# 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}
}
• Published in ICALP 26 February 2015
• Computer Science
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 Science
ACM 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, Mathematics
SODA
• 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
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 Science
SIAM 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, Mathematics
ICALP
• 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, Mathematics
ESA
• 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 Science
ACM 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.

## References

SHOWING 1-10 OF 27 REFERENCES
Improved Submatrix Maximum Queries in Monge Matrices
• Computer Science, Mathematics
ICALP
• 2014
A linear upper bound on the number of breakpoints in the upper envelope of a Monge partial matrix is given, which shows that the inverse Ackermann α(n) factor in the analysis of the data structure of Kaplan et.
Persistent predecessor search and orthogonal point location on the word RAM
This work presents a (partially) persistent data structure that supports predecessor search in a set of integers in {1,...,U} under an arbitrary sequence of n insertions and deletions, with O(log log U) expected query time and expected amortized update time, and O(n) space.
Orthogonal range searching on the RAM, revisited
• Computer Science, Mathematics
SoCG '11
• 2011
We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model: We present
On Space Efficient Two Dimensional Range Minimum Data Structures
• Computer Science, Mathematics
Algorithmica
• 2011
It is shown that every algorithm enabled to access A during the query and using a data structure of size O(N/c) bits requires Ω(c) query time, for any c where 1≤c≤N, and the lower bound holds for arrays of any dimension.
Succinct Indices for Range Queries with Applications to Orthogonal Range Maxima
• Computer Science, Mathematics
ICALP
• 2012
A structure is obtained that uses O(N) words and answers the above query in O(logN loglogN) time, a direct improvement of Chazelle's result from 1985.
Weighted Ancestors in Suffix Trees
• Computer Science, Mathematics
ESA
• 2014
It is shown that a suffix tree built for a text w[1..n] can be preprocessed using O(n) extra space, so that queries can be answered in O(1) time, which improves the running times of several applications.
Geometric applications of a matrix-searching algorithm
• Mathematics
SCG '86
• 1986
The Θ(m) bound on finding the maxima of wide totally monotone matrices is used to speed up several geometric algorithms by a factor of logn.
Time-space trade-offs for predecessor search
• Computer Science
STOC '06
• 2006
The first lower bound for an explicit problem which breaks this communication complexity barrier is given, and it is implied that van Emde Boas' classic data structure from [FOCS'75] is optimal in this case.
An Almost Linear Time Algorithm for Generalized Matrix Searching
• Mathematics, Computer Science
SIAM J. Discret. Math.
• 1990
How the algorithm can be modified to give an O( n \alpha ( n ) ) algorithm for a class of dynamic programming problems satisfying convex quadrangle inequalities results in faster algorithms for a number of problems arising in molecular biology, speech recognition, and geology.