Fragile Complexity of Adaptive Algorithms

The fragile complexity of a comparison-based algorithm is f(n) if each input element participates in O(f(n)) comparisons. In this paper, we explore the algorithms adaptive to various restrictions on input, i.e., with parameterized by quantity other than size n. We show that searching for predecessor sorted array has \(\varTheta (\log k)\), where k rank query element, both randomized and deterministic setting. For searches, also how optimally reduce amortized elements array. prove following results: Selecting kth smallest expected \(O(\log \log k)\) selected. Deterministically finding minimum (\mathrm {Inv}))\) {Runs}))\), \(\mathrm {Inv}\) number inversions sequence {Runs}\) increasing runs sequence. median {Runs}) + n)\) {Inv}))\). Deterministic sorting but it regardless runs.