Author: Danupon Nanongkai, Atish Das Sarma, Gopal Pandurangan
(Author names are NOT in alphabetical order. )

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Conference: Submitted

Journal: -

Abstract:

We consider the problem of performing a random walk in a distributed network. Given bandwidth constraints, the goal of the problem is to minimize the number of rounds required to obtain a random walk sample. Das Sarma et al. [PODC'10] show that a random walk of length $\ell$ on a network of diameter $D$ can be performed in $\tilde O(\sqrt{\ell D}+D)$ time. A major question left open is whether there exists a faster algorithm, especially whether the multiplication of $\sqrt{\ell}$ and $\sqrt{D}$ is necessary.
In this paper, we show a tight unconditional lower bound on the time complexity of distributed random walk computation. Specifically, we show that for any $n$, $D$, and $D\leq \ell \leq (n/(D^3\log n))^{1/4}$, performing a random walk of length $\Theta(\ell)$ on an $n$-node network of diameter $D$ requires $\Omega(\sqrt{\ell D}+D)$ time. This bound is {\em unconditional}, i.e., it holds for any (possibly randomized) algorithm. To the best of our knowledge, this is the first lower bound that the diameter plays a role of multiplicative factor. Our bound shows that the algorithm of Das Sarma et al. is time optimal.
Our proof technique introduces a new connection between {\em bounded-round} communication complexity and distributed algorithm lower bounds with $D$ as a trade-off parameter, strengthening the previous study by Das Sarma et al. [STOC'11]. In particular, we make use of the bounded-round communication complexity of the pointer chasing problem. Our technique can be of independent interest and may be useful in showing non-trivial lower bounds on the complexity of other fundamental distributed computing problems.

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6.10.2010: New link to the pdf posted. PPTX posted.

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Conference: ICDE 2011: the IEEE International Conference on Data Engineering [link]

Abstract:

The study of skylines and their variants has receivedconsiderable attention in recent years. Skylines are essentiallysets of most interesting (undominated) tuples in a database.However, since the skyline is often very large, much researcheffort has been devoted to identifying a smaller subset of (sayk) “representative skyline” points. Several different definitionsof representative skylines have been considered. Most of theseformulations are intuitive in that they try to achieve some kindof clustering “spread” over the entire skyline, with k points. Inthis work, we take a more principled approach in defining therepresentative skyline objective. One of our main contributionsis to formulate the problem of displaying k representative skylinepoints such that the probability that a random user would clickon one of them is maximized.

Two major research questions arise naturally from this formu-lation. First, how does one mathematically model the likelihoodwith which a user is interested in and will “click” on a certaintuple? Second, how does one negotiate the absence of theknowledge of an explicit set of target users; in particular whatdo we mean by “a random user”? To answer the first question,we model users based on a novel formulation of thresholdpreferences which we will motivate further in the paper. Toanswer the second question, we assume a probability distributionof users instead of a fixed set of users. While this makes theproblem harder, it lends more mathematical structures that canbe exploited as well, as one can now work with probabilities ofthresholds and handle cumulative density functions.

On the theoretical front, our objective is NP-hard. For thecase of a finite set of users with known thresholds, we presenta simple greedy algorithm that attains an approximation ratio of of the optimal. For the case of user distributions,we show that a careful yet similar greedy algorithm achieves thesame approximation ratio. Unfortunately, it turns out that thisalgorithm is rather involved and computationally expensive. Sowe present a threshold sampling based algorithm that is morecomputationally affordable and, for any fixed , has anapproximation ratio of . We perform experimentson both real and synthetic data to show that our algorithmsignificantly outperforms previously proposed approaches.

