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11
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to ..."
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Cited by 152 (13 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
Adwords and generalized online matching
 In FOCS ’05: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competit ..."
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Cited by 98 (5 self)
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How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1 − 1/e for this problem. 1
An O(log 2 k)competitive Algorithm for Metric Bipartite Matching
"... Abstract. We consider the online metric matching problem. In this problem, we are given a graph with edge weights satisfying the triangle inequality, and k vertices that are designated as the right side of the matching. Over time up to k requests arrive at an arbitrary subset of vertices in the grap ..."
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Cited by 4 (0 self)
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Abstract. We consider the online metric matching problem. In this problem, we are given a graph with edge weights satisfying the triangle inequality, and k vertices that are designated as the right side of the matching. Over time up to k requests arrive at an arbitrary subset of vertices in the graph and each vertex must be matched to a right side vertex immediately upon arrival. A vertex cannot be rematched to another vertex once it is matched. The goal is to minimize the total weight of the matching. We give a O(log 2 k) competitive randomized algorithm for the problem. This improves upon the best known guarantee of O(log 3 k) due to Meyerson, Nanavati and Poplawski [19]. It is well known that no deterministic algorithm can have a competitive less than 2k − 1, and that no randomized algorithm can have a competitive ratio of less than ln k. 1
The Online Matching Problem on a Line
, 2003
"... We study the online matching problem when the metric space is a single straight line. For this case, the offline matching problem is trivial but the online problem has been open and the best known competitive ratio was the trivial Θ(n) where n is the number of requests. It was conjectured that the g ..."
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Cited by 2 (0 self)
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We study the online matching problem when the metric space is a single straight line. For this case, the offline matching problem is trivial but the online problem has been open and the best known competitive ratio was the trivial Θ(n) where n is the number of requests. It was conjectured that the generalized Work Function Algorithm has constant competitive ratio for this problem. We show that it is in fact Ω(log n) and O(n), and make some progress towards proving a better upper bound by establishing some structural properties of the solutions. Our technique for the upper bound doesn’t use a potential function but it reallocates the online cost in a way that the comparison with the offline cost becomes more direct.
Simple Competitive Request Scheduling Strategies
 in 11th ACM Symposium on Parallel Architectures and Algorithms
, 1999
"... In this paper we study the problem of scheduling realtime requests in distributed data servers. We assume the time to be divided into time steps of equal length called rounds. During every round a set of requests arrives at the system, and every resource is able to fulfill one request per round. Ev ..."
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Cited by 2 (0 self)
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In this paper we study the problem of scheduling realtime requests in distributed data servers. We assume the time to be divided into time steps of equal length called rounds. During every round a set of requests arrives at the system, and every resource is able to fulfill one request per round. Every request specifies two (distinct) resources and requires to get access to one of them. Furthermore, every request has a deadline of d, i.e. a request that arrives in round t has to be fulfilled during round t +d 1 at the latest. The number of requests which arrive during some round and the two alternative resources of every request are selected by an adversary. The goal is to maximize the number of requests that are fulfilled before their deadlines expire. We examine the scheduling problem in an online setting, i.e. new requests continuously arrive at the system, and we have to determine online an assignment of the requests to the resources in such a way that every resource has to fulfil...
Online Matching for Scheduling Problems
 In Proceedings of the 16th Symposium on Theoretical Aspects in Computer Science, LNCS 1563
, 1999
"... . In this work an alternative online variant of the matching problem in bipartite graphs is presented. It is motivated by a scheduling problem in an online environment. In such an environment, a task is unknown up to its disclosure. However, in that moment it is not necessary to take a decision on t ..."
