Online Algorithms for Multilevel Aggregation

Bieńkowski, Marcin; Böhm, Martin; Byrka, Jarosław; Chrobák, Marek; Dürr, Christoph; Folwarczný, Lukáš; Jeż, Łukasz; Sgall, Jiřį́; Thắng, Nguyễn Kim; Veselý, Pavel

doi:10.1287/opre.2019.1847

Cited by 9 publications

(15 citation statements)

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“…1. An O(D 2 )-competitive deterministic algorithm for online multilevel aggregation with delay on a tree of depth D. This is an exponential improvement over the O(D 4 2 D )-competitive algorithm in [7].…”

Section: Our Resultsmentioning

confidence: 99%

General Framework for Metric Optimization Problems with Delay or with Deadlines

Azar

Touitou

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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In this paper, we present a framework used to construct and analyze algorithms for online optimization problems with deadlines or with delay over a metric space. Using this framework, we present algorithms for several di erent problems. We present an O(D 2 )-competitive deterministic algorithm for online multilevel aggregation with delay on a tree of depth D, an exponential improvement over the O(D 4 2 D )-competitive algorithm of Bienkowski et al. (ESA '16). We also present an O(log 2 n)competitive randomized algorithm for online service with delay over any general metric space of n points, improving upon the O(log 4 n)-competitive algorithm by Azar et al. (STOC '17).In addition, we present the problem of online facility location with deadlines. In this problem, requests arrive over time in a metric space, and need to be served until their deadlines by facilities that are opened momentarily for some cost. We also consider the problem of facility location with delay, in which the deadlines are replaced with arbitrary delay functions. For those problems, we present O(log 2 n)-competitive algorithms, with n the number of points in the metric space.The algorithmic framework we present includes techniques for the design of algorithms as well as techniques for their analysis. arXiv:1904.07131v1 [cs.DS] 15 Apr 20192. An O(log 2 n)-competitive randomized algorithm for online service with delay over a metric space with n points. This improves upon the O(log 4 n)-competitive randomized algorithm in [5].3. An O(log 2 n)-competitive randomized algorithm for online facility location with deadlines over a metric space with n points.4. An O(log 2 n)-competitive randomized algorithm for online facility location with delay over a metric space with n points.Our algorithms all share a common framework, which we present. The framework provides general structure to both the algorithm and its analysis.Such an improvement for the online multilevel aggregation problem is only known for the special case of deadlines, as given in [13].The algorithms for online facility location with deadlines and with delay can be easily extended to the case in which the cost of opening a facility is di erent for each point in the metric space. This changes the competitiveness of the algorithms to O(log 2 ∆ + log ∆ log n), where ∆ is the aspect ratio of the metric space. Our TechniquesAll of our algorithms are based on corresponding competitive algorithms for HSTs. The randomized algorithms for general metric spaces are obtained through randomized HST embedding. The O(D 2 )competitive deterministic algorithm for online multilevel aggregation with delay on a tree is based on decomposing the tree into a forest of HSTs. This decomposition is similar to that used in [13] for the case of deadlines.The framework -algorithm design. In designing algorithms for the problems over HSTs, we use a certain framework. In an algorithm designed using the framework, there is a counter for every node (in the case of facility location) or every edge (in the case of online m...

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Section: Our Resultsmentioning

confidence: 99%

General Framework for Metric Optimization Problems with Delay or with Deadlines

Azar

Touitou

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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“…There are many other problems that use this paradigm: most notably the ski-rental problem and its continuous counterpart, the spin-block problem [29], where a purchase decision can be delayed until renting cost becomes sufficiently large. Such rent-or-buy (wait-or-act) trade-offs are also found in other areas, for example in aggregating messages in computer networks [1,11,21,28,31,39], in aggregating orders in supply-chain management [9,10,14,15,17,18] or in some scheduling variants [6].…”

Section: Related Workmentioning

confidence: 85%

A Match in Time Saves Nine: Deterministic Online Matching with Delays

Bieńkowski

Kraska

Schmidt

2018

Approximation and Online Algorithms

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We consider the problem of online Min-cost Perfect Matching with Delays (MPMD) introduced by Emek et al. (STOC 2016). In this problem, an even number of requests appear in a metric space at different times and the goal of an online algorithm is to match them in pairs. In contrast to traditional online matching problems, in MPMD all requests appear online and an algorithm can match any pair of requests, but such decision may be delayed (e.g., to find a better match). The cost is the sum of matching distances and the introduced delays.We present the first deterministic online algorithm for this problem. Its competitive ratio is O(m log 2 5.5 ) = O(m 2.46 ), where 2m is the number of requests. This is polynomial in the number of metric space points if all requests are given at different points. In particular, the bound does not depend on other parameters of the metric, such as its aspect ratio. Unlike previous (randomized) solutions for the MPMD problem, our algorithm does not need to know the metric space in advance. time τ , an online algorithm may decide to match any pair of requests (p i , t i ) and (p j , t j ) that have already arrived (τ ≥ t i and τ ≥ t j ) and have not been matched yet. The cost incurred by such matching edge is dist(p i , p j ) + (τ − t i ) + (τ − t j ), i.e., is the sum of the connection cost and the waiting costs of these two requests.The goal is to eventually match all requests and minimize the total cost. We use a typical yardstick to measure the performance: a competitive ratio [13], defined as the maximum, over all inputs, of the ratios between the cost of an online algorithm and the cost of an optimal offline solution Opt that knows the entire input sequence in advance.

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“…The first competitive solution for arbitrary trees of depth D was given by Bienkowski et al [5,6] [8,9]. The best known lower bound is 3.618 [10], and for arbitrary trees the existence of an algorithm with competitive ratio independent of tree depth remains open.…”

Section: Previous Work On Online Algorithmsmentioning

confidence: 99%