How Long It Takes for an Ordinary Node with an Ordinary ID to Output?

In the context of distributed synchronous computing, processors perform in rounds, and the time-complexity of a distributed algorithm is classically defined as the number of rounds before all computing nodes have output. Hence, this complexity measure captures the running time of the slowest node(s). In this paper, we are interested in the running time of the ordinary nodes, to be compared with the running time of the slowest nodes. The node-averaged time-complexity of a distributed algorithm on a given instance is defined as the average, taken over every node of the instance, of the number of rounds before that node output. We compare the node-averaged time-complexity with the classical one in the standard LOCAL model for distributed network computing. We show that there can be an exponential gap between the node-averaged time-complexity and the classical time-complexity, as witnessed by, e.g., leader election. Our first main result is a positive one, stating that, in fact, the two time-complexities behave the same for a large class of problems on very sparse graphs. In particular, we show that, for LCL problems on cycles, the node-averaged time complexity is of the same order of magnitude as the slowest node time-complexity. In addition, in the LOCAL model, the time-complexity is computed as a worst case over all possible identity assignments to the nodes of the network. In this paper, we also investigate the ID-averaged time-complexity, when the number of rounds is averaged over all possible identity assignments. Our second main result is that the ID-averaged time-complexity is essentially the same as the expected time-complexity of randomized algorithms (where the expectation is taken over all possible random bits used by the nodes, and the number of rounds is measured for the worst-case identity assignment). Finally, we study the node-averaged ID-averaged time-complexity.Comment: (Submitted) Journal versio

Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

Distributed Symmetry Breaking in Hypergraphs

Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and $\Delta +1$-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in {\em hypergraphs}. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs. In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in $O(\log^2 n)$ rounds ($n$ is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper --- an $O(\Delta^{\epsilon}\text{polylog}(n))$-round hypergraph MIS algorithm in the CONGEST model where $\Delta$ is the maximum node degree of the hypergraph and $\epsilon > 0$ is any arbitrarily small constant. To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the {\em balanced minimal dominating set} problem (left open in Harris et al. [ICALP 2013]) and the {\em minimal connected dominating set problem}. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.Comment: Changes from the previous version: More references adde

On the Complexity of Local Distributed Graph Problems

This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and $(\Delta+1)$-vertex coloring), the randomized complexity is at most polylogarithmic in the size $n$ of the network, while the best deterministic complexity is typically $2^{O(\sqrt{\log n})}$. Understanding and narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms. We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in $\log^{O(1)} n$ rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and $(\Delta+1)$-coloring) in $\log^{O(1)} n$ rounds in the LOCAL model. We show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values

Bounds for self-stabilization in unidirectional networks

A distributed algorithm is self-stabilizing if after faults and attacks hit the system and place it in some arbitrary global state, the systems recovers from this catastrophic situation without external intervention in finite time. Unidirectional networks preclude many common techniques in self-stabilization from being used, such as preserving local predicates. In this paper, we investigate the intrinsic complexity of achieving self-stabilization in unidirectional networks, and focus on the classical vertex coloring problem. When deterministic solutions are considered, we prove a lower bound of $n$ states per process (where $n$ is the network size) and a recovery time of at least $n(n-1)/2$ actions in total. We present a deterministic algorithm with matching upper bounds that performs in arbitrary graphs. When probabilistic solutions are considered, we observe that at least $\Delta + 1$ states per process and a recovery time of $\Omega(n)$ actions in total are required (where $\Delta$ denotes the maximal degree of the underlying simple undirected graph). We present a probabilistically self-stabilizing algorithm that uses $\mathtt{k}$ states per process, where $\mathtt{k}$ is a parameter of the algorithm. When $\mathtt{k}=\Delta+1$, the algorithm recovers in expected $O(\Delta n)$ actions. When $\mathtt{k}$ may grow arbitrarily, the algorithm recovers in expected O(n) actions in total. Thus, our algorithm can be made optimal with respect to space or time complexity