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Book ChapterDOI

Selection in Monotone Matrices and Computing kth Nearest Neighbors

TL;DR: This work presents 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, which is the first subquadratic algorithm for selecting an item from a totally monOTone matrix.
Abstract: 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 totally monotone matrix. Our method also yields an algorithm for generalized row selection in monotone matrices of the same time complexity. Given a set S=p1, ..., pn of n points in convex position and a vector k=k1, ..., kn, we also present an O(n4/3 logO(1) n) algorithm to compute the k i th nearest neighbor of pi for every i≤n; c is an appropriate constant. This algorithm is considerably faster than the one based on a row-selection algorithm for monotone matrices. If the points of S are arbitrary, then the k i th nearest neighbor of pi, for all i≤n, can be computed in time O(n7/5 logcn), which also improves upon the previously best-known result.
Citations
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Journal ArticleDOI
TL;DR: This objective has been to dissect this dynamic behaviour in Corydoras paleatus, a benthic South American catfish that schools in two dimensions rather than three and displays a fission/fusion social behaviour that depends on group size.

14 citations


Cites background from "Selection in Monotone Matrices and ..."

  • ...…distance (kNND) for each individual (idealized as a point) is found by computing the Euclidean distance from that point to all others (or some efficient subset), ranking the distances from low to high, and choosing the distance having the kth rank (Kovalenko 1993; Agarwal & Sen 1996)....

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Journal ArticleDOI
TL;DR: Algorithmic results for combinatorial problems with cost arrays possessing certain algebraic Monge properties and how their algorithms can be modified to solve bottleneck shortest path problems, even though strict compatibility does not hold in that case.

10 citations


Cites methods from "Selection in Monotone Matrices and ..."

  • ...First, use the algorithm of Agarwal and Sen [1] to 3nd the j + i=2 th smallest entry in CT , which we call $....

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  • ...We use the result of Agarwal and Sen [1], who show how to 3nd the dth smallest entry in an m × n totally monotone array in O((m+ n) √ n log n) time....

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  • ...We use the algorithm of Agarwal and Sen [1] in our binary search, and for each dth smallest value T in the m×n cost array, we query to see if there exists a perfect matching that satis3es the threshold T ....

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  • ...First, use the algorithm of Agarwal and Sen [1] to 3nd the j + i=2 th smallest entry in C , which we call $....

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Journal Article
TL;DR: In this paper, it was shown that the monge-array results for two sum-of-edge costs shortest-path problems can also be extended to a general algebraic setting, provided the problems' ordered commutative semigroup satisfies one additional restriction.
Abstract: When restricted to cost arrays possessing the sum Monge property, many combinatorial optimization problems with sum objective functions become significantly easier to solve. The more general algebraic assignment and transportation problems are similarly easier to solve given cost arrays possessing the corresponding algebraic Monge property. We show that Monge-array results for two sum-of-edge-costs shortest-path problems can likewise be extended to a general algebraic setting, provided the problems' ordered commutative semigroup satisfies one additional restriction. In addition to this general result, we also show how our algorithms can be modified to solve certain bottleneck shortest-path problems, even though the ordered commutative semigroup naturally associated with bottleneck problems does not satisfy our additional restriction. We show how our bottleneck shortest-path techniques can be used to obtain fast algorithms for a variant of Hirschberg and Larmore's optimal paragraph formation problem, and a special case of the bottleneck traveling-salesman problem.
Book ChapterDOI
26 Aug 2002
TL;DR: It is shown how the bottleneck shortest-path techniques can be used to obtain fast algorithms for a variant of Hirschberg and Larmore's optimal paragraph formation problem, and a special case of the bottleneck traveling-salesman problem.
Abstract: When restricted to cost arrays possessing the sum Monge property, many combinatorial optimization problems with sum objective functions become significantly easier to solve The more general algebraic assignment and transportation problems are similarly easier to solve given cost arrays possessing the corresponding algebraic Monge property We show that Monge-array results for two sum-of-edge-costs shortest-path problems can likewise be extended to a general algebraic setting, provided the problems' ordered commutative semigroup satisfies one additional restriction In addition to this general result, we also show how our algorithms can be modified to solve certain bottleneck shortestpath problems, even though the ordered commutative semigroup naturally associated with bottleneck problems does not satisfy our additional restriction We show how our bottleneck shortest-path techniques can be used to obtain fast algorithms for a variant of Hirschberg and Larmore's optimal paragraph formation problem, and a special case of the bottleneck traveling-salesman problem

Cites methods or result from "Selection in Monotone Matrices and ..."

  • ...We use Agarwal and Sen [ 2 ] to find the � j+i 2 � -th smallest entryin CT , which we call τ . We use our queryalgorithm to test if the graph contains a k-edge 1 � → n path that uses onlyedges whose costs are less than or equal to τ . If the queryanswers “yes,” then we call Binary-Search(CT , i, � j+i 2 � ); otherwise we call Binary-...

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  • ...Proof. We use the result of Agarwal and Sen [ 2 ], who show how to find the d-th smallest entryin an m × n totallymonotone arrayin O((m + n) √ n lg n) time....

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  • ...Theorem 4.3. We use the algorithm of Agarwal and Sen [ 2 ] in our binarysearch, and for each d-th smallest value T in the m×n cost array, we query to see if there exists a perfect matching that satisfies the threshold T . The total running time is O(lg(mn)((m + n) √ n lg n + n)) = O(n 3/2 lg 2 n). Similarly, if the cost array is bitonic, then the running time is O(lg(mn)(m lg n + n)) = O(m lg 2 n + n lg n)....

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References
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Book
01 Jan 1976

4,719 citations


"Selection in Monotone Matrices and ..." refers methods in this paper

  • ...Dijkstra [ 15 ] gave a simple O(nlogn) time algorithm for computing the longest monotone subsequence U. Recently, Bar Yehuda and Fogel [11] presented an O(n 3/2) algorithm for partitioning U into a collection of at most 2v/n monotone subsequences....

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Book
01 Jan 1987
TL;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

Book ChapterDOI
TL;DR: In this paper, the present problem has been suggested by Miss Esther Klein in connection with the following proposition: "Our present problem is the same problem as the one suggested by the author of this paper."
Abstract: Our present problem has been suggested by Miss Esther Klein in connection with the following proposition.

1,556 citations

Journal ArticleDOI
TL;DR: The number of comparisons required to select the i-th smallest of n numbers is shown to be at most a linear function of n by analysis of a new selection algorithm-PICK.

1,384 citations

Journal ArticleDOI
TL;DR: It is pointed out that analyses of parallelism in computational problems have practical implications even when multi-processor machines are not available, and a unified framework for cases like this is presented.
Abstract: The goal of this paper is to point out that analyses of parallelism m computational problems have practical implications even when mult~processor machines are not available. This is true because, in many cases, a good parallel algorithm for one problem may turn out to be useful for designing an efficsent serial algorithm for another problem A unified framework for cases like this is presented. Particular cases, which axe discussed in this paper, provide motivation for examining parallelism in sorting, selecuon, minimum-spanning-tree, shortest route, max-flow, and matrix multiplication problems, as well as in scheduling and locational problems.

696 citations


"Selection in Monotone Matrices and ..." refers methods in this paper

  • ...We now apply the parametric searching technique, due to Megiddo [ 23 ], to the above data structure for answering queries of the following form: Given a point q E P and a parameter k < n, determine its k th nearest neighbor, ~k(q), in S. The basic idea is the same as described in [2], but we have to modify their technique a little....

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