Fast Routing Table Construction Using Small Messages

We describe a distributed randomized algorithm computing approximate distances and routes that approximate shortest paths. Let n denote the number of nodes in the graph, and let HD denote the hop diameter of the graph, i.e., the diameter of the graph when all edges are considered to have unit weight. Given 0 < eps <= 1/2, our algorithm runs in weak-O(n^(1/2 + eps) + HD) communication rounds using messages of O(log n) bits and guarantees a stretch of O(eps^(-1) log eps^(-1)) with high probability. This is the first distributed algorithm approximating weighted shortest paths that uses small messages and runs in weak-o(n) time (in graphs where HD in weak-o(n)). The time complexity nearly matches the lower bounds of weak-Omega(sqrt(n) + HD) in the small-messages model that hold for stateless routing (where routing decisions do not depend on the traversed path) as well as approximation of the weigthed diameter. Our scheme replaces the original identifiers of the nodes by labels of size O(log eps^(-1) log n). We show that no algorithm that keeps the original identifiers and runs for weak-o(n) rounds can achieve a polylogarithmic approximation ratio. Variations of our techniques yield a number of fast distributed approximation algorithms solving related problems using small messages. Specifically, we present algorithms that run in weak-O(n^(1/2 + eps) + HD) rounds for a given 0 < eps <= 1/2, and solve, with high probability, the following problems: - O(eps^(-1))-approximation for the Generalized Steiner Forest (the running time in this case has an additive weak-O(t^(1 + 2eps)) term, where t is the number of terminals); - O(eps^(-2))-approximation of weighted distances, using node labels of size O(eps^(-1) log n) and weak-O(n^(eps)) bits of memory per node; - O(eps^(-1))-approximation of the weighted diameter; - O(eps^(-3))-approximate shortest paths using the labels 1,...,n.Comment: 40 pages, 2 figures, extended abstract submitted to STOC'1

With Great Speed Come Small Buffers: Space-Bandwidth Tradeoffs for Routing

We consider the Adversarial Queuing Theory (AQT) model, where packet arrivals are subject to a maximum average rate $0\le\rho\le1$ and burstiness $\sigma\ge0$. In this model, we analyze the size of buffers required to avoid overflows in the basic case of a path. Our main results characterize the space required by the average rate and the number of distinct destinations: we show that $O(k d^{1/k})$ space suffice, where $d$ is the number of distinct destinations and $k=\lfloor 1/\rho \rfloor$; and we show that $\Omega(\frac 1 k d^{1/k})$ space is necessary. For directed trees, we describe an algorithm whose buffer space requirement is at most $1 + d' + \sigma$ where $d'$ is the maximum number of destinations on any root-leaf path

A Note on Distributed Stable Matching

We consider the distributed complexity of the stable mar-riage problem. In this problem, the communication graph is undirected and bipartite, and each node ranks its neigh-bors. Given a matching of the nodes, a pair of unmatched nodes is called blocking if they prefer each other to their assigned match. A matching is called stable if it does not induce any blocking pair. In the distributed model, nodes exchange messages in each round over the communication links, until they find a stable matching. We show that if mes-sages may contain at most B bits each, then any distributed algorithm that solves the stable marriage problem requires Ω( n/B log n) communication rounds in the worst case, even for graphs of diameter O(log n), where n is the num-ber of nodes in the graph. Furthermore, the lower bound holds even if we allow the output to contain O( n) block-ing pairs. We also consider ε-stability, where a pair is called ε-blocking if they can improve the quality of their match by more than an ε fraction, for some 0 ≤ ε ≤ 1. Our lower bound extends to ε-stability where ε is arbitrarily close to 1/2. We also present a simple distributed algorithm for ε-stability whose time complexity is O(n/ε).

