A new problem called monotone bipartitioning of a planar point set is identified which is found to be useful in VLSI layout design. Let F denote a rectangular floor containing a set A of n points. The portion of a str...
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A new problem called monotone bipartitioning of a planar point set is identified which is found to be useful in VLSI layout design. Let F denote a rectangular floor containing a set A of n points. The portion of a straight line formed by two points from the set A is called a line segment. A monotone increasing path (MP) in F is a connected and ordered sequence of line segments from the bottom-left corner of F to its top-right corner, such that the slope of each line segment is nonnegative, and each pair of consecutive line segments share a common point of A. An MP is said to be maximal (MMP) if no other point in A can be included in it preserving monotonicity. Let A(L) denote the subset of A corresponding to the end points of the line segments in an MMP, L. The path L partitions the set of points A n A L into two subsets lying on its two sides. The objective of monotone bipartitioning is to find an MMP L, such that the difference in the number of points in these two subsets is minimum. This problem can be formulated as finding a path between two designated vertices of an edge-weighted digraph (the weight of an edge being an integer lying in the range [ - n, n]), for which the absolute value of the algebraic sum of weights is minimized. An O( n x e) time algorithm is proposed for this problem, where e denotes the number of edges of the graph determined from the geometry of the point set. The monotone bipartitioning problem has various applications to image processing, facility location, and plant layout problems. A related problem arises while partitioning a VLSI floorplan. Given a floorplan with n rectangular blocks, the goal is to find a monotone staircase channel from one corner of the floor to its diagonally opposite corner such that the difference in the numbers of blocks lying on its two sides is minimum. The problem is referred to as the staircase bipartitioning problem. The proposed algorithm for a point set can be directly used to solve this problem in O(n(2
Several interior point algorithms have been proposed for solving nonlinear monotone complementarity problems. Some of them have polynomial worst-case complexity but have to confine to short steps, whereas some of the ...
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Several interior point algorithms have been proposed for solving nonlinear monotone complementarity problems. Some of them have polynomial worst-case complexity but have to confine to short steps, whereas some of the others can take long steps but no polynomial complexity is proven. This paper presents an algorithm which is both long-step and polynomial. In addition, the sequence generated by the algorithm, as well as the corresponding complementarity gap, converges quadratically. The proof of the polynomial complexity requires that the monotone mapping satisfies a scaled Lipschitz condition, while the quadratic rate of convergence is derived under the assumptions that the problem has a strictly complementary solution and that the Jacobian of the mapping satisfies certain regularity conditions
We examine in this paper a variant of the bin packing problem, where it is permissible to fragment the objects while packing them into bins of fixed capacity. We call this the Fragmentable Object Bin Packing problem (...
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We examine in this paper a variant of the bin packing problem, where it is permissible to fragment the objects while packing them into bins of fixed capacity. We call this the Fragmentable Object Bin Packing problem (FOBP). Fragmentation is associated with a cost, leading to the consumption of additional bin capacity. We show that the problem and its absolute approximation are both NP-complete. This is an interesting problem because if the cost of fragmentation is nullified then the problem can be easily solved optimally. If fragmentation is not permitted, then we get the usual bin packing problem. The application comes from a problem in data path synthesis where it is some times necessary to schedule data transfers, subject to restrictions arising from the underlying hardware. We show that FOBP reduces to a simplified version of this problem, thereby proving that it is also a similar hard problem. (C) 1998 Elsevier Science Ltd. All rights reserved.
We study invertibility of big n × n matrices. There exists a number of algorithms, especially in mathematical statistics and numerical mathematics, requiring to invert step by step large matrices which are closel...
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We present a theoretical result on a Path-following algorithm for convex programs. The algorithm employs a nonsmooth Newton subroutine. It starts from a near center of a restricted constraint set, performs a partial n...
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We present a theoretical result on a Path-following algorithm for convex programs. The algorithm employs a nonsmooth Newton subroutine. It starts from a near center of a restricted constraint set, performs a partial nonsmooth Newton step in each iteration, and converges to a Point whose cost is within epsilon accuracy of the optimal cost in O(square-root m\ ln epsilon\) iterations, where m is the number of constraints in the problem. Unlike other interior point methods, the analyzed algorithm only requires a first-order Lipschitzian condition and a generalized Hessian similarity condition on objective and constraint functions. Therefore, our result indicates the theoretical feasibility of applying interior point methods to certain C1-optimization problems instead of C2-problems. Since the complexity bound is unchanged compared with similar algorithms for C2-convex programming, the result shows that the smoothness of functions may not be a factor affecting the complexity of interior point methods.
