A. Aggarwal, M. M. Klawe, S. Moran, P. W. Shor, and R. Wilber. Geometric applications of a matrixsearching algorithm. Algorithmica, 2:195--208, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by executing only O(log n) calls to the decision procedure, so the running time of this matrix searching technique is O(log n) times the cost of the decision procedure. The technique, when applicable, is both efficient and simple compared to the standard parametric searching. Aggarwal et al. [23, 24, 25, 26] studied a different matrix searching technique for optimization problems. They gave a linear time algorithm for computing the minimum (or maximum) element of every row of a totally monotone matrix; a matrix A = fa i;j g is called totally monotone if a i 1 ;j 1 a i 1 ;j 2 implies that a i 2 ;j 1 ....

....1 i 2 m; 1 j 1 j 2 n. Totally monotone matrices arise in many geometric, as well as nongeometric, optimization problems. For example, the farthest neighbors of all vertices of a convex polygon and the geodesic diameter of a simple polygon can be computed in linear time, using such matrices [24, 151]. 4 Prune and Search Technique and Linear Programming Like parametric searching, the prune and search (or decimation) technique also performs an implicit binary search over the finite set of candidate values for , but, while doing so, it also tries to eliminate input objects that are ....

A. Aggarwal, M. M. Klawe, S. Moran, P. Shor, and R. Wilber, Geometric applications of a matrix-searching algorithm, Algorithmica, 2 (1987), 195--208.

....1.1 Background and previous work. Let A = A[i; j] be an m Theta n matrix. A is called totally monotone if for all i 1 i 2 and j 1 j 2 , A[i 1 ; j 1 ] A[i 1 ; j 2 ] implies A[i 2 ; j 1 ] A[i 2 ; j 2 ] Totally monotone matrices were introduced by Aggarwal, Klawe, Moran, Shor and Wilber [4]. These matrices arise naturally in the study of various problems in computational geometry, in the analysis of certain dynamic programming algorithms, and in other combinatorial problems related to VLSI Department of Mathematics, Raymond and Beverly Sackler, Faculty of Exact Sciences, Tel Aviv ....

....portion of this work was done while the author was in the department of Computer Science, Stanford University, CA 94305 2140, and supported by a Weizmann fellowship and contract ONR N00014 88 K 0166. circuit design. A wide variety of applications that use totally monotone matrices can be found in [4], 5] 9] 10] and their references. In most of the applications the problems are reduced to a selection or sorting problem in each row in an appropriate totally monotone matrix. The basic problem considered was row maxima (or row minima) i.e. the problem of finding the maximum (or minimum) ....

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A. Aggarwal, M. Klawe, S. Moran, P.Shor and R. Wilber, Geometric applications of a matrix searching algorithm, Algorithmica 2 (1987) pp. 195--208.

....d] for all a b and c d. 2. concave condition: M [a; c] M [b; c] M [a; d] M [b; d] for all a b and c d. Note that the Monge property implies total monotonicity, but the converse is not true. Therefore, both DIST and OUT are totally monotone by the concave condition. Aggarwal et al. [2] gave a recursive algorithm, nicknamed SMAWK in the literature, which can compute in O(n) time all row and column maxima of an n n totally monotone matrix, by querying only O(n) elements of the array. Hence, one can use SMAWK to compute the output row O by querying only O(n) elements of OUT. ....

Aggarwal, A., M. Klawe, S. Moran, P. Shor, and R. Wilber, Geometric Applications of a Matrix-Searching Algorithm, Algorithmica, 2, 195-208 (1987).

....analysis. The fastest data structure to date [27] runs in O(m log (m; n) time, where m is the number of operations. It is a slight improvement over Gabow s data structure [15] which runs in O(m (m;n) time. The problem of nding row maxima in totally monotone matrices has many applications [3]. For complete m n matrices the problem can be solved in O(m n) time [3] However, for partial matrices (where some elements may be missing) the complexity of the problem seems to depend on the shape of the non blank elements. Klawe [22] proved super linear lower bounds of n (n) on the ....

....time, where m is the number of operations. It is a slight improvement over Gabow s data structure [15] which runs in O(m (m;n) time. The problem of nding row maxima in totally monotone matrices has many applications [3] For complete m n matrices the problem can be solved in O(m n) time [3]. However, for partial matrices (where some elements may be missing) the complexity of the problem seems to depend on the shape of the non blank elements. Klawe [22] proved super linear lower bounds of n (n) on the time to nd row maxima in an n n matrix where the non blank elements are ....

