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104
Intrinsic Dimension Estimation Using Packing Numbers
, 2003
"... We propose a new algorithm to estimate the intrinsic dimension of data sets. The method is based on geometric properties of the data and requires neither parametric assumptions on the data generating model nor input parameters to set. The method is compared to a similar, widelyused algorithm from t ..."
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Cited by 68 (0 self)
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We propose a new algorithm to estimate the intrinsic dimension of data sets. The method is based on geometric properties of the data and requires neither parametric assumptions on the data generating model nor input parameters to set. The method is compared to a similar, widelyused algorithm from the same family of geometric techniques. Experiments show that our method is more robust in terms of the data generating distribution and more reliable in the presence of noise.
The Distance2 Matching Problem and its Relationship to the MACLayer Capacity of Ad Hoc Wireless Networks
 IEEE Journal on Selected Areas in Communications
, 2004
"... Abstract—We consider the problem of determining the maximum capacity of the media access (MAC) layer in wireless ad hoc networks. Due to spatial contention for the shared wireless medium, not all nodes can concurrently transmit packets to each other in these networks. The maximum number of possible ..."
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Cited by 61 (6 self)
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Abstract—We consider the problem of determining the maximum capacity of the media access (MAC) layer in wireless ad hoc networks. Due to spatial contention for the shared wireless medium, not all nodes can concurrently transmit packets to each other in these networks. The maximum number of possible concurrent transmissions is, therefore, an estimate of the maximum network capacity, and depends on the MAC protocol being used. We show that for a large class of MAC protocols based on virtual carrier sensing using RTS/CTS messages, which includes the popular IEEE 802.11 standard, this problem may be modeled as a maximum Distance2 matching (D2EMIS) in the underlying wireless network: Given a graph @ A, find a set of edges such that no two edges in are connected by another edge in. D2EMIS is NPcomplete. Our primary goal is to show that it
Maximizing Capacity in Arbitrary Wireless Networks in the SINR Model: Complexity and Game Theory
"... Abstract—In this paper we consider the problem of maximizing the number of supported connections in arbitrary wireless networks where a transmission is supported if and only if the signaltointerferenceplusnoise ratio at the receiver is greater than some threshold. The aim is to choose transmissi ..."
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Cited by 54 (3 self)
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Abstract—In this paper we consider the problem of maximizing the number of supported connections in arbitrary wireless networks where a transmission is supported if and only if the signaltointerferenceplusnoise ratio at the receiver is greater than some threshold. The aim is to choose transmission powers for each connection so as to maximize the number of connections for which this threshold is met. We believe that analyzing this problem is important both in its own right and also because it arises as a subproblem in many other areas of wireless networking. We study both the complexity of the problem and also present some game theoretic results regarding capacity that is achieved by completely distributed algorithms. We also feel that this problem is intriguing since it involves both continuous aspects (i.e. choosing the transmission powers) as well as discrete aspects (i.e. which connections should be supported).
PolynomialTime Approximation Schemes for Packing and Piercing Fat Objects
 JOURNAL OF ALGORITHMS
, 2001
"... We consider two problems: given a collection of n fat objects in a xed dimension, 1. (packing) nd the maximum subcollection of pairwise disjoint objects, and 2. (piercing) nd the minimum point set that intersects every object. Recently, Erlebach, Jansen, and Seidel gave a polynomialtime approxim ..."
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Cited by 52 (6 self)
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We consider two problems: given a collection of n fat objects in a xed dimension, 1. (packing) nd the maximum subcollection of pairwise disjoint objects, and 2. (piercing) nd the minimum point set that intersects every object. Recently, Erlebach, Jansen, and Seidel gave a polynomialtime approximation scheme (PTAS) for the packing problem, based on a shifted hierarchical subdivision method. Using shifted quadtrees, we describe a similar algorithm for packing but with a smaller time bound. Erlebach et al.'s algorithm requires polynomial space. We describe a dierent algorithm, based on geometric separators, that requires only linear space. This algorithm can also be applied to piercing, yielding the rst PTAS for that problem. Abbreviated title. Packing and Piercing Fat Objects.
Local approximation schemes for ad hoc and sensor networks
 In Proc. 3rd Joint Workshop on Foundations of Mobile Computing (DialMPOMC
, 2005
"... We present two local approaches that yield polynomialtime approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a (1 + ε)approximation to the problems at hand for any given ε> 0. ..."
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Cited by 39 (9 self)
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We present two local approaches that yield polynomialtime approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a (1 + ε)approximation to the problems at hand for any given ε> 0. The time complexity of both algorithms is O(TMIS + log ∗n/εO(1)), where TMIS is the time required to compute a maximal independent set in the graph, and n denotes the number of nodes. We then extend these results to a more general class of graphs in which the maximum number of pairwise independent nodes in every rneighborhood is at most polynomial in r. Such graphs of polynomially bounded growth are introduced as a more realistic model for wireless networks and they generalize existing models, such as unit disk graphs or coverage area graphs.
Strategyproof Auctions for Balancing Social Welfare and Fairness
 in Secondary Spectrum Markets,” in Proc. IEEE INFOCOM 2011, April 2011. et al.: DESIGNING TWODIMENSIONAL SPECTRUM AUCTIONS FOR MOBILE SECONDARY USERS 613
"... Abstract—Secondary spectrum access is emerging as a promising approach for mitigating the spectrum scarcity in wireless networks. Coordinated spectrum access for secondary users can be achieved using periodic spectrum auctions. Recent studies on such auction design mostly neglect the repeating natur ..."