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Cited by 2 (1 self)
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. In this work an alternative online variant of the matching problem in bipartite graphs is presented. It is motivated by a scheduling problem in an online environment. In such an environment, a task is unknown up to its disclosure. However, in that moment it is not necessary to take a decision on the service of that particular task. In reality, an online scheduler has to decide on how to use the current resources. Therefore, our problem is called online request server matching (ORSM). It differs substantially from the online bipartite matching problem of Karp et al. [KVV90]. Hence, the analysis of an optimal, deterministic online algorithm for the ORSM problem results in a smaller competitive ratio of 1:5. We also introduce an extension to a weighted bipartite matching problem. A lower bound of p 5+1 2 1:618 and an upper bound of 2 is given for the competitive ratio. 1 Introduction Motivation. The problem, which is investigated here, is motivated by online schedulin...
The Online Transportation Problem: On the Exponential Boost of One Extra
"... We present a polylogcompetitive deterministic online algorithm for the online transportation problem on hierarchically separated trees when the online algorithm has one extra server per site. Using metric embedding results in the literature, one can then obtain a polylogcompetitive randomized on ..."
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Cited by 1 (0 self)
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We present a polylogcompetitive deterministic online algorithm for the online transportation problem on hierarchically separated trees when the online algorithm has one extra server per site. Using metric embedding results in the literature, one can then obtain a polylogcompetitive randomized online algorithm for the online transportation on an arbitrary metric space when the online algorithm has one extra server per site. 1
Asymptotically Optimal Algorithm for Stochastic Adwords
, 2012
"... In this paper we consider the adwords problem in the unknown distribution model. We consider the case where the budget to bid ratio k is at least 2, and give improved competitive ratios. Earlier results had competitive ratios better than 1 − 1/e only for “large enough ” k, while our competitive rati ..."
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Cited by 1 (0 self)
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In this paper we consider the adwords problem in the unknown distribution model. We consider the case where the budget to bid ratio k is at least 2, and give improved competitive ratios. Earlier results had competitive ratios better than 1 − 1/e only for “large enough ” k, while our competitive ratio increases continuously with k. For k = 2 the competitive ratio we get is 0.729 and it is 0.9 for k = 16. We also improve the asymptotic competitive ratio for large k from 1 − O ( p log n/k) to 1 − O ( p 1/k), thus removing any dependence on n, the number of advertisers. This ratio is optimal, even with known distributions. That is, even if an algorithm is tailored to the distribution, it cannot get a competitive ratio of 1 − o ( p 1/k), whereas our algorithm does not depend on the distribution. The algorithm is rather simple, it computes a score for every advertiser based on his original budget, the remaining budget and the remaining number of steps in the algorithm and assigns a query to the advertiser with the highest bid plus his score. The analysis is based on a “hybrid argument ” that considers algorithms that are part actual, part hypothetical, to prove that our (actual) algorithm is better than a completely hypothetical algorithm whose performance is easy to analyze.
General
"... How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive an optimal algorithm achieving a competit ..."
Abstract
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How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive an optimal algorithm achieving a competitive ratio of 1 − 1/e for this problem.
Published In An O(log2 k)competitive Algorithm for Metric Bipartite Matching
"... Abstract. We consider the online metric matching problem. In this problem, we are given a graph with edge weights satisfying the triangle inequality, and k vertices that are designated as the right side of the matching. Over time up to k requests arrive at an arbitrary subset of vertices in the grap ..."
Abstract
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Abstract. We consider the online metric matching problem. In this problem, we are given a graph with edge weights satisfying the triangle inequality, and k vertices that are designated as the right side of the matching. Over time up to k requests arrive at an arbitrary subset of vertices in the graph and each vertex must be matched to a right side vertex immediately upon arrival. A vertex cannot be rematched to another vertex once it is matched. The goal is to minimize the total weight of the matching. We give a O(log2 k) competitive randomized algorithm for the problem. This improves upon the best known guarantee ofO(log3 k) due to Meyerson, Nanavati and Poplawski [19]. It is well known that no deterministic algorithm can have a competitive less than 2k − 1, and that no randomized algorithm can have a competitive ratio of less than ln k. 1