A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilber. Geometric applications of a matrix-searching algorithm. Algorithmica, 2:195-208, 1987.

....whether quadratic time is necessary, or if even more eOEcient optimization algorithms could be developed. The inherent complexity of the multisplitting task is an uncharted territory. However, mathematically similar problems have been encountered in computational geometry and string matching [1, 16, 17]. This paper reAEects that work to the multisplitting framework. It turns out that the optimal multisplitting algorithms solve as a subproblem an instance of the column minima problem [17] for which lower bound results are already known [1] Depending on the level of monotonicity of the function, ....

....in computational geometry and string matching [1, 16, 17] This paper reAEects that work to the multisplitting framework. It turns out that the optimal multisplitting algorithms solve as a subproblem an instance of the column minima problem [17] for which lower bound results are already known [1]. Depending on the level of monotonicity of the function, the inherent complexity of the column minima problem is Omega Gamma b ) Theta(b log b) or O(b) Unfortunately, it turns out that commonly used attribute evaluation functions do not fulll the required monotonicity properties. It is ....

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Aggarwal, A., Klawe, M. M., Moran, S., Shor, P., Wilber, R.: Geometric applications of a matrix searching algorithm, Algorithmica, 2(2), 1987.

.... this area is that a minimum area triangle can be found in time O(n by using geometric duality to transform the problem into one of searching a line arrangement [7, 8] Algorithms are also known for optimizing other functions including minimum perimeter [1, 5, 9] and maximum perimeter and area [2, 4]. For some time it remained open whether the minimum area triangle result could be generalized to finding minimum area k gons. There are actually four reasonable ways of generalizing this: one could search for (1) a minimum area k gon, 2) a minimum area convex k gon, 3) a minimum area empty ....

A. Aggarwal, M.M. Klawe, S. Moran, P. Shor and R. Wilber. Geometric applications of a matrix-searching algorithm. Algorithmica 2 (1987) 195--208.

....d] for all a b and c d. 2. concave condition: M [a; c] M [b; c] M [a; d] M [b; d] for all a b and c d. Note that the Monge property implies total monotonicity, but the converse is not true. Therefore, both DIST and OUT are totally monotone by the concave condition. Aggarwal et al. [1] gave a recursive algorithm, nicknamed SMAWK in the literature, which can compute in O(n) time all row and column maxima of an n n totally monotone matrix, by querying only O(n) elements of the array. Hence, one could use SMAWK to compute the output row O by querying only O(n) elements of OUT . ....

Aggarwal, A., M. Klawe, S. Moran, P. Shor, and R. Wilber, Geometric Applications of a Matrix-Searching Algorithm, Algorithmica, 2, 195-208 (1987).

....fi 2 ) time, but we describe two techniques which decrease the time complexity, each by a factor of Theta(n) The first technique transforms the search space into a larger, but more tractable, one. The second uses monotonematrix concepts, i.e. the Monge property [14] and the SMAWK algorithm [3]. Our approach requires a better understanding of the combinatorics of lopsided trees, which, in turn, requires introducing some definitions. Let ff; fi be positive integers, ff fi. A binary lopsided ff; fi tree (or just a lopsided tree, if ff and fi are understood) is a binary tree in which ....

....1; j 1) p j i ff Gammai 1 Gamma p j i ff Gammai 0 which completes the proof. 2 A 2 Theta 2 matrix A is defined to be monotone if either A 11 A 12 or A 21 A 22 . An n Theta m matrix A is defined to be totally monotone if every 2 Theta 2 submatrix of A is monotone. The SMAWK algorithm [3] takes as its input a function which computes the entries of an n Theta m totally monotone matrix A and gives as output a non decreasing function f , where 1 f(i) m for 1 i n, such that A i;f(i) is the minimum value of row i of A. The time complexity of the SMAWK algorithm is O(n m) ....

A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilber, Geometric applications of a matrixsearching algorithm, Algorithmica, 2 (1987), 195--208.

