Rumor Spreading on Random Regular Graphs and Expanders

Broadcasting algorithms are important building blocks of distributed systems. In this work we investigate the typical performance of the classical and well-studied push model. Assume that initially one node in a given network holds some piece of information. In each round, every one of the informed nodes chooses independently a neighbor uniformly at random and transmits the message to it. In this paper we consider random networks where each vertex has degree d, which is at least 3, i.e., the underlying graph is drawn uniformly at random from the set of all d-regular graphs with n vertices. We show that with probability 1 - o(1) the push model broadcasts the message to all nodes within (1 + o(1))C_d ln n rounds, where C_d = 1/ ln(2(1-1/d)) - 1/(d ln(1 - 1/d)). In particular, we can characterize precisely the effect of the node degree to the typical broadcast time of the push model. Moreover, we consider pseudo-random regular networks, where we assume that the degree of each node is very large. There we show that the broadcast time is (1+o(1))C ln n with probability 1 - o(1), where C= 1/ ln 2 + 1, is the limit of C_d as d grows.Comment: 18 page

Going after the k-SAT Threshold

Random $k$-SAT is the single most intensely studied example of a random constraint satisfaction problem. But despite substantial progress over the past decade, the threshold for the existence of satisfying assignments is not known precisely for any $k\geq3$. The best current results, based on the second moment method, yield upper and lower bounds that differ by an additive $k\cdot \frac{\ln2}2$, a term that is unbounded in $k$ (Achlioptas, Peres: STOC 2003). The basic reason for this gap is the inherent asymmetry of the Boolean value `true' and `false' in contrast to the perfect symmetry, e.g., among the various colors in a graph coloring problem. Here we develop a new asymmetric second moment method that allows us to tackle this issue head on for the first time in the theory of random CSPs. This technique enables us to compute the $k$-SAT threshold up to an additive $\ln2-\frac12+O(1/k)\approx 0.19$. Independently of the rigorous work, physicists have developed a sophisticated but non-rigorous technique called the "cavity method" for the study of random CSPs (M\'ezard, Parisi, Zecchina: Science 2002). Our result matches the best bound that can be obtained from the so-called "replica symmetric" version of the cavity method, and indeed our proof directly harnesses parts of the physics calculations

On the Insertion Time of Cuckoo Hashing

Cuckoo hashing is an efficient technique for creating large hash tables with high space utilization and guaranteed constant access times. There, each item can be placed in a location given by any one out of k different hash functions. In this paper we investigate further the random walk heuristic for inserting in an online fashion new items into the hash table. Provided that k > 2 and that the number of items in the table is below (but arbitrarily close) to the theoretically achievable load threshold, we show a polylogarithmic bound for the maximum insertion time that holds with high probability.Comment: 27 pages, final version accepted by the SIAM Journal on Computin

Robustness of Randomized Rumour Spreading

In this work we consider three well-studied broadcast protocols: Push, Pull and Push&Pull. A key property of all these models, which is also an important reason for their popularity, is that they are presumed to be very robust, since they are simple, randomized, and, crucially, do not utilize explicitly the global structure of the underlying graph. While sporadic results exist, there has been no systematic theoretical treatment quantifying the robustness of these models. Here we investigate this question with respect to two orthogonal aspects: (adversarial) modifications of the underlying graph and message transmission failures. We explore in particular the following notion of Local Resilience: beginning with a graph, we investigate up to which fraction of the edges an adversary has to be allowed to delete at each vertex, so that the protocols need significantly more rounds to broadcast the information. Our main findings establish a separation among the three models. It turns out that Pull is robust with respect to all parameters that we consider. On the other hand, Push may slow down significantly, even if the adversary is allowed to modify the degrees of the vertices by an arbitrarily small positive fraction only. Finally, Push&Pull is robust when no message transmission failures are considered, otherwise it may be slowed down. On the technical side, we develop two novel methods for the analysis of randomized rumour spreading protocols. First, we exploit the notion of self-bounding functions to facilitate significantly the round-based analysis: we show that for any graph the variance of the growth of informed vertices is bounded by its expectation, so that concentration results follow immediately. Second, in order to control adversarial modifications of the graph we make use of a powerful tool from extremal graph theory, namely Szemer\`edi's Regularity Lemma.Comment: version 2: more thorough literature revie