David Kempe

Gossip-based computation of aggregate information

Over the last decade, we have seen a revolution in connectivity between computers, and a resulting paradigm shift from centralized to highly distributed systems. With massive scale also comes massive instability, as node and link failures become the norm rather than the exception. For such highly volatile systems, decentralized gossip-based protocols are emerging as an approach to maintaining simplicity and scalability while achieving fault-tolerant information dissemination. In this paper, we study the problem of computing aggregates with gossip-style protocols. Our first contribution is an analysis of simple gossip-based protocols for the computation of sums, averages, random samples, quantiles, and other aggregate functions, and we show that our protocols converge exponentially fast to the true answer when using uniform gossip. Our second contribution is the definition of a precise notion of the speed with which a nodes data diffuses through the network. We show that this diffusion speed is at the heart of the approximation guarantees for all of the above problems. We analyze the diffusion speed of uniform gossip in the presence of node and link failures, as well as for flooding-based mechanisms. The latter expose interesting connections to random walks on graphs.

Influential nodes in a diffusion model for social networks

We study the problem of maximizing the expected spread of an innovation or behavior within a social network, in the presence of “word-of-mouth” referral. Our work builds on the observation that individuals’ decisions to purchase a product or adopt an innovation are strongly influenced by recommendations from their friends and acquaintances. Understanding and leveraging this influence may thus lead to a much larger spread of the innovation than the traditional view of marketing to individuals in isolation. In this paper, we define a natural and general model of influence propagation that we term the decreasing cascade model, generalizing models used in the sociology and economics communities. In this model, as in related ones, a behavior spreads in a cascading fashion according to a probabilistic rule, beginning with a set of initially “active” nodes. We study the target set selection problem: we wish to choose a set of individuals to target for initial activation, such that the cascade beginning with this active set is as large as possible in expectation. We show that in the decreasing cascade model, a natural greedy algorithm is a 1-1/ e-e approximation for selecting a target set of size k.

Auction-Based Multi-Robot Routing.

Recently, auction methods have been investigated as effective, decentralized methods for multi-robot coordination. Experimental research has shown great potential, but has not been complemented yet by theoretical analysis. In this paper we contribute a theoretical analysis of the performance of auction methods for multi-robot routing. We suggest a generic framework for auction-based multi-robot routing and analyze a variety of bidding rules for different team objectives. This is the first time that auction methods are shown to offer theoretical guarantees for such a variety of bidding rules and team objectives.

Spatial gossip and resource location protocols

The dynamic behavior of a network in which information is changing continuously over time requires robust and efficient mechanisms for keeping nodes updated about new information. Gossip protocols are mechanisms for this task in which nodes communicate with one another according to some underlying deterministic or randomized algorithm, exchanging information in each communication step. In a variety of contexts, the use of randomization to propagate information has been found to provide better reliability and scalability than more regimented deterministic approaches. In many settings --- consider a network of sensors, or a cluster of distributed computing hosts --- new information is generated at individual nodes, and is most “interesting” to nodes that are nearby. Thus, we propose distance-based propagation bounds as a performance measure for gossip algorithms: a node at distance d from the origin of a new piece of information should be able to learn about this information with a delay that grows slowly with d, and is independent of the size of the network. For nodes arranged with uniform density in Euclidean space, we present natural gossip algorithms that satisfy such a guarantee: new information is spread to nodes at distance \DIST, with high probability, in O(\log^{1 + \ve} \DIST) time steps. Such a bound combines the desirable qualitative features of uniform gossip, in which information is spread with a delay that is logarithmic in the full network size, and deterministic flooding, in which information is spread with a delay that is linear in the distance and independent of the network size. Our algorithms and their analysis resolve a conjecture of Demers et al. We show an application of our gossip algorithms to a basic resource location problem, in which nodes seek to rapidly

On the bias of traceroute sampling: or, power-law degree distributions in regular graphs

Understanding the structure of the Internet graph is a crucial step for building accurate network models and designing efficient algorithms for Internet applications. Yet, obtaining its graph structure is a surprisingly difficult task, as edges cannot be explicitly queried. Instead, empirical studies rely on traceroutes to build what are essentially single-source, all-destinations, shortest-path trees. These trees only sample a fraction of the networks edges, and a recent paper by Lakhina et al. found empirically that the resuting sample is intrinsically biased. For instance, the observed degree distribution under traceroute sampling exhibits a power law even when the underlying degree distribution is Poisson.In this paper, we study the bias of traceroute sampling systematically, and, for a very general class of underlying degree distributions, calculate the likely observed distributions explicitly. To do this, we use a continuous-time realization of the process of exposing the BFS tree of a random graph with a given degree distribution, calculate the expected degree distribution of the tree, and show that it is sharply concentrated. As example applications of our machinery, we show how traceroute sampling finds power-law degree distributions in both δ-regular and Poisson-distributed random graphs. Thus, our work puts the observations of Lakhina et al. on a rigorous footing, and extends them to nearly arbitrary degree distributions.

