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Recently Rothemund and Winfree 6] have considered the program size complexity of constructing squares by selfassembly. Here, we consider the time complexity o f s u c h constructions using a natural generalization of the Tile Assembly Model de ned in 6]. In the generalized model, the Rothemund-Winfree construction of n n squares requires time (n log n) and program size (log n). We present a new construction for assembling n n squares which uses optimal time (n) and program size ( log n log log n ). This program size is also optimal since it matches the bound dictated by Kolmogorov complexity. Our improved time is achieved by demonstrating a set of tiles for parallel selfassembly of binary counters. Our improved program size is achieved by demonstrating that self-assembling systems can compute changes in the base representation of numbers. Self-assembly is emerging as a useful paradigm for computation. In addition the development of a computational theory of self-assembly promises to provide a new conduit by w h i c h results and methods of theoretical computer science might be applied to problems of interest in biology and the physical sciences.

We present a truthful auction for pricing advertising slots on a web-page assuming that advertisements for different merchants must be ranked in decreasing order of their (weighted) bids. This captures both the Overture model where bidders are ranked in order of the submitted bids, and the Google model where bidders are ranked in order of the expected revenue (or utility) that their advertisement generates. Assuming separable click-through rates, we prove revenue-equivalence between our auction and the non-truthful next-price auctions currently in use.

In this paper, we analyze the efficiency of Monte Carlo methods for incremental computation of PageRank, personalized PageRank, and similar random walk based methods (with focus on SALSA), on large-scale dynamically evolving social networks. We assume that the graph of friendships is stored in distributed shared memory, as is the case for large social networks such as Twitter.For global PageRank, we assume that the social network has n nodes, and m adversarially chosen edges arrive in a random order. We show that with a reset probability of , the expected total work needed to maintain an accurate estimate (using the Monte Carlo method) of the PageRank of every node at all times is O( n ln m 2 ). This is significantly better than all known bounds for incremental PageRank. For instance, if we naively recompute the PageRanks as each edge arrives, the simple power iteration method needs Ω( ) total time and the Monte Carlo method needs O(mn/) total time; both are prohibitively expensive. We also show that we can handle deletions equally efficiently.We then study the computation of the top k personalized PageRanks starting from a seed node, assuming that personalized PageRanks follow a power-law with exponent α < 1. We show that if we store R > q ln n random walks starting from every node for large enough constant q (using the approach outlined for global PageRank), then the expected number of calls made to the distributed social network database is O(k/(R (1−α)/α )). We also present experimental results from the social networking site, Twitter, ver- * Work done while interning at Twitter † Work done while at Twitter.

Abstract. Wireless sensor networks (WSNs) are emerging as an effective means for environment monitoring. This paper investigates a strategy for energy efficient monitoring in WSNs that partitions the sensors into covers, and then activates the covers iteratively in a round-robin fashion. This approach takes advantage of the overlap created when many sensors monitor a single area. Our work builds upon previous work in [2], where the model is first formulated. We have designed three approximation algorithms for a variation of the SET K-COVER problem, where the objective is to partition the sensors into covers such that the number of covers that include an area, summed over all areas, is maximized. The first algorithm is randomized and partitions the sensors, in expectation, within a fraction 1 − 1 e (∼.63) of the optimum. We present two other deterministic approximation algorithms. One is a distributed greedy algorithm with a 1 2 approximation ratio and the other is a centralized greedy algorithm with a 1 − 1 e approximation ratio. We show that it is NP-Complete to guarantee better than 15 16 of the optimal coverage, indicating that all three algorithms perform well with respect to the best approximation algorithm possible. Simulations indicate that in practice, the deterministic algorithms perform far above their worst case bounds, consistently covering more than 72% of what is covered by an optimum solution. Simulations also indicate that the increase in longevity is proportional to the amount of overlap amongst the sensors. The algorithms are fast, easy to use, and according to simulations, significantly increase the longevity of sensor networks. The randomized algorithm in particular seems quite practical.

