Given an array arr[] consisting of N integers, the task is to maximize the resultant sum obtained after adding remaining elements after every removal of the array half with maximum sum.
Array can be divided into two non-empty halves left[] and right[] where left[] contains the elements from the indices [0, N/2) and right[] contains the elements from the indices [N/2, N)
Examples:
Input: arr[] = {6, 2, 3, 4, 5, 5}
Output: 18
Explanation:
The given array is arr[] = {6, 2, 3, 4, 5, 5}
Step 1:
Sum of left half = 6 + 2 + 3 = 11
Sum of right half = 4 + 5 + 5 = 12
Resultant sum S = 11.
Step 2:
Modified array is arr[] = {6, 2, 3}
Sum of left half = 6
Sum of right half = 2 + 3 = 5
Resultant sum S = 11 + 5 = 16
Step 3:
Modified array is arr[] = {2, 3}
Sum of left half = 2
Sum of right half = 3
Resultant sum S = 16 + 2 = 18.
Therefore, the resultant sum is 18.Input: arr[] = {4}
Output: 0
Naive Approach: The simplest approach to solve the problem is to use recursion. Below are the steps:
- Use the concept of prefix sum and initialize a variable, say res to store the final result.
- Create a dictionary.
- Traverse the array and store all the prefix sum in the dictionary.
- Now, iterate over the range [0, N] and store the prefix sum of left and right halves of the array as left and right respectively.
- Now, there are three possible conditions:
- left > right
- left < right
- left == right
- For all the above conditions, ignore the maximum sum and add the minimum among left and right sum to the resultant sum and continue the recursive calls.
- After all the recursive call ends, print the maximum value of the resultant sum.
Below is the implementation of the above approach:
C++14
// C++14 program to implement// the above approach#include <bits/stdc++.h>using namespace std;
// Function to find the maximum sumint maxweight(int s, int e,
unordered_map<int, int>& pre)
{ // Base case
// len of array is 1
if (s == e)
return 0;
// Stores the final result
int ans = 0;
// Traverse the array
for(int i = s; i < e; i++)
{
// Store left prefix sum
int left = pre[i] - pre[s - 1];
// Store right prefix sum
int right = pre[e] - pre[i];
// Compare the left and right
if (left < right)
ans = max(ans, left +
maxweight(s, i, pre));
// If both are equal apply
// the optimal method
if (left == right)
{
// Update with minimum
ans = max({ans, left +
maxweight(s, i, pre),
right +
maxweight(i + 1, e, pre)});
}
if (left > right)
ans = max(ans, right +
maxweight(i + 1, e, pre));
}
// Return the final ans
return ans;
}// Function to print maximum sumvoid maxSum(vector<int> arr)
{ // Dictionary to store prefix sums
unordered_map<int, int> pre;
pre[-1] = 0;
pre[0] = arr[0];
// Traversing the array
for(int i = 1; i < arr.size(); i++)
{
// Add prefix sum of the array
pre[i] = pre[i - 1] + arr[i];
}
cout << maxweight(0, arr.size() - 1, pre);
}// Driver Codeint main()
{ vector<int> arr = { 6, 2, 3, 4, 5, 5 };
// Function call
maxSum(arr);
return 0;
}// This code is contributed by mohit kumar 29 |
Java
// Java program to implement // the above approach import java.util.*;
class GFG{
// Function to find the maximum sumstatic int maxweight(int s, int e,
Map<Integer, Integer> pre)
{ // Base case
// len of array is 1
if (s == e)
return 0;
// Stores the final result
int ans = 0;
// Traverse the array
for(int i = s; i < e; i++)
{
// Store left prefix sum
int left = pre.get(i) - pre.get(s - 1);
// Store right prefix sum
int right = pre.get(e) - pre.get(i);
// Compare the left and right
if (left < right)
ans = Math.max(ans, left +
maxweight(s, i, pre));
// If both are equal apply
// the optimal method
if (left == right)
{
// Update with minimum
ans = Math.max(ans, Math.max(left +
maxweight(s, i, pre),
right + maxweight(i + 1,
e, pre)));
}
if (left > right)
ans = Math.max(ans, right +
maxweight(i + 1, e, pre));
}
// Return the final ans
return ans;
}// Function to print maximum sumstatic void maxSum(List<Integer> arr)
{ // To store prefix sums
Map<Integer, Integer> pre = new HashMap<>();
pre.put(-1, 0);
pre.put(0, arr.get(0));
// Traversing the array
for(int i = 1; i < arr.size(); i++)
{
// Add prefix sum of the array
pre.put(i, pre.getOrDefault(i - 1, 0) +
arr.get(i));
}
System.out.println(maxweight(0,
