No abstract

We show that every weighted connected graph G contains as a subgraph a spanning tree into which the edges of G can be embedded with average stretch O(log 2 n log log n). Moreover, we show that this tree can be constructed in time O(m log n + n log 2 n) in general, and in time O(m log n) if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique.Our new algorithm can be immediately used to improve the running time of the recent solver for symmetric diagonally dominant linear systems of Spielman and Teng from m2 (O( √ log n log log n)) to m log O(1) n, and to O(n log 2 n log log n) when the system is planar. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees.

We show that every weighted connected graph G contains as a subgraph a spanning tree into which the edges of G can be embedded with average stretch O(log 2 n log log n). Moreover, we show that this tree can be constructed in time O(m log n + n log 2 n) in general, and in time O(m log n) if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for symmetric diagonally dominant linear systems of Spielman and Teng from m2 (O(√ log n log log n)) to m log O(1) n, and to O(n log 2 n log log n) when the system is planar. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees.

Signaling is an important topic in the study of asymmetric information in economic settings. In particular, the transparency of information available to a seller in an auction setting is a question of major interest. We introduce the study of signaling when conducting a second price auction of a probabilistic good whose actual instantiation is known to the auctioneer but not to the bidders. This framework can be used to model impressions selling in display advertising. We establish several results within this framework. First, we study the problem of computing a signaling scheme that maximizes the auctioneer's revenue in a Bayesian setting. We show that this problem is polynomially solvable for some interesting special cases, but computationally hard in general. Second, we establish a tight bound on the minimum number of signals required to implement an optimal signaling scheme. Finally, we show that at least half of the maximum social welfare can be preserved within such a scheme.

The traditional models of distributed computing focus mainly on networks of computer-like devices that can exchange large messages with their neighbors and perform arbitrary local computations. Recently, there is a trend to apply distributed computing methods to networks of sub-microprocessor devices, e.g., biological cellular networks or networks of nano-devices. However, the suitability of the traditional distributed computing models to these types of networks is questionable: do tiny bio/nano nodes "compute" and/or "communicate" essentially the same as a computer? In this paper, we introduce a new model that depicts a network of randomized finite state machines operating in an asynchronous environment. Although the computation and communication capabilities of each individual device in the new model are, by design, much weaker than those of a computer, we show that some of the most important and extensively studied distributed computing problems can still be solved efficiently.

This paper studies a new online problem, referred to as min-cost perfect matching with delays (MPMD), defined over a finite metric space (i.e., a complete graph with positive edge weights obeying the triangle inequality) M that is known to the algorithm in advance. Requests arrive in a continuous time online fashion at the points of M and should be served by matching them to each other. The algorithm is allowed to delay its request matching commitments, but this does not come for free: the total cost of the algorithm is the sum of metric distances between matched requests plus the sum of times each request waited since it arrived until it was matched. A randomized online MPMD algorithm is presented whose competitive ratio is O(logwhere n is the number of points in M and ∆ is its aspect ratio. The analysis is based on a machinery developed in the context of a new stochastic process that can be viewed as two interleaved Poisson processes; surprisingly, this new process captures precisely the behavior of our algorithm. A related problem in which the algorithm is allowed to clear any unmatched request at a fixed penalty is also addressed. It is suggested that the MPMD problem is merely the tip of the iceberg for a general framework of online problems with delayed service that captures many more natural problems.

Multi-level marketing is a marketing approach that motivates its participants to promote a certain product among their friends. The popularity of this approach increases due to the accessibility of modern social networks, however, it existed in one form or the other long before the Internet age began (the infamous Pyramid scheme that dates back at least a century is in fact a special case of multi-level marketing). This paper lays foundations for the study of reward mechanisms in multi-level marketing within social networks. We provide a set of desired properties for such mechanisms and show that they are uniquely satisfied by geometric reward mechanisms. The resilience of mechanisms to falsename manipulations is also considered; while geometric reward mechanisms fail against such manipulations, we exhibit other mechanisms which are false-name-proof.

We consider broadcasting in radio networks, modeled as unit disk graphs (UDG). Such networks occur in wireless communication between sites (e.g., stations or sensors) situated in a terrain. Network stations are represented by points in the Euclidean plane, where a station is connected to all stations at distance at most 1 from it. A message transmitted by a station reaches all its neighbors, but a station hears a message (receives the message correctly) only if exactly one of its neighbors transmits at a given time step. One station of the network, called the source, has a message which has to be disseminated to all other stations. Stations are unaware of the network topology. Two broadcasting models are considered. In the conditional wake up model, the stations other than the source are initially idle and cannot transmit until they hear a message for the first time. In the spontaneous wake up model, all stations are awake (and may transmit messages) from the beginning.It turns out that broadcasting time depends on two parameters of the UDG network, namely, its diameter D and its granularity g, which is the inverse of the minimum dis- * tance between any two stations. We present a deterministic broadcasting algorithm which works in time O(Dg) under the conditional wake up model and prove that broadcasting in this model cannot be accomplished by any deterministic algorithm in time better than Ω(D √ g). For the spontaneous wake up model, we design two deterministic broadcasting algorithms: the first works in time O(D + g 2 ) and the second in time O(D log g). While none of these algorithms alone is optimal for all parameter values, we prove that the algorithm obtained by interleaving their steps, and thus working in time O`min˘D + g 2 , D log g¯´, turns out to be optimal by establishing a matching lower bound.