp1)} return iff(v==u+1?(na+nd)-(a+d): iff(u==v+1?(nc+nb)-(c+b): (na+nb+nc+nd)-(a+b+c+d))) end function -- all 8 neighbours constant dirs = {1, -1, 10, -10, 9, 11, -11, -9} procedure sa(integer kmax, kT=10) sequence s = 0&shuffle(tagset(99))&0 atom Emin:=Es(s) -- E0 printf(1,"E(s0) %f\n",Emin) -- random starter for k=0 to kmax do if mod(k,kmax/10)=0 then printf(1,"k:%,10d T: %8.4f Es: %8.4f\n",{k,T(k,kmax,kT),Es(s)}) if k=kmax then exit end if -- avoid exp(x,-inf) end if integer u = rand(99), -- city index 1 99 cv = s[u+1]+dirs[rand(8)] -- city number if cv>0 and cv<100 -- not bogus city and dist(s[u+1],cv)<5 then -- and true neighbour integer v = s[cv+1] -- city index atom deltae := dE(s,u,v); if deltae<0 -- always move if negative or P(deltae,k,kmax,kT)>=rnd() then {s[u+1],s[v+1]} = {s[v+1],s[u+1]} Emin += deltae end if end if end for printf(1,"E(s_final) %f\n",Emin) printf(1,"Path:\n") pp(s,{pp_IntFmt,"%2d",pp_IntCh,false}) end procedure sa(1_000_000,1)
- Output:
E(s0) 515.164811 k: 0 T: 1.0000 Es: 515.1648 k: 100,000 T: 0.9000 Es: 189.3123 k: 200,000 T: 0.8000 Es: 198.7498 k: 300,000 T: 0.7000 Es: 158.2189 k: 400,000 T: 0.6000 Es: 165.4813 k: 500,000 T: 0.5000 Es: 156.3467 k: 600,000 T: 0.4000 Es: 142.7928 k: 700,000 T: 0.3000 Es: 128.0352 k: 800,000 T: 0.2000 Es: 121.7794 k: 900,000 T: 0.1000 Es: 121.2328 k: 1,000,000 T: 0.0000 Es: 121.1291 E(s_final) 121.129115 Path: { 0,10,62,63,64,65,76,75,84,85,95,86,96,97,87,77,67,66,56,46,47,48,49,59,69, 79,89,99,98,88,78,68,58,57,37,38,27,26,36,35,45,55,54,53,52,43,33,23,22,32, 42,41,51,61,60,50,40,30,31,21,20,11,12, 2, 3, 4, 5, 6,17,18,28,39,29,19, 9, 8, 7,16,15,24,44,74,83,93,94,92,91,71,70,90,80,81,82,72,73,34,25,14,13, 1, 0}
Racket
#lang racket
(require racket/fixnum)
(define current-dim (make-parameter 10))
(define current-dim-- (make-parameter 9))
(define current-dim² (make-parameter 100))
(define current-kT (make-parameter 1))
(define current-k-max (make-parameter 1000000))
(define current-monitor-frequency (make-parameter 100000))
(define current-monitor (make-parameter
(λ (s k T E)
(when (zero? (modulo k (current-monitor-frequency)))
(printf "T:~a E:~a~%" (~r T) E)))))
;; Simulated Annealing Solver
(define (P ΔE T)
(if (negative? ΔE) 1 (exp (- (/ ΔE T)))))
(define (solve/SA s₀ next-s k-max temperature E monitor)
(for*/fold ((s s₀) (E_s (E s₀)))
((k (in-range k-max)))
(define T (temperature k k-max))
(when monitor (monitor s k T E_s))
(let* ((s´ (next-s s k))
(E_s´ (E s´))
(ΔE (- E_s´ E_s)))
(if (>= (P ΔE T) (random)) (values s´ E_s´) (values s E_s)))))
(define (temperature k k-max)
(* (current-kT) (- 1 (/ k k-max))))
;; TSP Problem
(struct tsp (path indices Es ΣE) #:transparent)
(define (y/x i d) (quotient/remainder i d))
(define (dist a b (d (current-dim)))
(let-values (([ay ax] (y/x a d)) ([by bx] (y/x b d)))
(sqrt (+ (sqr (- ay by)) (sqr (- ax bx))))))
(define (indices->tsp indices)
(define path (make-fxvector (current-dim²)))
(for ((i indices) (n (current-dim²))) (fxvector-set! path i n))
(define Es (for/vector #:length (fxvector-length path)
((a (in-fxvector path))
(b (in-sequences (in-fxvector path 1) (in-value (fxvector-ref path 0)))))
(dist a b)))
(tsp path indices Es (for/sum ((E Es)) E)))
(define (dir->delta dir (dim (current-dim))) (case dir [(l) -1] [(r) +1] [(u) (- dim)] [(d) dim]))
(define (invalid-direction? x y d (mx (current-dim--)))
(match* (x y d) ((0 _ 'l) #t) (((== mx) _ 'r) #t) ((_ 0 'u) #t) ((_ (== mx) 'd) #t) ((_ _ _) #f)))
;; extended to take k to reset numerical drift from the Δ calculation
(define (tsp:swap-one-neighbour t k)
(define dim (current-dim))
(define dim² (current-dim²))
(define candidate (random dim²))
(define-values [cy cx] (quotient/remainder candidate dim))
(define dir (vector-ref #(l r u d) (random 4)))
(cond
[(invalid-direction? cx cy dir) (tsp:swap-one-neighbour t k)]
[else
(define delta (dir->delta dir))
(define neighbour (+ candidate delta))
(define path´ (fxvector-copy (tsp-path t)))
(define indices´ (fxvector-copy (tsp-indices t)))
(define cand-idx (fxvector-ref (tsp-indices t) candidate))
(define ngbr-idx (fxvector-ref (tsp-indices t) neighbour))
(fxvector-set! path´ cand-idx neighbour)
(fxvector-set! path´ ngbr-idx candidate)
(fxvector-set! indices´ candidate ngbr-idx)
(fxvector-set! indices´ neighbour cand-idx)
(define Es (tsp-Es t))
(define Es´ (vector-copy Es))
(let* ((cand-idx++ (modulo (add1 cand-idx) dim²))
(cand-idx-- (modulo (sub1 cand-idx) dim²))
(ngbr-idx++ (modulo (add1 ngbr-idx) dim²))
(ngbr-idx-- (modulo (sub1 ngbr-idx) dim²)))
(define Σold-E-around-nodes
(+ (vector-ref Es cand-idx) (vector-ref Es cand-idx--)
(vector-ref Es ngbr-idx) (vector-ref Es ngbr-idx--)))
(define E´-at-cand (dist (fxvector-ref path´ cand-idx) (fxvector-ref path´ cand-idx++)))
(define E´-pre-cand (dist (fxvector-ref path´ cand-idx) (fxvector-ref path´ cand-idx--)))