arr.size() - 1, pre));
}// Driver codepublic static void main (String[] args)
{ List<Integer> arr = Arrays.asList(6, 2, 3,
4, 5, 5);
// Function call
maxSum(arr);
}}// This code is contributed by offbeat |
Python3
# Python3 program to implement# the above approach# Function to find the maximum sumdef maxweight(s, e, pre):
# Base case
# len of array is 1
if s == e:
return 0
# Stores the final result
ans = 0
# Traverse the array
for i in range(s, e):
# Store left prefix sum
left = pre[i] - pre[s - 1]
# Store right prefix sum
right = pre[e] - pre[i]
# Compare the left and right
if left < right:
ans = max(ans, left \
+ maxweight(s, i, pre))
# If both are equal apply
# the optimal method
if left == right:
# Update with minimum
ans = max(ans, left \
+ maxweight(s, i, pre),
right \
+ maxweight(i + 1, e, pre))
if left > right:
ans = max(ans, right \
+ maxweight(i + 1, e, pre))
# Return the final ans
return ans
# Function to print maximum sumdef maxSum(arr):
# Dictionary to store prefix sums
pre = {-1: 0, 0: arr[0]}
# Traversing the array
for i in range(1, len(arr)):
# Add prefix sum of the array
pre[i] = pre[i - 1] + arr[i]
print(maxweight(0, len(arr) - 1, pre))
# Drivers Codearr = [6, 2, 3, 4, 5, 5]
# Function CallmaxSum(arr) |
Javascript
<script>// js program to implement// the above approach// Function to find the maximum sumfunction maxweight(s, e, pre){
// Base case
// len of array is 1
if (s == e)
return 0;
// Stores the final result
let ans = 0;
// Traverse the array
for(let i = s; i < e; i++)
{
// Store left prefix sum
if(!pre[i])
pre[i] = 0;
if(!pre[e])
pre[e] = 0;
if(!pre[s-1])
pre[s-1] = 0;
let left = pre[i] - pre[s - 1];
// Store right prefix sum
let right = pre[e] - pre[i];
// Compare the left and right
if (left < right)
ans = Math.max(ans, left +
maxweight(s, i, pre));
// If both are equal apply
// the optimal method
if (left == right)
{
// Update with minimum
ans = Math.max(ans, Math.max(left +
maxweight(s, i, pre),
right +
maxweight(i + 1, e, pre)));
}
if (left > right)
ans = Math.max(ans, right +
maxweight(i + 1, e, pre));
}
// Return the final ans
return ans;
}// Function to print maximum sumfunction maxSum(arr)
{ // Dictionary to store prefix sums
let pre = new Map;
pre[-1] = 0;
pre[0] = arr[0];
// Traversing the array
for(let i = 1; i < arr.length; i++)
{
// Add prefix sum of the array
pre[i] = pre[i - 1] + arr[i];
}
document.write( maxweight(0, arr.length - 1, pre));
}// Driver Codearr = [ 6, 2, 3, 4, 5, 5 ];// Function call maxSum(arr);
</script> |
C#
// C# program to implement // the above approach using System;
using System.Collections.Generic;
class GFG{
// Function to find the maximum sumstatic int maxweight(int s, int e,
Dictionary<int,
int> pre)
{ // Base case
// len of array is 1
if (s == e)
return 0;
// Stores the
// readonly result
int ans = 0;
// Traverse the array
for(int i = s; i < e; i++)
{
// Store left prefix sum
int left = pre[i] - pre[s - 1];
// Store right prefix sum
int right = pre[e] - pre[i];
// Compare the left and right
if (left < right)
ans = Math.Max(ans, left +
maxweight(s, i, pre));
// If both are equal apply
// the optimal method
if (left == right)
{
// Update with minimum
ans = Math.Max(ans, Math.Max(left +
maxweight(s, i, pre),
right + maxweight(i + 1,
e, pre)));
}
if (left > right)
ans = Math.Max(ans, right +
maxweight(i + 1, e, pre));
}
// Return the readonly ans
return ans;
}// Function to print maximum sumstatic void maxSum(List<int> arr)
{ // To store prefix sums
Dictionary<int,
int> pre = new Dictionary<int,
int>();
pre.Add(-1, 0);
pre.Add(0, arr[0]);
// Traversing the array
for(int i = 1; i < arr.Count; i++)
{
// Add prefix sum of the array
if(pre[i - 1] != 0)
pre.Add(i, pre[i - 1] + arr[i]);
else
pre.Add(i, arr[i]);
}
Console.WriteLine(maxweight(0,
arr.Count - 1, pre));
}// Driver codepublic static void Main(String[] args)
{ List<int> arr = new List<int>();
arr.Add(6);
arr.Add(2);
arr.Add(3);
arr.Add(4);
arr.Add(5);
arr.Add(5);
// Function call
maxSum(arr);
}}// This code is contributed by gauravrajput1 |
Output:
18
Time Complexity: O(2N)
Auxiliary Space: O(N2)
Efficient Approach: To optimize the above approach, the idea is to observe that there are a number of repeated overlapping subproblems.
