Shortest path problem in rectangular complexes of global nonpositive curvature

CAT(0) metric spaces constitute a far-reaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a CAT(0) metric space are connected by a unique shortest path @c(x,y). In this paper, we present an efficient algorithm for answering two-point distance queries in CAT(0) rectangular complexes and two of theirs subclasses, ramified rectilinear polygons (CAT(0) rectangular complexes in which the links of all vertices are bipartite graphs) and squaregraphs (CAT(0) rectangular complexes arising from plane quadrangulations in which all inner vertices have degrees >=4). Namely, we show that for a CAT(0) rectangular complex K with n vertices, one can construct a data structure D of size O(n^2) so that, given any two points x,[email protected]?K, the shortest path @c(x,y) between x and y can be computed in O(d(p,q)) time, where p and q are vertices of two faces of K containing the points x and y, respectively, such that @c(x,y)@?K(I(p,q)) and d(p,q) is the distance between p and q in the underlying graph of K. If K is a ramified rectilinear polygon, then one can construct a data structure D of optimal size O(n) and answer two-point shortest path queries in O(d(p,q)[email protected]) time, where @D is the maximal degree of a vertex of G(K). Finally, if K is a squaregraph, then one can construct a data structure D of size O(nlogn) and answer two-point shortest path queries in O(d(p,q)) time.

[1]  David Eppstein,et al.  Ramified Rectilinear Polygons: Coordinatization by Dendrons , 2010, Discret. Comput. Geom..

[2]  Leonidas J. Guibas,et al.  Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons , 1987, Algorithmica.

[3]  David Eppstein,et al.  Media theory , 2007 .

[4]  David Eppstein,et al.  Combinatorics and Geometry of Finite and Infinite Squaregraphs , 2009, SIAM J. Discret. Math..

[5]  Louis J. Billera,et al.  Geometry of the Space of Phylogenetic Trees , 2001, Adv. Appl. Math..

[6]  Michah Sageev,et al.  Ends of Group Pairs and Non‐Positively Curved Cube Complexes , 1995 .

[7]  Joseph S. B. Mitchell,et al.  Shortest paths among obstacles in the plane , 1993, SCG '93.

[8]  Martyn Mulder,et al.  The structure of median graphs , 1978, Discret. Math..

[9]  H. Bandelt.  Networks with condorcet solutions , 1985 .

[10]  James A. Storer,et al.  Shortest paths in the plane with polygonal obstacles , 1994, JACM.

[11]  W. Imrich,et al.  Product Graphs: Structure and Recognition , 2000 .

[12]  Indira Chatterji,et al.  Kazhdan and Haagerup properties from the median viewpoint , 2007, 0704.3749.

[13]  John Hershberger,et al.  Computing Minimum Length Paths of a Given Homotopy Class (Extended Abstract) , 1991, WADS.

[14]  Feodor F. Dragan,et al.  Center and diameter problems in plane triangulations and quadrangulations , 2002, SODA '02.

[15]  G. Birkhoff.  Rings of sets , 1937 .

[16]  Leonidas J. Guibas,et al.  Optimal shortest path queries in a simple polygon , 1987, SCG '87.

[17]  Robert E. Tarjan,et al.  Fast Algorithms for Finding Nearest Common Ancestors , 1984, SIAM J. Comput..

[18]  G. Birkhoff,et al.  A ternary operation in distributive lattices , 1947 .

[19]  Christine T. Cheng.  A Poset-based Approach to Embedding Median Graphs in Hypercubes and Lattices , 2012, Order.

[20]  Van de M. L. J. Vel.  Theory of convex structures , 1993 .

[21]  R. Larson,et al.  Embeddings of Finite Distributive Lattices into Products of Chains , 1998 .

[22]  Robert Ghrist,et al.  The geometry and topology of reconfiguration , 2007, Adv. Appl. Math..

[23]  Hans-Jürgen Bandelt,et al.  Median algebras , 1983, Discret. Math..

[24]  J. Scott Provan,et al.  A Fast Algorithm for Computing Geodesic Distances in Tree Space , 2009, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[25]  Christine T. Cheng,et al.  Weak sense of direction labelings and graph embeddings , 2011, Discret. Appl. Math..

[26]  H. Bandelt,et al.  Embedding Topological Median Algebras in Products of Dendrons , 1989 .

[27]  David Eppstein.  The lattice dimension of a graph , 2005, Eur. J. Comb..

[28]  Subhash Suri,et al.  An Optimal Algorithm for Euclidean Shortest Paths in the Plane , 1999, SIAM J. Comput..

[29]  H. M. Mulder.  The interval function of a graph , 1980 .

[30]  Joseph S. B. Mitchell,et al.  Geometric Shortest Paths and Network Optimization , 2000, Handbook of Computational Geometry.

[31]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[32]  Feodor F. Dragan,et al.  Distance and routing labeling schemes for non-positively curved plane graphs , 2006, J. Algorithms.

[33]  Steven M. LaValle,et al.  Nonpositive Curvature and Pareto Optimal Coordination of Robots , 2006, SIAM J. Control. Optim..

[34]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[35]  Susan Holmes,et al.  Computational Tools for Evaluating Phylogenetic and Hierarchical Clustering Trees , 2010, Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America.

[36]  A. O. Houcine.  On hyperbolic groups , 2006 .

[37]  Bogdan Nica.  Cubulating spaces with walls , 2004 .

[38]  Suresh Venkatasubramanian,et al.  Computing Hulls And Centerpoints In Positive Definite Space , 2009, ArXiv.

[39]  H. Bandelt,et al.  Metric graph theory and geometry: a survey , 2006 .

[40]  Amin Coja-Oghlan,et al.  A heuristic for the Stacker Crane Problem on trees which is almost surely exact , 2003, J. Algorithms.

[41]  D. T. Lee,et al.  Euclidean shortest paths in the presence of rectilinear barriers , 1984, Networks.

[42]  R. P. Dilworth,et al.  A DECOMPOSITION THEOREM FOR PARTIALLY ORDERED SETS , 1950 .

[43]  Victor Chepoi,et al.  Graphs of Some CAT(0) Complexes , 2000, Adv. Appl. Math..

[44]  Bernard Chazelle.  Triangulating a simple polygon in linear time , 1991, Discret. Comput. Geom..

[45]  M. Bridson,et al.  Metric Spaces of Non-Positive Curvature , 1999 .

[46]  Leonidas J. Guibas,et al.  Optimal Point Location in a Monotone Subdivision , 1986, SIAM J. Comput..

[47]  Ravindra K. Ahuja,et al.  Network Flows , 2011 .