Patterns

Trade memory for speed: remember what you have seen in a hash map or set.

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Two indices walk the data from both ends (or at two speeds) to prune pairs.

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Grow a window to the right, shrink from the left while it is invalid.

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Last-in-first-out: match brackets, evaluate expressions, find the next greater element.

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Halve the search space each step: on sorted data or on a monotonic answer.

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Pointer surgery: dummy heads, fast/slow pointers, in-place reversal.

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Recursion that combines answers from the left and right subtrees.

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A prefix tree: each edge is a character, so shared prefixes are stored once.

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Always get the smallest (or largest) item in O(log n).

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Build candidates one choice at a time, undo the choice, try the next.

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Model entities as nodes and relations as edges; explore with BFS or DFS.

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Weighted shortest paths, minimum spanning trees, and Eulerian paths.

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The answer for i depends on a few earlier answers: memoize or tabulate.

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Two indices (two strings, a grid, or item and capacity) define the state.

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Take the locally best choice and prove it never hurts.

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Sort by start (or end), then sweep while tracking the current interval.

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Matrix rotation, spiral order, number tricks, and fast exponentiation.

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Character counting, palindromes, and parsing.

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Use XOR, masks, and shifts to work with bits directly.

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