Analysis of algorithms

Analysis of algorithms studies whether algorithms are correct and how their resource use grows with input size, typically through mathematical models of time and space.

Big O notation: A notation expressing an asymptotic upper bound on a function’s growth. It summarizes an algorithm’s resource growth while suppressing constant factors and smaller terms.
Loop invariant: A statement that remains true before and after each iteration of a loop. It supports correctness proofs by connecting initialization, preservation, and termination.
Merge sort: A comparison sorting algorithm that recursively sorts halves and merges them in linear time. Its recurrence yields a worst-case running time proportional to n log n.
Average-case complexity: The expected resource use of an algorithm under a specified distribution of inputs. It contrasts with worst-case analysis by depending on assumptions about input frequency.
Donald Knuth: An American computer scientist known for foundational work on algorithms and the analysis of programs. His writings helped establish rigorous analysis as a core part of algorithm design.
Big Omega notation: A notation expressing an asymptotic lower bound on a function’s growth. It describes guaranteed minimum growth, complementing upper-bound analyses.
Master theorem: A theorem giving asymptotic solutions for a class of divide-and-conquer recurrences. It quickly bounds many recursive algorithms by comparing work across recursion levels.
Binary search: A search algorithm that repeatedly halves a sorted range to locate a target. Its logarithmic search time exemplifies the effect of halving input size.
Best-case complexity: The minimum resource use of an algorithm over inputs of a given size. It describes the easiest inputs and rarely guarantees performance on arbitrary inputs.
Alonzo Church: An American mathematician whose work on lambda calculus and logic shaped foundations of computation. Formal models of computation underpin precise claims about what algorithms can do.

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