Therefore, for optimization, use Dynamic Programming. The idea is to use a dictionary and keep track of the result values so that when they are required in further computations it can be accessed without calculating them again.
Below is the implementation of the above approach:
C++
// C++ program to implement// the above approach#include <bits/stdc++.h>using namespace std;
int dp[100][100];
// Function to find the maximum sumint maxweight(int s, int e,
map<int, int> pre)
{ // Base Case
if (s == e)
return 0;
// Create a key to map
// the values
// Check if (mapped key is
// found in the dictionary
if (dp[s][e] != -1)
return dp[s][e];
int ans = 0;
// Traverse the array
for(int i = s; i < e; i++)
{
// Store left prefix sum
int left = pre[i] - pre[s - 1];
// Store right prefix sum
int right = pre[e] - pre[i];
// Compare the left and
// right values
if (left < right)
ans = max(
ans, (int)(left +
maxweight(s, i, pre)));
if (left == right)
ans = max(
ans, max(left + maxweight(s, i,
pre),
right + maxweight(i + 1,
e, pre)));
if (left > right)
ans = max(
ans, right + maxweight(i + 1, e, pre));
// Store the value in dp array
dp[s][e] = ans;
}
// Return the final answer
return dp[s][e];
}// Function to print maximum sumvoid maxSum(int arr[], int n)
{ // Stores prefix sum
map<int, int> pre;
pre[-1] = 0;
pre[0] = arr[0];
// Store results of subproblems
memset(dp, -1, sizeof dp);
// Traversing the array
for(int i = 0; i < n; i++)
// Add prefix sum of array
pre[i] = pre[i - 1] + arr[i];
// Print the answer
cout << (maxweight(0, n - 1, pre));
}// Driver Codeint main()
{ int arr[] = { 6, 2, 3, 4, 5, 5 };
// Function call
maxSum(arr, 6);
}// This code is contributed by grand_master |
Java
// Java program to implement// the above approachimport java.util.*;
class solution{
static int[][] dp = new int[100][100];
// Function to find the maximum sumstatic int maxweight(int s, int e,
HashMap<Integer, Integer> pre)
{ // Base Case
if (s == e)
return 0;
// Create a key to map
// the values
// Check if (mapped key is
// found in the dictionary
if (dp[s][e] != -1)
return dp[s][e];
int ans = 0;
// Traverse the array
for(int i = s; i < e; i++)
{
// Store left prefix sum
int left = pre.get(i) -
pre.get(s - 1);
// Store right prefix sum
int right = pre.get(e) -
pre.get(i);
// Compare the left and
// right values
if (left < right)
ans = Math.max(ans, (int)(left +
maxweight(s, i, pre)));
if (left == right)
ans = Math.max(ans,
Math.max(left + maxweight(s, i,
pre),
right + maxweight(i + 1,
e, pre)));
if (left > right)
ans = Math.max(ans, right + maxweight(i + 1,
e, pre));
// Store the value in dp array
dp[s][e] = ans;
}
// Return the final answer
return dp[s][e];
}// Function to print maximum sumstatic void maxSum(int arr[], int n)
{ // Stores prefix sum
HashMap<Integer,
Integer> pre = new HashMap<Integer,
Integer>();
pre.put(-1, 0);
pre.put(0, arr[0]);
// Store results of subproblems
for(int i = 0; i < 100; i++)
{
for(int j = 0; j < 100; j++)
dp[i][j] = -1;
}
// Traversing the array
for(int i = 0; i < n; i++)
// Add prefix sum of array
pre.put(i, pre.get(i - 1) + arr[i]);
// Print the answer
System.out.print((maxweight(0, n - 1, pre)));
}// Driver Codepublic static void main(String args[])
{ int []arr = { 6, 2, 3, 4, 5, 5 };
// Function call
maxSum(arr, 6);
}}// This code is contributed by Surendra_Gangwar |
Python3
# Python3 program to implement# the above approach# Function to find the maximum sumdef maxweight(s, e, pre, dp):
# Base Case
if s == e:
return 0
# Create a key to map
# the values
key = (s, e)
# Check if mapped key is
# found in the dictionary
if key in dp:
return dp[key]
ans = 0
# Traverse the array
for i in range(s, e):
# Store left prefix sum
left = pre[i] - pre[s-1]
# Store right prefix sum
right = pre[e] - pre[i]
# Compare the left and
# right values
if left < right:
ans = max(ans, left \
+ maxweight(s, i, pre, dp))
if left == right:
# Update with minimum
ans = max(ans, left \
+ maxweight(s, i, pre, dp),
