Coin Change | BFS Approach

Last Updated : 31 May, 2022

Given an integer X and an array arr[] of length N consisting of positive integers, the task is to pick minimum number of integers from the array such that they sum up to N. Any number can be chosen infinite number of times. If no answer exists then print -1.
Examples:

Input: X = 7, arr[] = {3, 5, 4}
Output:
The minimum number elements will be 2 as
3 and 4 can be selected to reach 7.

Input: X = 4, arr[] = {5}
Output: -1

Here, we will see a slightly different approach to solve this problem using BFS
Before that, let’s go ahead and define a state. A state SX can be defined as the minimum number of integers we would need to take from array to get a total of X.
Now, if we start looking at each state as a node in a graph such that each node is connected to (SX – arr[0], SX – arr[1], … SX – arr[N – 1])
Thus, we have to find the shortest path from state N to 0 in an unweighted and this can be done using BFS. BFS works here because the graph is unweighted.
Below is the implementation of the above approach:

C++

 `// C++ implementation of the approach` `#include ` `using` `namespace` `std;`   `// Function to find the minimum number` `// of integers required` `int` `minNumbers(``int` `x, ``int``* arr, ``int` `n)` `{` `    ``// Queue for BFS` `    ``queue<``int``> q;`   `    ``// Base value in queue` `    ``q.push(x);`   `    ``// Boolean array to check if a number has been` `    ``// visited before` `    ``unordered_set<``int``> v;`   `    ``// Variable to store depth of BFS` `    ``int` `d = 0;`   `    ``// BFS algorithm` `    ``while` `(q.size()) {`   `        ``// Size of queue` `        ``int` `s = q.size();` `        ``while` `(s--) {`   `            ``// Front most element of the queue` `            ``int` `c = q.front();`   `            ``// Base case` `            ``if` `(!c)` `                ``return` `d;` `            ``q.pop();` `            ``if` `(v.find(c) != v.end() or c < 0)` `                ``continue``;`   `            ``// Setting current state as visited` `            ``v.insert(c);`   `            ``// Pushing the required states in queue` `            ``for` `(``int` `i = 0; i < n; i++)` `                ``q.push(c - arr[i]);` `        ``}`   `        ``d++;` `    ``}`   `    ``// If no possible solution` `    ``return` `-1;` `}`   `// Driver code` `int` `main()` `{` `    ``int` `arr[] = { 3, 3, 4 };` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(``int``);` `    ``int` `x = 7;`   `    ``cout << minNumbers(x, arr, n);`   `    ``return` `0;` `}`

Java

 `// Java implementation of the approach` `import` `java.util.*;`   `class` `GFG` `{`   `// Function to find the minimum number` `// of integers required` `static` `int` `minNumbers(``int` `x, ``int` `[]arr, ``int` `n)` `{` `    ``// Queue for BFS` `    ``Queue q = ``new` `LinkedList<>();`   `    ``// Base value in queue` `    ``q.add(x);`   `    ``// Boolean array to check if ` `    ``// a number has been visited before` `    ``HashSet v = ``new` `HashSet();`   `    ``// Variable to store depth of BFS` `    ``int` `d = ``0``;`   `    ``// BFS algorithm` `    ``while` `(q.size() > ``0``) ` `    ``{`   `        ``// Size of queue` `        ``int` `s = q.size();` `        ``while` `(s-- > ``0``)` `        ``{`   `            ``// Front most element of the queue` `            ``int` `c = q.peek();`   `            ``// Base case` `            ``if` `(c == ``0``)` `                ``return` `d;` `            ``q.remove();` `            ``if` `(v.contains(c) || c < ``0``)` `                ``continue``;`   `            ``// Setting current state as visited` `            ``v.add(c);`   `            ``// Pushing the required states in queue` `            ``for` `(``int` `i = ``0``; i < n; i++)` `                ``q.add(c - arr[i]);` `        ``}` `        ``d++;` `    ``}`   `    ``// If no possible solution` `    ``return` `-``1``;` `}`   `// Driver code` `public` `static` `void` `main(String[] args)` `{` `    ``int` `arr[] = { ``3``, ``3``, ``4` `};` `    ``int` `n = arr.length;` `    ``int` `x = ``7``;`   `    ``System.out.println(minNumbers(x, arr, n));` `}` `}`   `// This code is contributed by Rajput-Ji`

