Section: New Results

Distributed computational complexities

Local Distributed Decision

Participants : Pierre Fraigniaud [CNRS LIAFA, University of Paris Diderot, France] , Amos Korman [CNRS LIAFA, University of Paris Diderot, France] , David Peleg [Weizmann Institute of Science, Israel] .

Inspired by sequential complexity theory, in [20] we focus on a complexity theory for distributed decision problems. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Our main result provides a sharp threshold for the impact of randomization on decision hereditary problems. In addition, we investigate the impact of non-determinism on local decision, and establish some structural results inspired by classical computational complexity theory. Specifically, we show that non-determinism does help, but that this help is limited, as there exist languages that cannot be decided non-deterministically. Perhaps surprisingly, it turns out that it is the combination of randomization with non-determinism that enables to decide all languages in constant time. Finally, we introduce the notion of local reduction, and establish some completeness results

Asynchronous Wait-free Decision

Participants : Pierre Fraigniaud [CNRS LIAFA, University of Paris Diderot, France] , Sergio Rajsbaum [Maths. Institute, University of Mexico, Mexico] , Corentin Travers [Technion, Israel] .

In order to capture the core of asynchronous distributed decision model, we address in [22] the wait-free model with crash failures. The set of tasks whose input is a pair $(s,t)$ and deciding whether $t\in \Delta \left(s\right)$, i.e. whether $t$ is a valid output for $s$, has been proven to be decidable in this model.

Mobile Distributed Decision

Participants : Pierre Fraigniaud [CNRS LIAFA, University of Paris Diderot, France] , Andrzej Pelc [UQO, University of Quebec, Canada] .

In [21] , we partially answer the question of decidability of any language for mobile agents in a 2D environment like telecom networks or robots. It is proven that, for every agent, verifying whether (i) he/she is alone or not and (ii) he/she is able to capture the environment, is associated with the question of pertaining to an equivalence class of a map. A positive answer helps in the non-deterministic decision for any language for mobile agent.

Approximating the Statistics of various Properties in Randomly Weighted Graphs

Participants : Yuval Emek [University of Tel Aviv, Israel] , Amos Korman [CNRS LIAFA, University of Paris Diderot, France] , Yuval Shavitt [University of Tel Aviv, Israel] .

In [19] , we consider the setting of randomly weighted graphs. Under this setting, weighted graph properties typically become random variables and we are interested in computing their statistical features. Unfortunately, this turns out to be computationally hard for some weighted graph properties albeit the problem of computing the properties per se in the traditional setting of algorithmic graph theory is tractable. For example, there are well known efficient algorithms that compute the diameter of a given weighted graph, yet, computing the expected diameter of a given randomly weighted graph is ♯P-hard even if the edge weights are identically distributed. In this paper, we define a family of weighted graph properties and show that for each property in this family, the problem of computing the $k$'th moment (and in particular, the expected value) of the corresponding random variable in a given randomly weighted graph $G$ admits a fully polynomial time randomized approximation scheme (FPRAS) for every fixed $k$. This family includes fundamental weighted graph properties such as the diameter of $G$, the radius of $G$ (with respect to any designated vertex) and the weight of a minimum spanning tree of $G$.

New bounds for the controller problem

Participants : Yuval Emek [University of Tel Aviv, Israel] , Amos Korman [CNRS LIAFA, University of Paris Diderot, France] .

In [10] , we establish two new lower bounds on the message complexity of the controller problem. We first prove a simple lower bound stating that any $(M,W)$-controller must send $\Omega (Nlog\frac{M}{W+1})$ messages. Second, for the important case when $W$ is proportional to $M$ (this is the common case in most applications), we use a surprising reduction from the (centralized) monotonic labeling problem to show that any $(M,W)$-controller must send $\Omega (NlogN)$ messages. In fact, under a long lasting conjecture regarding the complexity of the monotonic labeling problem, this lower bound is improved to a tight $\Omega \left(N{log}^{2}N\right)$.

Online computation with advice

Participants : Yuval Emek [University of Tel Aviv, Israel] , Pierre Fraigniaud [CNRS LIAFA, University of Paris Diderot, France] , Amos Korman [CNRS LIAFA, University of Paris Diderot, France] , Adi Rosén [CNRS LIAFA, University of Paris Diderot, France] .

In [9] , we consider a model for online computation in which the online algorithm receives, together with each request, some information regarding the future, referred to as advice. We are interested in the impact of such advice on the competitive ratio, and in particular, in the relation between the size $b$ of the advice, measured in terms of bits of information per request, and the (improved) competitive ratio. In this paper we propose the above model and illustrate its applicability by considering two of the most extensively studied online problems, namely, metrical task systems (MTS) and the $k$-server problem.

Tight Bounds For Distributed MST Verification

Participants : Liah Kor [Weizmann Institute of Science, Israel] , Amos Korman [CNRS LIAFA, University of Paris Diderot, France] , David Peleg [Weizmann Institute of Science, Israel] .

In [26] , we establishes tight bounds for the Minimum-weight Spanning Tree (MST) verification problem in the distributed setting. Specifically, we provide an MST verification algorithm that achieves simultaneously $\tilde{O}\left(\right|E\left|\right)$ messages and $\tilde{O}(\sqrt{n}+D)$ time, where $\left|E\right|$ is the number of edges in the given graph $G$ and $D$ is $G$'s diameter. On the negative side, we show that any MST verification algorithm must send $\Omega \left(\right|E\left|\right)$ messages and incur $\tilde{\Omega }(\sqrt{n}+D)$ time in worst case. Our upper bound result appears to indicate that the verification of an MST may be easier than its construction.

Distributed verification and hardness of distributed approximation

Participants : Atish Das Sarma [Google research, USA] , Stephan Holzer [ETH, Zurich, Switzerland] , Liah Kor [Weizmann Institute of Science, Israel] , Amos Korman [CNRS LIAFA, University of Paris Diderot, France] , Danupon Nanongkai [Nanyang Technological University, Singapore] , David Peleg [Weizmann Institute of Science, Israel] , Roger Wattenhofer [ETH, Zurich, Switzerland] .

In [30] , we initiate a systematic study of distributed verification, and give almost tight lower bounds on the running time of distributed verification algorithms for many fundamental problems such as connectivity, spanning connected subgraph, and $s-t$ cut verification. We then show applications of these results in deriving strong unconditional time lower bounds on the hardness of distributed approximation for many classical optimization problems including minimum spanning tree, shortest paths, and minimum cut. Many of these results are the first non-trivial lower bounds for both exact and approximate distributed computation and they resolve previous open questions. Moreover, our unconditional lower bound of approximating minimum spanning tree (MST) subsumes and improves upon the previous hardness of approximation bound of Elkin [STOC 2004] as well as the lower bound for (exact) MST computation of Peleg and Rubinovich [FOCS 1999]. Our result implies that there can be no distributed approximation algorithm for MST that is significantly faster than the current exact algorithm, for any approximation factor. Our lower bound proofs show an interesting connection between communication complexity and distributed computing which turns out to be useful in establishing the time complexity of exact and approximate distributed computation of many problems.