arXiv subjects

Gopal Pandurangan

Publications and source records attributed to Gopal Pandurangan.

62 records · Page 4Linked to original sources

arXiv2012

Near-Optimal Random Walk Sampling in Distributed Networks

Performing random walks in networks is a fundamental primitive that has found numerous applications in communication networks such as token management, load balancing, network topology discovery and construction, search, and peer-to-peer membership management. While several such algorithms are ubiquitous, and use numerous random walk samples, the walks themselves have always been performed naively. In this paper, we focus on the problem of performing random walk sampling efficiently in a distributed network. Given bandwidth constraints, the goal is to minimize the number of rounds and messages required to obtain several random walk samples in a continuous online fashion. We present the first round and message optimal distributed algorithms that present a significant improvement on all previous approaches. The theoretical analysis and comprehensive experimental evaluation of our algorithms show that they perform very well in different types of networks of differing topologies. In particular, our results show how several random walks can be performed continuously (when source nodes are provided only at runtime, i.e., online), such that each walk of length $\ell$ can be performed exactly in just $\tilde{O}(\sqrt{\ell D})$ rounds, (where $D$ is the diameter of the network), and $O(\ell)$ messages. This significantly improves upon both, the naive technique that requires $O(\ell)$ rounds and $O(\ell)$ messages, and the sophisticated algorithm of [DasSarma et al. PODC 2010] that has the same round complexity as this paper but requires $Ω(m\sqrt{\ell})$ messages (where $m$ is the number of edges in the network). Our theoretical results are corroborated through extensive experiments on various topological data sets. Our algorithms are fully decentralized, lightweight, and easily implementable, and can serve as building blocks in the design of topologically-aware networks.

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arXiv2011

Efficient Computation of Distance Sketches in Distributed Networks

Distance computation is one of the most fundamental primitives used in communication networks. The cost of effectively and accurately computing pairwise network distances can become prohibitive in large-scale networks such as the Internet and Peer-to-Peer (P2P) networks. To negotiate the rising need for very efficient distance computation, approximation techniques for numerous variants of this question have recently received significant attention in the literature. The goal is to preprocess the graph and store a small amount of information such that whenever a query for any pairwise distance is issued, the distance can be well approximated (i.e., with small stretch) very quickly in an online fashion. Specifically, the pre-processing (usually) involves storing a small sketch with each node, such that at query time only the sketches of the concerned nodes need to be looked up to compute the approximate distance. In this paper, we present the first theoretical study of distance sketches derived from distance oracles in a distributed network. We first present a fast distributed algorithm for computing approximate distance sketches, based on a distributed implementation of the distance oracle scheme of [Thorup-Zwick, JACM 2005]. We also show how to modify this basic construction to achieve different tradeoffs between the number of pairs for which the distance estimate is accurate and other parameters. These tradeoffs can then be combined to give an efficient construction of small sketches with provable average-case as well as worst-case performance. Our algorithms use only small-sized messages and hence are suitable for bandwidth-constrained networks, and can be used in various networking applications such as topology discovery and construction, token management, load balancing, monitoring overlays, and several other problems in distributed algorithms.

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arXiv2011

Information Spreading in Dynamic Networks

We study the fundamental problem of information spreading (also known as gossip) in dynamic networks. In gossip, or more generally, $k$-gossip, there are $k$ pieces of information (or tokens) that are initially present in some nodes and the problem is to disseminate the $k$ tokens to all nodes. The goal is to accomplish the task in as few rounds of distributed computation as possible. The problem is especially challenging in dynamic networks where the network topology can change from round to round and can be controlled by an on-line adversary. The focus of this paper is on the power of token-forwarding algorithms, which do not manipulate tokens in any way other than storing and forwarding them. We first consider a worst-case adversarial model first studied by Kuhn, Lynch, and Oshman~\cite{kuhn+lo:dynamic} in which the communication links for each round are chosen by an adversary, and nodes do not know who their neighbors for the current round are before they broadcast their messages. Our main result is an $Ω(nk/\log n)$ lower bound on the number of rounds needed for any deterministic token-forwarding algorithm to solve $k$-gossip. This resolves an open problem raised in~\cite{kuhn+lo:dynamic}, improving their lower bound of $Ω(n \log k)$, and matching their upper bound of $O(nk)$ to within a logarithmic factor. We next show that token-forwarding algorithms can achieve subquadratic time in the offline version of the problem where the adversary has to commit all the topology changes in advance at the beginning of the computation, and present two polynomial-time offline token-forwarding algorithms. Our results are a step towards understanding the power and limitation of token-forwarding algorithms in dynamic networks.

