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# Implementation of 0/1 Knapsack using Branch and Bound

We strongly recommend to refer below post as a prerequisite for this. Branch and Bound | Set 1 (Introduction with 0/1 Knapsack) We discussed different approaches to solve above problem and saw that the Branch and Bound solution is the best suited method when item weights are not integers. In this post implementation of Branch and Bound method for 0/1 knapsack problem is discussed.

How to find bound for every node for 0/1 Knapsack?

The idea is to use the fact that the Greedy approach provides the best solution for Fractional Knapsack problem. To check if a particular node can give us a better solution or not, we compute the optimal solution (through the node) using Greedy approach. If the solution computed by Greedy approach itself is more than the best so far, then we can’t get a better solution through the node.

Complete Algorithm:

1. Sort all items in decreasing order of ratio of value per unit weight so that an upper bound can be computed using Greedy Approach.
2. Initialize maximum profit, maxProfit = 0
3. Create an empty queue, Q.
4. Create a dummy node of decision tree and enqueue it to Q. Profit and weight of dummy node are 0.
5. Do following while Q is not empty.
• Extract an item from Q. Let the extracted item be u.
• Compute profit of next level node. If the profit is more than maxProfit, then update maxProfit.
• Compute bound of next level node. If bound is more than maxProfit, then add next level node to Q.
• Consider the case when next level node is not considered as part of solution and add a node to queue with level as next, but weight and profit without considering next level nodes.

Illustration:

`Input:// First thing in every pair is weight of item// and second thing is value of itemItem arr[] = {{2, 40}, {3.14, 50}, {1.98, 100},              {5, 95}, {3, 30}};Knapsack Capacity W = 10Output:The maximum possible profit = 235Below diagram shows illustration. Items are considered sorted by value/weight.`
`Note :  The image doesn't strictly follow the algorithm/code as there is no dummy node in theimage.`

Following is implementation of above idea.

## Cpp

 `// C++ program to solve knapsack problem using``// branch and bound``#include ``using` `namespace` `std;` `// Structure for Item which store weight and corresponding``// value of Item``struct` `Item``{``    ``float` `weight;``    ``int` `value;``};` `// Node structure to store information of decision``// tree``struct` `Node``{``    ``// level --> Level of node in decision tree (or index``    ``//             in arr[]``    ``// profit --> Profit of nodes on path from root to this``    ``//         node (including this node)``    ``// bound ---> Upper bound of maximum profit in subtree``    ``//         of this node/``    ``int` `level, profit, bound;``    ``float` `weight;``};` `// Comparison function to sort Item according to``// val/weight ratio``bool` `cmp(Item a, Item b)``{``    ``double` `r1 = (``double``)a.value / a.weight;``    ``double` `r2 = (``double``)b.value / b.weight;``    ``return` `r1 > r2;``}` `// Returns bound of profit in subtree rooted with u.``// This function mainly uses Greedy solution to find``// an upper bound on maximum profit.``int` `bound(Node u, ``int` `n, ``int` `W, Item arr[])``{``    ``// if weight overcomes the knapsack capacity, return``    ``// 0 as expected bound``    ``if` `(u.weight >= W)``        ``return` `0;` `    ``// initialize bound on profit by current profit``    ``int` `profit_bound = u.profit;` `    ``// start including items from index 1 more to current``    ``// item index``    ``int` `j = u.level + 1;``    ``int` `totweight = u.weight;` `    ``// checking index condition and knapsack capacity``    ``// condition``    ``while` `((j < n) && (totweight + arr[j].weight <= W))``    ``{``        ``totweight += arr[j].weight;``        ``profit_bound += arr[j].value;``        ``j++;``    ``}` `    ``// If k is not n, include last item partially for``    ``// upper bound on profit``    ``if` `(j < n)``        ``profit_bound += (W - totweight) * arr[j].value /``                                        ``arr[j].weight;` `    ``return` `profit_bound;``}` `// Returns maximum profit we can get with capacity W``int` `knapsack(``int` `W, Item arr[], ``int` `n)``{``    ``// sorting Item on basis of value per unit``    ``// weight.``    ``sort(arr, arr + n, cmp);` `    ``// make a queue for traversing the node``    ``queue Q;``    ``Node u, v;` `    ``// dummy node at starting``    ``u.level = -1;``    ``u.profit = u.weight = 0;``    ``Q.push(u);` `    ``// One by one extract an item from decision tree``    ``// compute profit of all children of extracted item``    ``// and keep saving maxProfit``    ``int` `maxProfit = 0;``    ``while` `(!Q.empty())``    ``{``        ``// Dequeue a node``        ``u = Q.front();``        ``Q.pop();` `        ``// If it is starting node, assign level 0``        ``if` `(u.level == -1)``            ``v.level = 0;` `        ``// If there is nothing on next level``        ``if` `(u.level == n-1)``            ``continue``;` `        ``// Else if not last node, then increment level,``        ``// and compute profit of children nodes.``        ``v.level = u.level + 1;` `        ``// Taking current level's item add current``        ``// level's weight and value to node u's``        ``// weight and value``        ``v.weight = u.weight + arr[v.level].weight;``        ``v.profit = u.profit + arr[v.level].value;` `        ``// If cumulated weight is less than W and``        ``// profit is greater than previous profit,``        ``// update maxprofit``        ``if` `(v.weight <= W && v.profit > maxProfit)``            ``maxProfit = v.profit;` `        ``// Get the upper bound on profit to decide``        ``// whether to add v to Q or not.``        ``v.bound = bound(v, n, W, arr);` `        ``// If bound value is greater than profit,``        ``// then only push into queue for further``        ``// consideration``        ``if` `(v.bound > maxProfit)``            ``Q.push(v);` `        ``// Do the same thing, but Without taking``        ``// the item in knapsack``        ``v.weight = u.weight;``        ``v.profit = u.profit;``        ``v.bound = bound(v, n, W, arr);``        ``if` `(v.bound > maxProfit)``            ``Q.push(v);``    ``}` `    ``return` `maxProfit;``}` `// driver program to test above function``int` `main()``{``    ``int` `W = 10; ``// Weight of knapsack``    ``Item arr[] = {{2, 40}, {3.14, 50}, {1.98, 100},``                ``{5, 95}, {3, 30}};``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);` `    ``cout << ``"Maximum possible profit = "``        ``<< knapsack(W, arr, n);` `    ``return` `0;``}`

