# Bicriteria Rectilinear Shortest Paths among Rectilinear Obstacles in the Plane

## TL;DR

This paper corrects and improves algorithms for computing bicriteria rectilinear shortest paths among disjoint obstacles in the plane, achieving faster query times and addressing errors in previous methods.

## Contribution

It identifies and corrects critical errors in existing algorithms and introduces more efficient solutions for bicriteria rectilinear shortest path problems.

## Key findings

- Corrected all previous algorithms for bicriteria rectilinear shortest paths.
- Developed faster algorithms with improved query times for one-point and two-point path queries.
- Achieved computational complexity improvements for specific path types and query scenarios.

## Abstract

Given a rectilinear domain $\mathcal{P}$ of $h$ pairwise-disjoint rectilinear obstacles with a total of $n$ vertices in the plane, we study the problem of computing bicriteria rectilinear shortest paths between two points $s$ and $t$ in $\mathcal{P}$. Three types of bicriteria rectilinear paths are considered: minimum-link shortest paths, shortest minimum-link paths, and minimum-cost paths where the cost of a path is a non-decreasing function of both the number of edges and the length of the path. The one-point and two-point path queries are also considered. Algorithms for these problems have been given previously. Our contributions are threefold. First, we find a critical error in all previous algorithms. Second, we correct the error in a not-so-trivial way. Third, we further improve the algorithms so that they are even faster than the previous (incorrect) algorithms when $h$ is relatively small. For example, for the minimum-link shortest paths, we obtain the following results. Our algorithm computes a minimum-link shortest $s$-$t$ path in $O(n+h\log^{3/2} h)$ time. For the one-point queries, we build a data structure of size $O(n+ h\log h)$ in $O(n+h\log^{3/2} h)$ time for a source point $s$, such that given any query point $t$, a minimum-link shortest $s$-$t$ path can be determined in $O(\log n)$ time. For the two-point queries, with $O(n+h^2\log^2 h)$ time and space preprocessing, a minimum-link shortest $s$-$t$ path can be determined in $O(\log n+\log^2 h)$ time for any two query points $s$ and $t$; alternatively, with $O(n+h^2\cdot \log^{2} h \cdot 4^{\sqrt{\log h}})$ time and $O(n+h^2\cdot \log h \cdot 4^{\sqrt{\log h}})$ space preprocessing, we can answer each two-point query in $O(\log n)$ time.