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39
A Theory for MemoryBased Learning
 Machine Learning
, 1994
"... A memorybased learning system is an extended memory management system that decomposes the input space either statically or dynamically into subregions for the purpose of storing and retrieving functional information. The main generalization techniques employed by memorybased learning systems are t ..."
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Cited by 7 (1 self)
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A memorybased learning system is an extended memory management system that decomposes the input space either statically or dynamically into subregions for the purpose of storing and retrieving functional information. The main generalization techniques employed by memorybased learning systems are the nearestneighbor search, space decomposition techniques, and clustering. Research on memorybased learning is still in its early stage. In particular, there are very few rigorous theoretical results regarding memory requirement, sample size, expected performance, and computational complexity. In this paper, we propose a model for memorybased learning and use it to analyze several methods fflcovering, hashing, clustering, treestructured clustering, and receptivefieldsfor learning smooth functions. The sample size and system complexity are derived for each method. Our model is built upon the generalized PAC learning model of Haussler (Haussler, 1989) and is closely related to the method of vector quantization in data compression. Our main result is that we can build memorybased learning systems using new clustering storage in typical situations.
Restricted Delivery Problems on a Network
, 1996
"... We consider a delivery problem on a network one is given a network in which nodes have supplies or demands for certain products, and arcs have lengths satisfying the Triangle Inequality. A vehicle of infinite capacity, travels through the network, carrying products to their destinations, and is limi ..."
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Cited by 6 (1 self)
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We consider a delivery problem on a network one is given a network in which nodes have supplies or demands for certain products, and arcs have lengths satisfying the Triangle Inequality. A vehicle of infinite capacity, travels through the network, carrying products to their destinations, and is limited in that it can carry only a single type of product at a time. The general problem asks for a shortest delivery route of all products from their origin to their destination. Here we consider certain restrictions on the delivery paths allowed, and compare the quality of the solution of the unrestricted problem to that of the restricted one. Both the general and restricted problems are NPhard, and we discuss approximation algorithms. We also give a constant factor approximation algorithm for the Clustered Traveling Salesman Problem. Keywords: Traveling salesman problem, approximation algorithm. 1 Introduction In this paper we are concerned with a delivery problem on a network. We are gi...
Placing Resources on a Growing Line
, 1998
"... We consider the problem of placing k identical resources in a graph where each vertex is associated with a nonnegative weight representing the frequency of requests issued by that vertex for the resource. We define the cost of a placement as the sum over all vertices of their distances to the closes ..."
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Cited by 6 (0 self)
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We consider the problem of placing k identical resources in a graph where each vertex is associated with a nonnegative weight representing the frequency of requests issued by that vertex for the resource. We define the cost of a placement as the sum over all vertices of their distances to the closest resource weighted by their weights. The optimal placement is the placement with least cost among all placements. We give an algorithm for placing optimally k resources on a "growing" line. The algorithm starts with an empty line. At each step a new vertex is appended to the line and the algorithm has to recompute the optimal placement of the k resources. Our algorithm processes each new vertex in O(k) amortized time. As a corollary, we obtain an algorithm that computes the optimal placement of k resources in an nvertex line in time O(kn), which is optimal for constant k.
Improved Bounds on Relaxations of a Parallel Machine Scheduling Problem
 Journal of Combinatorial Optimization
, 1998
"... We consider the problem of scheduling n jobs with release dates on m identical parallel machines to minimize the average completion time of the jobs. We prove that the ratio of the average completion time of the optimal nonpreemptive schedule to that of the optimal preemptive schedule is at most 7 ..."
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Cited by 4 (2 self)
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We consider the problem of scheduling n jobs with release dates on m identical parallel machines to minimize the average completion time of the jobs. We prove that the ratio of the average completion time of the optimal nonpreemptive schedule to that of the optimal preemptive schedule is at most 7 3 , improving a bound of (3 \Gamma 1 m ) due to Phillips, Stein and Wein. We then use our technique to give an improved bound on the quality of a linear programming relaxation of the problem considered by Hall, Schulz, Shmoys and Wein. A preliminary presentation of these results was given in the Proceedings of the 1996 International Colloquium on Automata, Languages and Programming [1]. y caphill@cs.sandia.gov. Sandia National Labs, Albuquerque, NM. This work was performed under U.S. Department of Energy contract number DEAC0476AL85000. z schulz@math.tuberlin.de. Department of Mathematics, Technical University of Berlin, 10623 Berlin, Germany. Research partially supported by the ...
Online Maintenance of kMedians and kCovers on a Line
"... The standard dynamic programming solution to finding k medians on a line with n nodes requires O(kn²) time. Dynamic programming speedup techniques, e.g., use of the quadrangle inequality or properties of totally monotone matrices, can reduce this to O(kn) time but these techniques are inherent ..."
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Cited by 4 (2 self)
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The standard dynamic programming solution to finding k medians on a line with n nodes requires O(kn²) time. Dynamic programming speedup techniques, e.g., use of the quadrangle inequality or properties of totally monotone matrices, can reduce this to O(kn) time but these techniques are inherently static. The major result of this paper is to show that we can maintain the dynamic programming speedup in an online setting where points are added from left to right on a line. Computing the
On the Integrality Gap of Capacitated Facility Location
"... We consider the facility location problem with hard nonuniform upper and lower bounds on the amount of demand that can be routed to any facility. We examine the natural integer programming formulation of this problem. First, for the version with just the upper bounds (i.e., with the lower bounds be ..."
