Affiliation:
1. University of Manitoba, Manitoba, Canada
2. Université du Québec en Outaouais, Gatineau, Quebec, Canada
3. Institute of Infrastructure Technology Research and Management, Ahmedabad, Gujarat, India
Abstract
Leader election is a fundamental task in distributed computing. It is a symmetry breaking problem, calling for one node of the network to become the
leader
, and for all other nodes to become
non-leaders
. We consider leader election in anonymous radio networks modeled as simple undirected connected graphs. Nodes communicate in synchronous rounds. In each round, a node can either transmit a message to all its neighbours, or stay silent and listen. A node
v
hears a message from a neighbour
w
in a given round if
v
listens in this round and if
w
is its only neighbour transmitting in this round. If
v
listens in a round in which more than one neighbour transmits, then
v
hears noise that is different from any message and different from silence.
We assume that nodes are identical (anonymous) and execute the same deterministic algorithm. Under this scenario, symmetry can be broken only in one way: by different wake-up times of the nodes. In which situations is it possible to break symmetry and elect a leader using time as symmetry breaker? In order to answer this question, we consider
configurations
. A configuration is the underlying graph with nodes tagged by non-negative integers with the following meaning. A node can either wake up spontaneously in the round shown on its tag, according to some global clock, or can be woken up hearing a message sent by one of its already awoken neighbours. The local clock of a node starts at its wakeup and nodes do not have access to the global clock determining their tags. A configuration is
feasible
if there exists a distributed algorithm that elects a leader for this configuration.
Our main result is a complete algorithmic characterization of feasible configurations. More precisely, we design a centralized decision algorithm, working in polynomial time, whose input is a configuration and which decides if the configuration is feasible. Using this algorithm we also provide a dedicated deterministic distributed leader election algorithm for each feasible configuration that elects a leader for this configuration in time
O
(
n
2
σ, where
n
is the number of nodes and σ is the difference between the largest and smallest tag of the configuration. We then ask the question whether there exists a universal deterministic distributed algorithm electing a leader for all feasible configurations. The answer turns out to be no, and we show that such a universal algorithm cannot exist even for the class of 4-node feasible configurations. We also prove that a distributed version of our decision algorithm cannot exist.
Funder
Natural Sciences and Engineering Research Council of Canada
Research Chair in Distributed Computing of the Université du Québec en Outaouais
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Reference42 articles.
1. Local and global properties in networks of processors (Extended Abstract)
2. Better computing on the anonymous ring
3. Computing on an anonymous ring
4. On the time-complexity of broadcast in multi-hop radio networks: An exponential gap between determinism and randomization
5. Paolo Boldi, Shella Shammah, Sebastiano Vigna, Bruno Codenotti, Peter Gemmell, and Janos Simon. 1996. Symmetry breaking in anonymous networks: Characterizations. In Fourth Israel Symposium on Theory of Computing and Systems, ISTCS 1996, Jerusalem, Israel, June 10–12, 1996, Proceedings. IEEE Computer Society, 16–26.