.... 15 7 17 6 13 11 32 28 23 37 32 44 45 88 85 76 92 90 100 110 31 30 34 53 51 66 73 6 9 13 32 32 52 62 18 29 35 43 55 78 95 1 C C C C C C C C C C C C C C A Figure 1: An example of a totally monotone matrix Totally monotone matrices were originally introduced by Aggarwal et al. [4]. They presented an O(m n) time algorithm for computing the leftmost maximal element in each row of an m Theta n totally monotone matrix, and applied it to a number of problems in computational geometry and VLSI routing, including all farthest neighbors of convex polygons. Since the publication ....

A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilber, Geometric applications of a matrix-searching algorithm, Algorithmica 2 (1987), 195--208.

....in the set, which varies between O(n 2 ) and O(n 3 ) 7] Boyce, Dobkin, Drysdale, and Guibas [5] treated the problems of finding maximum perimeter and maximum area convex k gons. Their algorithms work in linear space and O(kn log n n log 2 n) time. Aggarwal, Klawe, Moran, Shor, and Wilber [1] improved these results to O(kn n log n) In application like statistical clustering and pattern recognition minimization problems tend to play a more important role than maximization problems. Minimization problems seem to be computationally harder than maximization problems in this context. ....

....The next result follows immediately from Theorem 5.2. Corollary 5. 3 The minimum perimeter (1) convex k gon, 2) empty convex k gon, 3) convex hull of k points can be determined in time O(kn 3 ) # The method can also maximize area or perimeter but the bounds will be worse than the methods of [1]. Corollary 5.4 Given a set P of n points, the convex k gon with vertices in P containing the minimum or maximum number of points of P in its interior can be determined in time O(kn 3 ) Proof: From Theorem 4.1. it follows that G(n) O(n 3 ) # 15 All other weight functions listed above can ....

A. Aggarwal, M.M. Klawe, S. Moran, P. Shor and R. Wilber, Geometric applications of a matrix-searching algorithm, Algorithmica 2 (1987), 195--208.

....functions. Miller and Myers [53] independently solved the same problem in similar time bounds. Wilber [78] pointed out a resemblance between the least weight subsequence problem and a matrix searching technique that had been previously used to solve a number of problems in computational geometry [2]. He used this technique in an algorithm for solving the least weight subsequence problem in linear time. His 18 algorithm also extends to the generalization of the least weight subsequence problem expressed in recurrence 8 above. However in its application to sequence alignment, we require the ....

....Plass [38] with general penalty functions. Galil and Giancarlo [17] discovered algorithms for both the convex and concave cases which take time O(n log n) or linear time for some special cases. Miller and Myers [53] independently discovered a similar algorithm for the convex case. Aggarwal et al. [2] had previously given an algorithm which solves an o#ine version of the concave case, in which D does not depend on E, in time O(n) Wilber [78] extended this work to an ingenious O(n) algorithm for the online concave case; however as we shall see Wilber s algorithm has shortcomings that make it ....

[Article contains additional citation context not shown here]

Alok Aggarwal, Maria M. Klawe, Shlomo Moran, Peter Shor, and Robert Wilber, Geometric Applications of a Matrix-Searching Algorithm, Algorithmica 2, 1987, pp. 209--233.

....by executing only O(log n) calls to the decision procedure, so the running time of this matrix searching technique is O(log n) times the cost of the decision procedure. The technique, when applicable, is both efficient and simple compared to the standard parametric searching. Aggarwal et al. [22, 23, 24, 25] studied a different matrix searching technique for optimization problems. They gave a linear time algorithm for computing the minimum (or maximum) element of every row of a totally monotone matrix; a matrix A = fa i;j g is called totally monotone if a i 1 ;j 1 a i 1 ;j 2 implies that a i 2 ;j 1 ....

....1 i 2 m; 1 j 1 j 2 n. Totally monotone matrices arise in many geometric, as well as nongeometric, optimization problems. For example, the farthest neighbors of all vertices of a convex polygon and the geodesic diameter of a simple polygon can be computed in linear time, using such matrices [23, 129]. 4 Prune and Search Technique and Linear Programming Like parametric searching, the prune and search (or decimation) technique also performs an implicit binary search over the finite set of candidate values for , but, while doing so, it also tries to eliminate input objects that are ....

A. Aggarwal, M. M. Klawe, S. Moran, P. Shor, and R. Wilber, Geometric applications of a matrix-searching algorithm, Algorithmica, 2 (1987), 195--208.

