Computing the largest empty rectangle
SIAM Journal on Computing
A note on finding a maximum empty rectangle
Discrete Applied Mathematics
Fast algorithms for shortest paths in planar graphs, with applications
SIAM Journal on Computing
Fast algorithms for computing the largest empty rectangle
SCG '87 Proceedings of the third annual symposium on Computational geometry
Finding the upper envelope of n line segments in O(n log n) time
Information Processing Letters
An almost linear time algorithm for generalized matrix searching
SIAM Journal on Discrete Mathematics
Applications of generalized matrix searching to geometric algorithms
Discrete Applied Mathematics - Computational combinatiorics
Superlinear bounds for matrix searching problems
Journal of Algorithms
Dynamic algorithms for shortest paths in planar graphs
Theoretical Computer Science
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Perspectives of Monge properties in optimization
Discrete Applied Mathematics
Journal of the ACM (JACM)
Efficient Algorithms for Shortest Paths in Sparse Networks
Journal of the ACM (JACM)
Voronoi diagrams based on convex distance functions
SCG '85 Proceedings of the first annual symposium on Computational geometry
Shortest path queries in planar graphs
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
LATIN '00 Proceedings of the 4th Latin American Symposium on Theoretical Informatics
Location of the Largest Empty Rectangle among Arbitrary Obstacles
Proceedings of the 14th Conference on Foundations of Software Technology and Theoretical Computer Science
On-Line Algorithms for Shortest Path Problems on Planar Digraphs
WG '96 Proceedings of the 22nd International Workshop on Graph-Theoretic Concepts in Computer Science
Largest empty rectangle among a point set
Journal of Algorithms
Multiple-source shortest paths in planar graphs
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Computational Aspects of VLSI
Planar graphs, negative weight edges, shortest paths, and near linear time
Journal of Computer and System Sciences - Special issue on FOCS 2001
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Notes on searching in multidimensional monotone arrays
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Shortest paths in planar graphs with real lengths in O(n log2n/ log log n) time
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part II
Improved algorithms for min cut and max flow in undirected planar graphs
Proceedings of the forty-third annual ACM symposium on Theory of computing
Improved distance queries in planar graphs
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Multiple-Source Multiple-Sink Maximum Flow in Directed Planar Graphs in Near-Linear Time
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Exact distance oracles for planar graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Many Distances in Planar Graphs
Algorithmica
Finding the maximal empty disk containing a query point
Proceedings of the twenty-eighth annual symposium on Computational geometry
Localized geometric query problems
Computational Geometry: Theory and Applications
Hi-index | 0.00 |
We describe a data structure for submatrix maximum queries in Monge matrices or Monge partial matrices, where a query specifies a contiguous submatrix of the given matrix, and its output is the maximum element of that submatrix. Our data structure for an n x n Monge matrix takes O(n log n) space, O(n log2 n) preprocessing time, and can answer queries in O(log2 n) time. For a Monge partial matrix the space bound and the preprocessing time both grow by the small factor α(n), where α(n) is the inverse Ackermann function. Our design exploits an interpretation of the column maxima in a Monge matrix (resp., Monge partial matrix) as an upper envelope of pseudo-lines (resp., pseudo-segments). We give two applications for this data structure: (1) For a set of n points in a rectangle B in the plane, we build a data structure that, given a query point p, returns the largest-area empty axis-parallel rectangle contained in B and containing p, in O(log4 n) time. The preprocessing time is O(nα(n) log4 n), and the space required is O(nα(n) log3 n). This improves substantially a previous data structure of Augustine et al. [arXiv: 1004.0558] that requires quadratic space. (2) Given an n-node arbitrarily weighted planar digraph, with possibly negative edge weights, we build, in O(n log2 n/log log n) time, a linear-size data structure that supports edge-weight updates and distance queries between arbitrary pairs of nodes (where the distance is minimum weight of a path in the graph between the pair of nodes), in O(n2/3 log5/3n) time for each update and query. This improves the O(n4/5 log13/5n)-time bound of Fakcharoenphol and Rao [JCSS 72, 2006]. Our data structure has already been applied in a recent maximum flow algorithm for planar graphs of Borradaile et al. [FOCS 2011], and we believe it will find additional applications.