700241. Max sum path in two arrays
Max sum path in two arrays
Summary
The problem is to find the maximum sum path between two arrays. Given two arrays A and B, each containing non-negative integers, you need to find a sequence of elements from both arrays that adds up to the maximum possible value. This problem requires understanding of dynamic programming concepts and array manipulation techniques.
Detailed Explanation
To solve this problem, we will use dynamic programming to build a 2D table where each cell represents the maximum sum path ending at that point. We start by initializing a 2D table dp with dimensions (m+1) x (n+1) where m and n are the lengths of arrays A and B, respectively.
We iterate through both arrays simultaneously, and for each pair of elements (a, b) from A and B, we calculate the maximum sum path ending at that point. This is done by considering two options: either take the current element from array A or array B, and then recursively find the maximum sum path in the remaining arrays.
The time complexity of this approach is O(mn) where m and n are the lengths of arrays A and B, respectively. The space complexity is also O(mn) for storing the dynamic programming table.
Here's a step-by-step breakdown of the solution:
- Initialize a 2D table
dpwith dimensions(m+1) x (n+1)wheremandnare the lengths of arraysAandB, respectively. - Iterate through both arrays simultaneously, considering each pair of elements
(a, b)fromAandB. - For each pair
(a, b), calculate the maximum sum path ending at that point by considering two options:- Take the current element
afrom arrayAand recursively find the maximum sum path in the remaining arrays. - Take the current element
bfrom arrayBand recursively find the maximum sum path in the remaining arrays.
- Take the current element
- Store the maximum sum path ending at each point in the dynamic programming table
dp. - The maximum sum path is stored in the last cell of the dynamic programming table, which corresponds to the maximum sum path between the two arrays.
Optimized Solutions
Java
public int maxSumPath(int[] A, int[] B) {
int m = A.length;
int n = B.length;
int[][] dp = new int[m+1][n+1];
for (int i = 0; i <= m; i++) {
for (int j = 0; j <= n; j++) {
if (i == 0) {
dp[i][j] = B[j];
} else if (j == 0) {
dp[i][j] = A[i-1];
} else {
dp[i][j] = Math.max(A[i-1], B[j-1]) + dp[i-1][j-1];
}
}
}
return dp[m][n];
}