right \
+ maxweight(i + 1, e, pre, dp))
if left > right:
ans = max(ans, right \
+ maxweight(i + 1, e, pre, dp))
# Store the value in dp array
dp[key] = ans
# Return the final answer
return dp[key]
# Function to print maximum sumdef maxSum(arr):
# Stores prefix sum
pre = {-1: 0, 0: arr[0]}
# Store results of subproblems
dp = {}
# Traversing the array
for i in range(1, len(arr)):
# Add prefix sum of array
pre[i] = pre[i - 1] + arr[i]
# Print the answer
print(maxweight(0, len(arr) - 1, pre, dp))
# Driver Codearr = [6, 2, 3, 4, 5, 5]
# Function CallmaxSum(arr) |
Javascript
<script>// js program to implement// the above approachlet dp=[];for(let i = 0;i<100;i++){
dp[i] = [];for(let j = 0;j<100;j++){
dp[i][j] = 0;}}// Function to find the maximum sumfunction maxweight( s, e, pre)
{ // Base Case
if (s == e)
return 0;
// Create a key to map
// the values
// Check if (mapped key is
// found in the dictionary
if (dp[s][e] != -1)
return dp[s][e];
let ans = 0;
// Traverse the array
for(let i = s; i < e; i++)
{
// Store left prefix sum
let left = pre[i] - pre[s - 1];
// Store right prefix sum
let right = pre[e] - pre[i];
// Compare the left and
// right values
if (left < right)
ans = Math.max(
ans, Number(left +
maxweight(s, i, pre)));
if (left == right)
ans = Math.max(
ans,Math. max(left + maxweight(s, i,
pre),
right + maxweight(i + 1,
e, pre)));
if (left > right)
ans = Math.max(
ans, right + maxweight(i + 1, e, pre));
// Store the value in dp array
dp[s][e] = ans;
}
// Return the final answer
return dp[s][e];
}// Function to print maximum sumfunction maxSum(arr, n)
{ // Stores prefix sum
let pre = new Map();
pre[-1] = 0;
pre[0] = arr[0];
// Store results of subproblems
for(let i = 0;i<100;i++){
for(let j = 0;j<100;j++){
dp[i][j] = -1;
}} // Traversing the array
for(let i = 0; i < n; i++)
// Add prefix sum of array
pre[i] = pre[i - 1] + arr[i];
// Print the answer
document.write(maxweight(0, n - 1, pre));
}// Driver Codelet arr= [ 6, 2, 3, 4, 5, 5 ]; // Function call
maxSum(arr, 6);
</script> |
C#
// C# program to implement// the above approachusing System;
using System.Collections.Generic;
class GFG{
static int[,] dp = new int[100, 100];
// Function to find the maximum sumstatic int maxweight(int s, int e,
Dictionary<int, int> pre)
{ // Base Case
if (s == e)
return 0;
// Create a key to map
// the values
// Check if (mapped key is
// found in the dictionary
if (dp[s, e] != -1)
return dp[s, e];
int ans = 0;
// Traverse the array
for(int i = s; i < e; i++)
{
// Store left prefix sum
int left = pre[i] -
pre[s - 1];
// Store right prefix sum
int right = pre[e] -
pre[i];
// Compare the left and
// right values
if (left < right)
ans = Math.Max(ans, (int)(left +
maxweight(s, i, pre)));
if (left == right)
ans = Math.Max(ans,
Math.Max(left + maxweight(s, i,
pre),
right + maxweight(i + 1,
e, pre)));
if (left > right)
ans = Math.Max(ans, right + maxweight(i + 1,
e, pre));
// Store the value in dp array
dp[s, e] = ans;
}
// Return the readonly answer
return dp[s, e];
}// Function to print maximum sumstatic void maxSum(int []arr, int n)
{ // Stores prefix sum
Dictionary<int,
int> pre = new Dictionary<int,
int>();
pre.Add(-1, 0);
pre.Add(0, arr[0]);
// Store results of subproblems
for(int i = 0; i < 100; i++)
{
for(int j = 0; j < 100; j++)
dp[i, j] = -1;
}
// Traversing the array
for(int i = 1; i < n; i++)
// Add prefix sum of array
pre.Add(i, pre[i - 1] + arr[i]);
// Print the answer
Console.Write((maxweight(0, n - 1, pre)));
}// Driver Codepublic static void Main(String []args)
{ int []arr = { 6, 2, 3, 4, 5, 5 };
// Function call
maxSum(arr, 6);
}}// This code is contributed by Amit Katiyar |
Output:
18
Time Complexity: O(N3)
Auxiliary Space: O(N2)
Efficient approach : Using DP Tabulation method ( Iterative approach )
The approach to solve this problem is same but DP tabulation(bottom-up) method is better then Dp + memoization(top-down) because memoization method needs extra stack space of recursion calls.