## Java

 `import` `java.util.Arrays;``import` `java.util.Comparator;``import` `java.util.PriorityQueue;` `class` `Item {``    ``float` `weight;``    ``int` `value;` `    ``Item(``float` `weight, ``int` `value) {``        ``this``.weight = weight;``        ``this``.value = value;``    ``}``}` `class` `Node {``    ``int` `level, profit, bound;``    ``float` `weight;` `    ``Node(``int` `level, ``int` `profit, ``float` `weight) {``        ``this``.level = level;``        ``this``.profit = profit;``        ``this``.weight = weight;``    ``}``}` `public` `class` `KnapsackBranchAndBound {``    ``static` `Comparator itemComparator = (a, b) -> {``        ``double` `ratio1 = (``double``) a.value / a.weight;``        ``double` `ratio2 = (``double``) b.value / b.weight;``        ``// Sorting in decreasing order of value per unit weight``        ``return` `Double.compare(ratio2, ratio1);``    ``};` `    ``static` `int` `bound(Node u, ``int` `n, ``int` `W, Item[] arr) {``        ``if` `(u.weight >= W)``            ``return` `0``;` `        ``int` `profitBound = u.profit;``        ``int` `j = u.level + ``1``;``        ``float` `totalWeight = u.weight;` `        ``while` `(j < n && totalWeight + arr[j].weight <= W) {``            ``totalWeight += arr[j].weight;``            ``profitBound += arr[j].value;``            ``j++;``        ``}` `        ``if` `(j < n)``            ``profitBound += (``int``) ((W - totalWeight) * arr[j].value / arr[j].weight);` `        ``return` `profitBound;``    ``}` `    ``static` `int` `knapsack(``int` `W, Item[] arr, ``int` `n) {``        ``Arrays.sort(arr, itemComparator);``        ``PriorityQueue priorityQueue =``          ``new` `PriorityQueue<>((a, b) -> Integer.compare(b.bound, a.bound));``        ``Node u, v;` `        ``u = ``new` `Node(-``1``, ``0``, ``0``);``        ``priorityQueue.offer(u);` `        ``int` `maxProfit = ``0``;` `        ``while` `(!priorityQueue.isEmpty()) {``            ``u = priorityQueue.poll();` `            ``if` `(u.level == -``1``)``                ``v = ``new` `Node(``0``, ``0``, ``0``);``            ``else` `if` `(u.level == n - ``1``)``                ``continue``;``            ``else``                ``v = ``new` `Node(u.level + ``1``, u.profit, u.weight);` `            ``v.weight += arr[v.level].weight;``            ``v.profit += arr[v.level].value;` `            ``if` `(v.weight <= W && v.profit > maxProfit)``                ``maxProfit = v.profit;` `            ``v.bound = bound(v, n, W, arr);` `            ``if` `(v.bound > maxProfit)``                ``priorityQueue.offer(v);` `            ``v = ``new` `Node(u.level + ``1``, u.profit, u.weight);``            ``v.bound = bound(v, n, W, arr);` `            ``if` `(v.bound > maxProfit)``                ``priorityQueue.offer(v);``        ``}` `        ``return` `maxProfit;``    ``}` `    ``public` `static` `void` `main(String[] args) {``        ``int` `W = ``10``;``        ``Item[] arr = {``            ``new` `Item(``2``, ``40``),``            ``new` `Item(``3``.14f, ``50``),``            ``new` `Item(``1``.98f, ``100``),``            ``new` `Item(``5``, ``95``),``            ``new` `Item(``3``, ``30``)``        ``};``        ``int` `n = arr.length;` `        ``int` `maxProfit = knapsack(W, arr, n);``        ``System.out.println(``"Maximum possible profit = "` `+ maxProfit);``    ``}``}`