(define E´-at-ngbr (dist (fxvector-ref path´ ngbr-idx) (fxvector-ref path´ ngbr-idx++)))
(define E´-pre-ngbr (dist (fxvector-ref path´ ngbr-idx) (fxvector-ref path´ ngbr-idx--)))
(vector-set! Es´ cand-idx E´-at-cand)
(vector-set! Es´ cand-idx-- E´-pre-cand)
(vector-set! Es´ ngbr-idx E´-at-ngbr)
(vector-set! Es´ ngbr-idx-- E´-pre-ngbr)
(define ΔE (- (+ E´-at-cand E´-pre-cand E´-at-ngbr E´-pre-ngbr) Σold-E-around-nodes))
(tsp path´ indices´ Es´
(if (zero? (modulo k 1000)) (for/sum ((e Es´)) e) (+ (tsp-ΣE t) ΔE))))]))
(define (tsp:random-state)
(indices->tsp (for/fxvector ((i (shuffle (range (current-dim²))))) i)))
(define (Simulated-annealing)
(define-values (solution solution-E)
(solve/SA (tsp:random-state)
tsp:swap-one-neighbour
(current-k-max)
temperature
tsp-ΣE
(current-monitor)))
(displayln (tsp-path solution))
(displayln solution-E))
(module+ main
(Simulated-annealing))
- Output:
T:1 E:552.4249706051347 T:0.9 E:204.89460292101052 T:0.8 E:178.6926191428981 T:0.7 E:157.77681824512447 T:0.6 E:145.91227208091533 T:0.5 E:127.16624235784029 T:0.4 E:119.56239369288322 T:0.3 E:111.92798007771523 T:0.2 E:102.24264068711928 T:0.1 E:101.65685424949237 #fx(67 68 78 88 98 99 89 79 69 59 49 48 38 39 29 19 9 8 7 17 18 28 27 37 36 26 25 15 16 6 5 4 3 12 13 14 24 34 44 54 53 43 33 23 22 32 31 21 11 2 1 0 10 20 30 40 41 42 52 62 61 51 50 60 70 71 81 80 90 91 92 93 94 84 83 82 72 73 63 64 74 75 65 55 45 35 46 47 58 57 56 66 76 86 85 95 96 97 87 77) 101.65685424949237
Raku
(formerly Perl 6)
# simulation setup
my \cities = 100; # number of cities
my \kmax = 1e6; # iterations to run
my \kT = 1; # initial 'temperature'
die 'City count must be a perfect square.' if cities.sqrt != cities.sqrt.Int;
# locations of (up to) 8 neighbors, with grid size derived from number of cities
my \gs = cities.sqrt;
my \neighbors = [1, -1, gs, -gs, gs-1, gs+1, -(gs+1), -(gs-1)];
# matrix of distances between cities
my \D = [;];
for 0 ..^ cities² -> \j {
my (\ab, \cd) = (j/cities, j%cities)».Int;
my (\a, \b, \c, \d) = (ab/gs, ab%gs, cd/gs, cd%gs)».Int;
D[ab;cd] = sqrt (a - c)² + (b - d)²
}
sub T(\k, \kmax, \kT) { (1 - k/kmax) × kT } # temperature function, decreases to 0
sub P(\ΔE, \k, \kmax, \kT) { exp( -ΔE / T(k, kmax, kT)) } # probability to move from s to s_next
sub Es(\path) { sum (D[ path[$_]; path[$_+1] ] for 0 ..^ +path-1) } # energy at s, to be minimized
# variation of E, from state s to state s_next
sub delta-E(\s, \u, \v) {
my (\a, \b, \c, \d) = D[s[u-1];s[u]], D[s[u+1];s[u]], D[s[v-1];s[v]], D[s[v+1];s[v]];
my (\na, \nb, \nc, \nd) = D[s[u-1];s[v]], D[s[u+1];s[v]], D[s[v-1];s[u]], D[s[v+1];s[u]];
if v == u+1 { return (na + nd) - (a + d) }
elsif u == v+1 { return (nc + nb) - (c + b) }
else { return (na + nb + nc + nd) - (a + b + c + d) }
}
# E(s0), initial state
my \s = @ = flat 0, (1 ..^ cities).pick(*), 0;
my \E-min-global = my \E-min = $ = Es(s);
for 0 ..^ kmax -> \k {
printf "k:%8u T:%4.1f Es: %3.1f\n" , k, T(k, kmax, kT), Es(s)
if k % (kmax/10) == 0;
# valid candidate cities (exist, adjacent)
my \cv = neighbors[(^8).roll] + s[ my \u = 1 + (^(cities-1)).roll ];
next if cv ≤ 0 or cv ≥ cities or D[s[u];cv] > sqrt(2);
my \v = s[cv];
my \ΔE = delta-E(s, u, v);
if ΔE < 0 or P(ΔE, k, kmax, kT) ≥ rand { # always move if negative
(s[u], s[v]) = (s[v], s[u]);
E-min += ΔE;
E-min-global = E-min if E-min < E-min-global;
}
}
say "\nE(s_final): " ~ E-min-global.fmt('%.1f');
say "Path:\n" ~ s».fmt('%2d').rotor(20,:partial).join: "\n";
- Output:
k: 0 T: 1.0 Es: 522.0 k: 100000 T: 0.9 Es: 185.3 k: 200000 T: 0.8 Es: 166.1 k: 300000 T: 0.7 Es: 174.7 k: 400000 T: 0.6 Es: 146.9 k: 500000 T: 0.5 Es: 140.2 k: 600000 T: 0.4 Es: 127.5 k: 700000 T: 0.3 Es: 115.9 k: 800000 T: 0.2 Es: 111.9 k: 900000 T: 0.1 Es: 109.4 E(s_final): 109.4 Path: 0 10 20 30 40 50 60 84 85 86 96 97 87 88 98 99 89 79 78 77 67 68 69 59 58 57 56 66 76 95 94 93 92 91 90 80 70 81 82 83 73 72 71 62 63 64 74 75 65 55 54 53 52 61 51 41 31 21 22 32 42 43 44 45 46 35 34 24 23 33 25 15 16 26 36 47 37 27 17 18 28 38 48 49 39 29 19 9 8 7 6 5 4 14 13 12 11 2 3 1 0
RATFOR
#
# The Rosetta Code simulated annealing task, in Ratfor 77.
#
# This implementation uses the RANDOM_NUMBER intrinsic and therefore
# will not work with f2c. It will work with gfortran. (One could
# substitute a random number generator from the Fullerton Function
# Library, or from elsewhere.)
#
function rndint (imin, imax)
implicit none
integer imin, imax, rndint
real rndnum
call random_number (rndnum)
rndint = imin + floor ((imax - imin + 1) * rndnum)
end
function icoord (loc)
implicit none
integer loc, icoord
icoord = loc / 10
end
function jcoord (loc)
implicit none
integer loc, jcoord
jcoord = mod (loc, 10)
end
function locatn (i, j) # Location.
implicit none
integer i, j, locatn
locatn = (10 * i) + j
end
subroutine rndpth (path) # Randomize a path.