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[v1] November 14, 2010 (Conference version)

Author: Atish Das Sarma, Stephan Holzer, Liah Kor, Amos Korman, Danupon Nanongkai, Gopal Pandurangan, David Peleg, Roger Wattenhofer

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Conference: STOC 2011

Journal: -

Abstract:

We study the verification problem in distributed networks, stated as follows. Let be a subgraph of a network where each vertex of knows which edges incident on it are in . We would like to verify whether has some properties, e.g., if it is a tree or if it is connected. We would like to perform this verification in a decentralized fashion via a distributed algorithm. The time complexity of verification is measured as the number of rounds of distributed communication.
In this paper we initiate a systematic study of distributed verification, and give almost tight lower bounds on the running time of distributed verification algorithms for many fundamental problems such as connectivity, spanning connected subgraph, and cut verification. We then show applications of these results in deriving strong unconditional time lower bounds on the hardness of distributed approximation for many classical optimization problems including minimum spanning tree, shortest paths, and minimum cut. Many of these results are the first non-trivial lower bounds for both exact and approximate distributed computation and they resolve previous open questions. Moreover, our unconditional lower bound of approximating minimum spanning tree (MST) subsumes and improves upon the previous hardness of approximation bound of Elkin [STOC 2004] as well as the lower bound for (exact) MST computation of Peleg and Rubinovich [FOCS 1999]. Our result implies that there can be no distributed approximation algorithm for MST that is significantly faster than the current exact algorithm, for any approximation factor.
Our lower bound proofs show an interesting connection between communication complexity and distributed computing which turns out to be useful in establishing the time complexity of exact and approximate distributed computation of many problems.

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Conference: WINE 2010: 6th Workshop on Internet & Network Economics [wiki].  Published in LNCS Vol. 6484.

Abstract:

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Conference: VLDB 2010: 36th International Conference on Very Large Databases [wiki]

Abstract:

We propose the k-representative regret minimization query (k-regret) as an operation to support multi-criteria decision making. Like top-k, the k-regret query assumes that users have some utility or scoring functions; however, it never asks the users to provide such functions. Like skyline, it filters out a set of interesting points from a potentially large database based on the users’ criteria; however, it never overwhelms the users by outputting too many tuples.

In particular, for any number k and any class of utility functions, the k-regret query outputs k tuples from the database and tries to minimize the {\em maximum regret ratio}. This captures how disappointed a user could be had she seen k-representative tuples instead of the whole database. We focus on the class of linear utility functions, which is widely applicable.

The first challenge of this approach is that it is not clear if the maximum regret ratio can be small, or even bounded. We answer this question affirmatively. Theoretically, we prove that the maximum regret ratio can be bounded and this bound is independent of the database size. Moreover, our extensive experiments on real and synthetic datasets suggest that in practice the maximum regret ratio is reasonably small. Additionally, algorithms developed in this paper are practical as they run in linear time in the size of the database and the experiments show that their running time is small when they run on top of the skyline operation which means that these algorithm could be integrated into current database systems.

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[v1] June 28, 2010 (Conference version)

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Conference: Submitted

Abstract:

We  focus on  the problem of performing random walks efficiently in a distributed network. Given bandwidth constraints, the goal is to minimize the number of rounds required to obtain a random walk sample. We first present a fast sublinear time distributed algorithm for performing random walks whose time complexity is sublinear in the length of the walk. Our algorithm performs a random walk of length  in  rounds (with high probability) on an undirected  network, where is the diameter of the network. This improves over the previous best algorithm that ran in rounds (Das Sarma et al., PODC 2009). We further extend our algorithms to efficiently perform independent random walks in   rounds. We then show that there is a fundamental difficulty in improving the dependence on any further by proving a lower bound of under a general model of distributed random walk algorithms. Our random walk algorithms are useful in speeding up distributed algorithms for a variety of applications that use random walks as a subroutine. We present two main applications. First, we give a fast distributed algorithm for computing a random spanning tree (RST) in an arbitrary (undirected) network which runs in rounds (with high probability; here is the number of edges). Our second application is a fast decentralized algorithm for estimating mixing time and related parameters of the underlying network. Our algorithm is fully decentralized and can serve as a building block in the design of topologically-aware networks.

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Mar 03, 2009 (New version posted on ArXiv)
Nov 06, 2009 (Link to arXiv posted)
Feb 18, 2009 (New version posted)

Download: pdf, arXiv

Conference: -

Abstract:

Stackelberg Pricing Games is a two-level combinatorial pricing problem studied in the Economics, Operation Research, and Computer Science communities. In this paper, we consider the decade-old shortest path version of this problem which is the first and most studied problem in this family. The game is played on a graph (representing a network) consisting of fixed cost edges and pricable or variable cost edges. The fixed cost edges already have some fixed price (representing the competitor’s prices). Our task is to choose prices for the variable cost edges. After that, a client will buy the cheapest path from a node to a node , using any combination of fixed cost and variable cost edges. The goal is to maximize the revenue on variable cost edges.

