Knapsack problem/0-1

Knapsack problem/0-1
You are encouraged to solve this task according to the task description, using any language you may know.

See also: Knapsack problem/Unbounded, Knapsack problem/Bounded

A tourist wants to make a good trip at the weekend with his friends. They will go to the mountains to see the wonders of nature, so he needs to pack well for the trip. He has a good knapsack for carrying things, but knows that he can carry a maximum of only 4kg in it and it will have to last the whole day. He creates a list of what he wants to bring for the trip but the total weight of all items is too much. He then decides to add columns to his initial list detailing their weights and a numerical "value" representing how important the item is for the tour.

Here is the list:

Table of potential knapsack items

Item	Weight (dag)	Value
map	9	150
compass	13	35
water	153	200
sandwich	50	160
glucose	15	60
tin	68	45
banana	27	60
apple	39	40
cheese	23	30
beer	52	10
suntan cream	11	70
camera	32	30
t-shirt	24	15
trousers	48	10
umbrella	73	40
waterproof trousers	42	70
waterproof overclothes	43	75
note-case	22	80
sunglasses	7	20
towel	18	12
socks	4	50
book	30	10
Knapsack	<=400 dag	?

The tourist can choose to take any combination of items from the list, but only one of each item is available. He do not may cut the items, so he can only take whole units of any item.

Which items does the tourist carry in his knapsack so that their total weight does not exceed 4 kg, and their total value is maximised?

D

D V.2 code, from the Python solution. <lang d>import std.stdio: writeln; import std.algorithm: max; import std.typecons: tuple;

struct Item {

   string name;
   int weight, value;

}

Item[] knapsack01_din_prog(const Item[] items, int limit) {

   auto tab = new int[][](items.length + 1, limit + 1);

   foreach (j; 1 .. items.length + 1) {
       int wt = items[j-1].weight; // no handy tuple unpacking
       int val = items[j-1].value;
       foreach (w; 1 .. limit + 1)
           tab[j][w] = (wt > w) ? tab[j-1][w] : max(tab[j-1][w], tab[j-1][w-wt] + val);
   }

   Item[] result;
   int w = limit;
   for (int j = items.length; j > 0; j--) // foreach can't be used here
       if (tab[j][w] != tab[j-1][w]) {
           w -= items[j-1].weight;
           result ~= items[j-1];
       }

   return result;

}

/// Totalise a particular combination of items auto solution_value(const Item[] comb, int limit) {

   int tot_w, tot_v;
   foreach (item; comb) {
       tot_w += item.weight;
       tot_v += item.value;
   }
   return (tot_w <= limit) ? tuple(tot_v, tot_w) : tuple(0, 0);

}

void main() {

   enum Item[] items = [
       {"map", 9, 150},
       {"compass", 13, 35},
       {"water", 153, 200},
       {"sandwich", 50, 160},
       {"glucose", 15, 60},
       {"tin", 68, 45},
       {"banana", 27, 60},
       {"apple", 39, 40},
       {"cheese", 23, 30},
       {"beer", 52, 10},
       {"suntan cream", 11, 70},
       {"camera", 32, 30},
       {"t-shirt", 24, 15},
       {"trousers", 48, 10},
       {"umbrella", 73, 40},
       {"waterproof trousers", 42, 70},
       {"waterproof overclothes", 43, 75},
       {"note-case", 22, 80},
       {"sunglasses", 7, 20},
       {"towel", 18, 12},
       {"socks", 4, 50},
       {"book", 30, 10}];

   auto bagged = knapsack01_din_prog(items, 400);
   writeln("Bagged the following items:");
   foreach (item; bagged.sort)
       writeln(" ", item.name);
   auto val_wt = solution_value(bagged, 400);
   writeln("For a total value of ", val_wt.field[0],
           " and a total weight of ", val_wt.field[1]);

}</lang> Output:

Bagged the following items:
 map
 socks
 water
 banana
 compass
 glucose
 sandwich
 note-case
 sunglasses
 suntan cream
 waterproof trousers
 waterproof overclothes
For a total value of 1030 and a total weight of 396

Java

General dynamic solution after wikipedia.

