A new polynomial-time algorithm for linear programming
Combinatorica
There are planar graphs almost as good as the complete graph
Journal of Computer and System Sciences
A general approach to dominance in the plane
Journal of Algorithms
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Applying Parallel Computation Algorithms in the Design of Serial Algorithms
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The BOXEL framework for 2.5D data with applications to virtual drivethroughs and ray tracing
Computational Geometry: Theory and Applications
Geometric Spanner of Objects under L1 Distance
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Multiple polyline to polygon matching
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
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The dilation of a geometric graph is the maximum, over all pairs of points in the graph, of the ratio of the Euclidean length of the shortest path between them in the graph and their Euclidean distance. We consider a generalized version of this notion, where the nodes of the graph are not points but axis-parallel rectangles in the plane. The arcs in the graph are horizontal or vertical segments connecting a pair of rectangles, and the distance measure we use is the L1-distance. The dilation of a pair of points is then defined as the length of the shortest rectilinear path between them that stays within the union of the rectangles and the connecting segments, divided by their L1-distance. The dilation of the graph is the maximum dilation over all pairs of points in the union of the rectangles.We study the following problem: given n non-intersecting rectangles and a graph describing which pairs of rectangles are to be connected, we wish to place the connecting segments such that the dilation is minimized. We obtain four results on this problem: (i) for arbitrary graphs, the problem is NP-hard; (ii) for trees, we can solve the problem by linear programming on O(n2) variables and constraints; (iii) for paths, we can solve the problem in time O(n3 log n); (iv) for rectangles sorted vertically along a path, the problem can be solved in O(n2) time, and a (1 + ε)-approximation can be computed in linear time.