Given an array of N integers, using ‘+’ and ‘-‘ between the elements check if there is a way to form a sequence of numbers which evaluate to a number divisible by M

Examples:

Input : arr = {1, 2, 3, 4, 6} M = 4 Output : True, There is a valid sequence i. e., (1 - 2 + 3 + 4 + 6), which evaluates to 12 that is divisible by 4 Input : arr = {1, 3, 9} M = 2 Output : False There is no sequence which evaluates to a number divisible by M.

A **simple solution** is to recursively consider all possible scenarios ie either use a ;+’ or a ‘-‘ operator between the elements and maintain a variable sum which stores the result.If this result is divisible by M then return true else return false

Recursive implementation is as follows:

bool isPossible(int index, int sum) { // Base case if (index == n) { // check if sum is divisible by M if ((sum % M) == 0) return true; return false; } // recursively call by considering '+' // or '-' between index and index+1 // 1.Try placing '+' bool placeAdd = isPossible(index + 1, sum + arr[index]); // 2. Try placing '-' bool placeMinus = isPossible(index + 1, sum - arr[index]); if (placeAdd || placeMinus) return true; return false; }

There are overlapping subproblems as shown in the image below (Note: the image represents the recursion tree till index = 3)

So, now we will solve this using dynamic programming

**Method 1:**

We apply Dynamic Programming with two states :-

(i) index,

(ii) sum

So DP[index][sum] stores the current index we are at and sum stores the result of evaluation of the sequence formed till that index.

#include <bits/stdc++.h> using namespace std; const int MAX = 1000; bool isPossible(int n, int index, int sum, int M, int arr[], int dp[][MAX]) { // Base case if (index == n) { // check if sum is divisible by M if ((sum % M) == 0) return true; return false; } // check if the current state // is already computed if (dp[index][sum] != -1) return dp[index][sum]; // 1.Try placing '+' bool placeAdd = isPossible(n, index + 1, sum + arr[index], M, arr, dp); // 2. Try placing '-' bool placeMinus = isPossible(n, index + 1, sum - arr[index], M, arr, dp); // calculate value of res for recursive case bool res = (placeAdd || placeMinus); // store the value for res for current // states and return for parent call dp[index][sum] = res; return res; } int main() { int arr[] = { 1, 2, 3, 4, 6 }; int n = sizeof(arr)/sizeof(arr[0]); int M = 4; int dp[n + 1][MAX]; memset(dp, -1, sizeof(dp)); bool res; res = isPossible(n, 0, 0, M, arr, dp); cout << (res ? "True" : "False") << endl; return 0; }

The Complexity of this method is O(N*sum) where sum is the maximum possible sum for the sequence of integers and N is the number of elements in the array.

**Method 2(efficient):**

This is more efficient than Method 1. Here also, we apply Dynamic Programming but with two different states :-

(i) index,

(ii) modulo

So DP[index][modulo] stores the modulus of the result of evaluation of the sequence formed till that index, with M.

#include <bits/stdc++.h> using namespace std; const int MAX = 100; int isPossible(int n, int index, int modulo, int M, int arr[], int dp[][MAX]) { // Calculate modulo for this call modulo = ((modulo % M) + M) % M; // Base case if (index == n) { // check if sum is divisible by M if (modulo == 0) return 1; return 0; } // check if the current state is // already computed if (dp[index][modulo] != -1) return dp[index][modulo]; // 1.Try placing '+' int placeAdd = isPossible(n, index + 1, modulo + arr[index], M, arr, dp); // 2. Try placing '-' int placeMinus = isPossible(n, index + 1, modulo - arr[index], M, arr, dp); // calculate value of res for recursive // case bool res = (placeAdd || placeMinus); // store the value for res for current // states and return for parent call dp[index][modulo] = res; return res; } int main() { int arr[] = { 1, 2, 3, 4, 6 }; int n = sizeof(arr)/sizeof(arr[0]); int M = 4; // MAX is the Maximum value M can take int dp[n + 1][MAX]; memset(dp, -1, sizeof(dp)); bool res; res = isPossible(n, 0, 0, M, arr, dp); cout << (res ? "True" : "False") << endl; return 0; }

The Complexity of this method is O(N*M).

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