K-th Largest Sum Contiguous Subarray
Given an array of integers. Write a program to find the K-th largest sum of contiguous subarray within the array of numbers which has negative and positive numbers.
Examples:
Input: a[] = {20, -5, -1}
k = 3
Output: 14
Explanation: All sum of contiguous
subarrays are (20, 15, 14, -5, -6, -1)
so the 3rd largest sum is 14.
Input: a[] = {10, -10, 20, -40}
k = 6
Output: -10
Explanation: The 6th largest sum among
sum of all contiguous subarrays is -10.
A brute force approach is to store all the contiguous sums in another array and sort it and print the k-th largest. But in the case of the number of elements being large, the array in which we store the contiguous sums will run out of memory as the number of contiguous subarrays will be large (quadratic order)
An efficient approach is to store the pre-sum of the array in a sum[] array. We can find sum of contiguous subarray from index i to j as sum[j]-sum[i-1]
Now for storing the Kth largest sum, use a min heap (priority queue) in which we push the contiguous sums till we get K elements, once we have our K elements, check if the element is greater than the Kth element it is inserted to the min heap with popping out the top element in the min-heap, else not inserted. In the end, the top element in the min-heap will be your answer.
Below is the implementation of the above approach.
C++
// CPP program to find the k-th largest sum// of subarray#include <bits/stdc++.h>using namespace std;// function to calculate kth largest element// in contiguous subarray sumint kthLargestSum(int arr[], int n, int k){ // array to store predix sums int sum[n + 1]; sum[0] = 0; sum[1] = arr[0]; for (int i = 2; i <= n; i++) sum[i] = sum[i - 1] + arr[i - 1]; // priority_queue of min heap priority_queue<int, vector<int>, greater<int> > Q; // loop to calculate the contiguous subarray // sum position-wise for (int i = 1; i <= n; i++) { // loop to traverse all positions that // form contiguous subarray for (int j = i; j <= n; j++) { // calculates the contiguous subarray // sum from j to i index int x = sum[j] - sum[i - 1]; // if queue has less then k elements, // then simply push it if (Q.size() < k) Q.push(x); else { // it the min heap has equal to // k elements then just check // if the largest kth element is // smaller than x then insert // else its of no use if (Q.top() < x) { Q.pop(); Q.push(x); } } } } // the top element will be then kth // largest element return Q.top();}// Driver program to test above functionint main(){ int a[] = { 10, -10, 20, -40 }; int n = sizeof(a) / sizeof(a[0]); int k = 6; // calls the function to find out the // k-th largest sum cout << kthLargestSum(a, n, k); return 0;} |
Java
// Java program to find the k-th // argest sum of subarrayimport java.util.*;class KthLargestSumSubArray{ // function to calculate kth largest // element in contiguous subarray sum static int kthLargestSum(int arr[], int n, int k) { // array to store predix sums int sum[] = new int[n + 1]; sum[0] = 0; sum[1] = arr[0]; for (int i = 2; i <= n; i++) sum[i] = sum[i - 1] + arr[i - 1]; // priority_queue of min heap PriorityQueue<Integer> Q = new PriorityQueue<Integer> (); // loop to calculate the contiguous subarray // sum position-wise for (int i = 1; i <= n; i++) { // loop to traverse all positions that // form contiguous subarray for (int j = i; j <= n; j++) { // calculates the contiguous subarray // sum from j to i index int x = sum[j] - sum[i - 1]; // if queue has less then k elements, // then simply push it if (Q.size() < k) Q.add(x); else { // it the min heap has equal to // k elements then just check // if the largest kth element is // smaller than x then insert // else its of no use if (Q.peek() < x) { Q.poll(); Q.add(x); } } } } // the top element will be then kth // largest element return Q.poll(); } // Driver Code public static void main(String[] args) { int a[] = new int[]{ 10, -10, 20, -40 }; int n = a.length; int k = 6; // calls the function to find out the // k-th largest sum System.out.println(kthLargestSum(a, n, k)); }}/* This code is contributed by Danish Kaleem */ |
Python3
# Python program to find the k-th largest sum # of subarray import heapq# function to calculate kth largest element # in contiguous subarray sum def kthLargestSum(arr, n, k): # array to store predix sums sum = [] sum.append(0) sum.append(arr[0]) for i in range(2, n + 1): sum.append(sum[i - 1] + arr[i - 1]) # priority_queue of min heap Q = [] heapq.heapify(Q) # loop to calculate the contiguous subarray # sum position-wise for i in range(1, n + 1): # loop to traverse all positions that # form contiguous subarray for j in range(i, n + 1): x = sum[j] - sum[i - 1] # if queue has less then k elements, # then simply push it if len(Q) < k: heapq.heappush(Q, x) else: # it the min heap has equal to # k elements then just check # if the largest kth element is # smaller than x then insert # else its of no use if Q[0] < x: heapq.heappop(Q) heapq.heappush(Q, x) # the top element will be then kth # largest element return Q[0]# Driver program to test above function a = [10,-10,20,-40]n = len(a)k = 6# calls the function to find out the # k-th largest sum print(kthLargestSum(a,n,k))# This code is contributed by Kumar Suman |
C#
// C# program to find the k-th // argest sum of subarrayusing System;using System.Collections.Generic;public class KthLargestSumSubArray{ // function to calculate kth largest // element in contiguous subarray sum static int kthLargestSum(int []arr, int n, int k) { // array to store predix sums int []sum = new int[n + 1]; sum[0] = 0; sum[1] = arr[0]; for (int i = 2; i <= n; i++) sum[i] = sum[i - 1] + arr[i - 1]; // priority_queue of min heap List<int> Q = new List<int> (); // loop to calculate the contiguous subarray // sum position-wise for (int i = 1; i <= n; i++) { // loop to traverse all positions that // form contiguous subarray for (int j = i; j <= n; j++) { // calculates the contiguous subarray // sum from j to i index int x = sum[j] - sum[i - 1]; // if queue has less then k elements, // then simply push it if (Q.Count < k) Q.Add(x); else { // it the min heap has equal to // k elements then just check // if the largest kth element is // smaller than x then insert // else its of no use Q.Sort(); if (Q[0] < x) { Q.RemoveAt(0); Q.Add(x); } } Q.Sort(); } } // the top element will be then kth // largest element return Q[0]; } // Driver Code public static void Main(String[] args) { int []a = new int[]{ 10, -10, 20, -40 }; int n = a.Length; int k = 6; // calls the function to find out the // k-th largest sum Console.WriteLine(kthLargestSum(a, n, k)); }}// This code contributed by Rajput-Ji |
Output:
-10
Time complexity: O(n^2 log (k))
Auxiliary Space : O(k) for min-heap and we can store the sum array in the array itself as it is of no use.
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