Is Create Grid With Exactly K Paths I easy or hard?

The problem is generally considered medium difficulty because the challenge is conceptual rather than implementation-heavy. Understanding how to control path counts through construction takes more insight than standard grid traversal problems.

Create Grid With Exactly K Paths I Python/Java solution

Python solutions usually use lists and recursive or iterative DP for counting paths after constructing the grid. Java implementations often use arrays for deterministic memory usage and iterative state transitions. Both languages support the same O(log k) constructive strategy.

How to solve Create Grid With Exactly K Paths I in O(log k)?

Use a grid construction where path counts combine additively through controlled branching. By mapping powers of two to specific grid patterns, you can represent k using only logarithmic layers. Dynamic programming is then used to confirm the exact number of paths.

What is the best approach for Create Grid With Exactly K Paths I?

The strongest approach uses constructive design combined with dynamic programming. The idea is to build a compact grid where each branch contributes a predictable number of paths, often based on the binary representation of k. This reduces the construction size while keeping verification efficient.

Is Create Grid With Exactly K Paths I asked at Google/Amazon/Meta?

Constructive graph and path-counting problems appear frequently in interviews at companies like Google and Meta because they test DP, graph modeling, and combinational reasoning together. Variants of exact-path construction are common in competitive programming rounds as well.

What data structure is used in Create Grid With Exactly K Paths I?

Most solutions rely on 2D grids, dynamic programming tables, and graph-style traversal logic. Some optimized implementations also use adjacency lists or bit manipulation to encode the contribution of each branch efficiently.

What is the time complexity of Create Grid With Exactly K Paths I?

The optimal constructive solution typically runs in O(log k) time and space because each added structure represents a bit or partial contribution to the total path count. A separate DP verification step on the generated grid runs in O(m * n).

Description Hints 6 Video 32:26

3988. Create Grid With Exactly K Paths I

Medium33.1% acceptance

You are given three integers m, n, and k.

Construct any m x n grid consisting only of the characters '.' and '#', where:

'.' represents a free cell.
'#' represents an obstacle cell.

A valid path is a sequence of free cells that:

Starts at the top-left cell (0, 0).
Ends at the bottom-right cell (m - 1, n - 1).
Moves only:
- Right, from (i, j) to (i, j + 1), or
- Down, from (i, j) to (i + 1, j).

Return any grid such that there are exactly k valid paths from the top-left cell to the bottom-right cell. If no such grid exists, return an empty array.

Example 1:

Input: m = 2, n = 3, k = 2

Output: ["...","#.."]

Explanation:

There are exactly k = 2 valid paths from (0, 0) to (1, 2):

(0, 0) → (0, 1) → (0, 2) → (1, 2)
(0, 0) → (0, 1) → (1, 1) → (1, 2)

Example 2:

Input: m = 3, n = 3, k = 4

Output: ["..#","...","#.."]

Explanation:

There are exactly k = 4 valid paths from (0, 0) to (2, 2):

(0, 0) → (0, 1) → (1, 1) → (1, 2) → (2, 2)
(0, 0) → (0, 1) → (1, 1) → (2, 1) → (2, 2)
(0, 0) → (1, 0) → (1, 1) → (1, 2) → (2, 2)
(0, 0) → (1, 0) → (1, 1) → (2, 1) → (2, 2)

Example 3:

Input: m = 1, n = 4, k = 2

Output: []

Explanation:

No grid exists with exactly k = 2 valid paths for a 1 x 4 grid, so the answer is an empty array.

Constraints:

1 <= m, n <= 10
1 <= k <= 4

Topics

Hints open one at a time

Hint 1

If m == 1 or n == 1, the only possible positive number of valid paths is 1.

Hint 2

For m, n > 1, construct a small grid pattern with exactly k paths for each k from 1 to 4.

Hint 3

For k = 2, use a 2 x 2 open block. It has exactly two paths: right then down, or down then right.

Hint 4

For k = 3, use a 2 x 3 open block, or a 3 x 2 open block. An empty 2 x 3 grid has exactly three paths.

Hint 5

For k = 4, use either a fully open 2 x 4 block, a fully open 4 x 2 block, or a 3 x 3 block with the top-right and bottom-left cells blocked.

Hint 6

After building the small block, connect its bottom-right cell to (m - 1, n - 1) using a single corridor, and fill all other cells with obstacles. If the required block cannot fit in either orientation, return an empty array.

3988. Create Grid With Exactly K Paths I

Topics

Hints open one at a time

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