In this paper, we show that the problem is hard to approximate within , improving the previous APX-hardness result by Joret [to appear in Networks]. Our technique combines the existing ideas with a new insight into the price structure and its relation to the hardness of the instances.

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[v1] Oct 2, 2009 (Manuscript posted on arXiv)

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Conference: ICALP 2010

Abstract:

We consider the problem of finding semi-matching in bipartite graphs, a problem also extensively studied under various names in the scheduling literature. We give faster algorithms for both weighted and unweighted case.

For the weighted case, we give an -time algorithm, where is the number of vertices and is the number of edges, by exploiting geometric structure of the problem. This improves the classical algorithms by Horn [Operations Research 1973] and Brono, Coffman and Sethi [Communications of the ACM 1974].

For the unweighted case, the bound could be improved even further. We give a simple divide-and-conquer algorithm which runs in time , improving two previous -time algorithms by Abraham [MSc thesis, University of Glasgow 2003] and Harvey, Ladner, Lovasz and Tamir [WADS 2003 and Journal of Algorithms 2006]. We also extend this algorithm to solve the Balance Edge Cover problem in time , improving the previous -time algorithm by Harada, Ono, Sadakane and Yamashita [ISAAC 2008].

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[v1] Feb. 10, 2010 (Paper posted)

Author (ordered alphabetically): Atish Das Sarma, Ashwin Lall, Danupon Nanongkai, Jun Xu

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Journal: Soon

Conference: VLDB 2009: 35th International Conference on Very Large Databases [wiki]

Abstract:

We consider external algorithms for skyline computation without pre-processing. Our goal is to develop an algorithm with a good worst case guarantee while performing well on average. Due to the nature of disks, it is desirable that such algorithms access the input as a stream (even if in multiple passes). Using the tools of randomness, proved to be useful in many applications, we present an efficient multi-pass streaming algorithm, RAND, for skyline computation. As far as we are aware, RAND is the first randomized skyline algorithm in the literature.

RAND is near-optimal for the streaming model, which we prove via a simple lower bound. Additionally, our algorithm is distributable and can handle partially ordered domains on each attribute. Finally, we demonstrate the robustness of RAND via extensive experiments on both real and synthetic datasets. RAND is comparable to the existing algorithms in average case and additionally tolerant to simple modifications of the data, while other algorithms degrade considerably with such variation.

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[v1] August 20, 2009 (Conference version)

Authors:

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Journal: -

Conference: PODC 2009: 28th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing [wiki]

Abstract:

Performing random walks in networks is a fundamental primitive that has found applications in many areas of computer science, including distributed computing. In this paper, we focus on the problem of performing random walks efficiently in a distributed network. Given bandwidth constraints, the goal is to minimize the number of rounds required to obtain a random walk sample.

All previous algorithms that compute a random walk sample of length as a subroutine always do so naively, i.e., in rounds. The main contribution of this paper is a fast distributed
algorithm for performing random walks. We show that  a random walk sample of length can be computed in rounds on an undirected unweighted network, where is the diameter of the network. ( hides factors where is the number of nodes in the network and is the minimum degree.) For small diameter graphs, this is a significant improvement over the naive bound. We also show that our algorithm  can be applied to speedup the more general Metropolis-Hastings sampling.

We extend our algorithms to perform a large number, , of random walks efficiently. We show how destinations can be sampled in rounds if and rounds otherwise. We  also present faster algorithms for performing random walks of length larger than (or equal to) the mixing time of the underlying graph. Our techniques can be useful in speeding up distributed algorithms for a variety of applications that use random walks as a subroutine.

Keywords: Random walks, Random sampling, Distributed algorithm, Metropolis-Hastings sampling.

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[v1] May 31, 2009 (Conference version)