## Python

 `from` `Queue ``import` `Queue` `# Define an Item class to represent``# each item in the knapsack``class` `Item:``    ``def` `__init__(``self``, weight, value):``        ``self``.weight ``=` `weight``        ``self``.value ``=` `value` `# Define a Node class to represent each``# node in the branch and bound tree``class` `Node:``    ``def` `__init__(``self``, level, profit, bound, weight):``        ``self``.level ``=` `level``        ``self``.profit ``=` `profit``        ``self``.bound ``=` `bound``        ``self``.weight ``=` `weight` `# Define a compare function to sort items``# by their value-to-weight ratio in``# descending order``def` `compare(a, b):``    ``r1 ``=` `float``(a.value) ``/` `a.weight``    ``r2 ``=` `float``(b.value) ``/` `b.weight``    ``return` `r1 > r2` `# Define a function to calculate the``# maximum possible profit for a given node``def` `bound(u, n, W, arr):``    ``# If the node exceeds the knapsack's``    ``# capacity, its profit bound is 0``    ``if` `u.weight >``=` `W:``        ``return` `0` `    ``# Calculate the profit bound by adding``    ``# the profits of all remaining items``    ``# that can fit into the knapsack``    ``profitBound ``=` `u.profit``    ``j ``=` `u.level ``+` `1``    ``totWeight ``=` `int``(u.weight)` `    ``while` `j < n ``and` `totWeight ``+` `int``(arr[j].weight) <``=` `W:``        ``totWeight ``+``=` `int``(arr[j].weight)``        ``profitBound ``+``=` `arr[j].value``        ``j ``+``=` `1` `    ``# If there are still items remaining,``    ``# add a fraction of the next item's``    ``# profit proportional to the remaining``    ``# space in the knapsack``    ``if` `j < n:``        ``profitBound ``+``=` `int``((W ``-` `totWeight) ``*` `arr[j].value ``/` `arr[j].weight)` `    ``return` `profitBound` `# Define the knapsack_solution function``# that uses the Branch and Bound algorithm``# to solve the 0-1 Knapsack problem``def` `knapsack_solution(W, arr, n):` `    ``# Sort the items in descending order``    ``# of their value-to-weight ratio``    ``arr.sort(``cmp``=``compare, reverse``=``True``)` `    ``# Initialize a queue with a root node``    ``# of the branch and bound tree``    ``q ``=` `Queue()``    ``u ``=` `Node(``-``1``, ``0``, ``0``, ``0``)``    ``q.put(u)` `    ``# Initialize a variable to keep track``    ``# of the maximum profit found so far``    ``maxProfit ``=` `0` `    ``# Loop through each node in the``    ``# branch and bound tree``    ``while` `not` `q.empty():``        ``u ``=` `q.get()` `        ``# If the node is the root node,``        ``# add its child nodes to the queue``        ``if` `u.level ``=``=` `-``1``:``            ``v ``=` `Node(``0``, ``0``, ``0``, ``0``)` `        ``# If the node is a leaf node, skip it``        ``if` `u.level ``=``=` `n ``-` `1``:``            ``continue` `        ``# Calculate the child node that``        ``# includes the next item in knapsack``        ``v ``=` `Node(u.level ``+` `1``, u.profit ``+``                 ``arr[u.level ``+` `1``].value, ``0``, u.weight ``+` `arr[u.level ``+` `1``].weight)` `        ``# If the child node's weight is``        ``# less than or equal to the knapsack's``        ``# capacity and its profit is greater``        ``# than the maximum profit found so far,``        ``# update the maximum profit``        ``if` `v.weight <``=` `W ``and` `v.profit > maxProfit:``            ``maxProfit ``=` `v.profit` `        ``# Calculate the profit bound for the``        ``# child node and add it to the queue``        ``# if its profit bound is greater than``        ``# the maximum profit found so far``        ``v.bound ``=` `bound(v, n, W, arr)` `        ``if` `v.bound > maxProfit:``            ``q.put(v)` `        ``v ``=` `Node(u.level ``+` `1``, u.profit, ``0``, u.weight)` `        ``v.bound ``=` `bound(v, n, W, arr)` `        ``if` `v.bound > maxProfit:``            ``q.put(v)` `    ``return` `maxProfit`  `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ``W ``=` `10``    ``arr ``=` `[Item(``2``, ``40``), Item(``3.14``, ``50``), Item(``        ``1.98``, ``100``), Item(``5``, ``95``), Item(``3``, ``30``)]``    ``n ``=` `len``(arr)` `    ``print` `'Maximum possible profit ='``, knapsack_solution(W, arr, n)`

