What is the optimal time complexity of solving Min Cost Path?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Min Cost Path?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Min Cost Path?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Min Cost Path?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Competitive ProgrammingEasy

Min Cost Path

Learn how to solve the 'Min Cost Path' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function min_cost_path(cost, m, n) that finds the minimum cost to reach cell (m, n) from cell (0, 0) in a 2D cost matrix cost. You can only move down, right, and diagonally down-right from a cell.

Constraints

•1 <= len(cost), len(cost[0]) <= 100
•0 <= cost[i][j] <= 1000
•0 <= m < len(cost)
•0 <= n < len(cost[0])

Examples

Example 1

Input

min_cost_path([[1, 2, 3], [4, 8, 2], [1, 5, 3]], 2, 2)

Output

Explanation

The path with minimum cost is (0,0) -> (0,1) -> (1,2) -> (2,2) with total cost 1 + 2 + 2 + 3 = 8.

Example 2

Input

min_cost_path([[1, 2, 3], [4, 8, 2], [1, 5, 3]], 1, 1)

Output

Explanation

The path with minimum cost is (0,0) -> (1,1) with total cost 1 + 8 = 9.

Need a Hint?

Define subproblem states, establish the recurrence relation, and use memoization (top-down) or tabulation (bottom-up).

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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