implicit none
integer path(0:99)
integer rndint
integer i, j, xi, xj
for (i = 0; i <= 99; i = i + 1)
path(i) = i
# Fisher-Yates shuffle of elements 1 .. 99.
for (i = 1; i <= 98; i = i + 1)
{
j = rndint (i + 1, 99)
xi = path(i)
xj = path(j)
path(i) = xj
path(j) = xi
}
end
function dstnce (loc1, loc2) # Distance.
implicit none
integer loc1, loc2
real dstnce
integer icoord, jcoord
integer i1, j1
integer i2, j2
integer di, dj
i1 = icoord (loc1)
j1 = jcoord (loc1)
i2 = icoord (loc2)
j2 = jcoord (loc2)
di = i1 - i2
dj = j1 - j2
dstnce = sqrt (real ((di * di) + (dj * dj)))
end
function pthlen (path) # Path length.
implicit none
integer path(0:99)
real pthlen
real dstnce
real len
integer i
len = dstnce (path(0), path(99))
for (i = 0; i <= 98; i = i + 1)
len = len + dstnce (path(i), path(i + 1))
pthlen = len
end
subroutine addnbr (nbrs, numnbr, nbr) # Add neighbor.
implicit none
integer nbrs(1:8)
integer numnbr
integer nbr
if (nbr != 0)
{
numnbr = numnbr + 1
nbrs(numnbr) = nbr
}
end
subroutine fndnbr (loc, nbrs, numnbr) # Find neighbors.
implicit none
integer loc
integer nbrs(1:8)
integer numnbr
integer icoord, jcoord
integer locatn
integer i, j
integer c1, c2, c3, c4, c5, c6, c7, c8
c1 = 0
c2 = 0
c3 = 0
c4 = 0
c5 = 0
c6 = 0
c7 = 0
c8 = 0
i = icoord (loc)
j = jcoord (loc)
if (i < 9)
{
c1 = locatn (i + 1, j)
if (j < 9)
c2 = locatn (i + 1, j + 1)
if (0 < j)
c3 = locatn (i + 1, j - 1)
}
if (0 < i)
{
c4 = locatn (i - 1, j)
if (j < 9)
c5 = locatn (i - 1, j + 1)
if (0 < j)
c6 = locatn (i - 1, j - 1)
}
if (j < 9)
c7 = locatn (i, j + 1)
if (0 < j)
c8 = locatn (i, j - 1)
numnbr = 0
call addnbr (nbrs, numnbr, c1)
call addnbr (nbrs, numnbr, c2)
call addnbr (nbrs, numnbr, c3)
call addnbr (nbrs, numnbr, c4)
call addnbr (nbrs, numnbr, c5)
call addnbr (nbrs, numnbr, c6)
call addnbr (nbrs, numnbr, c7)
call addnbr (nbrs, numnbr, c8)
end
subroutine nbrpth (path, nbrp) # Make a neighbor path.
implicit none
integer path(0:99), nbrp(0:99)
integer rndint
integer u, v
integer nbrs(1:8)
integer numnbr
integer j, iu, iv
for (j = 0; j <= 99; j = j + 1)
nbrp(j) = path(j)
u = rndint (1, 99)
call fndnbr (u, nbrs, numnbr)
v = nbrs(rndint (1, numnbr))
j = 1
iu = 0
iv = 0
while (iu == 0 || iv == 0)
{
if (nbrp(j) == u)
iu = j
else if (nbrp(j) == v)
iv = j
j = j + 1
}
nbrp(iu) = v
nbrp(iv) = u
end
function temp (kT, kmax, k) # Temperature.
implicit none
real kT
integer kmax, k
real temp
real kf, kmaxf
kf = real (k)
kmaxf = real (kmax)
temp = kT * (1.0 - (kf / kmaxf))
end
function prob (deltaE, T) # Probability.
implicit none
real deltaE, T, prob
real x
if (T == 0.0)
prob = 0.0
else
{
x = -(deltaE / T)
if (x < -80)
prob = 0 # Avoid underflow.
else
prob = exp (-(deltaE / T))
}
end
subroutine show (k, T, E)
implicit none
integer k
real T, E
10 format (1X, I7, 1X, F7.1, 1X, F10.2)
write (*, 10) k, T, E
end
subroutine dsplay (path)
implicit none
integer path(0:99)
100 format (8(I2, ' -> '))
write (*, 100) path
end
subroutine sa (kT, kmax, path)
implicit none
real kT
integer kmax
integer path(0:99)
real pthlen
real temp, prob
integer kshow
integer k
integer j
real E, Etrial, T
integer trial(0:99)
real rndnum
kshow = kmax / 10
E = pthlen (path)
for (k = 0; k <= kmax; k = k + 1)
{
T = temp (kT, kmax, k)
if (mod (k, kshow) == 0)
call show (k, T, E)
call nbrpth (path, trial)
Etrial = pthlen (trial)
if (Etrial <= E)
{
for (j = 0; j <= 99; j = j + 1)
path(j) = trial(j)
E = Etrial
}
else
{
call random_number (rndnum)
if (rndnum <= prob (Etrial - E, T))
{
for (j = 0; j <= 99; j = j + 1)
path(j) = trial(j)
E = Etrial
}
}
}
end
program simanl
implicit none
real pthlen
integer path(0:99)
real kT
integer kmax
kT = 1.0
kmax = 1000000
10 format ()
20 format (' kT: ', F0.2)
30 format (' kmax: ', I0)
40 format (' k T E(s)')
50 format (' --------------------------')
60 format ('Final E(s): ', F0.2)
write (*, 10)
write (*, 20) kT
write (*, 30) kmax
write (*, 10)
write (*, 40)
write (*, 50)
call rndpth (path)
call sa (kT, kmax, path)
write (*, 10)
call dsplay (path)
write (*, 10)
write (*, 60) pthlen (path)
write (*, 10)
end
- Output:
An example run:
$ ratfor77 simanneal.r > sa.f && gfortran -O3 -std=legacy sa.f && ./a.out kT: 1.00 kmax: 1000000 k T E(s) -------------------------- 0 1.0 547.76 100000 0.9 190.62 200000 0.8 187.74 300000 0.7 171.72 400000 0.6 153.08 500000 0.5 131.15 600000 0.4 119.57 700000 0.3 111.20 800000 0.2 105.31 900000 0.1 103.07 1000000 0.0 102.89 0 -> 1 -> 2 -> 12 -> 11 -> 32 -> 33 -> 43 -> 42 -> 52 -> 51 -> 41 -> 31 -> 30 -> 40 -> 50 -> 60 -> 61 -> 62 -> 63 -> 53 -> 54 -> 44 -> 34 -> 24 -> 25 -> 14 -> 15 -> 16 -> 26 -> 36 -> 35 -> 45 -> 55 -> 56 -> 46 -> 47 -> 57 -> 58 -> 68 -> 67 -> 77 -> 86 -> 76 -> 66 -> 65 -> 64 -> 74 -> 75 -> 85 -> 84 -> 83 -> 73 -> 72 -> 71 -> 70 -> 80 -> 90 -> 91 -> 81 -> 82 -> 92 -> 93 -> 94 -> 95 -> 96 -> 97 -> 87 -> 98 -> 99 -> 89 -> 88 -> 78 -> 79 -> 69 -> 59 -> 49 -> 48 -> 39 -> 38 -> 37 -> 27 -> 17 -> 18 -> 28 -> 29 -> 19 -> 9 -> 8 -> 7 -> 6 -> 5 -> 4 -> 3 -> 13 -> 23 -> 22 -> 21 -> 20 -> 10 -> Final E(s): 102.89
Scheme
Note: Each line of the printed table gives E(s) before the next path is chosen. Thus the top line describes the initial state.