In a Multiple Attribute Tree (MAT) based data organization, the average case response to a specific range query depends on the structural properties of MAT. These structural properties depend very much on the interrel...
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In a Multiple Attribute Tree (MAT) based data organization, the average case response to a specific range query depends on the structural properties of MAT. These structural properties depend very much on the interrelationships among the data elements. Efficiency in searching can be achieved by exploiting the data properties in the construction of MAT. The order or ranking of attributes is a key factor in deciding the profile of the MAT for given data. In this paper, we estimate the average cost of a range query in MAT based data organization. We then prove that the average performance can be improved by ranking the attributes in such a way that the average size of the filial sets decreases towards the lower levels of the tree structure.
In this paper the utility and the difficulties of probabilistic analysis for optimization algorithms are discussed. Such an analysis is expected to deliver valuable criteria-better than the worst-case complexity-for t...
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In this paper the utility and the difficulties of probabilistic analysis for optimization algorithms are discussed. Such an analysis is expected to deliver valuable criteria-better than the worst-case complexity-for the efficiency of an algorithm in practice.
A certain nonlinear cost functional form which arises in optimal maintenance facility design is shown to take its minimum on the edges of corners of a hypercube. This results in a search procedure of complexity o (N 2...
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A certain nonlinear cost functional form which arises in optimal maintenance facility design is shown to take its minimum on the edges of corners of a hypercube. This results in a search procedure of complexity o (N 2 2 N ) . A novel algorithm is described which reduces the complexity to o (N 1n N) , by employing a judicious presort and a binary search.
In the paper a general approach for solving the balanced divide and conquer equation, which describes the complexity of the algorithms based on the “divide and conquer” principle, is presented. A method for solving ...
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In the paper a general approach for solving the balanced divide and conquer equation, which describes the complexity of the algorithms based on the “divide and conquer” principle, is presented. A method for solving this equation by the change of the index set is proposed. The explicit solutions for some special types of the “divide and conquer” equation are also given.
Day [3] describes an analytical model of minimum-length sequence (MLS) metrics measuring distances between partitions of a set. By selecting suitable values of model coordinates, a user may identify within the model t...
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Day [3] describes an analytical model of minimum-length sequence (MLS) metrics measuring distances between partitions of a set. By selecting suitable values of model coordinates, a user may identify within the model that metric most appropriate to his classification application. Users should understand that within the model similar metrics may nevertheless exhibit extreme differences in their computational complexities. For example, the asymptotic time complexities of two MLS metrics are known to be linear in the number of objects being partitioned; yet we establish below that the computational problem for a closely related MLS metric is NP-complete.","doi":"10.1109/TPAMI.1984.4767476","publicationTitle":"IEEE Transactions on Pattern Analysis and Machine Intelligence","startPage":"69","endPage":"73","rightsLink":"http://***/AppDispatchServlet?publisherName=ieee&publication=0162-8828&title=Extremes+in+the+complexity+of+Computing+Metric+Distances+Between+Partitions&isbn=&publicationDate=Jan.+1984&author=William+H.+E.+Day&ContentID=10.1109/TPAMI.1984.4767476&orderBeanReset=true&startPage=69&endPage=73&volumeNum=PAMI-6&issueNum=1","displayPublicationTitle":"IEEE Transactions on Pattern Analysis and Machine Intelligence","pdfPath":"/iel5/34/4767466/***","keywords":[{"type":"IEEE Keywords","kwd":["Multilevel systems","Polynomials","Analytical models","Computational complexity","Classification algorithms","Partitioning algorithms","NP-complete problem","Artificial intelligence","Pattern analysis","Extraterrestrial phenomena"]},{"type":"Author Keywords ","kwd":["partitions of a set","Comparison of nonhierarchic classifications","complexity of algorithms","metric measures of distance","minimum-length sequence metrics","NP-complete problems"]}],"allowComments":false,"pubLink":"/xpl/***?punumber=34","issueLink":"/xpl/***?isnumber=4767466","standardTitle":"Extremes in the complexity of Computing Metric Distances Between Partitions
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