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arXiv2011

Distributed Verification and Hardness of Distributed Approximation

We study the {\em verification} problem in distributed networks, stated as follows. Let $H$ be a subgraph of a network $G$ where each vertex of $G$ knows which edges incident on it are in $H$. We would like to verify whether $H$ 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 $s-t$ cut verification. We then show applications of these results in deriving strong unconditional time lower bounds on the {\em 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 {\em 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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arXiv2011

A Tight Lower Bound on Distributed Random Walk Computation

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 $Θ(\ell)$ on an $n$-node network of diameter $D$ requires $Ω(\sqrt{\ell D}+D)$ 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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arXiv2011

Xheal: Localized Self-healing using Expanders

We consider the problem of self-healing in reconfigurable networks (e.g. peer-to-peer and wireless mesh networks) that are under repeated attack by an omniscient adversary and propose a fully distributed algorithm, Xheal that maintains good expansion and spectral properties of the network, also keeping the network connected. Moreover, Xheal does this while allowing only low stretch and degree increase per node. Thus, the algorithm heals global properties while only doing local changes and using only local information. Our work improves over the self-healing algorithms 'Forgiving tree'[PODC 2008] and 'Forgiving graph'[PODC 2009] (using a similar model) in that we are able to give guarantees on degree and stretch, while at the same time preserving the expansion and spectral properties of the network. These repairs preserve the invariants in the following sense. At any point in the algorithm, the expansion of the graph will be either `better' than the expansion of the graph formed by considering only the adversarial insertions (not the adversarial deletions) or the expansion will be, at least, a constant. Also, the stretch i.e. the distance between any pair of nodes in the healed graph is no more than a $O(\log n)$ factor. Similarly, at any point, a node $v$ whose degree would have been $d$ in the graph with adversarial insertions only, will have degree at most $O(κd)$ in the actual graph, for a small parameter $κ$. We also provide bounds on the second smallest eigenvalue of the Laplacian which captures key properties such as mixing time, conductance, congestion in routing etc. Our distributed data structure has low amortized latency and bandwidth requirements.

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arXiv2010

Efficient Distributed Random Walks with Applications

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 $\ell$ in $\tilde{O}(\sqrt{\ell D})$ rounds (with high probability) on an undirected network, where $D$ is the diameter of the network. This improves over the previous best algorithm that ran in $\tilde{O}(\ell^{2/3}D^{1/3})$ rounds (Das Sarma et al., PODC 2009). We further extend our algorithms to efficiently perform $k$ independent random walks in $\tilde{O}(\sqrt{k\ell D} + k)$ rounds. We then show that there is a fundamental difficulty in improving the dependence on $\ell$ any further by proving a lower bound of $Ω(\sqrt{\frac{\ell}{\log \ell}} + D)$ 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 $\tilde{O}(\sqrt{m}D)$ rounds (with high probability; here $m$ 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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arXiv2010

Optimal Gossip-Based Aggregate Computation

We present the first provably almost-optimal gossip-based algorithms for aggregate computation that are both time optimal and message-optimal. Given a $n$-node network, our algorithms guarantee that all the nodes can compute the common aggregates (such as Min, Max, Count, Sum, Average, Rank etc.) of their values in optimal $O(\log n)$ time and using $O(n \log \log n)$ messages. Our result improves on the algorithm of Kempe et al. \cite{kempe} that is time-optimal, but uses $O(n \log n)$ messages as well as on the algorithm of Kashyap et al. \cite{efficient-gossip} that uses $O(n \log \log n)$ messages, but is not time-optimal (takes $O(\log n \log \log n)$ time). Furthermore, we show that our algorithms can be used to improve gossip-based aggregate computation in sparse communication networks, such as in peer-to-peer networks. The main technical ingredient of our algorithm is a technique called {\em distributed random ranking (DRR)} that can be useful in other applications as well. DRR gives an efficient distributed procedure to partition the network into a forest of (disjoint) trees of small size. Our algorithms are non-address oblivious. In contrast, we show a lower bound of $Ω(n\log n)$ on the message complexity of any address-oblivious algorithm for computing aggregates. This shows that non-address oblivious algorithms are needed to obtain significantly better message complexity. Our lower bound holds regardless of the number of rounds taken or the size of the messages used. Our lower bound is the first non-trivial lower bound for gossip-based aggregate computation and also gives the first formal proof that computing aggregates is strictly harder than rumor spreading in the address-oblivious model.

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