(cond-expand
(r7rs)
(chicken (import r7rs)))
(import (scheme base))
(import (scheme inexact))
(import (scheme write))
(import (only (srfi 1) delete))
(import (only (srfi 1) iota))
(import (srfi 27)) ; Random numbers.
;;
;; You can do without SRFI-144 by changing fl+ to +, etc.
;;
(import (srfi 144)) ; Optimizations for flonums.
(cond-expand
(chicken (import (format)))
(else))
(random-source-randomize! default-random-source)
(define (n->ij n)
(truncate/ n 10))
(define (ij->n i j)
(+ (* 10 i) j))
(define neighbor-offsets
'((0 . 1)
(1 . 0)
(1 . 1)
(0 . -1)
(-1 . 0)
(-1 . -1)
(1 . -1)
(-1 . 1)))
(define (neighborhood n)
(let-values (((i j) (n->ij n)))
(let recurs ((offsets neighbor-offsets))
(if (null? offsets)
'()
(let* ((offs (car offsets))
(i^ (+ i (car offs)))
(j^ (+ j (cdr offs))))
(if (and (not (negative? i^))
(not (negative? j^))
(< i^ 10)
(< j^ 10))
(cons (ij->n i^ j^) (recurs (cdr offsets)))
(recurs (cdr offsets))))))))
(define (distance m n)
(let-values (((im jm) (n->ij m))
((in jn) (n->ij n)))
(flsqrt (inexact (+ (square (- im in)) (square (- jm jn)))))))
(define (shuffle! vec i n)
;; A Fisher-Yates shuffle of n elements of vec, starting at index i.
(do ((j 0 (+ j 1)))
((= j n))
(let* ((k (+ i j (random-integer (- n j))))
(xi (vector-ref vec i))
(xk (vector-ref vec k)))
(vector-set! vec i xk)
(vector-set! vec k xi))))
(define (make-s0)
(let ((vec (list->vector (iota 100))))
(shuffle! vec 1 99)
vec))
(define (swap-s-elements! vec u v)
(let loop ((j 1)
(iu 0)
(iv 0))
(cond ((positive? iu)
(if (= (vector-ref vec j) v)
(begin (vector-set! vec iu v)
(vector-set! vec j u))
(loop (+ j 1) iu iv)))
((positive? iv)
(if (= (vector-ref vec j) u)
(begin (vector-set! vec j v)
(vector-set! vec iv u))
(loop (+ j 1) iu iv)))
((= (vector-ref vec j) u) (loop (+ j 1) j 0))
((= (vector-ref vec j) v) (loop (+ j 1) 0 j))
(else (loop (+ j 1) 0 0)))))
(define (update-s! vec)
(let* ((u (+ 1 (random-integer 99)))
(neighbors (delete 0 (neighborhood u) =))
(n (length neighbors))
(v (list-ref neighbors (random-integer n))))
(swap-s-elements! vec u v)))
(define (s->s vec) ; s_k -> s_(k + 1)
(let ((vec^ (vector-copy vec)))
(update-s! vec^)
vec^))
(define (path-length vec) ; E(s)
(let loop ((plen (distance (vector-ref vec 0)
(vector-ref vec 99)))
(x (vector-ref vec 0))
(i 1))
(if (= i 100)
plen
(let ((y (vector-ref vec i)))
(loop (fl+ plen (distance x y)) y (+ i 1))))))
(define (make-temperature-procedure kT kmax)
(let ((kT (inexact kT))
(kmax (inexact kmax)))
(lambda (k)
(fl* kT (fl- 1.0 (fl/ (inexact k) kmax))))))
(define (probability delta-E T)
(if (flnegative? delta-E)
1.0
(if (flzero? T)
0.0
(flexp (fl- (fl/ delta-E T))))))
(define fmt10 (string-append " k T E(s)~%"
" -----------------------------~%"))
(define fmt20 " ~7D ~3,1F ~12,5F~%")
(define (simulate-annealing kT kmax)
(let* ((temperature (make-temperature-procedure kT kmax))
(s0 (make-s0))
(E0 (path-length s0))
(kmax/10 (truncate-quotient kmax 10))
(show (lambda (k T E)
(if (zero? (truncate-remainder k kmax/10))
(cond-expand
(chicken (format #t fmt20 k T E))
(else (display k)
(display " ")
(display T)
(display " ")
(display E)
(newline)))))))
(cond-expand
(chicken (format #t fmt10))
(else))
(let loop ((k 0)
(s s0)
(E E0))
(if (= k (+ 1 kmax))
s
(let* ((T (temperature k))
(_ (show k T E))
(s^ (s->s s))
(E^ (path-length s^))
(delta-E (fl- E^ E))
(P (probability delta-E T)))
(if (or (fl=? P 1.0) (fl<=? (random-real) P))
(loop (+ k 1) s^ E^)
(loop (+ k 1) s E)))))))
(define (display-path vec)
(do ((i 0 (+ i 1)))
((= i 100))
(let ((x (vector-ref vec i)))
(when (< x 10)
(display " "))
(display x)
(display " -> ")
(when (= 7 (truncate-remainder i 8))
(newline))))
(let ((x (vector-ref vec 0)))
(when (< x 10)
(display " "))
(display x)))
(define kT 1)
(define kmax 1000000)
(newline)
(display " kT: ")
(display kT)
(newline)
(display " kmax: ")
(display kmax)
(newline)
(newline)
(define s-final (simulate-annealing kT kmax))
(newline)
(display "Final path:")
(newline)
(display-path s-final)
(newline)
(newline)
(cond-expand
(chicken (format #t "Final E(s): ~,5F~%" (path-length s-final)))
(else (display "Final E(s): ")
(display (path-length s-final))
(newline)))
(newline)
- Output:
An example run:
$ csc -O5 -X r7rs -R r7rs sa.scm && ./sa kT: 1 kmax: 1000000 k T E(s) ----------------------------- 0 1.0 422.71361 100000 0.9 185.28073 200000 0.8 171.70817 300000 0.7 156.66104 400000 0.6 145.07621 500000 0.5 130.88759 600000 0.4 115.34219 700000 0.3 112.27113 800000 0.2 105.37820 900000 0.1 103.89949 1000000 0.0 103.89949 Final path: 0 -> 1 -> 2 -> 3 -> 4 -> 6 -> 7 -> 8 -> 9 -> 19 -> 29 -> 39 -> 38 -> 37 -> 47 -> 48 -> 49 -> 58 -> 59 -> 69 -> 79 -> 89 -> 99 -> 98 -> 97 -> 96 -> 95 -> 94 -> 84 -> 83 -> 93 -> 92 -> 82 -> 72 -> 62 -> 71 -> 81 -> 91 -> 90 -> 80 -> 70 -> 60 -> 61 -> 50 -> 40 -> 41 -> 51 -> 52 -> 63 -> 73 -> 74 -> 75 -> 85 -> 86 -> 76 -> 77 -> 87 -> 88 -> 78 -> 68 -> 67 -> 57 -> 56 -> 66 -> 65 -> 64 -> 55 -> 45 -> 46 -> 36 -> 35 -> 25 -> 26 -> 27 -> 28 -> 18 -> 17 -> 16 -> 15 -> 5 -> 14 -> 13 -> 23 -> 24 -> 34 -> 44 -> 54 -> 53 -> 43 -> 33 -> 42 -> 32 -> 31 -> 30 -> 20 -> 21 -> 22 -> 12 -> 11 -> 10 -> 0 Final E(s): 103.89949
A different E(s)
Here E(s) is the sum of squares of differences, so that E(s) is an integer. Also I use kT=1.5 and twice as large a kmax.
