Online Sorting via Searching and Selection

TL;DR

This paper introduces a flexible framework for online selection and search in unsorted lists, which progressively sorts the list and achieves efficient query handling through parameterized algorithms based on Quickselect.

Contribution

The paper presents a novel, adaptable framework that combines online selection and search with incremental sorting, improving efficiency and flexibility over existing methods.

Findings

Answers q≤n queries in O(n log q) average time

List becomes fully sorted after Ω(n) queries

Repeated queries are answered in constant or logarithmic time

Abstract

In this paper, we present a framework based on a simple data structure and parameterized algorithms for the problems of finding items in an unsorted list of linearly ordered items based on their rank (selection) or value (search). As a side-effect of answering these online selection and search queries, we progressively sort the list. Our algorithms are based on Hoare's Quickselect, and are parameterized based on the pivot selection method. For example, if we choose the pivot as the last item in a subinterval, our framework yields algorithms that will answer q<=n unique selection and/or search queries in a total of O(n log q) average time. After q=\Omega(n) queries the list is sorted. Each repeated selection query takes constant time, and each repeated search query takes O(log n) time. The two query types can be interleaved freely. By plugging different pivot selection methods into our framework, these results can, for example, become randomized expected time or deterministic worst-case time. Our methods are easy to implement, and we show they perform well in practice.