Communication Complexity: Basics

Symmetry-breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open. In this article we work in the LOCAL model (where the input graph and underlying distributed network are identical) and study the randomized complexity of four fundamental symmetry-breaking problems on graphs: computing MISs (maximal independent sets), maximal matchings, vertex colorings, and ruling sets. A small sample of our results includes the following: —An MIS algorithm running in O(log2Δ + 2o(√log log n)) time, where Δ is the maximum degree. This is the first MIS algorithm to improve on the 1986 algorithms of Luby and Alon, Babai, and Itai, when log n L Δ L 2√log n, and comes close to the Ω(log Δ / log log Δ lower bound of Kuhn, Moscibroda, and Wattenhofer. —A maximal matching algorithm running in O(log Δ + log 4log n) time. This is the first significant improvement to the 1986 algorithm of Israeli and Itai. Moreover, its dependence on Δ is nearly optimal. —A (Δ + 1)-coloring algorithm requiring O(log Δ + 2o(√log log n) time, improving on an O(log Δ + √log n)-time algorithm of Schneider and Wattenhofer. —A method for reducing symmetry-breaking problems in low arboricity/degeneracy graphs to low-degree graphs. (Roughly speaking, the arboricity or degeneracy of a graph bounds the density of any subgraph.) Corollaries of this reduction include an O(√log n)-time maximal matching algorithm for graphs with arboricity up to 2√log n and an O(log 2/3n)-time MIS algorithm for graphs with arboricity up to 2(log n)1/3. Each of our algorithms is based on a simple but powerful technique for reducing a randomized symmetry-breaking task to a corresponding deterministic one on a poly(log n)-size graph.

https://www.cs.bgu.ac.il/~elkinm/symmetry_pettie.pdf

The Locality of Distributed Symmetry Breaking

Computational and biological systems are often distributed so that processors (cells) jointly solve a task, without any of them receiving all inputs or observing all outputs. Maximal independent set (MIS) selection is a fundamental distributed computing procedure that seeks to elect a set of local leaders in a network. A variant of this problem is solved during the development of the fly's nervous system, when sensory organ precursor (SOP) cells are chosen. By studying SOP selection, we derived a fast algorithm for MIS selection that combines two attractive features. First, processors do not need to know their degree; second, it has an optimal message complexity while only using one-bit messages. Our findings suggest that simple and efficient algorithms can be developed on the basis of biologically derived insights.

/pdf/a-biological-solution-to-a-fundamental-distributed-computing-4qt00mobu6.pdf

A Biological Solution to a Fundamental Distributed Computing Problem

In this study, random incomplete coloring (RIC) with low time-complexity and high spatial reuse is proposed to overcome in-between wireless-body-area-networks (WBAN) interference, which can cause serious throughput degradation and energy waste. Interference-avoidance scheduling of wireless networks can be modeled as a problem of graph coloring. For instance, high spatial-reuse scheduling for a dense sensor network is mapped to high spatial-reuse coloring; fast convergence scheduling for a mobile ad hoc network (MANET) is mapped to low time-complexity coloring. However, for a dense and mobile WBAN, inter-WBAN scheduling (IWS) should simultaneously satisfy both of the following requirements: 1) high spatial-reuse and 2) fast convergence, which are tradeoffs in conventional coloring. By relaxing the coloring rule, the proposed distributed coloring algorithm RIC avoids this tradeoff and satisfies both requirements. Simulation results verify that the proposed coloring algorithm effectively overcomes inter-WBAN interference and invariably supports higher system throughput in various mobile WBAN scenarios compared to conventional colorings.

https://ir.nctu.edu.tw:443/bitstream/11536/20778/1/000312839100004.pdf

Coloring-Based Inter-WBAN Scheduling for Mobile Wireless Body Area Networks

Exploring the similarities and differences between distributed computations in biological and computational systems.

