Clean Citation Style
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Bläsius, Thomas; Rutter, Ignaz Simultaneous PQOrdering with Applications to Constrained Embedding Problems. Symposium on Discrete Algorithms (SODA) 2013: 10301043
In this article, we define and study the new problem of SIMULTANEOUS PQORDERING. Its input consists of a set of PQtrees, which represent sets of circular orders of their leaves, together with a set of childparent relations between these PQtrees, such that the leaves of the child form a subset of the leaves of the parent. SIMULTANEOUS PQORDERING asks whether orders of the leaves of each of the trees can be chosen simultaneously; that is, for every childparent relation, the order chosen for the parent is an extension of the order chosen for the child. We show that SIMULTANEOUS PQORDERING is NPcomplete in general, and we identify a family of instances that can be solved efficiently, the 2fixed instances. We show that this result serves as a framework for several other problems that can be formulated as instances of SIMULTANEOUS PQORDERING. In particular, we give lineartime algorithms for recognizing simultaneous interval graphs and extending partial interval representations. Moreover, we obtain a lineartime algorithm for PARTIALLY PQCONSTRAINED PLANARITY for biconnected graphs, which asks for a planar embedding in the presence of 16 PQtrees that restrict the possible orderings of edges around vertices, and a quadratictime algorithm for SIMULTANEOUS EMBEDDING WITH FIXED EDGES for biconnected graphs with a connected intersection. Both results can be extended to the case where the input graphs are not necessarily biconnected but have the property that each cutvertex is contained in at most two nontrivial blocks. This includes, for example, the case where both graphs have a maximum degree of 5.

Friedrich, Tobias; Gairing, Martin; Sauerwald, Thomas Quasirandom Load Balancing. Symposium on Discrete Algorithms (SODA) 2010: 16201629
We propose a simple distributed algorithm for balancing indivisible tokens on graphs. The algorithm is completely deterministic, though it tries to imitate (and enhance) a randomized algorithm by keeping the accumulated rounding errors as small as possible. Our new algorithm, surprisingly, closely approximates the idealized process (where the tokens are divisible) on important network topologies. On \(d\)dimensional torus graphs with \(n\) nodes it deviates from the idealized process only by an additive constant. In contrast, the randomized rounding approach of Friedrich and Sauerwald [Proceedings of the 41st Annual ACM Symposium on Theory of Computing, 2009, pp. 121130] can deviate up to \(\Omega(\text{polylog(n))\), and the deterministic algorithm of Rabani, Sinclair, and Wanka [Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, 1998, pp. 694705] has a deviation of \(\Omega(n^{1/d})\). This makes our quasirandom algorithm the first known algorithm for this setting, which is optimal both in time and achieved smoothness. We further show that on the hypercube as well, our algorithm has a smaller deviation from the idealized process than the previous algorithms. To prove these results, we derive several combinatorial and probabilistic results that we believe to be of independent interest. In particular, we show that firstpassage probabilities of a random walk on a path with arbitrary weights can be expressed as a convolution of independent geometric probability distributions.

Bradonjic, Milan; Elsässer, Robert; Friedrich, Tobias; Sauerwald, Thomas; Stauffer, Alexandre Efficient Broadcast on Random Geometric Graphs. Symposium on Discrete Algorithms (SODA) 2010: 14121421
A Random Geometric Graph (RGG) in two dimensions is constructed by distributing \(n\) nodes independently and uniformly at random in \([0, \sqrt n]^2\) and creating edges between every pair of nodes having Euclidean distance at most \(r\), for some prescribed \(r\). We analyze the following randomized broadcast algorithm on RGGs. At the beginning, only one node from the largest connected component of the RGG is informed. Then, in each round, each informed node chooses a neighbor independently and uniformly at random and informs it. We prove that with probability \(1  O(n^{1})\) this algorithm informs every node in the largest connected component of an RGG within \(O(\sqrt n / r + \log n)\) rounds. This holds for any value of \(r\) larger than the critical value for the emergence of a connected component with \(\Omega(n)\) nodes. In order to prove this result, we show that for any two nodes sufficiently distant from each other in \([0, \sqrt n]^2\), the length of the shortest path between them in the RGG, when such a path exists, is only a constant factor larger than the optimum. This result has independent interest and, in particular, gives that the diameter of the largest connected component of an RGG is \(\Theta(\sqrt n / r\)), which surprisingly has been an open problem so far.