We study the issue of polarization in society through a model of opinion formation. We say an opinion formation process is polarizing if it results in increased divergence of opinions. Empirical studies have shown that homophily, i.e., greater interaction between like-minded individuals, results in polarization. However, we show that DeGroot's well-known model of opinion formation based on repeated averaging can never be polarizing, even if individuals are arbitrarily homophilous. We generalize DeGroot's model to account for a phenomenon well known in social psychology as biased assimilation: When presented with mixed or inconclusive evidence on a complex issue, individuals draw undue support for their initial position, thereby arriving at a more extreme opinion. We show that in a simple model of homophilous networks, our biased opinion formation process results in polarization if individuals are sufficiently biased. In other words, homophily alone, without biased assimilation, is not sufficient to polarize society. Quite interestingly, biased assimilation also provides a framework to analyze the polarizing effect of Internet-based recommender systems that show us personalized content.T he issue of polarization in society has been extensively studied and vigorously debated in the academic literature as well as the popular press over the last few decades. In particular, are we as a society getting more polarized? If so, why, and how can we fix it? Different empirical studies arrive at different answers to this question depending on the context and the metric used to measure polarization.Evidence of polarization in politics has been found in the increasingly partisan voting patterns of the members of Congress (1, 2) and in the extreme policies adopted by candidates for political office (3). McCarty et al. (4) claim via rigorous analysis that America is polarized in terms of political attitudes and beliefs. Phenomena such as segregation in urban residential neighborhoods (5-7), the rising popularity of overtly partisan television news networks (8, 9), and the readership and linking patterns of blogs along partisan lines (10-13) can all be viewed as further evidence of polarization. On the other hand, it has also been argued on the basis of detailed surveys of public opinion that society as a whole is not polarized, even though the media and the politicians make it seem so (14, 15). We adopt the view that polarization is not a property of a state of society; instead it is a property of the dynamics through which individuals form opinions. We say that opinion formation dynamics are polarizing if they result in an increased divergence of opinions.It has been argued using empirical studies that homophily, i.e., greater interaction between like-minded individuals, results in polarization (12,16,17). This argument has been used to claim that the rise of cable news, talk radio, and the Internet has contributed to polarization: the increased diversity of information sources coupled with the increased ability to narrowly ta...

Internet routers require buffers to hold packets during times of congestion. The buffers need to be fast, and so ideally they should be small enough to use fast memory technologies such as SRAM or all-optical buffering. Unfortunately, a widely used rule-of-thumb says we need a bandwidth-delay product of buffering at each router so as not to lose link utilization. This can be prohibitively large. In a recent paper, Appenzeller et al. challenged this rule-of-thumb and showed that for a backbone network, the buffer size can be divided by √ N without sacrificing throughput, where N is the number of flows sharing the bottleneck. In this paper, we explore how buffers in the backbone can be significantly reduced even more, to as little as a few dozen packets, if we are willing to sacrifice a small amount of link capacity. We argue that if the TCP sources are not overly bursty, then fewer than twenty packet buffers are sufficient for high throughput. Specifically, we argue that O(log W ) buffers are sufficient, where W is the window size of each flow. We support our claim with analysis and a variety of simulations. The change we need to make to TCP is minimal-each sender just needs * This work was supported under DARPA/MTO DOD-N award no. W911NF-04-0001/KK4118 (LASOR PROJECT) and the Buffer Sizing Grant no. W911NF-05-1-0224. † Research also supported by an NSF career grant and an Alfred P. Sloan faculty fellowship. ‡ Research also supported in part by ONR grant N00014-04-1-0725.to pace packet injections from its window. Moreover, there is some evidence that such small buffers are sufficient even if we don't modify the TCP sources so long as the access network is much slower than the backbone, which is true today and likely to remain true in the future.We conclude that buffers can be made small enough for alloptical routers with small integrated optical buffers.

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