Fast algorithms for finding nearest common ancestors
SIAM Journal on Computing
Optimal shortest path queries in a simple polygon
Journal of Computer and System Sciences
Triangulating a simple polygon in linear time
Discrete & Computational Geometry
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
Shortest paths in the plane with polygonal obstacles
Journal of the ACM (JACM)
Computing minimum length paths of a given homotopy class
Computational Geometry: Theory and Applications
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
SIAM Journal on Computing
Regular Article: Graphs of Some CAT(0) Complexes
Advances in Applied Mathematics
Regular Article: Geometry of the Space of Phylogenetic Trees
Advances in Applied Mathematics
Center and diameter problems in plane triangulations and quadrangulations
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
The lattice dimension of a graph
European Journal of Combinatorics
Distance and routing labeling schemes for non-positively curved plane graphs
Journal of Algorithms
Nonpositive Curvature and Pareto Optimal Coordination of Robots
SIAM Journal on Control and Optimization
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
A Fast Algorithm for Computing Geodesic Distances in Tree Space
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Weak sense of direction labelings and graph embeddings
Discrete Applied Mathematics
Combinatorics and Geometry of Finite and Infinite Squaregraphs
SIAM Journal on Discrete Mathematics
| Hi-index | 0.00 |
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,y@?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)log@D) 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.