<lang java> package hu.pj.alg.test;

import hu.pj.alg.ZeroOneKnapsack; import hu.pj.obj.Item; import java.util.*; import java.text.*;

public class ZeroOneKnapsackForTourists {

   public ZeroOneKnapsackForTourists() {
       ZeroOneKnapsack zok = new ZeroOneKnapsack(400); // 400 dkg = 400 dag = 4 kg

       // making the list of items that you want to bring
       zok.add("map", 9, 150);
       zok.add("compass", 13, 35);
       zok.add("water", 153, 200);
       zok.add("sandwich", 50, 160);
       zok.add("glucose", 15, 60);
       zok.add("tin", 68, 45);
       zok.add("banana", 27, 60);
       zok.add("apple", 39, 40);
       zok.add("cheese", 23, 30);
       zok.add("beer", 52, 10);
       zok.add("suntan cream", 11, 70);
       zok.add("camera", 32, 30);
       zok.add("t-shirt", 24, 15);
       zok.add("trousers", 48, 10);
       zok.add("umbrella", 73, 40);
       zok.add("waterproof trousers", 42, 70);
       zok.add("waterproof overclothes", 43, 75);
       zok.add("note-case", 22, 80);
       zok.add("sunglasses", 7, 20);
       zok.add("towel", 18, 12);
       zok.add("socks", 4, 50);
       zok.add("book", 30, 10);

       // calculate the solution:
       List<Item> itemList = zok.calcSolution();

       // write out the solution in the standard output
       if (zok.isCalculated()) {
           NumberFormat nf  = NumberFormat.getInstance();

           System.out.println(
               "Maximal weight           = " +
               nf.format(zok.getMaxWeight() / 100.0) + " kg"
           );
           System.out.println(
               "Total weight of solution = " +
               nf.format(zok.getSolutionWeight() / 100.0) + " kg"
           );
           System.out.println(
               "Total value              = " +
               zok.getProfit()
           );
           System.out.println();
           System.out.println(
               "You can carry te following materials " +
               "in the knapsack:"
           );
           for (Item item : itemList) {
               if (item.getInKnapsack() == 1) {
                   System.out.format(
                       "%1$-23s %2$-3s %3$-5s %4$-15s \n",
                       item.getName(),
                       item.getWeight(), "dag  ",
                       "(value = " + item.getValue() + ")"
                   );
               }
           }
       } else {
           System.out.println(
               "The problem is not solved. " +
               "Maybe you gave wrong data."
           );
       }

   }

   public static void main(String[] args) {
       new ZeroOneKnapsackForTourists();
   }

} // class </lang>

<lang java> package hu.pj.alg;

import hu.pj.obj.Item; import java.util.*;

public class ZeroOneKnapsack {

   protected List<Item> itemList  = new ArrayList<Item>();
   protected int maxWeight        = 0;
   protected int solutionWeight   = 0;
   protected int profit           = 0;
   protected boolean calculated   = false;

   public ZeroOneKnapsack() {}

   public ZeroOneKnapsack(int _maxWeight) {
       setMaxWeight(_maxWeight);
   }

   public ZeroOneKnapsack(List<Item> _itemList) {
       setItemList(_itemList);
   }

   public ZeroOneKnapsack(List<Item> _itemList, int _maxWeight) {
       setItemList(_itemList);
       setMaxWeight(_maxWeight);
   }

   // calculte the solution of 0-1 knapsack problem with dynamic method:
   public List<Item> calcSolution() {
       int n = itemList.size();

       setInitialStateForCalculation();
       if (n > 0  &&  maxWeight > 0) {
           List< List<Integer> > c = new ArrayList< List<Integer> >();
           List<Integer> curr = new ArrayList<Integer>();

           c.add(curr);
           for (int j = 0; j <= maxWeight; j++)
               curr.add(0);
           for (int i = 1; i <= n; i++) {
               List<Integer> prev = curr;
               c.add(curr = new ArrayList<Integer>());
               for (int j = 0; j <= maxWeight; j++) {
                   if (j > 0) {
                       int wH = itemList.get(i-1).getWeight();
                       curr.add(
                           (wH > j)
                           ?
                           prev.get(j)
                           :
                           Math.max(
                               prev.get(j),
                               itemList.get(i-1).getValue() + prev.get(j-wH)
                           )
                       );
                   } else {
                       curr.add(0);
                   }
               } // for (j...)
           } // for (i...)
           profit = curr.get(maxWeight);

           for (int i = n, j = maxWeight; i > 0  &&  j >= 0; i--) {
               int tempI   = c.get(i).get(j);
               int tempI_1 = c.get(i-1).get(j);
               if (
                   (i == 0  &&  tempI > 0)
                   ||
                   (i > 0  &&  tempI != tempI_1)
               )
               {
                   Item iH = itemList.get(i-1);
                   int  wH = iH.getWeight();
                   iH.setInKnapsack(1);
                   j -= wH;
                   solutionWeight += wH;
               }
           } // for()
           calculated = true;
       } // if()
       return itemList;
   }

