Anna Huber

Reliable Broadcasting in Random Networks and the Effect of Density

Broadcasting algorithms are of fundamental importance for distributed systems engineering. In this paper we revisit the classical and well-studied push protocol for message broadcasting and we investigate a faulty version of it. Assuming that initially only one node has some piece of information, at each stage every one of the informed nodes chooses randomly and independently one of its neighbors and passes the message to it with some probability q that is, it fails to do so with probability 1-q. The performance of the push protocol on a fully connected network, where each node is joined by a link to every other node, with q=1 is very well understood. In particular, Frieze and Grimmett proved that with probability 1-o(1) the push protocol completes the broadcasting of the message within (1±ε) (log_2 n + ln n) stages, where n is the number of nodes in the network. However, there are no tight bounds for the broadcast time on networks that are significantly sparser than the complete graph. In this work we consider random networks on n nodes, where every edge is present with probability p, independently of every other edge. We show that if p≥ α(n)ln n/n, where α(n) is any function that tends to infinity as n grows, then the push protocol with faulty transmissions broadcasts the message within (1±±ε)(log_{1+q} n + 1/q ln n) stages with probability 1-o(1). In other words, in almost every network of density d such that d ≥ α(n) ln n, the push protocol broadcasts a message as fast as in a fully connected network and the speed is only affected by the success probability q. This is quite surprising in the sense that the time needed remains essentially unaffected by the fact that most of the links are missing. Our results are accompanied by experimental evaluation.

Towards minimizing k -submodular functions

In this paper we investigate k-submodular functions. This natural family of discrete functions includes submodular and bisubmodular functions as the special cases k=1 and k=2 respectively. In particular we generalize the known Min-Max-Theorem for submodular and bisubmodular functions. This theorem asserts that the minimum of the (bi)submodular function can be found by solving a maximization problem over a (bi)submodular polyhedron. We define a k-submodular polyhedron, prove a Min-Max-Theorem for k-submodular functions, and give a greedy algorithm to construct the vertices of the polyhedron.

Skew Bisubmodularity and Valued CSPs

An instance of the (finite-)valued constraint satisfaction problem (VCSP) is given by a finite set of variables, a finite domain of values, and a sum of (rational-valued) functions, with each function depending on a subset of the variables. The goal is to find an assignment of values to the variables that minimizes the sum. We study (assuming that

Quasirandom Rumor Spreading on the Complete Graph Is as Fast as Randomized Rumor Spreading

{PTIME}\neq{NP}

Strong Robustness of Randomized Rumor Spreading Protocols

) how the complexity of this very general problem depends on the functions allowed in the instances. The case when the variables can take only two values was classified by Cohen et al.: essentially, submodular functions give rise to the only tractable case, and any non--submodular function can be used to express, in a certain specific sense, the NP-hard Max Cut problem. We investigate the case when the variables can take three values. We identify a new infinite family of conditions that includes bisubmodularity as a special case and which can collectively be called skew bisubmodularity. By a recent result of Thapper and Živný, this condition imp...

ORACLE TRACTABILITY OF SKEW BISUBMODULAR FUNCTIONS

In this paper, we provide a detailed comparison between a fully randomized protocol for rumor spreading on a complete graph and a quasirandom protocol introduced by Doerr, Friedrich, and Sauerwald [Quasirandom rumor spreading, in Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2008, pp. 773-781]. In the former, initially there is one vertex which holds a piece of information, and during each round every one of the informed vertices chooses uniformly at random and independently one of its neighbors and informs it. In the quasirandom version of this method (cf. Doerr, Friedrich, and Sauerwald) each vertex has a cyclic list of its neighbors. Once a vertex has been informed, it chooses uniformly at random only one neighbor. In the following round, it informs this neighbor, and at each subsequent round it picks the next neighbor from its list and informs it. We give a precise analysis of the evolution of the quasirandom protocol on the complete graph with

Quasirandom broadcasting on the complete graph is as fast as randomized broadcasting

n

Brief announcement: the speed of broadcasting in random networks - density does not matter

vertices and show that it evolves essentially in the same way as the randomized protocol. In particular, if

Tight Bounds for Quasirandom Rumor Spreading

S(n)

Strong robustness of randomized rumor spreading protocols

denotes the number of rounds that are needed until all vertices are informed, we show that for any slowly growing function