Python3

 `# Python3 implementation of the approach `   `# Function to find the minimum number ` `# of integers required ` `def` `minNumbers(x, arr, n) : `   `    ``q ``=` `[] `   `    ``# Base value in queue ` `    ``q.append(x) `   `    ``v ``=` `set``([]) `   `    ``d ``=` `0`   `    ``while` `(``len``(q) > ``0``) : `   `        ``s ``=` `len``(q) ` `        ``while` `(s) : ` `            ``s ``-``=` `1` `            ``c ``=` `q[``0``] ` `            ``#print(q)` `            ``if` `(c ``=``=` `0``) : ` `                ``return` `d ` `            ``q.pop(``0``) ` `            ``if` `((c ``in` `v) ``or` `c < ``0``) :` `                ``continue`   `            ``# Setting current state as visited ` `            ``v.add(c) `   `            ``# Pushing the required states in queue ` `            ``for` `i ``in` `range``(n) : ` `                ``q.append(c ``-` `arr[i])             ` `            `  `        ``d ``+``=` `1` `        ``#print()` `        ``#print(d,c)`   `    ``# If no possible solution ` `    ``return` `-``1`   `arr ``=` `[ ``1``, ``4``,``6` `]` `n ``=` `len``(arr)` `x ``=` `20` `print``(minNumbers(x, arr, n))`   `# This code is contributed by divyeshrabadiya07` `# Improved by nishant.k108`

C#

 `// C# implementation of the approach` `using` `System;` `using` `System.Collections.Generic; ` `    `  `class` `GFG` `{`   `// Function to find the minimum number` `// of integers required` `static` `int` `minNumbers(``int` `x, ``int` `[]arr, ``int` `n)` `{` `    ``// Queue for BFS` `    ``Queue<``int``> q = ``new` `Queue<``int``>();`   `    ``// Base value in queue` `    ``q.Enqueue(x);`   `    ``// Boolean array to check if ` `    ``// a number has been visited before` `    ``HashSet<``int``> v = ``new` `HashSet<``int``>();`   `    ``// Variable to store depth of BFS` `    ``int` `d = 0;`   `    ``// BFS algorithm` `    ``while` `(q.Count > 0) ` `    ``{`   `        ``// Size of queue` `        ``int` `s = q.Count;` `        ``while` `(s-- > 0)` `        ``{`   `            ``// Front most element of the queue` `            ``int` `c = q.Peek();`   `            ``// Base case` `            ``if` `(c == 0)` `                ``return` `d;` `            ``q.Dequeue();` `            ``if` `(v.Contains(c) || c < 0)` `                ``continue``;`   `            ``// Setting current state as visited` `            ``v.Add(c);`   `            ``// Pushing the required states in queue` `            ``for` `(``int` `i = 0; i < n; i++)` `                ``q.Enqueue(c - arr[i]);` `        ``}` `        ``d++;` `    ``}`   `    ``// If no possible solution` `    ``return` `-1;` `}`   `// Driver code` `public` `static` `void` `Main(String[] args)` `{` `    ``int` `[]arr = { 3, 3, 4 };` `    ``int` `n = arr.Length;` `    ``int` `x = 7;`   `    ``Console.WriteLine(minNumbers(x, arr, n));` `}` `}`   `// This code is contributed by Rajput-Ji`

Javascript

 ``

Output:

`2`

Time complexity: O(N * X)
Auxiliary Space: O(N), since N extra space has been taken.

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