Problem Statement: Given an integer array arr, find the contiguous subarray (containing at least one number) which
has the largest sum and return its sum and print the subarray.
Examples:
Example 1: Input: arr = [-2,1,-3,4,-1,2,1,-5,4] Output: 6 Explanation: [4,-1,2,1] has the largest sum = 6. Examples 2: Input: arr = [1] Output: 1 Explanation: Array has only one element and which is giving positive sum of 1.
Solution
Disclaimer: Don’t jump directly to the solution, try it out yourself first.
Solution 1: Naive Approach
Approach: Using three for loops, we will get all possible subarrays in two loops and their sum in another loop, and then return the maximum of them.
Code:
C++ Code
#include<bits/stdc++.h>
using namespace std;
int maxSubArray(vector < int > & nums, vector < int > & subarray) {
int max_sum = 0;
int n = nums.size();
if (n == 1) {
return nums[0];
}
int i, j;
for (i = 0; i <= n - 1; i++) {
for (j = i; j <= n - 1; j++) {
int sum = 0;
for (int k = i; k <= j; k++)
sum = sum + nums[k];
if (sum > max_sum) {
subarray.clear();
max_sum = sum;
subarray.push_back(i);
subarray.push_back(j);
}
}
}
return max_sum;
}
int main() {
vector<int> arr{-2,1,-3,4,-1,2,1,-5,4};
vector < int > subarray;
int lon = maxSubArray(arr, subarray);
cout << "The longest subarray with maximum sum is " << lon << endl;
cout << "The subarray is " << endl;
for (int i = subarray[0]; i <= subarray[1]; i++) {
cout << arr[i] << " ";
}
}Output:
The longest subarray with maximum sum is 6
The subarray is
4 -1 2 1
Time Complexity: O(N^3)
Space Complexity: O(1)
Java Code
import java.util.*;
class TUF {
public static int maxSubArray(int[] nums, ArrayList < Integer > subarray) {
int max_sum = 0;
int n = nums.length;
if (n == 1) {
return nums[0];
}
for (int i = 0; i <= n - 1; i++) {
for (int j = i; j <= n - 1; j++) {
int sum = 0;
for (int k = i; k <= j; k++)
sum = sum + nums[k];
if (sum > max_sum) {
subarray.clear();
max_sum = sum;
subarray.add(i);
subarray.add(j);
}
}
}
return max_sum;
}
public static void main(String args[]) {
int arr[]={-2,1,-3,4,-1,2,1,-5,4};
ArrayList < Integer > subarray = new ArrayList < > ();
int lon = maxSubArray(arr, subarray);
System.out.println("The longest subarray with maximum sum is " + lon);
System.out.println("The subarray is ");
for (int i = subarray.get(0); i <= subarray.get(1); i++) {
System.out.print(arr[i] + " ");
}
}
}Output:
The longest subarray with maximum sum is 6
The subarray is
4 -1 2 1
Time Complexity: O(N^3)- TLE
Space Complexity: O(1)
Solution 2: A better approach
Approach :
We can also do this problem using only two for loop, starting the function with ( max_sum ) variable which will have the final answer. We can get the sum of all possible subarrays in this way and then return the maximum so far.