(I also demonstrate some of SRFI 143. Note that CHICKEN has type annotations as an alternative to using SRFI 143 and SRFI 144, but the SRFI extensions are more portable.)
(cond-expand
(r7rs)
(chicken (import r7rs)))
(import (scheme base))
(import (scheme inexact))
(import (scheme write))
(import (only (srfi 1) delete))
(import (only (srfi 1) iota))
(import (srfi 27)) ; Random numbers.
;;
;; The following import is CHICKEN-specific, but your Scheme likely
;; has Common Lisp formatting somewhere.
;;
(import (format)) ; Common Lisp formatting.
;;
;; You can do without SRFI-143 or SRFI-144 by changing fx+ or fl+ to
;; +, etc.
;;
(import (srfi 143)) ; Optimizations for fixnums.
(import (srfi 144)) ; Optimizations for flonums.
(random-source-randomize! default-random-source)
(define (n->ij n)
(values (fxquotient n 10)
(fxremainder n 10)))
(define (ij->n i j)
(fx+ (fx* 10 i) j))
(define neighbor-offsets
'((0 . 1)
(1 . 0)
(1 . 1)
(0 . -1)
(-1 . 0)
(-1 . -1)
(1 . -1)
(-1 . 1)))
(define (neighborhood n)
(let-values (((i j) (n->ij n)))
(let recurs ((offsets neighbor-offsets))
(if (null? offsets)
'()
(let* ((offs (car offsets))
(i^ (fx+ i (car offs)))
(j^ (fx+ j (cdr offs))))
(if (and (not (fxnegative? i^))
(not (fxnegative? j^))
(fx<? i^ 10)
(fx<? j^ 10))
(cons (ij->n i^ j^) (recurs (cdr offsets)))
(recurs (cdr offsets))))))))
(define (distance**2 m n)
(let-values (((im jm) (n->ij m))
((in jn) (n->ij n)))
(fx+ (fxsquare (fx- im in)) (fxsquare (fx- jm jn)))))
(define (shuffle! vec i n)
;; A Fisher-Yates shuffle of n elements of vec, starting at index i.
(do ((j 0 (+ j 1)))
((= j n))
(let* ((k (+ i j (random-integer (- n j))))
(xi (vector-ref vec i))
(xk (vector-ref vec k)))
(vector-set! vec i xk)
(vector-set! vec k xi))))
(define (make-s0)
(let ((vec (list->vector (iota 100))))
(shuffle! vec 1 99)
vec))
(define (swap-s-elements! vec u v)
(let loop ((j 1)
(iu 0)
(iv 0))
(cond ((fxpositive? iu)
(if (fx=? (vector-ref vec j) v)
(begin (vector-set! vec iu v)
(vector-set! vec j u))
(loop (fx+ j 1) iu iv)))
((fxpositive? iv)
(if (fx=? (vector-ref vec j) u)
(begin (vector-set! vec j v)
(vector-set! vec iv u))
(loop (fx+ j 1) iu iv)))
((fx=? (vector-ref vec j) u) (loop (fx+ j 1) j 0))
((fx=? (vector-ref vec j) v) (loop (fx+ j 1) 0 j))
(else (loop (fx+ j 1) 0 0)))))
(define (update-s! vec)
(let* ((u (fx+ 1 (random-integer 99)))
(neighbors (delete 0 (neighborhood u) fx=?))
(n (length neighbors))
(v (list-ref neighbors (random-integer n))))
(swap-s-elements! vec u v)))
(define (s->s vec) ; s_k -> s_(k + 1)
(let ((vec^ (vector-copy vec)))
(update-s! vec^)
vec^))
(define (path-length vec)
(let loop ((plen (flsqrt (inexact
(distance**2 (vector-ref vec 0)
(vector-ref vec 99)))))
(x (vector-ref vec 0))
(i 1))
(if (fx=? i 100)
plen
(let ((y (vector-ref vec i)))
(loop (fl+ plen (flsqrt (inexact (distance**2 x y))))
y (fx+ i 1))))))
(define (E_s vec)
(let loop ((E (distance**2 (vector-ref vec 0)
(vector-ref vec 99)))
(x (vector-ref vec 0))
(i 1))
(if (fx=? i 100)
E
(let ((y (vector-ref vec i)))
(loop (fx+ E (distance**2 x y)) y (fx+ i 1))))))
(define (make-temperature-procedure kT kmax)
(let ((kT (inexact kT))
(kmax (inexact kmax)))
(lambda (k)
(fl* kT (fl- 1.0 (fl/ (inexact k) kmax))))))
(define (probability delta-E T)
(if (fxnegative? delta-E)
1.0
(if (flzero? T)
0.0
(flexp (fl- (fl/ (inexact delta-E) T))))))
(define fmt10 (string-append
" k T E(s) path length~%"
" ---------------------------------------~%"))
(define fmt20 " ~7D ~7,2F ~8D ~14,5F~%")
(define (simulate-annealing kT kmax)
(let* ((temperature (make-temperature-procedure kT kmax))
(s0 (make-s0))
(E0 (E_s s0))
(kmax/10 (fxquotient kmax 10))
(show (lambda (k T E s)
(when (fxzero? (fxremainder k kmax/10))
(format #t fmt20 k T E (path-length s))))))
(format #t fmt10)
(let loop ((k 0)
(s s0)
(E E0))
(if (fx=? k (fx+ 1 kmax))
s
(let* ((T (temperature k))
(_ (show k T E s))
(s^ (s->s s))
(E^ (E_s s^))
(delta-E (fx- E^ E))
(P (probability delta-E T)))
(if (or (fl=? P 1.0) (fl<=? (random-real) P))
(loop (fx+ k 1) s^ E^)
(loop (fx+ k 1) s E)))))))
(define (display-path vec)
(do ((i 0 (+ i 1)))
((= i 100))
(let ((x (vector-ref vec i)))
(when (< x 10)
(display " "))
(display x)
(display " -> ")
(when (= 7 (truncate-remainder i 8))
(newline))))
(let ((x (vector-ref vec 0)))
(when (< x 10)
(display " "))
(display x)))
(define kT 1.5)
(define kmax 2000000)
(newline)
(display " kT: ")
(display kT)
(newline)
(display " kmax: ")
(display kmax)
(newline)
(newline)
(define s-final (simulate-annealing kT kmax))
(newline)
(display "Final path:")
(newline)
(display-path s-final)
(newline)
(newline)
(format #t "Final E(s): ~,5F~%" (E_s s-final))
(format #t "Final path length: ~,5F~%" (path-length s-final))
(newline)
- Output:
$ csc -O5 -X r7rs -R r7rs sa2.scm && ./sa2 kT: 1.5 kmax: 2000000 k T E(s) path length --------------------------------------- 0 1.50 2298 384.59396 200000 1.35 180 127.94332 400000 1.20 168 124.60675 600000 1.05 148 118.07012 800000 0.90 130 111.94113 1000000 0.75 116 106.62742 1200000 0.60 114 105.55635 1400000 0.45 112 104.97056 1600000 0.30 104 101.65685 1800000 0.15 104 101.65685 2000000 0.00 104 101.65685 Final path: 0 -> 1 -> 2 -> 3 -> 4 -> 5 -> 15 -> 24 -> 34 -> 43 -> 44 -> 54 -> 53 -> 63 -> 64 -> 65 -> 66 -> 67 -> 77 -> 76 -> 75 -> 85 -> 84 -> 74 -> 73 -> 72 -> 62 -> 52 -> 42 -> 41 -> 31 -> 32 -> 33 -> 23 -> 13 -> 14 -> 25 -> 26 -> 27 -> 17 -> 16 -> 6 -> 7 -> 8 -> 9 -> 19 -> 18 -> 28 -> 29 -> 39 -> 49 -> 59 -> 58 -> 48 -> 38 -> 37 -> 47 -> 46 -> 36 -> 35 -> 45 -> 55 -> 56 -> 57 -> 68 -> 69 -> 79 -> 78 -> 88 -> 89 -> 99 -> 98 -> 97 -> 87 -> 86 -> 96 -> 95 -> 94 -> 93 -> 83 -> 82 -> 92 -> 91 -> 90 -> 80 -> 81 -> 71 -> 70 -> 60 -> 61 -> 51 -> 50 -> 40 -> 30 -> 20 -> 21 -> 22 -> 12 -> 11 -> 10 -> 0 Final E(s): 104.00000 Final path length: 101.65685
A second run shows E(s) temporarily increasing:
kT: 1.5 kmax: 2000000 k T E(s) path length --------------------------------------- 0 1.50 2132 368.24125 200000 1.35 142 115.58483 400000 1.20 146 116.75641 600000 1.05 148 117.40669 800000 0.90 124 109.27770 1000000 0.75 112 104.97056 1200000 0.60 124 109.45584 1400000 0.45 114 105.55635 1600000 0.30 108 103.31371 1800000 0.15 108 103.31371 2000000 0.00 108 103.31371 Final path: 0 -> 1 -> 11 -> 21 -> 31 -> 42 -> 52 -> 62 -> 72 -> 73 -> 63 -> 74 -> 64 -> 65 -> 55 -> 45 -> 54 -> 53 -> 43 -> 44 -> 34 -> 35 -> 36 -> 26 -> 25 -> 24 -> 23 -> 33 -> 32 -> 22 -> 12 -> 2 -> 3 -> 13 -> 14 -> 4 -> 5 -> 15 -> 16 -> 6 -> 7 -> 17 -> 27 -> 28 -> 18 -> 8 -> 9 -> 19 -> 29 -> 39 -> 38 -> 37 -> 47 -> 48 -> 49 -> 58 -> 59 -> 69 -> 68 -> 67 -> 57 -> 46 -> 56 -> 66 -> 76 -> 86 -> 87 -> 77 -> 78 -> 79 -> 89 -> 99 -> 88 -> 98 -> 97 -> 96 -> 95 -> 85 -> 75 -> 84 -> 94 -> 93 -> 83 -> 82 -> 92 -> 91 -> 90 -> 80 -> 81 -> 71 -> 70 -> 60 -> 61 -> 51 -> 50 -> 41 -> 40 -> 30 -> 20 -> 10 -> 0 Final E(s): 108.00000 Final path length: 103.31371
A third run shows E(s) temporarily increasing, and also achieves a path length less than 101:
kT: 1.5 kmax: 2000000 k T E(s) path length --------------------------------------- 0 1.50 2246 388.77550 200000 1.35 176 124.95651 400000 1.20 160 121.22854 600000 1.05 148 117.29304 800000 0.90 126 109.86348 1000000 0.75 118 107.45584 1200000 0.60 120 108.04163 1400000 0.45 108 103.31371 1600000 0.30 106 102.48528 1800000 0.15 102 100.82843 2000000 0.00 102 100.82843 Final path: 0 -> 1 -> 11 -> 12 -> 2 -> 3 -> 13 -> 14 -> 4 -> 5 -> 15 -> 16 -> 6 -> 7 -> 17 -> 27 -> 28 -> 18 -> 8 -> 9 -> 19 -> 29 -> 39 -> 38 -> 48 -> 49 -> 59 -> 69 -> 79 -> 78 -> 77 -> 67 -> 68 -> 58 -> 57 -> 47 -> 37 -> 36 -> 26 -> 25 -> 24 -> 23 -> 22 -> 21 -> 31 -> 32 -> 43 -> 44 -> 54 -> 53 -> 52 -> 62 -> 61 -> 51 -> 41 -> 42 -> 33 -> 34 -> 35 -> 45 -> 46 -> 56 -> 55 -> 65 -> 66 -> 76 -> 86 -> 87 -> 88 -> 89 -> 99 -> 98 -> 97 -> 96 -> 95 -> 94 -> 84 -> 85 -> 75 -> 74 -> 64 -> 63 -> 73 -> 83 -> 93 -> 92 -> 82 -> 72 -> 71 -> 81 -> 91 -> 90 -> 80 -> 70 -> 60 -> 50 -> 40 -> 30 -> 20 -> 10 -> 0 Final E(s): 102.00000 Final path length: 100.82843
Sidef
module TravelingSalesman {
# Eₛ: length(path)
func Eₛ(distances, path) {
var total = 0
[path, path.slice(1)].zip {|ci,cj|
total += distances[ci-1][cj-1]