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Cited by 32 (10 self)
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Abstract—Secondary spectrum access is emerging as a promising approach for mitigating the spectrum scarcity in wireless networks. Coordinated spectrum access for secondary users can be achieved using periodic spectrum auctions. Recent studies on such auction design mostly neglect the repeating nature of such auctions, and focus on greedily maximizing social welfare. Such auctions can cause subsets of users to experience starvation in the long run, reducing their incentive to continue participating in the auction. It is desirable to increase the diversity of users allocated spectrum in each auction round, so that a tradeoff between social welfare and fairness is maintained. We study truthful mechanisms towards this objective, for both local and global fairness criteria. For local fairness, we introduce randomization into the auction design, such that each user is guaranteed a minimum probability of being assigned spectrum. Computing an optimal, interferencefree spectrum allocation is NPHard; we present an approximate solution, and tailor a payment scheme to guarantee truthful bidding is a dominant strategy for all secondary users. For global fairness, we adopt the classic maxmin fairness criterion. We tailor another auction by applying linear programming techniques for striking the balance between social welfare and maxmin fairness, and for finding feasible channel allocations. In particular, a pair of primal and dual linear programs are utilized to guide the probabilistic selection of feasible allocations towards a desired tradeoff in expectation. I.
Geometric Separation and Exact Solutions for the Parameterized Independent Set Problem on Disk Graphs
, 2002
"... We consider the parameterized problem, whether for a given set D of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k nonintersecting disks. We expose an algorithm running in time n , that isto our knowledgethe rst algorithm for this problem with running t ..."
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Cited by 27 (2 self)
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We consider the parameterized problem, whether for a given set D of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k nonintersecting disks. We expose an algorithm running in time n , that isto our knowledgethe rst algorithm for this problem with running time bounded by an exponential with a sublinear exponent. For precision disk graphs of bounded radius ratio, we show that the problem is xed parameter tractable with respect to parameter k.
Minimumcost coverage of point sets by disks
"... We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given bytheir centers (t j) and radii (r j) that cover a given set of demand points Y ae R2 at the smallest possible cost. We consider costfunctions of the form a* j f (r j), where f (r) ..."
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Cited by 27 (5 self)
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We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given bytheir centers (t j) and radii (r j) that cover a given set of demand points Y ae R2 at the smallest possible cost. We consider costfunctions of the form a* j f (r j), where f (r) = ra is the cost of transmission to radius r. Special cases arise for a = 1 (sum ofradii) and a = 2 (total area); power consumption models in wireless network design often use an exponent a> 2. Different scenarios arise according to possible restrictions on the transmission centers t j, which may be constrained to belong to a givendiscrete set or to lie on a line, etc. We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points t j on agiven line in order to cover demand points Y ae R2; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y; (c) a proof of NPhardness for a discrete set oftransmission points in R2 and any fixed a> 1; and (d) a polynomialtime approximation scheme for the problem of computinga minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of
Simple Heuristics and PTASs for Intersection Graphs in Wireless Ad Hoc Networks
 in Wireless Ad Hoc Networks, in DialM’02
, 2002
"... In wireless ad hoc networks, each wireless device has a transmission range, which is usually modeled as a disk centered at this node. A wireless node can send message directly to all nodes lying inside this disk. We present several intersection graphs to model the wireless networks. Then we present ..."
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Cited by 25 (10 self)
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In wireless ad hoc networks, each wireless device has a transmission range, which is usually modeled as a disk centered at this node. A wireless node can send message directly to all nodes lying inside this disk. We present several intersection graphs to model the wireless networks. Then we present some simple heuristics and/or PTASs to approximate the maximum independent set, the minimum vertex cover and the minimum graph coloring in these graph models.
Maximum Independent Set of Rectangles
"... We study the Maximum Independent Set of Rectangles (MISR) problem: given a collection R of n axisparallel rectangles, find a maximumcardinality subset of disjoint rectangles. MISR is a special case of the classical Maximum Independent Set problem, where the input is restricted to intersection grap ..."
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Cited by 25 (0 self)
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We study the Maximum Independent Set of Rectangles (MISR) problem: given a collection R of n axisparallel rectangles, find a maximumcardinality subset of disjoint rectangles. MISR is a special case of the classical Maximum Independent Set problem, where the input is restricted to intersection graphs of axisparallel rectangles. Due to its many applications, ranging from map labeling to data mining, MISR has received a significant amount of attention from various research communities. Since the problem is NPhard, the main focus has been on the design of approximation algorithms. Several groups of researches have independently suggested O(log n)approximation algorithms for MISR, and this remained the best currently known approximation factor for the problem. The main result of our paper is an O(log log n)approximation algorithm for MISR. Our algorithm combines existing approaches for solving special cases of the problem, in which the input set of rectangles is restricted to containing specific intersection types, with new insights into the combinatorial structure of sets of intersecting rectangles in the plane. We also consider a generalization of MISR to higher dimensions, where rectangles are replaced by ddimensional hyperrectangles. Our results for MISR imply an O((log n) d−2 log log n)approximation algorithm for this problem, improving upon the best previously known O((log n) d−1)approximation.