....a simple polygon, improving the previous best result by a factor of O(logn) in each case. Key Words: Shortest paths, matrix searching, geodesic diameter, farthest neighbors, geometric matching. 1 1 Introduction Matrix searching is the popular term for a technique introduced by Aggarwal et al. [2] for computing row wise maxima in a totally monotone matrix. A matrix M is called totally monotone if M(i; k) M(i; l) M(j; k) M(j; l) for any i j and k l. Aggarwal et al. 2] discovered the importance of totally monotone matrices and showed that many problems in computational ....

....1 1 Introduction Matrix searching is the popular term for a technique introduced by Aggarwal et al. 2] for computing row wise maxima in a totally monotone matrix. A matrix M is called totally monotone if M(i; k) M(i; l) M(j; k) M(j; l) for any i j and k l. Aggarwal et al. [2] discovered the importance of totally monotone matrices and showed that many problems in computational geometry can be formulated as finding maxima in a totally monotone matrix. The problem of finding farthest neighbors in a convex polygon is a prototypical application of this technique; we repeat ....

[Article contains additional citation context not shown here]

A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilber. Geometric applications of a matrix searching algorithm. Algorithmica, 2:195--208, 1987.

....cost matching for a 74 node graph on the the circle with Euclidian distance as the cost function. Dynamic programming problems based on cost functions which satisfy the (inverse) quadrangle inequality and some closely related matrix search problems have been studied by many authors, including [2, 3, 4, 7, 8, 9, 12, 15, 16, 17, 19, 24, 25]. However, there seems to be no direct connection between our quasi convex matching problem and the problems solved by these authors. The notion of a Monge array [13] is related to that of quasi convexity, but the Monge condition is a stronger (i.e. quasi convexity is strictly more general) ....

A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilber, Geometric applications of a matrix-searching algorithm, Algorithmica, 2 (1987), pp. 195--208.

....the index of the leftmost column containing the minimum value in row i of A. A is called monotone if i 1 #i 2 implies that j#i 1 # # j#i 2 #. A is totally monotone if every submatrix of A is monotone, or equivalently,ifa i 1 k 2 #a i 1 k 1 implies a i 2 k 2 #a i 2 k 1 for i 1 #i 2and k 1 #k 2 . Aggarwal et al. #1987# have shown that all row minima can be computed in O#j p# time if the matrix A is totally monotone. To prove that A =#a ik # is totally monotone, we show that A has even a stronger property, the so called Monge property: a i 1 k 1 a i 2 k 2 # a i 1 k 2 a i 2 k 1 , for i 1 #i 2 and k 1 #k 2 ....

Aggarwal, A., M. M. Klawe, S. Moran, P. Shor, and R. Wilber #1987#, Geometric applications of a matrix-searching algorithm, Algorithmica 2, 195#208.

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A. Aggarwal, M. M. Klawe, S. Moran, P. W. Shor, and R. Wilber. Geometric applications of a matrixsearching algorithm. Algorithmica, 2:195--208, 1987.

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A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilber, Geometric applications of a matrix searching algorithm, Algorithmica 2(1987), 195-208.

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A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilber, Geometric applications of a matrix-searching algorithm, Algorithmica, 2 (1987), pp. 195--208.

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A. Aggarwal, M.M. Klawe, S. Moran, and R. Wilber. Geometric applications of a matrixsearching algorithm. Algorithmica, 2:195-208, 1987.

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A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilber, Geometric applications of a matrix-searching algorithm, Algorithmica, 2 (1987), pp. 195--208.

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A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilber. "Geometric applications of a matrix-searching algorithm," Algorithmica, 2(2), 1987, pp. 195-208.

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Alok Aggarwal, Maria M. Klawe, Shlomo Moran, Peter Shor, and Robert Wilber. Geometric applications of a matrix-searching algorithm. Algorithmica, 2:195--208, 1987.

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A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilber, Geometric applications of a matrix searching algorithm, Algorithmica 2(1987), 195-208.

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Aggarwal, A.; Klawe, M.; Moran, S.; Shor, P.; Wilber, R. Geometric applications of a matrix searching algorithm. Algorithmica, 1987, 2, 195--208.

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A. Aggarwal, M. M. Klawe, S. Moran, P. Shor, and R. Wilber. Geometric applications of a matrix-searching algorithm. Algorithmica, 2(2):195--208, 1987.

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