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Recognize "strings that avoid a fixed pattern" with a DFA that tracks the longest matched prefix of the pattern and drops into an absorbing non-accepting trap state the instant the whole pattern appears.

A binary heap is a complete binary tree stored in an array; MAX-HEAPIFY repairs one node by sifting it down in Θ(lg n) , while BUILD-MAX-HEAP turns an unordered array into a heap bottom-up in Θ(n) — not Θ(nlg n) .

Schedule unit-time jobs for maximum profit by taking them in decreasing-profit order and placing each in the latest free slot before its deadline — provably optimal because the feasible sets form a matroid.

Sort items by value-to-weight ratio and fill the knapsack greedily, taking a fraction of the last item; this is provably optimal for the fractional knapsack, unlike 0/1.