Topological Sort on grid

Question

Apply the DFS algorithm on a grid to solve this optimization problem.

Codemia · Accepted Answer

Start by understanding the input/output format and constraints. Identify the core problem pattern. Ask clarifying questions: Is the input sorted? Can there be duplicates? What are the size constraints? What should we return for edge cases (empty input, single element)?

Think about what data structure would give you efficient access for the operations this problem requires.

Always start with the simplest correct solution, even if it's not optimal. This shows you can solve the problem and gives you a baseline to optimize from.

The brute force typically involves nested loops or exhaustive enumeration. Analyze its time complexity (often O(n^2) or O(2^n)) and space complexity. Explain why this isn't efficient enough for the given constraints.

The key insight is to identify the right algorithmic pattern. Common patterns to consider:

- **Hash Map**: For O(1) lookups to avoid nested loops (two-sum pattern)
- **Two Pointers**: For sorted arrays or when searching for pairs
- **Sliding Window**: For contiguous subarray problems with a constraint
- **BFS/DFS**: For graph/tree traversal and shortest path
- **Dynamic Programming**: For optimization problems with overlapping subproblems
- **Binary Search**: When the answer space is monotonic

Choose the pattern that matches the problem structure and implement it step by step.

**Time Complexity**: Analyze the number of operations as a function of input size. Consider the worst case. For nested loops it's O(n^2), for divide-and-conquer it's usually O(n log n), for hash map solutions it's typically O(n).

**Space Complexity**: Count the additional memory used beyond the input. Hash maps add O(n) space, recursion adds O(depth) stack space, in-place algorithms use O(1) extra space.

Always verify your solution handles these cases:

- Empty input (empty array, empty string)
- Single element
- All elements are the same (duplicates)
- Input is already sorted (ascending and descending)
- Negative numbers or zero
- Maximum size input (performance test)
- The answer doesn't exist (return appropriate default)

Topological Sort on grid

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