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27
Connected Sensor Cover: SelfOrganization of Sensor Networks for Efficient Query Execution
 MOBIHOC'03
, 2003
"... Spatial query execution is an essential functionality of a sensor network, where a query gathers sensor data within a specific geographic region. Redundancy within a sensor network can be exploited to rv uce the communication cost incurv1 in execution of such quer ies. Anyr eduction in communicatio ..."
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Cited by 166 (6 self)
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Spatial query execution is an essential functionality of a sensor network, where a query gathers sensor data within a specific geographic region. Redundancy within a sensor network can be exploited to rv uce the communication cost incurv1 in execution of such quer ies. Anyr eduction in communication cost wouldr esult in an e#cient use of the batter y ener gy, which is ver y limited in sensor s. One appr oach to r educe the communication cost of a quer y is to selfor ganize the networ# inr esponse to a quer , into a topology that involves only a small subset of the sensor s su#cient to pr ocess the quer y. The quer y is then executed using only the sensor in the constr ucted topology. In thisar icle, we design and analyze algor thms for such selfor"/0 zation of asensor networ tor educe enerP consumption. In par icular we develop the notion of a connected sensor cover and design a centr alized appr oximation algor thm that constr ucts a topology in ol ing anear optimal connected sensor co er . We pr o e that the size of the const rst ed topology is within an O(log n)factor ofthe optimal size, wher n is the networ size. We also de elop a distr ibuted selfor$1" zationer" on ofour algor thm, and prv ose seer/ optimizations tor educe the communication oer"E1 of the algorithm. Finally, we evaluate the distributed algorithm using simulations and show that our approach results in significant communication cost reduction.
Improved Approximation Algorithms for Geometric Set Cover
, 2005
"... Given a collection S of subsets of some set U, and M ⊂ U, the set cover problem is to find the smallest subcollection C ⊂ S such that M is a subset of the union of the sets in C. While the general problem is NPhard to solve, even approximately, here we consider some geometric special cases, where u ..."
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Cited by 76 (6 self)
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Given a collection S of subsets of some set U, and M ⊂ U, the set cover problem is to find the smallest subcollection C ⊂ S such that M is a subset of the union of the sets in C. While the general problem is NPhard to solve, even approximately, here we consider some geometric special cases, where usually U = ℜ d. Extending prior results[BG95], we show that approximation algorithms with provable performance exist, under a certain general condition: that for a random subset R ⊂ S and function f(), there is a decomposition of the complement U \ ∪Y∈RY into an expected f(R) regions, each region of a particular simple form. We show that under this condition, a cover of size O(f(C)) can be found. Our proof involves the generalization of shallow cuttings [Mat92] to more general geometric situations. We obtain constantfactor approximation algorithms for covering by unit cubes in ℜ³, for guarding a onedimensional terrain, and for covering by similarsized fat triangles in ℜ². We also obtain improved approximation guarantees for fat triangles, of arbitrary size, and for a class of fat objects.
Approximation schemes for NPhard geometric optimization problems: A survey
 Mathematical Programming
, 2003
"... NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (di ..."
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Cited by 45 (2 self)
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NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (distance between nodes (x1, y1) and (x2, y2) is ((x1−x2) 2 +(y1−y2) 2) 1/2) then the problem is called Euclidean TSP. More generally the distance could be defined using other norms, such as ℓp norms for any p> 1. All these are subcases of the more general notion of a geometric norm or Minkowski norm. We will refer to the version of the problem with a general geometric norm as geometric TSP. Some other NPhard geometric optimization problems are Minimum Steiner Tree (“Given n points, find the smallest network connecting them,”), kTSP(“Given n points and a number k, find the shortest salesman tour that visits k points”), kMST (“Given n points and a number k, find the shortest tree that contains k points”), vehicle routing, degree restricted minimum
Fast exact and heuristic methods for role minimization problems
, 2008
"... We describe several bottomup approaches to problems in role engineering for RoleBased Access Control (RBAC). The salient problems are all NPcomplete, even to approximate, yet we find that in instances that arise in practice these problems can be solved in minutes. We first consider role minimiza ..."