Distributed information processing in biological and computational systems

We present a randomized distributed maximal independent set (MIS) algorithm for arbitrary graphs of size n that halts in time O(log n) with probability 1 − o(n −1 ), and only needs messages containing 1 bit. Thus, its bit complexity par channel is O(log n). We assume that the graph is anonymous: unique identities are not available to distinguish between the processes; we only assume that each vertex distinguishes between its neighbours by locally known channel names. Furthermore we do not assume that the size (or an upper bound on the size) of the graph is known. This algorithm is optimal (modulo a multi- plicative constant) for the bit complexity and improves the best previous randomized distributed MIS algorithms (deducedfromtherandomizedPRAMalgorithmduetoLuby (SIAM J. Comput. 15:1036-1053, 1986)) for general graphs which is O(log 2 n) per channel (it halts in time O(log n)

An optimal bit complexity randomized distributed MIS algorithm

We present a randomised distributed maximal independent set (MIS) algorithm for arbitrary graphs of size $n$ that halts in time $O(\log n)$ with probability $1-o(n^{-1})$, each message containing $1$ bit: thus its bit complexity per channel is $O(\log n)$ (the bit complexity is the number of bits we need to solve a distributed task, it measures the communication complexity). We assume that the graph is anonymous: unique identities are not available to distinguish the processes; we only assume that each vertex distinguishes between its neighbours by locally known channel names. Furthermore we do not assume that the size (or an upper bound on the size) of the graph is known. This algorithm is optimal (modulo a multiplicative constant) for the bit complexity and improves the best previous randomised distributed MIS algorithms (deduced from the randomised PRAM algorithm due to Luby) for general graphs which is $O(\log^2 n)$ per channel (it halts in time $O(\log n)$ and the size of each message is $\log n$). This result is based on a powerful and general technique for converting unrealistic exchanges of messages containing real numbers drawn at random on each vertex of a network into exchanges of bits. Then we consider a natural question: what is the impact of a vertex inclusion in the MIS on distant vertices? We prove that this impact vanishes rapidly as the distance grows for bounded-degree vertices. We provide a counter-example that shows this result does not hold in general. We prove also that these results remain valid for Luby's algorithm presented by Lynch and by Wattenhofer. This question remains open for the variant given by Peleg.

An optimal bit complexity randomised distributed MIS algorithm

We present and analyse a very simple randomised distributed vertex colouring algorithm for arbitrary graphs of size n that halts in time O(logn) with probability 1-o(n^-^1). Each message containing 1 bit, its bit complexity per channel is O(logn). From this algorithm, we deduce and analyse a randomised distributed vertex colouring algorithm for arbitrary graphs of maximum degree @D and size n that uses at most @D+1 colours and halts in time O(logn) with probability 1-o(n^-^1). We also obtain a partition algorithm for arbitrary graphs of size n that builds a spanning forest in time O(logn) with probability 1-o(n^-^1). We study some parameters such as the number, the size and the radius of trees of the spanning forest.

About randomised distributed graph colouring and graph partition algorithms

In this paper we introduce and study a handshake algorithm based on random delays. This algorithm can also be considered as a probabilistic distributed algorithm to find a (maximal) matching. Each process of the network generates a random time, chosen uniformly in the real interval [0,1] for each of its neighboring processes. The generated number is supposed to be a possible time for a handshake, if both processes are available at that time. Initially the number of potentially possible times proposed by the processes is twice the number of links between them. Whenever the clock reaches the smallest generated item, there will be a handshake between the process that proposes this time and the requested process and they cancel all other data from their timetable, informing their other neighboring processes, which in turn remove the times proposed by the two processes. The process continues in the remaining processes equipped with the updated timetable until the unit time 1 has expired. This algorithm is substantially more efficient than previous ones known to us, in that the expected number of handshakes per round is larger.

Analysis of a Randomized Dynamic Timetable Handshake Algorithm

We design and analyze a randomized one-passage election algorithm in trees based on a result of Angluin in [Proceedings of the 12th Symposium on Theory of Computing, 1980, pp. 82]. The election process is a distributed elimination scheme which removes leaves one-by-one reducing the tree to a single vertex, called the leader (or elected vertex). We define a locally computable parameter guiding randomly the elimination process. As a particular instance, we provide a parameter assignment in a Markovian type random process in which all vertices have the same chance of being elected.

Nasser Saheb-Djahromi

Papers

An optimal bit complexity randomized distributed MIS algorithm

An optimal bit complexity randomised distributed MIS algorithm

About randomised distributed graph colouring and graph partition algorithms

Analysis of a Randomized Dynamic Timetable Handshake Algorithm

Locally guided randomized elections in trees: The totally fair case