Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Cutting hyperplanes for divide-and-conquer
Discrete & Computational Geometry
Motion segmentation and qualitative dynamic scene analysis from an image sequence
International Journal of Computer Vision
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Rectangular matrix multiplication revisited
Journal of Complexity
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Journal of Complexity
Fast context-free grammar parsing requires fast boolean matrix multiplication
Journal of the ACM (JACM)
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
An output-sensitive variant of the baby steps/giant steps determinant algorithm
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
A Group-Theoretic Approach to Fast Matrix Multiplication
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Computing all large sums-of-pairs in Rn and the discrete planar two-watchtower problem
Information Processing Letters
Learning functions of k relevant variables
Journal of Computer and System Sciences - Special issue: STOC 2003
Guarding a terrain by two watchtowers
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Group-theoretic Algorithms for Matrix Multiplication
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Answering distance queries in directed graphs using fast matrix multiplication
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
More algorithms for all-pairs shortest paths in weighted graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
ESA'07 Proceedings of the 15th annual European conference on Algorithms
More Algorithms for All-Pairs Shortest Paths in Weighted Graphs
SIAM Journal on Computing
Graph expansion analysis for communication costs of fast rectangular matrix multiplication
MedAlg'12 Proceedings of the First Mediterranean conference on Design and Analysis of Algorithms
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In a Batched Colored Intersection Searching Problem (CI), one is given a set of n geometric objects (of a certain class). Each object is colored by one of c colors, and the goal is to report all pairs of colors (c1,c2) such that there are two objects, one colored c1 and one colored c2, that intersect each other. We also consider the bipartite version of the problem, where we are interested in intersections between objects of one class with objects of another class (e.g., points and halfspaces).In a Sparse Rectangular Matrix Multiplication Problem (SRMM), one is given an n1×n2 matrix A and an n2×n3 matrix B, each containing at most m non-zero entries, and the goal is to compute their product AB.In this paper we present a technique for solving CI problems over a wide range of classes of geometric objects. The basic idea is first to use some decomposition method, such as geometric cuttings, to represent the intersection graph of the objects as a union of bi-cliques. Then, in each of these bi-cliques, contract all vertices of the same color. Finally, use an algorithm for sparse matrix multiplication (adapted from Yuster and Zwick [20]) to compute the union of the bi-cliques. We apply the technique to segments in R1, to segments in R2, to points and halfplanes in R2, and, more generally, to points and halfspaces in Rd, for any fixed d. However, the technique extends to colored intersection searching in any class (or pair of classes) of geometric objects of constant descriptive complexity.In particular, using our technique we obtain an algorithm that reports all the pairs of intersecting colors for n points and n halfplanes in R2, that are colored by c colors, in O(n4/3c0.46) time when n ≥ c1.44, and in O(n1.04c0.9 + c2) time when n≤c1.44.The algorithms that we give for CI use the algorithm for SRMM as a black box, which means that any improved algorithm for SRMM immediately leads to an improved algorithm for all colored intersection problems that our method applies to. We also show that the complexity of computing all intersecting colors in a set of segments on the real line is identical, up to a polylogarithmic multiplicative factor, to the complexity of SRMM with the appropriate parameters.