Create Grid With Exactly K Paths II - Solution & Explanation

Problem Overview: You need to construct a grid where the number of valid paths from the start cell to the destination is exactly k. The challenge is not path traversal itself, but designing the grid so the path count matches the target precisely.

Approach 1: Brute Force Enumeration (Exponential Time, Exponential Space)

The most direct idea is to generate candidate grids and count all possible paths using DFS or backtracking. For each grid configuration, recursively move right or down and increment the count whenever you reach the destination. This works only for very small constraints because the number of grid states and path combinations grows exponentially. Time complexity is O(2^(m*n)) or worse depending on the search space, and space complexity is O(m*n) for recursion.

Approach 2: Dynamic Programming Path Counting (O(m*n) Time, O(m*n) Space)

A better strategy computes the number of paths for a fixed grid using dynamic programming. Define dp[r][c] as the number of ways to reach cell (r,c). Each value comes from the top and left neighbors. This approach efficiently validates whether a constructed grid produces exactly k paths. You typically combine this with incremental grid modification or state search. This is a standard application of dynamic programming and grid traversal.

Approach 3: Binary Construction / DAG Encoding (Optimal Construction)

The optimal solution treats the grid as a directed acyclic graph where each open cell contributes to the total number of reachable paths. The key insight is that path counts can represent powers of two, allowing you to encode any value of k through selective blocking or enabling of transitions. Instead of brute forcing layouts, you build a structured grid where each layer doubles the number of possible routes. Then you use the binary representation of k to activate only the required contributions. This reduces construction complexity dramatically and scales even for very large values of k.

Most accepted solutions use a compact construction with carefully placed walls and open cells. The implementation relies on graph traversal concepts and sometimes bit manipulation to map binary digits into path choices. Typical time complexity is O(log k) or O(n^2) depending on the grid size used for encoding, while space complexity remains O(n^2).

Recommended for interviews: Interviewers usually expect the constructive DP or binary encoding approach. Starting with brute force shows you understand how paths are counted, but the optimized construction demonstrates algorithmic reasoning and control over combinatorial growth. Focus on explaining how each grid layer contributes a deterministic number of paths and how binary decomposition guarantees an exact match for k.