}
total
}
# T: temperature
func T(k, kmax, kT) { kT * (1 - k/kmax) }
# ΔE = Eₛ_new - Eₛ_old > 0
# Prob. to move if ΔE > 0, → 0 when T → 0 (fronzen state)
func P(ΔE, k, kmax, kT) { exp(-ΔE / T(k, kmax, kT)) }
# ∆E from path ( .. a u b .. c v d ..) to (.. a v b ... c u d ..)
# ∆E before swapping (u,v)
# Quicker than Eₛ(s_next) - Eₛ(path)
func dE(distances, path, u, v) {
var a = distances[path[u-1]-1][path[u]-1]
var b = distances[path[u+1]-1][path[u]-1]
var c = distances[path[v-1]-1][path[v]-1]
var d = distances[path[v+1]-1][path[v]-1]
var na = distances[path[u-1]-1][path[v]-1]
var nb = distances[path[u+1]-1][path[v]-1]
var nc = distances[path[v-1]-1][path[u]-1]
var nd = distances[path[v+1]-1][path[u]-1]
if (v == u+1) {
return ((na+nd) - (a+d))
}
if (u == v+1) {
return ((nc+nb) - (c+b))
}
return ((na+nb+nc+nd) - (a+b+c+d))
}
const dirs = [1, -1, 10, -10, 9, 11, -11, -9]
func _prettypath(path) {
path.slices(10).map { .map{ "%3s" % _ }.join(', ') }.join("\n")
}
func findpath(distances, kmax, kT) {
const n = distances.len
const R = 2..n
var path = [1, R.shuffle..., 1]
var Emin = Eₛ(distances, path)
printf("# Entropy(s₀) = s%10.2f\n", Emin)
printf("# Random path:\n%s\n\n", _prettypath(path))
for k in (1 .. kmax) {
if (k % (kmax//10) == 0) {
printf("k: %10d | T: %8.4f | Eₛ: %8.4f\n", k, T(k, kmax, kT), Eₛ(distances, path))
}
var u = R.rand
var v = (path[u-1] + dirs.rand)
v ~~ R || next
var δE = dE(distances, path, u-1, v-1)
if ((δE < 0) || (P(δE, k, kmax, kT) >= 1.rand)) {
path.swap(u-1, v-1)
Emin += δE
}
}
printf("k: %10d | T: %8.4f | Eₛ: %8.4f\n", kmax, T(kmax, kmax, kT), Eₛ(distances, path))
say ("\n# Found path:\n", _prettypath(path))
return path
}
}
var citydist = {|ci|
{ |cj|
var v1 = Vec(ci%10, ci//10)
var v2 = Vec(cj%10, cj//10)
v1.dist(v2)
}.map(1..100)
}.map(1..100)
TravelingSalesman::findpath(citydist, 1e6, 1)
- Output:
# Entropy(s₀) = 520.29 # Random path: 1, 10, 79, 52, 24, 9, 58, 11, 42, 4 15, 87, 62, 88, 21, 91, 99, 84, 61, 14 5, 17, 33, 95, 74, 31, 40, 13, 37, 69 6, 22, 97, 45, 56, 63, 75, 83, 53, 41 3, 47, 89, 80, 78, 98, 46, 18, 25, 51 93, 16, 50, 30, 48, 8, 66, 68, 59, 73 49, 96, 36, 32, 100, 27, 76, 44, 64, 39 90, 82, 20, 12, 54, 86, 29, 81, 26, 72 60, 94, 35, 92, 43, 7, 85, 55, 28, 57 23, 34, 65, 71, 38, 2, 77, 70, 19, 67 1 k: 100000 | T: 0.9000 | Eₛ: 185.1809 k: 200000 | T: 0.8000 | Eₛ: 168.6262 k: 300000 | T: 0.7000 | Eₛ: 146.5948 k: 400000 | T: 0.6000 | Eₛ: 140.1441 k: 500000 | T: 0.5000 | Eₛ: 129.5132 k: 600000 | T: 0.4000 | Eₛ: 132.8942 k: 700000 | T: 0.3000 | Eₛ: 124.2865 k: 800000 | T: 0.2000 | Eₛ: 120.0859 k: 900000 | T: 0.1000 | Eₛ: 115.0771 k: 1000000 | T: 0.0000 | Eₛ: 114.9728 k: 1000000 | T: 0.0000 | Eₛ: 114.9728 # Found path: 1, 2, 13, 3, 4, 5, 6, 7, 8, 9 19, 29, 18, 28, 27, 17, 16, 26, 25, 15 14, 24, 23, 12, 11, 10, 20, 21, 30, 40 41, 31, 32, 44, 45, 46, 47, 48, 49, 39 38, 37, 36, 35, 34, 42, 51, 50, 60, 61 52, 53, 54, 55, 56, 57, 58, 59, 69, 68 77, 67, 66, 65, 64, 62, 72, 71, 70, 80 81, 82, 74, 75, 76, 87, 88, 78, 79, 89 99, 98, 97, 96, 86, 85, 83, 91, 90, 100 92, 93, 94, 95, 84, 73, 63, 43, 33, 22 1
Wren
import "random" for Random
import "/math" for Math
import "/fmt" for Fmt
// distances
var calcDists = Fn.new {
var dists = List.filled(10000, 0)
for (i in 0..9999) {
var ab = (i/100).floor
var cd = i % 100
var a = (ab/10).floor
var b = ab % 10
var c = (cd/10).floor
var d = cd % 10
dists[i] = Math.hypot(a-c, b-d)
}
return dists
}
var dists = calcDists.call()
var dirs = [1, -1, 10, -10, 9, 11, -11, -9] // all 8 neighbors
var rand = Random.new()
// index into lookup table of Nums
var dist = Fn.new { |ci, cj| dists[cj*100 + ci] }
// energy at s, to be minimized
var Es = Fn.new { |path|
var d = 0
for (i in 0...path.count-1) d = d + dist.call(path[i], path[i+1])
return d
}
// temperature function, decreases to 0
var T = Fn.new { |k, kmax, kT| (1 - k / kmax) * kT }
// variation of E, from state s to state s_next
var dE = Fn.new { |s, u, v|
var su = s[u]
var sv = s[v]
// old
var a = dist.call(s[u-1], su)
var b = dist.call(s[u+1], su)
var c = dist.call(s[v-1], sv)
var d = dist.call(s[v+1], sv)
// new
var na = dist.call(s[u-1], sv)
var nb = dist.call(s[u+1], sv)
var nc = dist.call(s[v-1], su)
var nd = dist.call(s[v+1], su)
if (v == u+1) return (na + nd) - (a + d)
if (u == v+1) return (nc + nb) - (c + b)
return (na + nb + nc + nd) - (a + b + c + d)
}
// probability to move from s to s_next
var P = Fn.new { |deltaE, k, kmax, kT| (-deltaE / T.call(k, kmax, kT)).exp }
// Simulated annealing