   // add an item to the item list
   public void add(String name, int weight, int value) {
       if (name.equals(""))
           name = "" + (itemList.size() + 1);
       itemList.add(new Item(name, weight, value));
       setInitialStateForCalculation();
   }

   // add an item to the item list
   public void add(int weight, int value) {
       add("", weight, value); // the name will be "itemList.size() + 1"!
   }

   // remove an item from the item list
   public void remove(String name) {
       for (Iterator<Item> it = itemList.iterator(); it.hasNext(); ) {
           if (name.equals(it.next().getName())) {
               it.remove();
           }
       }
       setInitialStateForCalculation();
   }

   // remove all items from the item list
   public void removeAllItems() {
       itemList.clear();
       setInitialStateForCalculation();
   }

   public int getProfit() {
       if (!calculated)
           calcSolution();
       return profit;
   }

   public int getSolutionWeight() {return solutionWeight;}
   public boolean isCalculated() {return calculated;}
   public int getMaxWeight() {return maxWeight;}

   public void setMaxWeight(int _maxWeight) {
       maxWeight = Math.max(_maxWeight, 0);
   }

   public void setItemList(List<Item> _itemList) {
       if (_itemList != null) {
           itemList = _itemList;
           for (Item item : _itemList) {
               item.checkMembers();
           }
       }
   }

   // set the member with name "inKnapsack" by all items:
   private void setInKnapsackByAll(int inKnapsack) {
       for (Item item : itemList)
           if (inKnapsack > 0)
               item.setInKnapsack(1);
           else
               item.setInKnapsack(0);
   }

   // set the data members of class in the state of starting the calculation:
   protected void setInitialStateForCalculation() {
       setInKnapsackByAll(0);
       calculated     = false;
       profit         = 0;
       solutionWeight = 0;
   }

} // class </lang>

<lang java> package hu.pj.obj;

public class Item {

   protected String name    = "";
   protected int weight     = 0;
   protected int value      = 0;
   protected int bounding   = 1; // the maximal limit of item's pieces
   protected int inKnapsack = 0; // the pieces of item in solution

   public Item() {}

   public Item(Item item) {
       setName(item.name);
       setWeight(item.weight);
       setValue(item.value);
       setBounding(item.bounding);
   }

   public Item(int _weight, int _value) {
       setWeight(_weight);
       setValue(_value);
   }

   public Item(int _weight, int _value, int _bounding) {
       setWeight(_weight);
       setValue(_value);
       setBounding(_bounding);
   }

   public Item(String _name, int _weight, int _value) {
       setName(_name);
       setWeight(_weight);
       setValue(_value);
   }

   public Item(String _name, int _weight, int _value, int _bounding) {
       setName(_name);
       setWeight(_weight);
       setValue(_value);
       setBounding(_bounding);
   }

   public void setName(String _name) {name = _name;}
   public void setWeight(int _weight) {weight = Math.max(_weight, 0);}
   public void setValue(int _value) {value = Math.max(_value, 0);}

   public void setInKnapsack(int _inKnapsack) {
       inKnapsack = Math.min(getBounding(), Math.max(_inKnapsack, 0));
   }

   public void setBounding(int _bounding) {
       bounding = Math.max(_bounding, 0);
       if (bounding == 0)
           inKnapsack = 0;
   }

   public void checkMembers() {
       setWeight(weight);
       setValue(value);
       setBounding(bounding);
       setInKnapsack(inKnapsack);
   }

   public String getName() {return name;}
   public int getWeight() {return weight;}
   public int getValue() {return value;}
   public int getInKnapsack() {return inKnapsack;}
   public int getBounding() {return bounding;}

} // class </lang>

output:

Maximal weight           = 4 kg
Total weight of solution = 3,96 kg
Total value              = 1030

You can carry te following materials in the knapsack:
map                     9   dag   (value = 150)   
compass                 13  dag   (value = 35)    
water                   153 dag   (value = 200)   
sandwich                50  dag   (value = 160)   
glucose                 15  dag   (value = 60)    
banana                  27  dag   (value = 60)    
suntan cream            11  dag   (value = 70)    
waterproof trousers     42  dag   (value = 70)    
waterproof overclothes  43  dag   (value = 75)    
note-case               22  dag   (value = 80)    
sunglasses              7   dag   (value = 20)    
socks                   4   dag   (value = 50)    

