Improved Algorithms for Computing $k$-Sink on Dynamic Path Networks

TL;DR

This paper introduces faster algorithms for computing the $k$-sink in dynamic path networks, significantly improving efficiency over previous methods for various edge capacity scenarios.

Contribution

It presents new algorithms with improved time complexities for the $k$-sink problem on dynamic path networks, covering general and uniform edge capacities.

Findings

Algorithms run faster than previous methods for all $k$ values.

New algorithms achieve optimal or near-optimal time complexities.

Performance improvements are demonstrated for both general and uniform capacity cases.

Abstract

We present a novel approach to finding the $k$-sink on dynamic path networks with general edge capacities. Our first algorithm runs in $O(n \log n + k^2 \log^4 n)$ time, where $n$ is the number of vertices on the given path, and our second algorithm runs in $O(n \log^3 n)$ time. Together, they improve upon the previously most efficient $O(kn \log^2 n)$ time algorithm due to Arumugam et al. for all values of $k$. In the case where all the edges have the same capacity, we again present two algorithms that run in $O(n + k^2 \log^2n)$ time and $O(n \log n)$ time, respectively, and they together improve upon the previously best $O(kn)$ time algorithm due to Higashikawa et al. for all values of $k$.