Linear-time algorithms for max flow and multiple-source shortest paths in unit-weight planar graphs

Abstract

We give simple linear-time algorithms for two problems in planar graphs: max st-flow in directed graphs with unit capacities, and multiple-source shortest paths in undirected graphs with unit lengths.

Holiest minimum-cost paths and flows in surface graphs

“…To more easily support directed graphs, we will assume G is connected and that its embedding is given as a rotation system. These embeddings are sometimes referred to as combinatorial embeddings as well (see, for example, Eisenstat and Klein [24]). Let E denote a collection of "directed edges" we refer to as darts.…”

“…The most commonly applied consequence of this assumption is that the shortest paths entering (or leaving) a common vertex do not cross one another. From this consequence, one can prove near-linear running time bounds for a variety of problems, including the computation of maximum flows [4,5,24,26] and global minimum cuts [53] in directed planar (genus 0) graphs as well as the computation of minimum cut oracles in planar and more general embedded graphs [3,6] (see also ).This assumption is also used in algorithms for the multiple-source shortest paths problem introduced for planar graphs by Klein [46]. In the multiple-source shortest paths problem, one is given a surface embedded graph G = (V, E, F ) of genus g with vertices V , edges E, and faces F .…”

“…However, these fast implementations require some fairly heavy machinery, and even implementing Dijkstra's [22] algorithm for single-source shortest paths requires that same log n factor increase in the running time and the use of relatively complex dynamic tree data structures [38,62,63]; see Cabello et al [11, Section 6.2].Parametric shortest paths and the leafmost rule. Several algorithms for multiple-source shortest paths in embedded graphs [11,24,46] and maximum flows in planar graphs [4,24,26] rely (at least by some interpretations) on the parametric shortest paths framework introduced by Karp and Orlin [43,69]. In short, these algorithms redefine the length of a subset of edges to increase or decrease by an amount equal to some parameter λ.…”

“…Parametric shortest paths and the leafmost rule. Several algorithms for multiple-source shortest paths in embedded graphs [11,24,46] and maximum flows in planar graphs [4,24,26] rely (at least by some interpretations) on the parametric shortest paths framework introduced by Karp and Orlin [43,69]. In short, these algorithms redefine the length of a subset of edges to increase or decrease by an amount equal to some parameter λ.…”

“…That said, one can sometimes avoid the need for unique shortest paths by utilizing properties of planar embeddings. Klein [46], Borradaile and Klein [4], and Eisenstat and Klein [24] all give efficient algorithms that successfully use the parametric shortest path framework without doing anything explicit to the edge lengths to guarantee unique shortest paths. In particular, Eisenstat and Klein [24] give linear-time maximum flow and multiple-source shortest paths algorithms that cannot take advantage of the perturbation schemes mentioned above, because they crucially rely on the edge capacities/lengths being small non-negative integers.…”