Given an Array[] of N elements and a number K. ( 1 <= K <= N ) . Split the given array into K subarrays (they must cover all the elements). The maximum subarray sum achievable out of K subarrays formed, must be the minimum possible. Find that possible subarray sum.
Examples:
Input : Array[] = {1, 2, 3, 4}, K = 3
Output : 4
Optimal Split is {1, 2}, {3}, {4} . Maximum sum of all subarrays is 4, which is minimum possible for 3 splits.
Input : Array[] = {1, 1, 2} K = 2
Output : 2
Naive approach:
- Idea is to find out all the possibilities.
- To find out all possibilities we use BACKTRACKING.
- At each step, we divide the array into sub-array and find the sum of the sub-array and update the maximum sum
Below is the implementation of the above approach:
C++
#include <bits/stdc++.h>using namespace std;int ans = 100000000;// the answer is stored in ans// we call this function solvevoid solve(int a[], int n, int k, int index, int sum, int maxsum){ // K=1 is the base Case if (k == 1) { maxsum = max(maxsum, sum); sum = 0; for (int i = index; i < n; i++) { sum += a[i]; } // we update maxsum maxsum = max(maxsum, sum); // the answer is stored in ans ans = min(ans, maxsum); return; } sum = 0; // using for loop to divide the array into K-subarray for (int i = index; i < n; i++) { sum += a[i]; // for each subarray we calculate sum ans update // maxsum maxsum = max(maxsum, sum); // calling function again solve(a, n, k - 1, i + 1, sum, maxsum); }}// Driver Codeint main(){ int arr[] = { 1, 2, 3, 4 }; int k = 3; // K divisions int n = 4; // Size of Array solve(arr, n, k, 0, 0, 0); cout << ans << "\n";} |
C
#include <stdio.h>int ans = 100000000;// the answer is stored in ans// we call this function solve// max function is used to find max of two elementsint max(int a, int b) { return a > b ? a : b; }// min function is used to find min of two elementsint min(int a, int b) { return a < b ? a : b; }void solve(int a[], int n, int k, int index, int sum, int maxsum){ // K=1 is the base Case if (k == 1) { maxsum = max(maxsum, sum); sum = 0; for (int i = index; i < n; i++) { sum += a[i]; } // we update maxsum maxsum = max(maxsum, sum); // the answer is stored in ans ans = min(ans, maxsum); return; } sum = 0; // using for loop to divide the array into K-subarray for (int i = index; i < n; i++) { sum += a[i]; // for each subarray we calculate sum ans update // maxsum maxsum = max(maxsum, sum); // calling function again solve(a, n, k - 1, i + 1, sum, maxsum); }}// Driver Codeint main(){ int arr[] = { 1, 2, 3, 4 }; int k = 3; // K divisions int n = 4; // Size of Array solve(arr, n, k, 0, 0, 0); printf("%d", ans);} |
Java
class GFG { public static int ans = 10000000; public static void solve(int a[], int n, int k, int index, int sum, int maxsum) { // K=1 is the base Case if (k == 1) { maxsum = Math.max(maxsum, sum); sum = 0; for (int i = index; i < n; i++) { sum += a[i]; } // we update maxsum maxsum = Math.max(maxsum, sum); // the answer is stored in ans ans = Math.min(ans, maxsum); return; } sum = 0; // using for loop to divide the array into // K-subarray for (int i = index; i < n; i++) { sum += a[i]; // for each subarray we calculate sum ans update // maxsum maxsum = Math.max(maxsum, sum); // calling function again solve(a, n, k - 1, i + 1, sum, maxsum); } } public static void main(String[] args) { int arr[] = { 1, 2, 3, 4 }; int k = 3; // K divisions int n = 4; // Size of Array solve(arr, n, k, 0, 0, 0); System.out.println(ans + "\n"); }} |
Python3
ans = 10000000def solve(a, n, k, index, sum, maxsum): global ans # K=1 is the base Case if (k == 1): maxsum = max(maxsum, sum) sum = 0 for i in range(index,n): sum += a[i] # we update maxsum maxsum = max(maxsum, sum) # the answer is stored in ans ans = min(ans, maxsum) return sum = 0 # using for loop to divide the array into # K-subarray for i in range(index, n): sum += a[i] # for each subarray we calculate sum ans update # maxsum maxsum = max(maxsum, sum) # calling function again solve(a, n, k - 1, i + 1, sum, maxsum) # driver code arr = [ 1, 2, 3, 4 ]k = 3 # K divisionsn = 4 # Size of Arraysolve(arr, n, k, 0, 0, 0)print(ans) # this code is contributed by shinjanpatra |
C#