Oz

Using constraint programming. <lang oz>declare

 %% maps items to pairs of Weight(hectogram) and Value
 Problem = knapsack('map':9#150
                    'compass':13#35
                    'water':153#200
                    'sandwich':50#160
                    'glucose':15#60
                    'tin':68#45 
                    'banana':27#60 
                    'apple':39#40 
                    'cheese':23#30 
                    'beer':52#10 
                    'suntan cream':11#70 
                    'camera':32#30 
                    't-shirt':24#15 
                    'trousers':48#10 
                    'umbrella':73#40 
                    'waterproof trousers':42#70 
                    'waterproof overclothes':43#75 
                    'note-case':22#80 
                    'sunglasses':7#20 
                    'towel':18#12 
                    'socks':4#50 
                    'book':30#10
                   )

 %% item -> Weight
 Weights = {Record.map Problem fun {$ X} X.1 end}
 %% item -> Value
 Values =  {Record.map Problem fun {$ X} X.2 end}

 proc {Knapsack Solution}
    %% a solution maps items to finite domain variables
    %% with the domain {0,1}
    Solution = {Record.map Problem fun {$ _} {FD.int 0#1} end}
    %% no more than 400 hectograms
    {FD.sumC Weights Solution '=<:' 400} 
    %% search through valid solutions
    {FD.distribute naive Solution}
 end

 proc {PropagateLargerValue Old New}
    %% propagate that new solutions must yield a higher value
    %% than previously found solutions (essential for performance)
    {FD.sumC Values New '>:' {Value Old}} 
 end

 fun {Value Candidate}
    {Record.foldL {Record.zip Candidate Values Number.'*'} Number.'+' 0}
 end
 
 fun {Weight Candidate}
    {Record.foldL {Record.zip Candidate Weights Number.'*'} Number.'+' 0}
 end

 [Best] = {SearchBest Knapsack PropagateLargerValue}

in

 {System.showInfo "Items: "}
 {ForAll
    {Record.arity {Record.filter Best fun {$ T} T == 1 end}}
    System.showInfo}
 {System.printInfo "\n"}
 {System.showInfo "total value: "#{Value Best}}
 {System.showInfo "total weight: "#{Weight Best}}</lang>

Output:

Items: 
banana
compass
glucose
map
note-case
sandwich
socks
sunglasses
suntan cream
water
waterproof overclothes
waterproof trousers

total value: 1030
total weight: 396

Typically runs in less than 150 milliseconds.

Python

Dumb, brute force algorithm: <lang python>from itertools import combinations

def anycomb(items):

   ' return combinations of any length from the items '
   return ( comb
            for r in range(1, len(items)+1)
            for comb in combinations(items, r)
            )

def totalvalue(comb):

   ' Totalise a particular combination of items'
   totwt = totval = 0
   for item, wt, val in comb:
       totwt  += wt
       totval += val
   return (totval, -totwt) if totwt <= 400 else (0, 0)

items = (

   ("map", 9, 150), ("compass", 13, 35), ("water", 153, 200), ("sandwich", 50, 160),
   ("glucose", 15, 60), ("tin", 68, 45), ("banana", 27, 60), ("apple", 39, 40),
   ("cheese", 23, 30), ("beer", 52, 10), ("suntan cream", 11, 70), ("camera", 32, 30),
   ("t-shirt", 24, 15), ("trousers", 48, 10), ("umbrella", 73, 40), ("waterproof trousers", 42, 70),
   ("waterproof overclothes", 43, 75), ("note-case", 22, 80), ("sunglasses", 7, 20), ("towel", 18, 12),
   ("socks", 4, 50), ("book", 30, 10),
   )

bagged = max( anycomb(items), key=totalvalue) # max val or min wt if values equal print("Bagged the following items\n " +

     '\n  '.join(sorted(item for item,_,_ in bagged)))

val, wt = totalvalue(bagged) print("for a total value of %i and a total weight of %i" % (val, -wt))</lang> Sample output:

Bagged the following items
  banana
  compass
  glucose
  map
  note-case
  sandwich
  socks
  sunglasses
  suntan cream
  water
  waterproof overclothes
  waterproof trousers
for a total value of 1030 and a total weight of 396