Randomized Proof-Labeling Schemes

International audienceA proof-labeling scheme, introduced by Korman, Kutten and Peleg [PODC 2005], is a mechanism enabling to certify the legality of a network configuration with respect to a boolean predicate. Such a mechanism finds applications in many frameworks, including the design of fault-tolerant distributed algorithms. In a proof-labeling scheme, the verification phase consists of exchanging labels between neighbors. The size of these labels depends on the network predicate to be checked. There are predicates requiring large labels, of poly-logarithmic size (e.g., MST), or even polynomial size (e.g., Symmetry). In this paper, we introduce the notion of randomized proof-labeling schemes. By reduction from deterministic schemes, we show that randomization enables the amount of communication to be exponentially reduced. As a consequence, we show that checking any network predicate can be done with probability of correctness as close to one as desired by exchanging just a logarithmic number of bits between neighbors. Moreover, we design a novel space lower bound technique that applies to both deterministic and randomized proof-labeling schemes. Using this technique, we establish several tight bounds on the verification complexity of classical distributed computing problems, such as MST construction, and of classical predicates such as acyclicity, connectivity, and cycle length

On the Probe Complexity of Local Computation Algorithms

In the Local Computation Algorithms (LCA) model, the algorithm is asked to compute a part of the output by reading as little as possible from the input. For example, an LCA for coloring a graph is given a vertex name (as a "query"), and it should output the color assigned to that vertex after inquiring about some part of the graph topology using "probes"; all outputs must be consistent with the same coloring. LCAs are useful when the input is huge, and the output as a whole is not needed simultaneously. Most previous work on LCAs was limited to bounded-degree graphs, which seems inevitable because probes are of the form "what vertex is at the other end of edge i of vertex v?". In this work we study LCAs for unbounded-degree graphs. In particular, such LCAs are expected to probe the graph a number of times that is significantly smaller than the maximum, average, or even minimum degree. We show that there are problems that have very efficient LCAs on any graph - specifically, we show that there is an LCA for the weak coloring problem (where a coloring is legal if every vertex has a neighbor with a different color) that uses log^* n+O(1) probes to reply to any query. As another way of dealing with large degrees, we propose a more powerful type of probe which we call a strong probe: given a vertex name, it returns a list of its neighbors. Lower bounds for strong probes are stronger than ones in the edge probe model (which we call weak probes). Our main result in this model is that roughly Omega(sqrt{n}) strong probes are required to compute a maximal matching. Our findings include interesting separations between closely related problems. For weak probes, we show that while weak 3-coloring can be done with probe complexity log^* n+O(1), weak 2-coloring has probe complexity Omega(log n/log log n). For strong probes, our negative result for maximal matching is complemented by an LCA for (1-epsilon)-approximate maximum matching on regular graphs that uses O(1) strong probes, for any constant epsilon>0

Stable Secretaries

We define and study a new variant of the secretary problem. Whereas in the classic setting multiple secretaries compete for a single position, we study the case where the secretaries arrive one at a time and are assigned, in an on-line fashion, to one of multiple positions. Secretaries are ranked according to talent, as in the original formulation, and in addition positions are ranked according to attractiveness. To evaluate an online matching mechanism, we use the notion of blocking pairs from stable matching theory: our goal is to maximize the number of positions (or secretaries) that do not take part in a blocking pair. This is compared with a stable matching in which no blocking pair exists. We consider the case where secretaries arrive randomly, as well as that of an adversarial arrival order, and provide corresponding upper and lower bounds.Comment: Accepted for presentation at the 18th ACM conference on Economics and Computation (EC 2017

Randomized proof-labeling schemes

International audienceProof-labeling schemes, introduced by Korman et al. (Distrib Comput 22(4):215–233, 2010. https://doi.org/10.1007/s00446-010-0095-3), are a mechanism to certify that a network configuration satisfies a given boolean predicate. Such mechanisms find applications in many contexts, e.g., the design of fault-tolerant distributed algorithms. In a proof-labeling scheme, predicate verification consists of neighbors exchanging labels, whose contents depends on the predicate. In this paper, we introduce the notion of randomized proof-labeling schemes where messages are randomized and correctness is probabilistic. We show that randomization reduces verification complexity exponentially while guaranteeing probability of correctness arbitrarily close to one. We also present a novel message-size lower bound technique that applies to deterministic as well as randomized proof-labeling schemes. Using this technique, we establish several tight bounds on the verification complexity of MST, acyclicity, connectivity, and longest cycle size