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Linear time single-source shortest path algorithms in Euclidean graph classes

Joachim Gudmundsson, Yuan Sha, Sampson Wong

Abstract

In the celebrated paper of Henzinger, Klein, Rao and Subramanian (1997), it was shown that planar graphs admit a linear time single-source shortest path algorithm. Their algorithm unfortunately does not extend to Euclidean graph classes. We give criteria and prove that any Euclidean graph class satisfying the criteria admits a linear time single-source shortest path algorithm. As a main ingredient, we show that the contracted graphs of these Euclidean graph classes admit sublinear separators.

Linear time single-source shortest path algorithms in Euclidean graph classes

Abstract

In the celebrated paper of Henzinger, Klein, Rao and Subramanian (1997), it was shown that planar graphs admit a linear time single-source shortest path algorithm. Their algorithm unfortunately does not extend to Euclidean graph classes. We give criteria and prove that any Euclidean graph class satisfying the criteria admits a linear time single-source shortest path algorithm. As a main ingredient, we show that the contracted graphs of these Euclidean graph classes admit sublinear separators.
Paper Structure (15 sections, 21 theorems, 2 equations, 2 figures)

This paper contains 15 sections, 21 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Techniques
  4. Related work
  5. Paper Organization
  6. Preliminaries
  7. The main theorem
  8. The contracted graphs and their representative graphs
  9. Separators of the contracted graphs
  10. Recursive division
  11. SSSP for lanky-graphs
  12. Sparsity of the contracted graphs
  13. SSSP for the intersection graph of k-ply neighborhood system
  14. Sparsity of the contracted graphs
  15. SSSP for the intersection graph of kap-thick cubical neighborhood system

Key Result

Lemma 11

Let $H$ be any subgraph of $G_i$ with $k=\Omega(\beta_{i-1})$ vertices. A balanced separator of $H$ of size $O((c_1c_3^{1-1/d}+c_4)k^{(1-\frac{1}{3d-2})})$We did not try to optimize the size of the separator. can be computed in $O(c_2c_3 k)$ time given $H$ and $RH$.

Figures (2)

  • Figure 1: Illustration of $r$-division. An $r$-division of the graph divides the graph into regions (enclosed in blue lines) with interior vertices (black) and boundary vertices (red). Modified figure from shortest-path-planar_Frederickson1987.
  • Figure 2: (a) $cset_{G_{i-1}}(u)$ is enclosed in red, $cset_{G_{i-1}}(v)$ is enclosed in blue. The representative edge of $(u,v)$ in $G_i$ is the purple edge. (b) Illustrating the proof of Lemma \ref{['lem:separator-contracted-graph']}. The closed surface $\xi^*$ is the dashed circle. The red vertex is added to $S$ since its $\Gamma(\cdot)$ has one point inside $\mathbf{b}(o,r^*)$ and two points outside $\xi^*$.

Theorems & Definitions (28)

  • Definition 5: $r$-division
  • Definition 6: $(r,s)$-division
  • Definition 7: recursive division SSSP-planar_HenzingerKRS1997
  • Definition 8
  • Lemma 11
  • Lemma 12
  • Lemma 13
  • Lemma 14
  • Theorem 15
  • Corollary 16
  • ...and 18 more