Maximum triangle path sum

Starting from the top of a pyramid of numbers like this, you can walk down going one step on the right or on the left, until you reach the bottom row:

                          55
                        94 48
                       95 30 96
                     77 71 26 67

One of such walks is 55 - 94 - 30 - 26. You can compute the total of the numbers you have seen in such walk, in this case it's 205.

Your problem is to find the maximum total among all possible paths from the top to the bottom row of the triangle. In the little example above it's 321.

Task: find the maximum total in the triangle below:

                          55
                        94 48
                       95 30 96
                     77 71 26 67
                    97 13 76 38 45
                  07 36 79 16 37 68
                 48 07 09 18 70 26 06
               18 72 79 46 59 79 29 90
              20 76 87 11 32 07 07 49 18
            27 83 58 35 71 11 25 57 29 85
           14 64 36 96 27 11 58 56 92 18 55
         02 90 03 60 48 49 41 46 33 36 47 23
        92 50 48 02 36 59 42 79 72 20 82 77 42
      56 78 38 80 39 75 02 71 66 66 01 03 55 72
     44 25 67 84 71 67 11 61 40 57 58 89 40 56 36
   85 32 25 85 57 48 84 35 47 62 17 01 01 99 89 52
  06 71 28 75 94 48 37 10 23 51 06 48 53 18 74 98 15
27 02 92 23 08 71 76 84 15 52 92 63 81 10 44 10 69 93

Such numbers can be included in the solution code, or read from a "triangle.txt" file.

This task is derived from the Euler Problem #18.

Haskell

<lang haskell>parse = map (map read . words) . lines f x y z = x + max y z g xs ys = zipWith3 f xs ys $ tail ys solve = head . foldr1 g main = readFile "triangle.txt" >>= print . solve . parse</lang>

Output:

The Z+ and Zmax are examples of the zipwith metaoperator. We ought to be able to use [Z+]= as an assignment operator here, but rakudo has a bug. Note also we can use the Zmax metaoperator form because max is define as an infix in Perl 6. <lang perl6>my @rows = slurp("triangle.txt").lines.map: { [.words] }

while @rows > 1 {

   my @last := @rows.pop;
   @rows[*-1] = @rows[*-1] Z+ (@last Zmax @last[1..*]);

}

say @rows;</lang>

Output:

Here's a more FPish version with the same output. We define our own operator and the use it in the reduction metaoperator form, [op], which turns any infix into a list operator. <lang perl6>sub infix:<op>(@a,@b) { (@a Zmax @a[1..*]) Z+ @b }

say [op] slurp("triangle.txt").lines.reverse.map: { [.words] }</lang> Instead of using reverse, one could also define the op as right-associative. <lang perl6>sub infix:<op>(@a,@b) is assoc('right') { @a Z+ (@b Zmax @b[1..*]) }

say [op] slurp("triangle.txt").lines.map: { [.words] }</lang>

Python

A simple mostly imperative solution: <lang python>def solve(tri):

   while len(tri) > 1:
       t0 = tri.pop()
       t1 = tri.pop()
       tri.append([max(t0[i], t0[i+1]) + t for i,t in enumerate(t1)])
   return tri[0][0]

def main():

   data = """\
                         55
                       94 48
                      95 30 96
                    77 71 26 67
                   97 13 76 38 45
                 07 36 79 16 37 68
                48 07 09 18 70 26 06
              18 72 79 46 59 79 29 90
             20 76 87 11 32 07 07 49 18
           27 83 58 35 71 11 25 57 29 85
          14 64 36 96 27 11 58 56 92 18 55
        02 90 03 60 48 49 41 46 33 36 47 23
       92 50 48 02 36 59 42 79 72 20 82 77 42
     56 78 38 80 39 75 02 71 66 66 01 03 55 72
    44 25 67 84 71 67 11 61 40 57 58 89 40 56 36
  85 32 25 85 57 48 84 35 47 62 17 01 01 99 89 52
 06 71 28 75 94 48 37 10 23 51 06 48 53 18 74 98 15

27 02 92 23 08 71 76 84 15 52 92 63 81 10 44 10 69 93"""

   print solve([map(int, row.split()) for row in data.splitlines()])

main()</lang>

Output:

A more functional version, similar to the Haskell entry (same output): <lang python>from itertools import imap

f = lambda x, y, z: x + max(y, z) g = lambda xs, ys: list(imap(f, ys, xs, xs[1:])) data = [map(int, row.split()) for row in open("triangle.txt")][::-1] print reduce(g, data)[0]</lang>

REXX

<lang rexx>/*REXX program finds the max sum of a "column" of numbers in a triangle.*/ @. = @.1 = 55 @.2 = 94 48 @.3 = 95 30 96 @.4 = 77 71 26 67 @.5 = 97 13 76 38 45 @.6 = 07 36 79 16 37 68 @.7 = 48 07 09 18 70 26 06 @.8 = 18 72 79 46 59 79 29 90 @.9 = 20 76 87 11 32 07 07 49 18 @.10 = 27 83 58 35 71 11 25 57 29 85 @.11 = 14 64 36 96 27 11 58 56 92 18 55 @.12 = 02 90 03 60 48 49 41 46 33 36 47 23 @.13 = 92 50 48 02 36 59 42 79 72 20 82 77 42 @.14 = 56 78 38 80 39 75 02 71 66 66 01 03 55 72 @.15 = 44 25 67 84 71 67 11 61 40 57 58 89 40 56 36 @.16 = 85 32 25 85 57 48 84 35 47 62 17 01 01 99 89 52 @.17 = 06 71 28 75 94 48 37 10 23 51 06 48 53 18 74 98 15 @.18 = 27 02 92 23 08 71 76 84 15 52 92 63 81 10 44 10 69 93

. = 0

        do r=1  while  @.r\==       /*build a version of the triangle*/
            do k=1  for words(@.r)    /*build a row, number by number. */
            #.r.k=word(@.r,k)         /*assign a number to an array num*/
            end   /*k*/
        end       /*r*/

rows=r-1 /*compute the number of rows. */

        do r=rows  by -1 to 2; p=r-1  /*traipse through triangle rows. */
            do k=1  for p;     kn=k+1 /*re-calculate the previous row. */
            #.p.k=max(#.r.k,#.r.kn)+#.p.k   /*maybe replace the prev #.*/
            end   /*k*/
        end       /*r*/

say 'maximum path sum:' #.1.1 /*display the top (row 1) number.*/

                                      /*stick a fork in it, we're done.*/</lang>

output using the data within the REXX program:

maximum path sum: 1320

D

Haskell

Perl 6

Python

REXX