Probabilistic approximation of metric spaces and its algorithmic applications

Bartal, Yair

doi:10.1109/sfcs.1996.548477

Cited by 468 publications

(625 citation statements)

References 29 publications

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“…More specifically, we deduce that any n-point metric space with spread 1 ∆ can be embedded on-line into p with distortion O((log ∆) 1/p log n). Similarly, we also obtain an analog of a theorem due to Bartal [2] for embedding into ultrametrics. More precisely, we give an on-line probabilistic embedding of an input metric into a distribution over ultrametrics with distortion O(log n · log ∆).…”

Section: Results and Motivationmentioning

confidence: 81%

“…We observe that Bartal's embedding [2] can be easily modified to work in the on-line setting. We remark that this observation was also made independently by Englert, Räcke, and Westermann [6].…”

Section: Results and Motivationmentioning

confidence: 99%

“…We observe that Bartal's algorithm [2] can be interpreted as an on-line algorithm for constructing probabilistic partitions. The input to the problem is a metric M = (X, D), and a parameter r. In the first step of Bartal's algorithm, every edge of length less than r/n is contracted.…”

Section: Embedding General Metrics Into Ultrametrics and Into Pmentioning

confidence: 98%

“…We also give an on-line probabilistic embedding into a distribution over ultrametrics with distortion O(log n · log ∆). Both algorithms are on-line versions of the algorithm of Bartal [2], for embedding metrics into a distribution of dominating HSTs, with distortion O(log 2 n). Before we describe the algorithm we need to introduce some notation.…”

Section: Embedding General Metrics Into Ultrametrics and Into Pmentioning

confidence: 99%

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Online Embeddings

Indyk

Magen

Sidiropoulos

et al. 2010

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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We initiate the study of on-line metric embeddings. In such an embedding we are given a sequence of n points X = x 1 , . . . , x n one by one, from a metric space M = (X, D). Our goal is to compute a low-distortion embedding of M into some host space, which has to be constructed in an on-line fashion, so that the image of each x i depends only on x 1 , . . . , x i . We prove several results translating existing embeddings to the on-line setting, for the case of embedding into p spaces, and into distributions over ultrametrics.

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Section: Results and Motivationmentioning

confidence: 81%

Section: Results and Motivationmentioning

confidence: 99%

Section: Embedding General Metrics Into Ultrametrics and Into Pmentioning

confidence: 98%

Section: Embedding General Metrics Into Ultrametrics and Into Pmentioning

confidence: 99%

See 2 more Smart Citations

Online Embeddings

Indyk

Magen

Sidiropoulos

et al. 2010

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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“…To show this lower bound, we need to specify an instance of the problem such that no algorithm can come with in a factor of Ω log N log log N of the optimal offline cost, where N is the total number of customers. The proof is by construction, and we start by building a hierarchically well-separated binary tree (Bartal [6]) of depth h (Figure 1), where h will be specified later. The root node will be level 0, its children will be level 1, and so on.…”

Section: Lower Bound For Facility Location Problem With Online Custommentioning

confidence: 99%

From Cost Sharing Mechanisms to Online Selection Problems

Elmachtoub

Levi

2015

Mathematics of OR

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We consider a general class of online optimization problems, called online selection problems, where customers arrive sequentially, and one has to decide upon arrival whether to accept or reject each customer. If a customer is rejected, then a rejection cost is incurred. The accepted customers are served with minimum possible cost, either online or after all customers have arrived. The goal is to minimize the total production costs for the accepted customers plus the rejection costs for the rejected customers. These selection problems are related to online variants of offline prize collecting combinatorial optimization problems that have been widely studied in the computer science literature. In this paper, we provide a general framework to develop online algorithms for this class of selection problems. In essence the algorithmic framework leverages any cost sharing mechanism with certain properties into a poly-logarithmic competitive online algorithm for the respective problem; the competitive ratios are shown to be near-optimal. We believe that the general and transparent connection we establish between cost sharing mechanisms and online algorithms could lead to additional online algorithms for problems beyond the ones studied in this paper.

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A polynomial algorithm for thep-centdian problem on a tree

1998

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The most common problems studied in network location theory are the p-median and the p-center models. The p-median problem on a network is concerned with the location of p points (medians) on the network, such that the total (weighted) distance of all the nodes to their respective nearest points is minimized. The p center problem is concerned with the location of p-points (centers) on the network, such that the maximum (weighted) distance of all the nodes to their respective nearest points is minimized.To capture more real world problems, and obtain a good way to trade-off minisum (efficiency) and minimax (equity) approaches, Halpern [5,6,7], introduced the centdian model, where the objective is to minimize a convex combination of the objective functions of the center and the median problems. In this paper we study the p-centdian problem on tree networks, and present the first polynomial time algorithm for this problem.

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Probabilistic approximation of metric spaces and its algorithmic applications

Cited by 468 publications

References 29 publications

Online Embeddings

Online Embeddings

From Cost Sharing Mechanisms to Online Selection Problems

A polynomial algorithm for thep-centdian problem on a tree

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