using System;class GFG { public static int ans = 10000000; public static void solve(int []a, int n, int k, int index, int sum, int maxsum) { // K=1 is the base Case if (k == 1) { maxsum = Math.Max(maxsum, sum); sum = 0; for (int i = index; i < n; i++) { sum += a[i]; } // we update maxsum maxsum = Math.Max(maxsum, sum); // the answer is stored in ans ans = Math.Min(ans, maxsum); return; } sum = 0; // using for loop to divide the array into // K-subarray for (int i = index; i < n; i++) { sum += a[i]; // for each subarray we calculate sum ans update // maxsum maxsum = Math.Max(maxsum, sum); // calling function again solve(a, n, k - 1, i + 1, sum, maxsum); } } // Driver code public static void Main(String[] args) { int []arr = { 1, 2, 3, 4 }; int k = 3; // K divisions int n = 4; // Size of Array solve(arr, n, k, 0, 0, 0); Console.Write(ans + "\n"); }}// This code is contributed by shivanisinghss2110 |
Javascript
<script>var ans = 10000000;function solve(a, n, k, index, sum, maxsum) { // K=1 is the base Case if (k == 1) { maxsum = Math.max(maxsum, sum); sum = 0; for (var i = index; i < n; i++) { sum += a[i]; } // we update maxsum maxsum = Math.max(maxsum, sum); // the answer is stored in ans ans = Math.min(ans, maxsum); return; } sum = 0; // using for loop to divide the array into // K-subarray for (var i = index; i < n; i++) { sum += a[i]; // for each subarray we calculate sum ans update // maxsum maxsum = Math.max(maxsum, sum); // calling function again solve(a, n, k - 1, i + 1, sum, maxsum); } } var arr = [ 1, 2, 3, 4 ]; var k = 3; // K divisions var n = 4; // Size of Array solve(arr, n, k, 0, 0, 0); document.write(ans + "\n"); // this code is contributed by shivanisinghss2110</script> |
Output
4
Time Complexity: O((N−1)c(K−1) (NOTE: ‘c’ here depicts combinations i.e. ((n-1)!/((n-k)!*(k-1)!)
Where N is the number of elements of the array and K is the number of divisions.
Space Complexity: O(K)
Efficient Approach :
- Idea is to use Binary Search to find an optimal solution.
- For binary search minimum sum can be 1 and the maximum sum can be the sum of all the elements.
- To check if mid is the maximum subarray sum possible. Maintain a count of sub-arrays, include all possible elements in subarray until their sum is less than mid. After this evaluation, if the count is less than or equal to K, then mid is achievable else not. (Since if the count is less than K, we can further divide any subarray its sum will never increase mid ).
- Find the minimum possible value of mid which satisfies the condition.
Below is the implementation of the above approach:
C++
// C++ implementation of the above approach#include <bits/stdc++.h>using namespace std;// Function to check if mid can// be maximum sub - arrays sumbool check(int mid, int array[], int n, int K){ int count = 0; int sum = 0; for (int i = 0; i < n; i++) { // If individual element is greater // maximum possible sum if (array[i] > mid) return false; // Increase sum of current sub - array sum += array[i]; // If the sum is greater than // mid increase count if (sum > mid) { count++; sum = array[i]; } } count++; // Check condition if (count <= K) return true; return false;}// Function to find maximum subarray sum// which is minimumint solve(int array[], int n, int K){ int* max = max_element(array, array + n); int start = *max; //Max subarray sum, considering subarray of length 1 int end = 0; for (int i = 0; i < n; i++) { end += array[i]; //Max subarray sum, considering subarray of length n } // Answer stores possible // maximum sub array sum int answer = 0; while (start <= end) { int mid = (start + end) / 2; // If mid is possible solution // Put answer = mid; if (check(mid, array, n, K)) { answer = mid; end = mid - 1; } else { start = mid + 1; } } return answer;}// Driver Codeint main(){ int array[] = { 1, 2, 3, 4 }; int n = sizeof(array) / sizeof(array[0]); int K = 3; cout << solve(array, n, K);} |
Java
// Java implementation of the above approachclass GFG { // Function to check if mid can // be maximum sub - arrays sum static boolean check(int mid, int array[], int n, int K) { int count = 0; int sum = 0; for (int i = 0; i < n; i++) { // If individual element is greater // maximum possible sum if (array[i] > mid) return false; // Increase sum of current sub - array sum += array[i]; // If the sum is greater than // mid increase count if (sum > mid) { count++; sum = array[i]; } } count++; // Check condition if (count <= K) return true; return false; } // Function to find maximum subarray sum // which is minimum static int solve(int array[], int n, int K) { int start = 1; for (int i = 0; i < n; ++i) { if (array[i] > start) start = array[i]; //Max subarray sum, considering subarray of length 1 } int end = 0; for (int i = 0; i < n; i++) { end += array[i]; //Max subarray sum, considering subarray of length n } // Answer stores possible // maximum sub array sum int answer = 0; while (start <= end) { int mid = (start + end) / 2; // If mid is possible solution // Put answer = mid; if (check(mid, array, n, K)) { answer = mid; end = mid - 1; } else { start = mid + 1; } } return answer; } // Driver Code public static void main(String[] args) { int array[] = { 1, 2, 3, 4 }; int n = array.length; int K = 3; System.out.println(solve(array, n, K)); }}// This code is contributed by AnkitRai01 |
