# Geometric applications of a matrix-searching algorithm

TL;DR: 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.

Abstract: 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 ifi
1 >i
2 implies thatj(i
1) ≥J(i
2).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

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TL;DR: In this paper, linear-time algorithms for solving a class of problems that involve transforming a cost function on a grid using spatial information are described, where the binary image is replaced by an arbitrary function on the grid.

Abstract: We describe linear-time algorithms for solving a class of problems that involve transforming a cost function on a grid using spatial information. These problems can be viewed as a generalization of classical distance transforms of binary images, where the binary image is replaced by an arbitrary function on a grid. Alternatively they can be viewed in terms of the minimum convolution of two functions, which is an important operation in grayscale morphology. A consequence of our techniques is a simple and fast method for computing the Euclidean distance transform of a binary image. Our algorithms are also applicable to Viterbi decoding, belief propagation, and optimal control.

925 citations

### Cites background from "Geometric applications of a matrix-..."

...(1.1) Here d(p,q) is some measure of the distance (not necessarily a metric) between p and q. Intuitively, for each point p we find a point q that is close to p, and for which f (q) is small....

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TL;DR: In this article, a simple forward algorithm for the generalized Wagner-Whitin model is proposed, which solves the general model in 0n log n time and 0n space, as opposed to the well known shortest path algorithm advocated over the last 30 years with 0n2 time.

Abstract: This paper is concerned with the general dynamic lot size model, or generalized Wagner-Whitin model. Let n denote the number of periods into which the planning horizon is divided. We describe a simple forward algorithm which solves the general model in 0n log n time and 0n space, as opposed to the well-known shortest path algorithm advocated over the last 30 years with 0n2 time.
A linear, i.e., 0n-time and space algorithm is obtained for two important special cases: a models without speculative motives for carrying stock, i.e., where in each interval of time the per unit order cost increases by less than the cost of carrying a unit in stock; b models with nondecreasing setup costs.
We also derive conditions for the existence of monotone optimal policies and relate these to known planning horizon and other results from the literature.

403 citations

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TL;DR: This paper shows that for many of these concave cost economic lot size problems, the dynamic programming formulation of the problem gives rise to a special kind of array, called a Monge array, and shows how the structure of Monge arrays can be exploited to obtain significantly faster algorithms for these economic lots size problems.

Abstract: Many problems in inventory control, production planning, and capacity planning can be formulated in terms of a simple economic lot size model proposed independently by A. S. Manne (1958) and by H. M. Wagner and T. M. Whitin (1958). The Manne-Wagner-Whitin model and its variants have been studied widely in the operations research and management science communities, and a large number of algorithms have been proposed for solving various problems expressed in terms of this model, most of which assume concave costs and rely on dynamic programming. In this paper, we show that for many of these concave cost economic lot size problems, the dynamic programming formulation of the problem gives rise to a special kind of array, called a Monge array. We then show how the structure of Monge arrays can be exploited to obtain significantly faster algorithms for these economic lot size problems. We focus on uncapacitated problems, i.e., problems without bounds on production, inventory, or backlogging; capacitated problem...

349 citations

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TL;DR: This paper presents a survey on Monge matrices and related Monge properties and their role in combinatorial optimization, and deals with the following three main topics: fundamental combinatorsial properties of Monge structures, applications of MonGE properties to optimization problems and recognition ofMonge properties.

321 citations

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IBM

^{1}TL;DR: This work discusses data mining based on association rules for two numeric attributes and one Boolean attribute, which implies that bank customers whose ages and balances fall in a planar region P tend to use card loan with a high probability.

Abstract: We discuss data mining based on association rules for two numeric attributes and one Boolean attribute For example, in a database of bank customers, "Age" and "Balance" are two numeric attributes, and "CardLoan" is a Boolean attribute Taking the pair (Age, Balance) as a point in two-dimensional space, we consider an association rule of the form((Age, Balance) ∈ P) ⇒ (CardLoan = Yes),which implies that bank customers whose ages and balances fall in a planar region P tend to use card loan with a high probability We consider two classes of regions, rectangles and admissible (ie connected and x-monotone) regions For each class, we propose efficient algorithms for computing the regions that give optimal association rules for gain, support, and confidence, respectively We have implemented the algorithms for admissible regions, and constructed a system for visualizing the rules

316 citations

##### References

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13 Oct 1975

TL;DR: 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.

Abstract: A number of seemingly unrelated problems involving the proximity of N points in the plane are studied, such as finding a Euclidean minimum spanning tree, the smallest circle enclosing the set, k nearest and farthest neighbors, the two closest points, and a proper straight-line triangulation. For most of the problems considered a lower bound of O(N log N) is shown. For all of them the best currently-known upper bound is O(N2) or worse. 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. The Voronoi diagram is used to obtain O(N log N) algorithms for all of the problems.

1,140 citations

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TL;DR: A fully dynamic maintenance algorithm for convex hulls that can process insertions and deletions of single points in only O(log* n) steps per transaction, where n is the number of points currently in the set.

505 citations

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05 May 1975

TL;DR: 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.

Abstract: The complexity of a number of fundamental problems in computational geometry is examined and a number of new fast algorithms are presented and analyzed. General methods for obtaining results in geometric complexity are given and upper and lower bounds are obtained for problems involving sets of points, lines, and polygons in the plane. 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.

287 citations

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Brown University

^{1}TL;DR: A divide-and-conquer approach similar to the ones used by 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.

Abstract: We consider the following problem: Given a rectangle containing N points, find the largest area subrectangle with sides parallel to those of the original rectangle which contains none of the given points. If the rectangle is a piece of fabric or sheet metal and the points are flaws, this problem is finding the largest-area rectangular piece which can be salvaged. A previously known result [13] takes $O(N^2 )$ worst-case and $O(N\log ^2 N)$ expected time. This paper presents an $O(N\log ^3 N)$ time, $O(N\log N)$ space algorithm to solve this problem. It uses a divide-and-conquer approach similar to the ones used by Bentley [1] and introduces a new notion of Voronoi diagram along with a method for efficient computation of certain functions over paths of a tree.

133 citations

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TL;DR: Given n points in the plane, algorithms for finding maximum perimeter or area convex k-gons with vertices k of the given n points are presented and given for the special case of k = 3.

Abstract: Given n points in the plane, we present algorithms for finding maximum perimeter or area convex k-gons with vertices k of the given n points. Our algorithms work in linear space and time $O(kn\lg n + n\lg ^2 n)$. For the special case $k = 3$ we give $O(n\lg n)$ algorithms for these problems. Several related issues are discussed.

104 citations