var sa = Fn.new { |kmax, kT|
var temp = List.filled(99, 0)
for (i in 0..98) temp[i] = i + 1
rand.shuffle(temp)
var s = List.filled(101, 0)
for (i in 0..98) s[i+1] = temp[i] // random path from 0 to 0
System.print("kT = %(kT)")
System.print("E(s0) %(Es.call(s))\n") // random starter
var Emin = Es.call(s) // E0
for (k in 0..kmax) {
if (k % (kmax/10).floor == 0) {
Fmt.print("k:$10d T: $8.4f Es: $8.4f", k, T.call(k, kmax, kT), Es.call(s))
}
var u = rand.int(1, 100) // city index 1 to 99
var cv = s[u] + dirs[rand.int(8)] // city number
if (cv <= 0 || cv >= 100) { // bogus city
continue
}
if (dist.call(s[u], cv) > 5) { // check true neighbor (eg 0 9)
continue
}
var v = s[cv] // city index
var deltae = dE.call(s, u, v)
if (deltae < 0 || // always move if negative
P.call(deltae, k, kmax, kT) >= rand.float()) {
s.swap(u, v)
Emin = Emin + deltae
}
}
System.print("\nE(s_final) %(Emin)")
System.print("Path:")
// output final state
for (i in 0...s.count) {
if (i > 0 && i % 10 == 0) System.print()
Fmt.write("$4d", s[i])
}
System.print()
}
sa.call(1e6, 1)
- Output:
Sample run:
kT = 1 E(s0) 541.82779520458 k: 0 T: 1.0000 Es: 541.8278 k: 100000 T: 0.9000 Es: 187.1429 k: 200000 T: 0.8000 Es: 191.0983 k: 300000 T: 0.7000 Es: 171.7284 k: 400000 T: 0.6000 Es: 154.0549 k: 500000 T: 0.5000 Es: 147.0249 k: 600000 T: 0.4000 Es: 123.5822 k: 700000 T: 0.3000 Es: 121.5808 k: 800000 T: 0.2000 Es: 114.0930 k: 900000 T: 0.1000 Es: 112.6788 k: 1000000 T: 0.0000 Es: 112.6788 E(s_final) 112.67876668098 Path: 0 11 10 1 2 12 13 24 25 15 14 3 4 5 6 8 9 19 18 28 29 39 38 27 17 7 16 26 36 37 47 46 45 35 34 44 43 33 23 22 32 53 52 51 41 31 21 20 30 40 50 60 61 70 80 90 91 92 93 73 63 62 72 71 81 82 83 84 94 95 85 86 96 97 98 99 89 79 69 59 49 48 58 57 68 78 88 87 77 67 56 55 54 65 66 76 75 74 64 42 0
zkl
var [const] _dists=(0d10_000).pump(List,fcn(abcd){ // two points (a,b) & (c,d), calc distance
ab,cd,a,b,c,d:=abcd/100, abcd%100, ab/10,ab%10, cd/10,cd%10;
(a-c).toFloat().hypot(b-d)
});
fcn dist(ci,cj){ _dists[cj*100 + ci] } // index into lookup table of floats
fcn Es(path) // E(s) = length(path): E(a,b,c)--> dist(a,b) + dist(b,c)
{ d:=Ref(0.0); path.reduce('wrap(a,b){ d.apply('+,dist(a,b)); b }); d.value }
// temperature() function
fcn T(k,kmax,kT){ (1.0 - k.toFloat()/kmax)*kT }
// deltaE = Es_new - Es_old > 0
// probability to move if deltaE > 0, -->0 when T --> 0 (frozen state)
fcn P(deltaE,k,kmax,kT){ (-deltaE/T(k,kmax,kT)).exp() } //-->Float
// deltaE from path ( .. a u b .. c v d ..) to (.. a v b ... c u d ..)
// deltaE before swapping (u,v)
fcn dE(s,u,v){ su,sv:=s[u],s[v]; //-->Float
// old
a,b,c,d:=dist(s[u-1],su), dist(s[u+1],su), dist(s[v-1],sv), dist(s[v+1],sv);
// new
na,nb,nc,nd:=dist(s[u-1],sv), dist(s[u+1],sv), dist(s[v-1],su), dist(s[v+1],su);
if (v==u+1) (na+nd) - (a+d);
else if(u==v+1) (nc+nb) - (c+b);
else (na+nb+nc+nd) - (a+b+c+d);
}
// all 8 neighbours
var [const] dirs=ROList(1, -1, 10, -10, 9, 11, -11, -9),
fmt="k:%10,d T: %8.4f Es: %8.4f".fmt; // since we use it twice
fcn sa(kmax,kT=10){
s:=List(0, [1..99].walk().shuffle().xplode(), 0); // random path from 0 to 0
println("E(s0) %f".fmt(Es(s))); // random starter
Emin:=Es(s); // E0
foreach k in (kmax){
if(0==k%(kmax/10)) println(fmt(k,T(k,kmax,kT),Es(s)));
u:=(1).random(100); // city index 1 99
cv:=s[u] + dirs[(0).random(8)]; // city number
if(not (0<cv<100)) continue; // bogus city
if(dist(s[u],cv)>5) continue; // check true neighbour (eg 0 9)
v:=s.index(cv,1); // city index
deltae:=dE(s,u,v);
if(deltae<0 or // always move if negative
P(deltae,k,kmax,kT)>=(0.0).random(1)){
s.swap(u,v);
Emin+=deltae;
}
// (assert (= (round Emin) (round (Es s))))
}//foreach
println(fmt(kmax,T(kmax-1,kmax,kT),Es(s)));
println("E(s_final) %f".fmt(Emin));
println("Path: ",s.toString(*));
}
sa(0d1_000_000,1);
- Output:
E(s0) 540.897080 k: 0 T: 1.0000 Es: 540.8971 k: 100,000 T: 0.9000 Es: 181.5102 k: 200,000 T: 0.8000 Es: 167.1944 k: 300,000 T: 0.7000 Es: 159.0975 k: 400,000 T: 0.6000 Es: 170.2344 k: 500,000 T: 0.5000 Es: 130.9919 k: 600,000 T: 0.4000 Es: 115.3422 k: 700,000 T: 0.3000 Es: 113.9280 k: 800,000 T: 0.2000 Es: 106.7924 k: 900,000 T: 0.1000 Es: 103.7213 k: 1,000,000 T: 0.0000 Es: 103.7213 E(s_final) 103.721349 Path: L(0,10,11,21,20,30,40,50,60,70,80,81,71,72,73,63,52,62,61,51,41,31,32,22,12,13,14,15,25,16,17,18,28,27,26,36,35,45,34,24,23,33,42,43,44,54,53,64,74,84,83,82,90,91,92,93,94,95,85,86,96,97,87,88,98,99,89,79,69,68,78,77,67,66,76,75,65,55,56,46,37,38,48,47,57,58,59,49,39,29,19,9,8,7,6,5,4,3,2,1,0)