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… 
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References

SHOWING 1-10 OF 27 REFERENCES
Improved Submatrix Maximum Queries in Monge Matrices
TLDR
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.
Two dimensional range minimum queries and Fibonacci lattices
Persistent predecessor search and orthogonal point location on the word RAM
TLDR
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
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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.
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