Steps to solve this problem :
- Create a table DP to store the solution of the subproblems.
- Initialize the table with base cases.
- Now Iterate over subproblems to get the value of current problem form previous computation of subproblems stored in DP.
- Return the final solution stored in dp[0][n-1].
Implementation :
C++
// C++ code for above approach#include <bits/stdc++.h>using namespace std;
int maxSum(int arr[], int n)
{ // Create a prefix sum array
int pre[n+1];
pre[0] = 0;
for (int i = 1; i <= n; i++)
pre[i] = pre[i-1] + arr[i-1];
// Create a 2D dp table
int dp[n][n];
// Fill the diagonal elements with 0
for (int i = 0; i < n; i++)
dp[i][i] = 0;
// Fill the remaining elements
for (int len = 2; len <= n; len++) {
for (int i = 0; i <= n-len; i++) {
int j = i + len - 1;
dp[i][j] = INT_MIN;
// iterate over subproblems and get
// the current value from previous computation
for (int k = i; k < j; k++) {
int left_sum = pre[k+1] - pre[i];
int right_sum = pre[j+1] - pre[k+1];
// update current value with
// respect to different cases
if (left_sum < right_sum)
dp[i][j] = max(dp[i][j], left_sum + dp[i][k]);
else if (left_sum > right_sum)
dp[i][j] = max(dp[i][j], right_sum + dp[k+1][j]);
else
dp[i][j] = max(dp[i][j], max(left_sum +
dp[i][k], right_sum + dp[k+1][j]));
}
}
}
// Return the maximum sum
return dp[0][n-1];
}int main()
{ int arr[] = {6, 2, 3, 4, 5, 5};
int n = sizeof(arr)/sizeof(arr[0]);
// function call
cout << maxSum(arr, n) << endl;
return 0;
}// this code is contributed by bhardwajji |
Java
import java.util.*;
public class Main {
public static int maxSum(int[] arr, int n) {
// Create a prefix sum array
int[] pre = new int[n+1];
pre[0] = 0;
for (int i = 1; i <= n; i++)
pre[i] = pre[i-1] + arr[i-1];
// Create a 2D dp table
int[][] dp = new int[n][n];
// Fill the diagonal elements with 0
for (int i = 0; i < n; i++)
dp[i][i] = 0;
// Fill the remaining elements
for (int len = 2; len <= n; len++) {
for (int i = 0; i <= n-len; i++) {
int j = i + len - 1;
dp[i][j] = Integer.MIN_VALUE;
// iterate over subproblems and get
// the current value from previous computation
for (int k = i; k < j; k++) {
int left_sum = pre[k+1] - pre[i];
int right_sum = pre[j+1] - pre[k+1];
// update current value with
// respect to different cases
if (left_sum < right_sum)
dp[i][j] = Math.max(dp[i][j], left_sum + dp[i][k]);
else if (left_sum > right_sum)
dp[i][j] = Math.max(dp[i][j], right_sum + dp[k+1][j]);
else
dp[i][j] = Math.max(dp[i][j], Math.max(left_sum +
dp[i][k], right_sum + dp[k+1][j]));
}
}
}
// Return the maximum sum
return dp[0][n-1];
}
public static void main(String[] args) {
int[] arr = {6, 2, 3, 4, 5, 5};
int n = arr.length;
// function call
System.out.println(maxSum(arr, n));
}
} |
Python3
def maxSum(arr, n):
# Create a prefix sum array
pre = [0] * (n+1)
for i in range(1, n+1):
pre[i] = pre[i-1] + arr[i-1]
# Create a 2D dp table
dp = [[0 for j in range(n)] for i in range(n)]
# Fill the diagonal elements with 0
for i in range(n):
dp[i][i] = 0
# Fill the remaining elements
for length in range(2, n+1):
for i in range(n-length+1):