Python3
# Python 3 implementation of the above approach# Function to check if mid can# be maximum sub - arrays sumdef check(mid, array, n, K): count = 0 sum = 0 for i in range(n): # If individual element is greater # maximum possible sum if (array[i] > mid): return False # Increase sum of current sub - array sum += array[i] # If the sum is greater than # mid increase count if (sum > mid): count += 1 sum = array[i] count += 1 # Check condition if (count <= K): return True return False# Function to find maximum subarray sum# which is minimumdef solve(array, n, K): start = max(array) #Max subarray sum, considering subarray of length 1 end = 0 for i in range(n): end += array[i] #Max subarray sum, considering subarray of length n # Answer stores possible # maximum sub array sum answer = 0 while (start <= end): mid = (start + end) // 2 # If mid is possible solution # Put answer = mid; if (check(mid, array, n, K)): answer = mid end = mid - 1 else: start = mid + 1 return answer# Driver Codeif __name__ == '__main__': array = [1, 2, 3, 4] n = len(array) K = 3 print(solve(array, n, K)) # This code is contributed by# Surendra_Gangwar |
C#
// C# implementation of the above approachusing System; class GFG { // Function to check if mid can // be maximum sub - arrays sum static Boolean check(int mid, int []array, int n, int K) { int count = 0; int sum = 0; for (int i = 0; i < n; i++) { // If individual element is greater // maximum possible sum if (array[i] > mid) return false; // Increase sum of current sub - array sum += array[i]; // If the sum is greater than // mid increase count if (sum > mid) { count++; sum = array[i]; } } count++; // Check condition if (count <= K) return true; return false; } // Function to find maximum subarray sum // which is minimum static int solve(int []array, int n, int K) { int start = 1; for (int i = 0; i < n; ++i) { if (array[i] > start) start = array[i]; //Max subarray sum, considering subarray of length 1 } int end = 0; for (int i = 0; i < n; i++) { end += array[i]; //Max subarray sum, considering subarray of length n } // Answer stores possible // maximum sub array sum int answer = 0; while (start <= end) { int mid = (start + end) / 2; // If mid is possible solution // Put answer = mid; if (check(mid, array, n, K)) { answer = mid; end = mid - 1; } else { start = mid + 1; } } return answer; } // Driver Code public static void Main (String[] args) { int []array = { 1, 2, 3, 4 }; int n = array.Length ; int K = 3; Console.WriteLine(solve(array, n, K)); }}// This code is contributed by Princi Singh |
Javascript
<script>// Javascript implementation of the above approach// Function to check if mid can// be maximum sub - arrays sumfunction check(mid, array, n, K){ var count = 0; var sum = 0; for (var i = 0; i < n; i++) { // If individual element is greater // maximum possible sum if (array[i] > mid) return false; // Increase sum of current sub - array sum += array[i]; // If the sum is greater than // mid increase count if (sum > mid) { count++; sum = array[i]; } } count++; // Check condition if (count <= K) return true; return false;}// Function to find maximum subarray sum// which is minimumfunction solve(array, n, K){ var max = array.reduce((a,b)=>Math.max(a,b)); var start = max; //Max subarray sum, considering subarray of length 1 var end = 0; for (var i = 0; i < n; i++) { end += array[i]; //Max subarray sum, considering subarray of length n } // Answer stores possible // maximum sub array sum var answer = 0; while (start <= end) { var mid = parseInt((start + end) / 2); // If mid is possible solution // Put answer = mid; if (check(mid, array, n, K)) { answer = mid; end = mid - 1; } else { start = mid + 1; } } return answer;}// Driver Codevar array = [1, 2, 3, 4];var n = array.length;var K = 3;document.write( solve(array, n, K));</script> |
Output
4
Time Complexity : O(N*log(Sum))
Where N is the number of elements of the array and Sum is the sum of all the elements of the array.
Auxiliary Space: O(1) as no extra space has been used.
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