Selection from heaps, row-sorted matrices and $X+Y$ using soft heaps

TL;DR

This paper introduces simplified, optimal algorithms using soft heaps for selection problems in heaps, sorted lists, and sum sets, improving classical results with output-sensitive efficiency.

Contribution

It presents new, simpler algorithms leveraging soft heaps for selection tasks, extending and improving classical methods for various data structures.

Findings

Optimal algorithms for selecting the $k$-th smallest item from heaps and sorted lists.

An output-sensitive algorithm with $O(m+ extstyle ext{sum}_{i=1}^m ext{log}(k_i+1))$ comparisons.

Improved bounds matching or extending classical selection results.

Abstract

We use soft heaps to obtain simpler optimal algorithms for selecting the $k$-th smallest item, and the set of~$k$ smallest items, from a heap-ordered tree, from a collection of sorted lists, and from $X+Y$, where $X$ and $Y$ are two unsorted sets. Our results match, and in some ways extend and improve, classical results of Frederickson (1993) and Frederickson and Johnson (1982). In particular, for selecting the $k$-th smallest item, or the set of~$k$ smallest items, from a collection of~$m$ sorted lists we obtain a new optimal "output-sensitive" algorithm that performs only $O(m+\sum_{i=1}^m \log(k_i+1))$ comparisons, where $k_i$ is the number of items of the $i$-th list that belong to the overall set of~$k$ smallest items.