Is Paths in Matrix Whose Sum Is Divisible by K easy or hard?

Paths in Matrix Whose Sum Is Divisible by K is classified as a Hard problem on LeetCode. The difficulty comes from combining grid path DP with modular arithmetic and managing an extra remainder dimension in the DP state.

Paths in Matrix Whose Sum Is Divisible by K Python/Java solution

Python and Java implementations typically build a 3D DP array where dp[i][j][r] represents path counts with remainder r. Each cell updates states from the top and left neighbors while applying modulo k arithmetic. The final answer is the number of paths ending at the bottom-right cell with remainder 0.

How to solve Paths in Matrix Whose Sum Is Divisible by K in O(n)?

The problem cannot be solved in strict O(n) time because every grid cell and remainder state must be processed. The optimal approach processes m*n*k states using dynamic programming. Space can be optimized using rolling arrays to reduce memory usage, but the time complexity remains O(m*n*k).

What is the best approach for Paths in Matrix Whose Sum Is Divisible by K?

The best approach uses dynamic programming with a 3D DP state tracking the remainder of the path sum modulo k. Each state dp[i][j][r] stores the number of ways to reach cell (i,j) with remainder r. Transitions come from the top and left cells while updating the remainder. This solution runs in O(m*n*k) time and O(m*n*k) space.

Is Paths in Matrix Whose Sum Is Divisible by K asked at Google/Amazon/Meta?

Grid dynamic programming problems with modular constraints frequently appear in interviews at large tech companies including Google, Amazon, and Meta. Variations involving path counting, modulo arithmetic, and DP state compression are common interview themes.

What data structure is used in Paths in Matrix Whose Sum Is Divisible by K?

The solution uses a 3D dynamic programming array or equivalent memoization structure. The DP state tracks grid position (row, column) and the current remainder modulo k. This structure efficiently aggregates counts of valid paths.

What is the time complexity of Paths in Matrix Whose Sum Is Divisible by K?

The optimal dynamic programming solution runs in O(m*n*k) time, where m and n are the grid dimensions and k is the divisor. Each cell processes k remainder states and combines transitions from two directions. Space complexity is also O(m*n*k) for the DP table.

Description Hints 3 Solution Video 21:50

2435. Paths in Matrix Whose Sum Is Divisible by K

Hard58.7% acceptance4 companies

You are given a 0-indexed m x n integer matrix grid and an integer k. You are currently at position (0, 0) and you want to reach position (m - 1, n - 1) moving only down or right.

Return the number of paths where the sum of the elements on the path is divisible by k. Since the answer may be very large, return it modulo 10⁹ + 7.

Example 1:

Input: grid = [[5,2,4],[3,0,5],[0,7,2]], k = 3
Output: 2
Explanation: There are two paths where the sum of the elements on the path is divisible by k.
The first path highlighted in red has a sum of 5 + 2 + 4 + 5 + 2 = 18 which is divisible by 3.
The second path highlighted in blue has a sum of 5 + 3 + 0 + 5 + 2 = 15 which is divisible by 3.

Example 2:

Input: grid = [[0,0]], k = 5
Output: 1
Explanation: The path highlighted in red has a sum of 0 + 0 = 0 which is divisible by 5.

Example 3:

Input: grid = [[7,3,4,9],[2,3,6,2],[2,3,7,0]], k = 1
Output: 10
Explanation: Every integer is divisible by 1 so the sum of the elements on every possible path is divisible by k.

Constraints: