Expert Test - Algorithms

Kruskal sorts edges and adds the smallest that does not form a cycle, using a union-find structure, running in O(E log E) time. Prim grows a tree from one vertex, always adding the cheapest crossing edge with a priority queue, running in O((V+E) log V). Both yield a minimum spanning tree.

An NP-complete problem is in NP and is at least as hard as every other problem in NP, so any NP problem reduces to it in polynomial time. No polynomial-time algorithm is known for any of them, and finding one would prove P equals NP. Examples include SAT and the traveling salesman decision problem.

The Knuth-Morris-Pratt algorithm precomputes a prefix or failure table that tells it how far to shift the pattern after a mismatch without rechecking matched characters. This avoids re-scanning the text, giving O(n + m) time for a text of length n and pattern of length m, far better than the naive O(n times m).

A Fenwick, or binary indexed, tree supports prefix sums and point updates in O(log n) with less memory and simpler code than a segment tree. Choose it for cumulative frequency and prefix-sum tasks. Prefer a segment tree when you need range minimum, range maximum, or more general range queries and lazy updates.

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