Author

# Robert E. Wilber

Bio: Robert E. Wilber is an academic researcher from IBM. The author has contributed to research in topics: Matrix (mathematics) & Convex polygon. The author has an hindex of 2, co-authored 2 publications receiving 607 citations.

##### Papers

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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

506 citations

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01 Aug 1986TL;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 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

127 citations

##### Cited by

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01 Jan 1987TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.

Abstract: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.

2,284 citations

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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

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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