# Geometric applications of a matrix-searching algorithm

@article{Aggarwal1986GeometricAO, title={Geometric applications of a matrix-searching algorithm}, author={Alok Aggarwal and Maria M. Klawe and Shlomo Moran and Peter W. Shor and Robert E. Wilber}, journal={Algorithmica}, year={1986}, volume={2}, pages={195-208} }

LetA be a matrix with real entries and letj(i) be the index of the leftmost column containing the maximum value in rowi ofA.A is said to bemonotone ifi1 >i2 implies thatj(i1) ≥J(i2).A istotally monotone if all of its submatrices are monotone. We show that finding the maximum entry in each row of an arbitraryn xm monotone matrix requires Θ(m logn) time, whereas if the matrix is totally monotone the time is Θ(m) whenm≥n and is Θ(m(1 + log(n/m))) whenm<n. The problem of finding the maximum value…

## 396 Citations

Selection in Monotone Matrices and Computing kth Nearest Neighbors

- Computer Science, MathematicsSWAT
- 1994

We present an O(m+n√n log n) time algorithm to select the kth smallest item from an m×n totally monotone matrix for any k≤mn. This is the first subquadratic algorithm for selecting an item from a…

An efficient algorithm for row minima computations in monotone matrices

- Computer ScienceProceedings of the 1996 ICPP Workshop on Challenges for Parallel Processing
- 1996

Every algorithm that solves the problem of computing the minimum of an n/spl times/n matrix must take /spl Omega/(log log n) time, which is the previously best known lower bound for selection of the reconfigurable mesh.

Selection and sorting in totally monotone arrays

- MathematicsSODA '90
- 1990

The algorithm for sorting the rows of a totally monotone array is applied to the neighbor-ranking problem for the vertices of a convex polygonP and this technique is extended to arbitrary point sets.

Improved Selection on Totally Monotone Arrays

- Mathematics, Computer ScienceFSTTCS
- 1991

An O(n lg m)-time algorithm for computing an approximate median in each row of an m×n totally monotone array; this approximate median is an entry whose rank in its row lies between [ n/4] and [3n/4].

An Eecient Algorithm for Row Minima Computations on Basic Reconngurable Meshes

- Computer Science
- 1998

An (log log n) time lower bound is obtained for the problem of selecting the k-th smallest item in a monotone matrix, thus extending the best previously-known lower bound for selection on the reconngurable mesh.

Applications of generalized matrix searching to geometric algorithms

- MathematicsDiscret. Appl. Math.
- 1990

(Near-)Linear-Time Randomized Algorithms for Row Minima in Monge Partial Matrices and Related Problems

- Mathematics, Computer ScienceSODA
- 2021

We revisit classical problems about searching in totally monotone and Monge matrices, which have many applications in computational geometry and other areas. We present a number of new results,…

Near-Optimal Randomized Algorithms for Selection in Totally Monotone Matrices

- Mathematics, Computer ScienceSODA
- 2021

The selection algorithm implies an O(n polylog n) algorithm for computing incidences between n points and n pseudo-lines in the plane, and improves, extends, and simplifies a previous method by Agarwal and Sharir.

An Efficient Algorithm for On-Line Searching of Minima in Monge Path-Decomposable Tridimensional Arrays

- Computer Science, MathematicsInf. Process. Lett.
- 1998

## References

SHOWING 1-10 OF 26 REFERENCES

Computing the Largest Empty Rectangle

- Computer ScienceSTACS
- 1984

A divide-and-conquer approach similar to the ones used by Strong and Bentley is used and a new notion of Voronoi diagram is introduced along with a method for efficient computation of certain functions over paths of a tree.

Maintenance of Configurations in the Plane

- Mathematics, Computer ScienceJ. Comput. Syst. Sci.
- 1981

Closest-point problems

- Computer Science16th Annual Symposium on Foundations of Computer Science (sfcs 1975)
- 1975

The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space, and is used to obtain O(N log N) algorithms for most of the problems considered.

Optimal wiring between rectangles

- Computer ScienceSTOC '81
- 1981

Although the theoretical model implies that there can be great gains for the two-layer strategy, even in cases where no crossovers are required, when the typical design rules for laying out VLSI circuits are considered, there is no substantial advantage to the twolayer approach over the one-layer approach.

Finding extremal polygons

- MathematicsSTOC '82
- 1982

Algorithms for finding maximum perimeter or area convex-gons with vertices of the given n points in the plane with linear space and time are presented.

Geometric complexity

- Mathematics, Computer ScienceSTOC
- 1975

An effort is made to recast classical theorems into a useful computational form and analogies are developed between constructibility questions in Euclidean geometry and computability questions in modern computational complexity.