Dynamic programming solution

<lang python>try:

   xrange

except:

   xrange = range

def totalvalue(comb):

   ' Totalise a particular combination of items'
   totwt = totval = 0
   for item, wt, val in comb:
       totwt  += wt
       totval += val
   return (totval, -totwt) if totwt <= 400 else (0, 0)

items = (

   ("map", 9, 150), ("compass", 13, 35), ("water", 153, 200), ("sandwich", 50, 160),
   ("glucose", 15, 60), ("tin", 68, 45), ("banana", 27, 60), ("apple", 39, 40),
   ("cheese", 23, 30), ("beer", 52, 10), ("suntan cream", 11, 70), ("camera", 32, 30),
   ("t-shirt", 24, 15), ("trousers", 48, 10), ("umbrella", 73, 40), ("waterproof trousers", 42, 70),
   ("waterproof overclothes", 43, 75), ("note-case", 22, 80), ("sunglasses", 7, 20), ("towel", 18, 12),
   ("socks", 4, 50), ("book", 30, 10),
   )

def knapsack01_dp(items, limit):

   table = [[0 for w in range(limit + 1)] for j in xrange(len(items) + 1)]

   for j in xrange(1, len(items) + 1):
       item, wt, val = items[j-1]
       for w in xrange(1, limit + 1):
           if wt > w:
               table[j][w] = table[j-1][w]
           else:
               table[j][w] = max(table[j-1][w],
                                 table[j-1][w-wt] + val)

   result = []
   w = limit
   for j in range(len(items), 0, -1):
       was_added = table[j][w] != table[j-1][w]

       if was_added:
           item, wt, val = items[j-1]
           result.append(items[j-1])
           w -= wt

   return result

bagged = knapsack01_dp(items, 400) print("Bagged the following items\n " +

     '\n  '.join(sorted(item for item,_,_ in bagged)))

val, wt = totalvalue(bagged) print("for a total value of %i and a total weight of %i" % (val, -wt))</lang>

Tcl

As the saying goes, “when in doubt, try brute force”. Since there's only 22 items we can simply iterate over all possible choices. <lang tcl># The list of items to consider, as list of lists set items {

   {map			9	150}
   {compass			13	35}
   {water			153	200}
   {sandwich			50	160}
   {glucose			15	60}
   {tin			68	45}
   {banana			27	60}
   {apple			39	40}
   {cheese			23	30}
   {beer			52	10}
   {{suntan cream}		11	70}
   {camera			32	30}
   {t-shirt			24	15}
   {trousers			48	10}
   {umbrella			73	40}
   {{waterproof trousers}	42	70}
   {{waterproof overclothes}	43	75}
   {note-case			22	80}
   {sunglasses			7	20}
   {towel			18	12}
   {socks			4	50}
   {book			30	10}

}

1. Simple extraction functions

proc names {chosen} {

   set names {}
   foreach item $chosen {lappend names [lindex $item 0]}
   return $names

} proc weight {chosen} {

   set weight 0
   foreach item $chosen {incr weight [lindex $item 1]}
   return $weight

} proc value {chosen} {

   set value 0
   foreach item $chosen {incr value [lindex $item 2]}
   return $value

}

1. Recursive function for searching over all possible choices of items

proc knapsackSearch {items {chosen {}}} {

   # If we've gone over the weight limit, stop now
   if {[weight $chosen] > 400} {

return

   }
   # If we've considered all of the items (i.e., leaf in search tree)
   # then see if we've got a new best choice.
   if {[llength $items] == 0} {

global best max set v [value $chosen] if {$v > $max} { set max $v set best $chosen } return

   }
   # Branch, so recurse for chosing the current item or not
   set this [lindex $items 0]
   set rest [lrange $items 1 end]
   knapsackSearch $rest $chosen
   knapsackSearch $rest [lappend chosen $this]

}

1. Initialize a few global variables

set best {} set max 0

1. Do the brute-force search

knapsackSearch $items

1. Pretty-print the results

puts "Best filling has weight of [expr {[weight $best]/100.0}]kg and score [value $best]" puts "Best items:\n\t[join [lsort [names $best]] \n\t]"</lang> Output:

Best filling has weight of 3.96kg and score 1030
Best items:
	banana
	compass
	glucose
	map
	note-case
	sandwich
	socks
	sunglasses
	suntan cream
	water
	waterproof overclothes
	waterproof trousers