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Cited by 26 (0 self)
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We describe several bottomup approaches to problems in role engineering for RoleBased Access Control (RBAC). The salient problems are all NPcomplete, even to approximate, yet we find that in instances that arise in practice these problems can be solved in minutes. We first consider role minimization, the process of finding a smallest collection of roles that can be used to implement a preexisting usertopermission relation. We introduce fast graph reductions that allow recovery of the solution from the solution to a problem on a input graph. For our test cases, these reductions either solve the problem, or reduce the problem enough that we find the optimum solution with a (worstcase) exponential method. We introduce lower bounds that are sharp for seven of nine test cases and are within 3.4 % on the other two. We introduce and test a new polynomialtime approximation that on average yields 2% more roles than the optimum. We next consider the related problem of minimizing the number of connections between roles and users or permissions, and we develop effective heuristic methods for this problem as well. Finally, we propose methods for several related problems.
Hardness of Set Cover with Intersection 1
, 2000
"... We consider a restricted version of the general Set Covering problem in which each set in the given set system intersects with any other set in at most 1 element. We show that the Set Covering problem with intersection 1 cannot be approximated within a o(log n) factor in random polynomial time u ..."
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Cited by 19 (1 self)
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We consider a restricted version of the general Set Covering problem in which each set in the given set system intersects with any other set in at most 1 element. We show that the Set Covering problem with intersection 1 cannot be approximated within a o(log n) factor in random polynomial time unless NP ` ZT IME(n ). We also observe that the main challenge in derandomizing this reduction lies in find a hitting set for large volume combinatorial rectangles satisfying certain intersection properties. These properties are not satisfied by current methods of hitting set construction. An example
Epsilon nets and union complexity
 Proc. 25th Annu. Sympos. Comput. Geom
, 2009
"... We consider the following combinatorial problem: given a set of n objects (for example, disks in the plane, triangles), and an integer L ≥ 1, what is the size of the smallest subset of these n objects that covers all points that are in at least L of the objects? This is the classic question about th ..."
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Cited by 12 (1 self)
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We consider the following combinatorial problem: given a set of n objects (for example, disks in the plane, triangles), and an integer L ≥ 1, what is the size of the smallest subset of these n objects that covers all points that are in at least L of the objects? This is the classic question about the size of an L nnet for these objects. It is well known that for fairly general classes of geometric objects the size of an Lnet is n O ( n n log). There are some instances where this general L L bound can be improved, and this improvement is usually due to bounds on the combinatorial complexity (size) of the boundary of the union of these objects. Thus, the boundary of the union of m disks has size O(m), and this translates to an O ( n L) bound on the size of annet for disks. For m fat L n triangles, the size of the union boundary is O(m log log m), and this yields L n nnets of size O ( log log n L L). Improved nets directly translate into an upper bound on the ratio between the optimal integral solution and the optimal fractional solution for the corresponding geometric set cover problem. Thus, for covering k points by disks, this ratio is O(1); and for covering k points by fat triangles, this ratio is O(log log k). This connection to approximation algorithms for geometric set cover is a major motivation for attempting to improve bounds on nets. Our main result is an argument that in some cases yields nets that are smaller than those previously obtained from the size of the union boundary. Thus for fat triangles, for instance, we obtain nets of size O ( n log log log n). We use L this to obtain a randomized polynomial time algorithm that gives an O(log log log k)approximation for the problem of covering k points by the smallest subset of a given set of triangles.
Concise descriptions of subsets of structured sets
 In PODS
, 2003
"... We study the problem of economical representation of subsets of structured sets, that is, sets equipped with a set cover. Given a structured set U, and a language L whose expressions define subsets of U, the problem of Minimum Description Length in L (LMDL) is: “given a subset V of U, find a shorte ..."
