Selection on $X_1 + X_1 + \cdots X_m$ via Cartesian product tree

TL;DR

This paper introduces a new, efficient algorithm for selection on the sum of multiple sets using Cartesian product trees and layer-ordered heaps, improving performance over previous methods.

Contribution

It combines a novel LOH-based selection algorithm with a layered tree structure to achieve faster selection on multiple sums without soft heaps.

Findings

The new algorithm outperforms existing methods in empirical tests.

It achieves better asymptotic complexity for multi-set selection.

Performance improvements are demonstrated through empirical comparisons.

Abstract

Selection on the Cartesian product is a classic problem in computer science. Recently, an optimal algorithm for selection on $X+Y$, based on soft heaps, was introduced. By combining this approach with layer-ordered heaps (LOHs), an algorithm using a balanced binary tree of $X+Y$ selections was proposed to perform $k$-selection on $X_1+X_2+\cdots+X_m$ in $o(n\cdot m + k\cdot m)$, where $X_i$ have length $n$. Here, that $o(n\cdot m + k\cdot m)$ algorithm is combined with a novel, optimal LOH-based algorithm for selection on $X+Y$ (without a soft heap). Performance of algorithms for selection on $X_1+X_2+\cdots+X_m$ are compared empirically, demonstrating the benefit of the algorithm proposed here.