Code:
C++ Code
#include<bits/stdc++.h>
using namespace std;
int maxSubArray(vector < int > & nums, vector < int > & subarray) {
int max_sum = INT_MIN;
for (int i = 0; i < nums.size(); i++) {
int curr_sum = 0;
for (int j = i; j < nums.size(); j++) {
curr_sum += nums[j];
if (curr_sum > max_sum) {
subarray.clear();
max_sum = curr_sum;
subarray.push_back(i);
subarray.push_back(j);
}
}
}
return max_sum;
}
int main() {
vector<int> arr{-2,1,-3,4,-1,2,1,-5,4};
vector < int > subarray;
int lon = maxSubArray(arr, subarray);
cout << "The longest subarray with maximum sum is " << lon << endl;
cout << "The subarray is " << endl;
for (int i = subarray[0]; i <= subarray[1]; i++) {
cout << arr[i] << " ";
}
}Output:
The longest subarray with maximum sum is 6
The subarray is
4 -1 2 1
Time Complexity: O(N^2)
Space Complexity: O(1)
Java Code
import java.util.*;
class TUF {
public static int maxSubArray(int[] nums, ArrayList < Integer > subarray) {
int max_sum = Integer.MIN_VALUE;
for (int i = 0; i < nums.length; i++) {
int curr_sum = 0;
for (int j = i; j < nums.length; j++) {
curr_sum += nums[j];
if (curr_sum > max_sum) {
subarray.clear();
max_sum = curr_sum;
subarray.add(i);
subarray.add(j);
}
}
}
return max_sum;
}
public static void main(String args[]) {
int arr[]={-2,1,-3,4,-1,2,1,-5,4};
ArrayList < Integer > subarray = new ArrayList < > ();
int lon = maxSubArray(arr, subarray);
System.out.println("The longest subarray with maximum sum is " + lon);
System.out.println("The subarray is ");
for (int i = subarray.get(0); i <= subarray.get(1); i++) {
System.out.print(arr[i] + " ");
}
}
}Output:
The longest subarray with maximum sum is 6
The subarray is
4 -1 2 1
Time Complexity: O(N^2)
Space Complexity: O(1)
Solution : 3 Optimal Solution: Kadane’s Algorithm
Intuition: Basically this problem can be done in linear time complexity with Kadane’s algorithm along with that will also get the subarray which is giving the largest positive-sum.
Approach:
-> We will have the following variables in the beginning :
msf – max so far, meh – max end here, start – to get the starting index of ans’s subarray, end – to get the ending index of ans’s subarray.
-> Traverse the array starting with adding the ith element with max_end_here(meh) , now we will check that adding the element gives greater value than max_so_far(msf) , if yes then we will update our meh and also update the starting and ending index .
for(int i=0;i<nums.size();i++){
meh+=nums[i];
if(meh>msf){ msf=meh; start=s; end=i; }
if(meh<0){meh=0; s=i+1;}
}
->Now in this step, we will print the answer subarray using the start and end variables.
->Return the largest subarray sum that is:- msf.
Code:
C++ Code
#include<bits/stdc++.h>
using namespace std;
int maxSubArray(vector < int > & nums, vector < int > & subarray) {
int msf = nums[0], meh = 0;
int s = 0;
for (int i = 0; i < nums.size(); i++) {
meh += nums[i];
if (meh > msf) {
subarray.clear();
msf = meh;
subarray.push_back(s);
subarray.push_back(i);
}
if (meh < 0) {
meh = 0;
s = i + 1;
}
}
return msf;
}
int main() {
vector<int> arr{-2,1,-3,4,-1,2,1,-5,4};
vector < int > subarray;
int lon = maxSubArray(arr, subarray);
cout << "The longest subarray with maximum sum is " << lon << endl;
cout << "The subarray is " << endl;
for (int i = subarray[0]; i <= subarray[1]; i++) {
cout << arr[i] << " ";
}
}Output:
The longest subarray with maximum sum is 6
The subarray is
4 -1 2 1
Time Complexity: O(N)
Space Complexity:O(1)
Java Code
import java.util.*;
class TUF{
public static int maxSubArray(int[] nums,ArrayList<Integer> subarray) {
int msf=nums[0] , meh=0 ;
int s=0;
for(int i=0;i<nums.length;i++){
meh+=nums[i];
if(meh>msf)
{
subarray.clear();
msf=meh;
subarray.add(s);
subarray.add(i);
}
if(meh<0)
{
meh=0;
s=i+1;
}
}
return msf;
}
public static void main(String args[])
{
int arr[]={-2,1,-3,4,-1,2,1,-5,4};
ArrayList<Integer> subarray=new ArrayList<>();
int lon=maxSubArray(arr,subarray);
System.out.println("The longest subarray with maximum sum is "+lon);
System.out.println("The subarray is ");
for(int i=subarray.get(0);i<=subarray.get(1);i++)
{
System.out.print(arr[i]+" ");
}
}
}Output:
The longest subarray with maximum sum is 6
The subarray is
4 -1 2 1
Time Complexity: O(N)
Space Complexity:O(1)
Special thanks to Abhay Rai for contributing to this article on takeUforward. If you also wish to share your knowledge with the takeUforward fam, please check out this article