Shortest paths in Euclidean graphs
Algorithmica
Finding small simple cycle separators for 2-connected planar graphs
Journal of Computer and System Sciences
Fibonacci heaps and their uses in improved network optimization algorithms
Journal of the ACM (JACM)
A faster approximation algorithm for the Steiner problem in graphs
Information Processing Letters
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Triangulating a simple polygon in linear time
Discrete & Computational Geometry
Applications of random sampling to on-line algorithms in computational geometry
Discrete & Computational Geometry
An optimal algorithm for intersecting line segments in the plane
Journal of the ACM (JACM)
A deterministic linear time algorithm for geometric separators and its applications
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
A randomized linear-time algorithm to find minimum spanning trees
Journal of the ACM (JACM)
Planar separators and parallel polygon triangulation
Journal of Computer and System Sciences - Special issue on selected papers presented at the 24th annual ACM symposium on the theory of computing (STOC '92)
Separator based sparsification I.: planarity testing and minimum spanning trees
Journal of Computer and System Sciences
Disk packings and planar separators
Proceedings of the twelfth annual symposium on Computational geometry
Separators for sphere-packings and nearest neighbor graphs
Journal of the ACM (JACM)
Recent results on the single-source shortest paths problem
ACM SIGACT News
Faster shortest-path algorithms for planar graphs
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
Geometric Separators for Finite-Element Meshes
SIAM Journal on Scientific Computing
Scaling algorithms for the shortest paths problem
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Undirected single-source shortest paths with positive integer weights in linear time
Journal of the ACM (JACM)
A minimum spanning tree algorithm with inverse-Ackermann type complexity
Journal of the ACM (JACM)
Constructing Planar Cuttings in Theory and Practice
SIAM Journal on Computing
Single-source shortest-paths on arbitrary directed graphs in linear average-case time
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Introduction to Algorithms
SIAM Journal on Computing
WADS '93 Proceedings of the Third Workshop on Algorithms and Data Structures
Shortest Path Algorithms: An Evaluation Using Real Road Networks
Transportation Science
Computing the shortest path: A search meets graph theory
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Combining speed-up techniques for shortest-path computations
Journal of Experimental Algorithmics (JEA)
Algorithms for Reporting and Counting Geometric Intersections
IEEE Transactions on Computers
Highway hierarchies hasten exact shortest path queries
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Going off-road: transversal complexity in road networks
Proceedings of the 17th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings
SIAM Journal on Computing
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We provide linear-time algorithms for geometric graphs with sublinearly many crossings. That is, we provide algorithms running in O(n) time on connected geometric graphs having n vertices and k crossings, where k is smaller than n by an iterated logarithmic factor. Specific problems we study include Voronoi diagrams and single-source shortest paths. Our algorithms all run in linear time in the standard comparison-based computational model; hence, we make no assumptions about the distribution or bit complexities of edge weights, nor do we utilize unusual bit-level operations on memory words. Instead, our algorithms are based on a planarization method that "zeroes in" on edge crossings, together with methods for extending planar separator decompositions to geometric graphs with sublinearly many crossings. Incidentally, our planarization algorithm also solves an open computational geometry problem of Chazelle for triangulating a self-intersecting polygonal chain having n segments and k crossings in linear time, for the case when k is sublinear in n by an iterated logarithmic factor.