A Knapsack Secretary Problem with Applications

We consider situations in which a decision-maker with a fixed budget faces a sequence of options, each with a cost and a value, and must select a subset of them online so as to maximize the total value. Such situations arise in many contexts, e.g., hiring workers, scheduling jobs, and bidding in sponsored search auctions. This problem, often called the online knapsack problem, is known to be inapproximable. Therefore, we make the enabling assumption that elements arrive in a randomorder. Hence our problem can be thought of as a weighted version of the classical secretary problem, which we call the knapsack secretary problem. Using the random-order assumption, we design a constant-competitive algorithm for arbitrary weights and values, as well as a e-competitive algorithm for the special case when all weights are equal (i.e., the multiple-choice secretary problem). In contrast to previous work on online knapsack problems, we do not assume any knowledge regarding the distribution of weights and values beyond the fact that the order is random.

A Cascade Model for Externalities in Sponsored Search

One of the most important yet insufficiently studied issues in online advertising is the externality effect among ads: the value of an ad impression on a page is affected not just by the location that the ad is placed in, but also by the set of other ads displayed on the page. For instance, a high quality competing ad can detract users from another ad, while a low quality ad could cause the viewer to abandon the page altogether. In this paper, we propose and analyze a model for externalities in sponsored search ads . Our model is based on the assumption that users will visually scan the list of ads from the top to the bottom. After each ad, they make independent random decisions with ad-specific probabilities on whether to continue scanning. We then generalize the model in two ways: allowing for multiple separate blocks of ads, and allowing click probabilities to explicitly depend on ad positions as well. For the most basic model, we present a polynomial-time incentive-compatible auction mechanism for allocating and pricing ad slots. For the generalizations, we give approximation algorithms for the allocation of ads.

Online auctions and generalized secretary problems

We present generalized secretary problems as a framework for online auctions. Elements, such as potential employees or customers, arrive one by one online. After observing the value derived from an element, but without knowing the values of future elements, the algorithm has to make an irrevocable decision whether to retain the element as part of a solution, or reject it. The way in which the secretary framework differs from traditional online algorithms is that the elements arrive in uniformly random order. Many natural online auction scenarios can be cast as generalized secretary problems, by imposing natural restrictions on the feasible sets. For many such settings, we present surprisingly strong constant factor guarantees on the expected value of solutions obtained by online algorithms. The framework is also easily augmented to take into account time-discounted revenue and incentive compatibility. We give an overview of recent results and future research directions.

Maximizing the Spread of Influence through a Social Network

Models for the processes by which ideas and influence propagate through a social network have been studied in a number of domains, including the diffusion of medical and technological innovations, the sudden and widespread adoption of various strategies in game-theoretic settings, and the effects of “word of mouth” in the promotion of new products. Motivated by the design of viral marketing strategies, Domingos and Richardson posed a fundamental algorithmic problem for such social network processes: if we can try to convince a subset of individuals to adopt a new product or innovation, and the goal is to trigger a large cascade of further adoptions, which set of individuals should we target? We consider this problem in several of the most widely studied models in social network analysis. The optimization problem of selecting the most influential nodes is NP-hard here. The two conference papers upon which this article is based (KDD 2003 and ICALP 2005) provide the first provable approximation guarantees for efficient algorithms. Using an The present article is an expanded version of two conference papers [51, 52], which appeared in KDD 2003 and ICALP 2005, respectively. ∗Supported in part by an Intel Graduate Fellowship and an NSF Graduate Research Fellowship, an NSF CAREER Award, an ONR Young Investigator Award, and a Sloan Fellowship. †Supported in part by a David and Lucile Packard Foundation Fellowship and NSF ITR/IM Grant IIS-0081334. ‡Supported in part by NSF ITR grant CCR-011337, and ONR grant N00014-98-1-0589. ACM Classification: F.2.2, G.3 AMS Classification: 68W25, 90C59, 68Q25, 68Q17

Multi-robot forest coverage

One of the main applications of mobile robots is terrain coverage: visiting each location in known terrain. Terrain coverage is crucial for lawn mowing, cleaning, harvesting, search-and-rescue, intrusion detection and mine clearing. Naturally, coverage can be sped up with multiple robots. In this paper, we describe multi-robot forest coverage, a new multi-robot coverage algorithm based on an algorithm by Even et al. (2004) for finding a tree cover with trees of balanced weights. The cover time of multi-robot forest coverage is at most eight times larger than optimal, and our experiments show it to perform significantly better than existing multi-robot coverage algorithms.