j = i + length - 1
dp[i][j] = float('-inf')
# iterate over subproblems and get
# the current value from previous computation
for k in range(i, j):
left_sum = pre[k+1] - pre[i]
right_sum = pre[j+1] - pre[k+1]
# update current value with
# respect to different cases
if left_sum < right_sum:
dp[i][j] = max(dp[i][j], left_sum + dp[i][k])
elif left_sum > right_sum:
dp[i][j] = max(dp[i][j], right_sum + dp[k+1][j])
else:
dp[i][j] = max(dp[i][j], max(left_sum + dp[i][k], right_sum + dp[k+1][j]))
# Return the maximum sum
return dp[0][n-1]
# Driver codearr = [6, 2, 3, 4, 5, 5]
n = len(arr)
# Function callprint(maxSum(arr, n))
|
C#
using System;
public class MaxSumSubsequence
{ public static int MaxSum(int[] arr, int n)
{
// Create a prefix sum array
int[] pre = new int[n+1];
pre[0] = 0;
for (int i = 1; i <= n; i++)
pre[i] = pre[i-1] + arr[i-1];
// Create a 2D dp table
int[,] dp = new int[n,n];
// Fill the diagonal elements with 0
for (int i = 0; i < n; i++)
dp[i,i] = 0;
// Fill the remaining elements
for (int len = 2; len <= n; len++) {
for (int i = 0; i <= n-len; i++) {
int j = i + len - 1;
dp[i,j] = int.MinValue;
// iterate over subproblems and get
// the current value from previous computation
for (int k = i; k < j; k++) {
int left_sum = pre[k+1] - pre[i];
int right_sum = pre[j+1] - pre[k+1];
// update current value with
// respect to different cases
if (left_sum < right_sum)
dp[i,j] = Math.Max(dp[i,j], left_sum + dp[i,k]);
else if (left_sum > right_sum)
dp[i,j] = Math.Max(dp[i,j], right_sum + dp[k+1,j]);
else
dp[i,j] = Math.Max(dp[i,j], Math.Max(left_sum + dp[i,k], right_sum + dp[k+1,j]));
}
}
}
// Return the maximum sum
return dp[0,n-1];
}
public static void Main()
{
int[] arr = {6, 2, 3, 4, 5, 5};
int n = arr.Length;
// function call
Console.WriteLine(MaxSum(arr, n));
}
} |
Javascript
// Javascript code for above approachfunction maxSum(arr, n) {
// Create a prefix sum array
let pre = new Array(n + 1);
pre[0] = 0;
for (let i = 1; i <= n; i++) {
pre[i] = pre[i - 1] + arr[i - 1];
}
// Create a 2D dp table
let dp = new Array(n);
for (let i = 0; i < n; i++) {
dp[i] = new Array(n);
}
// Fill the diagonal elements with 0
for (let i = 0; i < n; i++) {
dp[i][i] = 0;
}
// Fill the remaining elements
for (let len = 2; len <= n; len++) {
for (let i = 0; i <= n - len; i++) {
let j = i + len - 1;
dp[i][j] = Number.MIN_SAFE_INTEGER;
// iterate over subproblems and get
// the current value from previous computation
for (let k = i; k < j; k++) {
let left_sum = pre[k + 1] - pre[i];
let right_sum = pre[j + 1] - pre[k + 1];
// update current value with
// respect to different cases
if (left_sum < right_sum) {
dp[i][j] = Math.max(dp[i][j], left_sum + dp[i][k]);
} else if (left_sum > right_sum) {
dp[i][j] = Math.max(dp[i][j], right_sum + dp[k + 1][j]);
} else {
dp[i][j] = Math.max(dp[i][j], Math.max(left_sum + dp[i][k], right_sum + dp[k + 1][j]));
}
}
}
}
// Return the maximum sum
return dp[0][n - 1];
}let arr = [6, 2, 3, 4, 5, 5];let n = arr.length;console.log(maxSum(arr, n)); |
Output
18
Time Complexity: O(N3)
Auxiliary Space: O(N2)