Distributional convergence for the number of symbol comparisons used by QuickSelect

TL;DR

This paper analyzes the distributional convergence of symbol comparison costs in QuickSelect, considering probabilistic key representations, and derives formulas for expected costs, extending prior comparison-based analyses.

Contribution

It introduces a detailed probabilistic model for symbol-based costs in QuickSelect and derives limiting distributions and expectations for these costs.

Findings

Identifies limiting distributions for symbol comparison costs.

Provides integral and series formulas for expected costs.

Recovers known results on key comparison counts.

Abstract

When the search algorithm QuickSelect compares keys during its execution in order to find a key of target rank, it must operate on the keys' representations or internal structures, which were ignored by the previous studies that quantified the execution cost for the algorithm in terms of the number of required key comparisons. In this paper, we analyze running costs for the algorithm that take into account not only the number of key comparisons but also the cost of each key comparison. We suppose that keys are represented as sequences of symbols generated by various probabilistic sources and that QuickSelect operates on individual symbols in order to find the target key. We identify limiting distributions for the costs and derive integral and series expressions for the expectations of the limiting distributions. These expressions are used to recapture previously obtained results on the number of key comparisons required by the algorithm.