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Cited by 10 (0 self)
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We study the problem of economical representation of subsets of structured sets, that is, sets equipped with a set cover. Given a structured set U, and a language L whose expressions define subsets of U, the problem of Minimum Description Length in L (LMDL) is: “given a subset V of U, find a shortest string in L that defines V ”. We show that the simple set cover is enough to model a number of realistic database structures. We focus on two important families: hierarchical and multidimensional organizations. The former is found in the context of semistructured data such as XML, the latter in the context of statistical and OLAP databases. In the case of general OLAP databases, data organization is a mixture of multidimensionality and hierarchy, which can also be viewed naturally as a structured set. We study the complexity of the LMDL problem in several settings, and provide an efficient algorithm for the hierarchical case. Finally, we illustrate the application of the theory to summarization of large result sets, (multi) query optimization for ROLAP queries, and XML queries. 1.
Turning clusters into patterns: Rectanglebased discriminative data description
 IEEE International Conference on Data Mining
, 2006
"... The ultimate goal of data mining is to extract knowledge from massive data. Knowledge is ideally represented as humancomprehensible patterns from which endusers can gain intuitions and insights. Yet not all data mining methods produce such readily understandable knowledge, e.g., most clustering al ..."
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Cited by 9 (3 self)
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The ultimate goal of data mining is to extract knowledge from massive data. Knowledge is ideally represented as humancomprehensible patterns from which endusers can gain intuitions and insights. Yet not all data mining methods produce such readily understandable knowledge, e.g., most clustering algorithms output sets of points as clusters. In this paper, we perform a systematic study of cluster description that generates interpretable patterns from clusters. We introduce and analyze novel description formats leading to more expressive power, motivate and define novel description problems specifying different tradeoffs between interpretability and accuracy. We also present effective heuristic algorithms together with their empirical evaluations. 1.
Hyperrectanglebased discriminative data generalization and applications in data mining
, 2007
"... The ultimate goal of data mining is to extract knowledge from massive data. Knowledge is ideally represented as humancomprehensible patterns from which endusers can gain intuitions and insights. Axisparallel hyperrectangles provide interpretable generalizations for multidimensional data points ..."
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Cited by 5 (2 self)
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The ultimate goal of data mining is to extract knowledge from massive data. Knowledge is ideally represented as humancomprehensible patterns from which endusers can gain intuitions and insights. Axisparallel hyperrectangles provide interpretable generalizations for multidimensional data points with numerical attributes. In this dissertation, we study the fundamental problem of rectanglebased discriminative data generalization in the context of several useful data mining applications: cluster description, rule learning, and Nearest Rectangle classification. Clustering is one of the most important data mining tasks. However, most clustering methods output sets of points as clusters and do not generalize them into interpretable patterns. We perform a systematic study of cluster description, where we propose novel description formats leading to enhanced expressive power and introduce novel description problems specifying different tradeoffs between interpretability and accuracy. We also present efficient heuristic algorithms for the introduced problems in the proposed formats. Ifthen rules are
An Approximation Algorithm for Minimum Convex Cover with Logarithmic Performance Guarantee
, 2001
"... The problem Minimum Convex Cover of covering a given polygon with a minimum number of (possibly overlapping) convex polygons is known to be NPhard, even for polygons without holes [3]. We propose a polynomialtime approximation algorithm for this problem for polygons with or without holes that ..."
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Cited by 5 (1 self)
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The problem Minimum Convex Cover of covering a given polygon with a minimum number of (possibly overlapping) convex polygons is known to be NPhard, even for polygons without holes [3]. We propose a polynomialtime approximation algorithm for this problem for polygons with or without holes that achieves an approximation ratio of O(log n), where n is the number of vertices in the input polygon. To obtain this result, we first show that an optimum solution of a restricted version of this problem, where the vertices of the convex polygons may only lie on a certain grid, contains at most three times as many convex polygons as the optimum solution of the unrestricted problem. As a second step, we use dynamic programming to obtain a convex polygon which is maximum with respect to the number of "basic triangles" that are not yet covered by another convex polygon. We obtain a solution that is at most a logarithmic factor o# the optimum by iteratively applying our dynamic programming algorithm. Furthermore, we show that Minimum Convex Cover is APXhard, i.e., there exists a constant #>0 such that no polynomialtime algorithm can achieve an approximation ratio of 1 + #. We obtain this result by analyzing and slightly modifying an already existing reduction [3].