arXiv subjects

Bernhard Haeupler

Publications and source records attributed to Bernhard Haeupler.

114 records · Page 7Linked to original sources

arXiv2010

Analyzing Network Coding Gossip Made Easy

We give a new technique to analyze the stopping time of gossip protocols that are based on random linear network coding (RLNC). Our analysis drastically simplifies, extends and strengthens previous results. We analyze RLNC gossip in a general framework for network and communication models that encompasses and unifies the models used previously in this context. We show, in most settings for the first time, that it converges with high probability in the information-theoretically optimal time. Most stopping times are of the form O(k + T) where k is the number of messages to be distributed and T is the time it takes to disseminate one message. This means RLNC gossip achieves "perfect pipelining". Our analysis directly extends to highly dynamic networks in which the topology can change completely at any time. This remains true even if the network dynamics are controlled by a fully adaptive adversary that knows the complete network state. Virtually nothing besides simple O(kT) sequential flooding protocols was previously known for such a setting. While RLNC gossip works in this wide variety of networks its analysis remains the same and extremely simple. This contrasts with more complex proofs that were put forward to give less strong results for various special cases.

cs.DC↗

arXiv2010

Satisfiability Thresholds for k-CNF Formula with Bounded Variable Intersections

We determine the thresholds for the number of variables, number of clauses, number of clause intersection pairs and the maximum clause degree of a k-CNF formula that guarantees satisfiability under the assumption that every two clauses share at most $α$ variables. More formally, we call these formulas $α$-intersecting and define, for example, a threshold $μ_i(k,α)$ for the number of clause intersection pairs $i$, such that every $α$-intersecting k-CNF formula in which at most $μ_i(k,α)$ pairs of clauses share a variable is satisfiable and there exists an unsatisfiable $α$-intersecting k-CNF formula with $μ_m(k,α)$ such intersections. We provide a lower bound for these thresholds based on the Lovasz Local Lemma and a nearly matching upper bound by constructing an unsatisfiable k-CNF to show that $μ_i(k,α) = \tildeΘ(2^{k(2+1/α)})$. Similar thresholds are determined for the number of variables ($μ_n = \tildeΘ(2^{k/α})$) and the number of clauses ($μ_m = \tildeΘ(2^{k(1+\frac{1}α)})$) (see [Scheder08] for an earlier but independent report on this threshold). Our upper bound construction gives a family of unsatisfiable formula that achieve all four thresholds simultaneously.

cs.DM↗

arXiv2009

Robust Regulatory Networks

One of the characteristic features of genetic networks is their inherent robustness, that is, their ability to retain functionality in spite of the introduction of random errors. In this paper, we seek to better understand how robustness is achieved and what functionalities can be maintained robustly. Our goal is to formalize some of the language used in biological discussions in a reasonable mathematical framework, where questions can be answered in a rigorous fashion. These results provide basic conceptual understanding of robust regulatory networks that should be valuable independent of the details of the formalism. We model the gene regulatory network as a boolean network, a general and well-established model introduced by Stuart Kauffman. A boolean network is said to be in a viable configuration if the node states of the network at its fixpoint satisfy some given constraint. We specify how mutations affect the behavior of the boolean network. A network is then said to be robust if most random mutations to the network reach a viable configuration. The main question investigated in our study is: given a constraint on the fixpoint configuration, does there exist a network that is robust with respect to it and, if so, what is its structure? We demonstrate both explicit constructions of robust networks as well as negative results disproving their existence.

q-bio.MN↗

arXiv2009

Heaps Simplified

The heap is a basic data structure used in a wide variety of applications, including shortest path and minimum spanning tree algorithms. In this paper we explore the design space of comparison-based, amortized-efficient heap implementations. From a consideration of dynamic single-elimination tournaments, we obtain the binomial queue, a classical heap implementation, in a simple and natural way. We give four equivalent ways of representing heaps arising from tournaments, and we obtain two new variants of binomial queues, a one-tree version and a one-pass version. We extend the one-pass version to support key decrease operations, obtaining the {\em rank-pairing heap}, or {\em rp-heap}. Rank-pairing heaps combine the performance guarantees of Fibonacci heaps with simplicity approaching that of pairing heaps. Like pairing heaps, rank-pairing heaps consist of trees of arbitrary structure, but these trees are combined by rank, not by list position, and rank changes, but not structural changes, cascade during key decrease operations.

cs.DS↗

arXiv2008

Incremental Topological Ordering and Strong Component Maintenance

We present an on-line algorithm for maintaining a topological order of a directed acyclic graph as arcs are added, and detecting a cycle when one is created. Our algorithm takes O(m^{1/2}) amortized time per arc, where m is the total number of arcs. For sparse graphs, this bound improves the best previous bound by a logarithmic factor and is tight to within a constant factor for a natural class of algorithms that includes all the existing ones. Our main insight is that the bidirectional search method of previous algorithms does not require an ordered search, but can be more general. This allows us to avoid the use of heaps (priority queues) entirely. Instead, the deterministic version of our algorithm uses (approximate) median-finding. The randomized version of our algorithm avoids this complication, making it very simple. We extend our topological ordering algorithm to give the first detailed algorithm for maintaining the strong components of a directed graph, and a topological order of these components, as arcs are added. This extension also has an amortized time bound of O(m^{1/2}) per arc.

cs.DS↗

arXiv2007

Finding a Feasible Flow in a Strongly Connected Network

We consider the problem of finding a feasible single-commodity flow in a strongly connected network with fixed supplies and demands, provided that the sum of supplies equals the sum of demands and the minimum arc capacity is at least this sum. A fast algorithm for this problem improves the worst-case time bound of the Goldberg-Rao maximum flow method by a constant factor. Erlebach and Hagerup gave an linear-time feasible flow algorithm. We give an arguably simpler one.

cs.DS↗