Timothy M. Chan's Publications: Geometric shortest paths

Approximate shortest paths and distance oracles in weighted unit-disk graphs

(with Dimitrios Skrepetos)

We present the first near-linear-time (1+epsilon)-approximation algorithm for the diameter of a weighted unit-disk graph of n vertices, running in O(n log^2 n) time, for any constant epsilon > 0, improving the near-O(n^{3/2})-time algorithm of Gao and Zhang [STOC 2003]. Using similar ideas, we can construct a (1+epsilon)-approximate distance oracle for weighted unit-disk graphs with O(1) query time, with a similar improvement in the preprocessing time, from near O(n^{3/2}) to O(n log^3 n). We also obtain new results for a number of other related problems in the weighted unit-disk graph metric, such as the radius and bichromatic closest pair.

As a further application, we use our new distance oracle, along with additional ideas, to solve the (1+epsilon)-approximate all-pairs bounded-leg shortest paths problem for a set of n planar points, with near O(n^{2.579}) preprocessing time, O(n^2 log n) space, and O(log log n) query time, improving thus the near-cubic preprocessing bound by Roditty and Segal [SODA 2007].

All-pairs shortest paths in geometric intersection graphs

(with Dimitrios Skrepetos)

We address the All-Pairs Shortest Paths (APSP) problem for a number of unweighted, undirected geometric intersection graphs. We present a general reduction of the problem to static, offline intersection searching (specifically detection). As a consequence, we can solve APSP for intersection graphs of n arbitrary disks in O(n^2 log n) time, axis-aligned line segments in O(n^2 loglog n) time, arbitrary line segments in O(n^{7/3} log^{1/3} n) time, d-dimensional axis-aligned boxes in O(n^2 log^{d-1.5} n) time for d >= 2, and d-dimensional axis-aligned unit hypercubes in O(n^2 loglog n) time for d=3 and O(n^2 log^{d-3} n) time for d >= 4.

In addition, we show how to solve the Single-Source Shortest Paths (SSSP) problem in unweighted intersection graphs of axis-aligned line segments in O(n log n) time, by a reduction to dynamic orthogonal point location.

All-pairs shortest paths in unit disk graphs in slightly subquadratic time

(with Dimitrios Skrepetos)

In this paper we study the all-pairs shortest paths problem in (unweighted) unit-disk graphs. The previous best solution for this problem required O(n^2 log n) time, by running the O(n log n)-time single-source shortest path algorithm of Cabello and Jejcic (2015) from every source vertex, where n is the number of vertices. We not only manage to eliminate the logarithmic factor, but also obtain the first (slightly) subquadratic algorithm for the problem, running in O(n^2 sqrt{loglog n/log n}) time. Our algorithm computes an implicit representation of all the shortest paths, and, in the same amount of time, can also compute the diameter of the graph.

More algorithms for all-pairs shortest paths in weighted graphs

In the first part of the paper, we reexamine the all-pairs shortest paths (APSP) problem and present a new algorithm with running time approaching O(n^3 log^3log n / log^2 n), which improves all known algorithms for general real-weighted dense graphs.

In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of "geometrically weighted" graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n^{3-(3-w)/(2d+4)}), where w < 2.376; in two dimensions, this is O(n^{2.922}). Our framework greatly extends the previously considered case of small-integer-weighted graphs, and incidentally also yields the first truly subcubic result (near O(n^{3-(3-w)/4}) = O(n^{2.844}) time) for APSP in real-vertex-weighted graphs, as well as an improved result (near O(n^{(3+w)/2}) = O(n^{2.688}) time) for the all-pairs lightest shortest path problem for small-integer-weighted graphs.

Fly cheaply: on the minimum fuel consumption problem

Alon Efrat)

In planning a flight, stops at intermediate airports are sometimes necessary to minimize fuel consumption, even if a direct flight is available. We investigate the problem of finding the cheapest path from one airport to another, given a set of n airports in R^2 and a function l: R^2 x R^2 -> R^+ representing the cost of a direct flight between any pair.

Given a source airport s, the cheapest-path map is a subdivision of R^2 where two points lie in the same region iff their cheapest paths from s use the same sequence of intermediate airports. We show a quadratic lower bound on the combinatorial complexity of this map for a class of cost functions. Nevertheless, we are able to obtain subquadratic algorithms to find the cheapest path from s to all other airports for any well-behaved cost function l: our general algorithm runs in O(n^{4/3+eps}) time, and a simpler, more practical variant runs in O(n^{3/2+eps}) time, while a special class of cost functions requires just O(n log n) time.

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Timothy Chan (Last updated April 2018)