## C#

 `using` `System;``using` `System.Collections.Generic;` `public` `struct` `Item``{``    ``public` `float` `weight;``    ``public` `int` `value;``}` `public` `struct` `Node``{``    ``public` `int` `level;``    ``public` `int` `profit;``    ``public` `int` `bound;``    ``public` `float` `weight;``}` `public` `class` `Knapsack``{``    ``public` `static` `bool` `Compare(Item a, Item b)``    ``{``        ``double` `r1 = (``double``)a.value / a.weight;``        ``double` `r2 = (``double``)b.value / b.weight;``        ``return` `r1 > r2;``    ``}` `    ``public` `static` `int` `Bound(Node u, ``int` `n, ``int` `W, Item[] arr)``    ``{``        ``if` `(u.weight >= W)``            ``return` `0;` `        ``int` `profitBound = u.profit;``        ``int` `j = u.level + 1;``        ``int` `totWeight = (``int``)u.weight;` `        ``while` `(j < n && totWeight + (``int``)arr[j].weight <= W)``        ``{``            ``totWeight += (``int``)arr[j].weight;``            ``profitBound += arr[j].value;``            ``j++;``        ``}` `        ``if` `(j < n)``            ``profitBound += (``int``)((W - totWeight) * arr[j].value / arr[j].weight);` `        ``return` `profitBound;``    ``}` `    ``public` `static` `int` `KnapsackSolution(``int` `W, Item[] arr, ``int` `n)``    ``{``        ``Array.Sort(arr, Compare);` `        ``Queue q = ``new` `Queue();``        ``Node u, v;``        ``int` `maxProfit = 0;` `        ``u.level = -1;``        ``u.profit = 0;``        ``u.weight = 0;``        ``q.Enqueue(u);` `        ``while` `(q.Count > 0)``        ``{``            ``u = q.Dequeue();` `            ``if` `(u.level == -1)``                ``v.level = 0;` `            ``if` `(u.level == n - 1)``                ``continue``;` `            ``v.level = u.level + 1;``            ``v.weight = u.weight + arr[v.level].weight;``            ``v.profit = u.profit + arr[v.level].value;` `            ``if` `(v.weight <= W && v.profit > maxProfit)``                ``maxProfit = v.profit;` `            ``v.bound = Bound(v, n, W, arr);` `            ``if` `(v.bound > maxProfit)``                ``q.Enqueue(v);` `            ``v.weight = u.weight;``            ``v.profit = u.profit;``            ``v.bound = Bound(v, n, W, arr);` `            ``if` `(v.bound > maxProfit)``                ``q.Enqueue(v);``        ``}` `        ``return` `maxProfit;``    ``}` `    ``public` `static` `void` `Main(``string``[] args)``    ``{``        ``int` `W = 10;``        ``Item[] arr = ``new` `Item[5] { ``new` `Item { weight = 2, value = 40 },``                                   ``new` `Item { weight = 3.14f, value = 50 },``                                   ``new` `Item { weight = 1.98f, value = 100 },``                                   ``new` `Item { weight = 5, value = 95 },``                                   ``new` `Item { weight = 3, value = 30 } };``        ``int` `n = arr.Length;` `        ``Console.WriteLine(``"Maximum possible profit = {0}"``, KnapsackSolution(W, arr, n));``    ``}``}`

