A heap is the reflex answer to K-th largest problems, but when the whole array is already in memory, Quick Select is faster on average. The size-K heap and Quick Select mechanics, and how the shape of the data decides between them.
Union-Find gives each group a root representative and decides whether two elements share a group. The unoptimized version stretches find to O(N) on chain trees; path compression and union by rank make it practically constant.
Kahn’s BFS for topological sort repeatedly takes nodes with zero in-degree, and cycle detection comes along without extra logic. The mechanism, the Course Schedule problems, and the selection criteria against DFS post-order.
Subarray-sum problems collapse from O(N²) to O(N) along two paths — sliding window or prefix sum with a hash. Negative inputs or strict-equality conditions break monotonicity and shift the work from sliding window to prefix sum.
When the answer isn’t sitting in any array, you can still binary-search the answer itself. The parametric-search recipe — define the decision function, fix the direction of monotonicity, justify lo and hi.