# Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search

@inproceedings{Gawrychowski2015SubmatrixMQ, title={Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search}, author={Pawel Gawrychowski and Shay Mozes and Oren Weimann}, booktitle={ICALP}, year={2015} }

We present an optimal data structure for submatrix maximum queries in n x n Monge matrices. Our result is a two-way reduction showing that the problem is equivalent to the classical predecessor problem in a universe of polynomial size. This gives a data structure of O(n) space that answers submatrix maximum queries in O(loglogn) time. It also gives a matching lower bound, showing that O(loglogn) query-time is optimal for any data structure of size O(n polylog(n)). Our result concludes a line of…

## 7 Citations

Submatrix Maximum Queries in Monge and Partial Monge Matrices Are Equivalent to Predecessor Search

- Computer ScienceACM Trans. Algorithms
- 2020

The result is a two-way reduction showing that the problem is equivalent to the classical predecessor problem in a universe of polynomial size, and gives a data structure of O(n) space that answers submatrix maximum queries in O(log log n) time, as well as a matching lower bound.

Near-Optimal Compression for the Planar Graph Metric

- Computer Science, MathematicsSODA
- 2018

An unexpected and decisive proof that weights can make planar graphs inherently more complex is presented, and a new compression of the planar graph metric into [Equation] bits is introduced, which is optimal up to log factors.

Data structures and dynamic algorithms for planar graphs

- Computer Science
- 2019

This thesis shows an optimal data structure maintaining a planar graph subject to edge contractions that explicitly maintains individual vertices’ neighbors lists and supports constant-time adjacency queries on the stored graph and studies decremental reachability algorithms for planar directed graphs.

Minimum Cuts and Shortest Cycles in Directed Planar Graphs via Noncrossing Shortest Paths

- Computer ScienceSIAM J. Discret. Math.
- 2017

An O(n\log\log n)-time algorithm for computing noncrossing shortest paths among nodes well ordered on a common face of a directed plane graph, extended from the algorithm of Italiano, Nussbaum, Sankowski, and Wulff-Nilsen for an undirect...

Improved Bounds for Shortest Paths in Dense Distance Graphs

- Computer Science, MathematicsICALP
- 2018

The first improvement to date over FR-Dijkstra for the important case when $r$ is polynomial in $n$ is shown, which implies improved upper bounds for such planar graph problems as multiple-source multiple-sink maximum flow, single-source all-sinksmaximum flow, and (dynamic) exact distance oracles.

Near-Optimal Distance Emulator for Planar Graphs

- Computer Science, MathematicsESA
- 2018

The result implies that, on any unweighted undirected planar graph, one can compute all-pairs shortest path distances among $k$ terminals in $\tilde O(n)$ time when $k=O(n^{1/3})$.

Submatrix Maximum Queries in Monge Matrices and Partial Monge Matrices, and Their Applications

- Computer ScienceACM Trans. Algorithms
- 2017

The design exploits an interpretation of the column maxima in a Monge (partial Monge, respectively) matrix as an upper envelope of pseudo-lines (pseudo-segments) in Monge matrices or partial MongeMatrices, where a query seeks the maximum element in a contiguous submatrix of the given matrix.

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