Fast Space Optimal Leader Election in Population Protocols

The model of population protocols refers to the growing in popularity theoretical framework suitable for studying pairwise interactions within a large collection of simple indistinguishable entities, frequently called agents. In this paper the emphasis is on the space complexity in fast leader election via population protocols governed by the random scheduler, which uniformly at random selects pairwise interactions within the population of n agents. The main result of this paper is a new fast and space optimal leader election protocol. The new protocol utilises O(log^2 n) parallel time (which is equivalent to O(n log^2 n) sequential pairwise interactions), and each agent operates on O(log log n) states. This double logarithmic space usage matches asymptotically the lower bound 1/2 log log n on the minimal number of states required by agents in any leader election algorithm with the running time o(n/polylog n). Our solution takes an advantage of the concept of phase clocks, a fundamental synchronisation and coordination tool in distributed computing. We propose a new fast and robust population protocol for initialisation of phase clocks to be run simultaneously in multiple modes and intertwined with the leader election process. We also provide the reader with the relevant formal argumentation indicating that our solution is always correct, and fast with high probability.Comment: 21 pages, 2 figures, published in SODA 2018 proceeding

Self-stabilizing sorting on linear networks

A self-stabilizing system has the ability to recover from an arbitrary (possibly faulty) state to a normal state without any manual intervention. A self-stabilizing algorithm does not require any initialization. Starting from an arbitrary state, it is guaranteed to satisfy its specification in finite number of steps; We propose a self-stabilizing distributed sorting algorithm on an oriented linear network with n nodes. Each node holds some initial value(s) drawn from an arbitrary set. We assume that we start with at most k items in the network. Each node has a local memory whose space is restricted to O(k * L ) where L is the maximum number of bits to store one item. A node may collect more than one value during the process of sorting. The stabilizing time for sorting is O(n) rounds where a round is the duration for all the enabled processes to execute at least one enabled step. We claim that our algorithm is self-stabilizing for the following reasons:;If any node starts in a faulty state (meaning its value is not sorted with respect to its neighbors), the algorithm guarantees that the node will reach the legitimate state (where a legitimate state is a state in which the values are in sorted order) in a finite amount of time, and will remain in the legitimate state until another fault occurs. Each node repeatedly communicates with its neighbors to check if the values of its neighbors are sorted with respect to its own value. If the values are not in order, either the node or one of its neighbors will eventually be enabled to execute so that in finite amount of time the values will be sorted

Almost Logarithmic-Time Space Optimal Leader Election in Population Protocols

The model of population protocols refers to a large collection of simple indistinguishable entities, frequently called {\em agents}. The agents communicate and perform computation through pairwise interactions. We study fast and space efficient leader election in population of cardinality $n$ governed by a random scheduler, where during each time step the scheduler uniformly at random selects for interaction exactly one pair of agents. We propose the first $o(\log^2 n)$-time leader election protocol. Our solution operates in expected parallel time $O(\log n\log\log n)$ which is equivalent to $O(n \log n\log\log n)$ pairwise interactions. This is the fastest currently known leader election algorithm in which each agent utilises asymptotically optimal number of $O(\log\log n)$ states. The new protocol incorporates and amalgamates successfully the power of assorted {\em synthetic coins} with variable rate {\em phase clocks}