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Cited by 4 (0 self)
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We consider the facility location problem with hard nonuniform upper and lower bounds on the amount of demand that can be routed to any facility. We examine the natural integer programming formulation of this problem. First, for the version with just the upper bounds (i.e., with the lower bounds being zero), we show that for every constant factor blowup in capacities, the integrality gap of the LP relaxation is a constant. We present a smooth tradeoff for the cost versus the blowup in capacities. Secondly, we show how to incorporate lower bounds into any approximation algorithm for the version with just the upper bounds. Nonuniform capacities make the problem significantly more difficult than the case involving uniform capacities.
Steiner Trees and Beyond: Approximation Algorithms for Network Design
, 1993
"... We present approximation algorithms for several NPhard optimization problems arising in network design. Almost all of our algorithms run in polynomial time and output solutions with a worstcase performance guarantee on the quality of the output solution. A typical problem that we consider can be s ..."
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Cited by 3 (1 self)
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We present approximation algorithms for several NPhard optimization problems arising in network design. Almost all of our algorithms run in polynomial time and output solutions with a worstcase performance guarantee on the quality of the output solution. A typical problem that we consider can be stated as follows: given an undirected graph and certain connectivity requirements, find a subgraph that satisfies these requirements and has minimum cost. In this thesis, we address many different connectivity requirements such as spanning trees, Steiner trees, generalized Steiner forests, and twoconnected networks. The cost criteria that we consider range from the total cost of the edges in the network, the total cost of the nodes in the network, the maximum degree of any node in the network, the maximum cost of any edge in the network to some combination of these. We also address the maximumleaf spanning tree problem and provide the first approximation algorithms for this problem. In t...
Compact Location Problems
"... ) V. Radhakrishnan S.O. Krumke y M.V. Marathe D.J. Rosenkrantz S. S. Ravi Department of Computer Science University at Albany  SUNY Albany, NY 12222 Abstract We consider the problem of placing a specified number (p) of facilities on the nodes of a network so as to minimize some measure of the ..."
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) V. Radhakrishnan S.O. Krumke y M.V. Marathe D.J. Rosenkrantz S. S. Ravi Department of Computer Science University at Albany  SUNY Albany, NY 12222 Abstract We consider the problem of placing a specified number (p) of facilities on the nodes of a network so as to minimize some measure of the distances between facilities. This type of problem models a number of problems arising in facility location, statistical clustering, pattern recognition, and also a processor allocation problem in multiprocessor systems. We consider the problem under three different objectives, namely minimizing the diameter, minimizing the average distance, and minimizing the variance. In general, the problem is NPhard under any of the objectives. We observe that in general, even obtaining a relative approximation for any of the objectives is NPhard. We present a general framework for approximating the minimum cost compact location problem for each of the above measures. We present efficient approximation...
Placing Resources in a Tree: Dynamic and Static Algorithms
 Proc. of the 22nd International Colloquium on Automata, Languages, and Programming (ICALP 95). To appear in Lecture Notes in Computer Science
"... . We study the classical problem of optimally placing resources in a tree. We give dynamic algorithms that recompute the optimal solution after a weight change in polylogarithmic time for the case of one resource in a general tree and for any number of resources in a complete tree. Our algorithms ar ..."
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Cited by 1 (1 self)
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. We study the classical problem of optimally placing resources in a tree. We give dynamic algorithms that recompute the optimal solution after a weight change in polylogarithmic time for the case of one resource in a general tree and for any number of resources in a complete tree. Our algorithms are the first dynamic algorithms for this problem. We also give lineartime algorithms for the static version of the problem for two resources. Previously known algorithms run in time quadratic in the number of vertices. Finally, in the case the tree is a line, we present an amortized constant time algorithm for the solution of the online version of the problem, for any number of resources. We also discuss some new open problems. 1 Introduction We study the dynamic version of the classical problem of optimally placing resources in a tree that can be described as follows. Let T be a tree with N vertices and, for each vertex v, let w(v) be the weight of v and we want to place a constant number...
Online Medians via Online Bribery (Extended Abstract)
"... We then consider the competitive ratio with respect to size. An algorithm is ssizecompetitive if, for each k, the cost of Fk is at most the minimum cost of any set of k facilities, while the size of Fk is at most sk. We present optimally competitive algorithms for this problem. Our proofs reduce o ..."
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We then consider the competitive ratio with respect to size. An algorithm is ssizecompetitive if, for each k, the cost of Fk is at most the minimum cost of any set of k facilities, while the size of Fk is at most sk. We present optimally competitive algorithms for this problem. Our proofs reduce online medians to the following online bribery problem: faced with some unknown threshold T 2 R+, an algorithm must submit "bids " b 2 R+ until it submits a bid as large as T. The algorithm pays the sum of its bids. We describe optimally competitive algorithms for online bribery. Our results on costcompetitive online medians extend to approximately metric distance functions, online fractional medians, and online bicriteria approximation.