## Javascript

 `// JavaScript program to solve knapsack problem using``// branch and bound` `// Structure for Item which store weight and corresponding value of Item``class Item {``    ``constructor(weight, value) {``        ``this``.weight = weight;``        ``this``.value = value;``    ``}``}` `// Node structure to store information of decision tree``class Node {``    ``constructor(level, profit, weight, bound) {``        ``this``.level = level; ``// Level of node in decision tree (or index in arr[])``        ``this``.profit = profit; ``// Profit of nodes on path from root to this node (including this node)``        ``this``.weight = weight; ``// Weight of nodes on path from root to this node (including this node)``        ``this``.bound = bound; ``// Upper bound of maximum profit in subtree of this node``    ``}``}` `// Comparison function to sort Item according to val/weight ratio``function` `cmp(a, b) {``    ``let r1 = a.value / a.weight;``    ``let r2 = b.value / b.weight;``    ``return` `r1 < r2;``}` `// Returns bound of profit in subtree rooted with u.``// This function mainly uses Greedy solution to find``// an upper bound on maximum profit.``function` `bound(u, n, W, arr) {``    ``// if weight overcomes the knapsack capacity, return 0 as expected bound``    ``if` `(u.weight >= W) {``        ``return` `0;``    ``}` `    ``// initialize bound on profit by current profit``    ``let profit_bound = u.profit;` `    ``// start including items from index 1 more to current item index``    ``let j = u.level + 1;``    ``let totweight = u.weight;` `    ``// checking index condition and knapsack capacity condition``    ``while` `(j < n && totweight + arr[j].weight <= W) {``        ``totweight += arr[j].weight;``        ``profit_bound += arr[j].value;``        ``j++;``    ``}` `    ``// If k is not n, include last item partially for upper bound on profit``    ``if` `(j < n) {``        ``profit_bound += (W - totweight) * arr[j].value / arr[j].weight;``    ``}` `    ``return` `profit_bound;``}` `// Returns maximum profit we can get with capacity W``function` `knapsack(W, arr, n) {``    ``// sorting Item on basis of value per unit weight.``    ``arr.sort(cmp);` `    ``// make a queue for traversing the node``    ``let Q = [];``    ``let u = ``new` `Node(-1, 0, 0, 0);``    ``let v;` `    ``// dummy node at starting``    ``Q.push(u);` `    ``// One by one extract an item from decision tree``    ``// compute profit of all children of extracted item``    ``// and keep saving maxProfit``    ``let maxProfit = 0;``    ``while` `(Q.length > 0) {``        ``// Dequeue a node``        ``u = Q.shift();` `        ``// If it is starting node, assign level 0``        ``if` `(u.level == -1) {``            ``v = ``new` `Node(0, 0, 0, 0);``        ``}` `        ``// If there is nothing on next level``        ``if` `(u.level == n - 1) {``            ``continue``;``        ``}` `        ``// Else if not last node, then increment level,``        ``// and compute profit of children nodes.``        ``v = ``new` `Node(u.level + 1, u.profit, u.weight, 0);` `        ``// Taking current level's item add current``        ``// level's weight``        ``v.weight = u.weight + arr[v.level].weight;``        ``v.profit = u.profit + arr[v.level].value;``            ``// If cumulated weight is less than W and``    ``// profit is greater than previous profit,``    ``// update maxprofit``    ``if` `(v.weight <= W && v.profit > maxProfit) {``        ``maxProfit = v.profit;``    ``}` `    ``// Get the upper bound on profit to decide``    ``// whether to add v to Q or not.``    ``v.bound = bound(v, n, W, arr);` `    ``// If bound value is greater than profit,``    ``// then only push into queue for further``    ``// consideration``    ``if` `(v.bound > maxProfit) {``        ``Q.push(v);``    ``}` `    ``// Do the same thing, but Without taking``    ``// the item in knapsack``    ``v = ``new` `Node(u.level + 1, u.profit, u.weight, 0);``    ``v.bound = bound(v, n, W, arr);``    ``if` `(v.bound > maxProfit) {``        ``Q.push(v);``    ``}``}` `return` `maxProfit;``}` `// driver program to test above function``function` `main() {``const W = 10; ``// Weight of knapsack``const arr = [``{ weight: 2, value: 40 },``{ weight: 3.14, value: 50 },``{ weight: 1.98, value: 100 },``{ weight: 5, value: 95 },``{ weight: 3, value: 30 },``];``const n = arr.length;` `console.log(`Maximum possible profit = \${knapsack(W, arr, n)}`);``}``main();`

Output :

`Maximum possible profit = 235`

Time complexity: O(n^2) because the while loop runs n times, and inside the while loop, there is another loop that also runs n times in the worst case.

Space complexity : O(n) because the queue stores the nodes, and in the worst case, all nodes are stored, so the size of the queue is proportional to the number of items, which is n.

This article is contributed Utkarsh Trivedi. If you likeGeeksforGeeks and would like to contribute, you can also write an article and mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.