Deterministic distributed resource discovery (brief announcement)

ABSTRACT

The resource discovery problem was introduced by Harchol-Balter, Leighton and Lewin in [HLL99], as a part of their work on web caching. They developed a randomized algorithm for the problem in the weakly connected directed graph model, that was implemented within LCS at MIT, and then licensed to Akamai Technologies. The directed graph is a logical one (as opposed to the underlying communication graph). It represents the nodes' “knowledge” about the topology of the underlying communication network. In [HLL99] the notion of topology “knowledge” is simplified, by modeling it as a “knowledge” of an ID of another node. In the general case such a “knowledge” may a include whole route, as well as any additional information needed in order to establish a connection (e.g. a cryptographic public key).It is assumed (here, and in [HLL99]) that the logical graph G is weakly connected. This can result from topology changes: e.g., a loss of a connection to a name server, or a gain of new knowledge that is uneven between different nodes, since it is obtained by distributed algorithms, and in non-homogeneous network. Note that weak connectivity is a necessary condition for the solvability of the problem. Dealing efficiently with the weakly connected graph was, in fact, the main contribution in [HLL99]. The alternative of transforming the graph into an undirected one, and then solving the problem on the resulting undirected graph, is possible. If E0 = O(n) it even leads to efficient solutions: A time optimal algorithm (for undirected graphs) appears in [SV82], It was designed for CRCW PRAM, but can be translated into the model of [HLL99] using O(|E0| log n) messages. The assumption that |E0| = O(n) may be suitable if topology changes are assumed to be limited. Many practical distributed systems, however, attempt to deal with the case that the network may be partitioned. Thus, E0 should be as big as possible, to enable the disconnected components to regain connectivity. The current paper proposes a deterministic algorithm for the problem in the same model as that of [HLL99]. Our algorithm is near optimal in all the measures: time, message, and communication complexities. Each previous algorithm had a complexity that was higher at least in one of the measures. Specifically, previous deterministic solutions required either time linear in the diameter of the initial network, or communication complexity O(n3) (with message complexity O(n2)), or message complexity O(|E0| log n) (where E0 is the edge set of the initial graph). Compared to the main randomized algorithm of the Harchol-Balter, Leighton, and Lewin, the time complexity is reduced from O(log2n) to O(log n log* n), the message complexity from O(n log 2n) to O(n log n log * n), and the communication complexity from O(n2 log3 n) to O(|E0|log2 n + n2 log n).

REFERENCES

• HLL99: Mor Harchol-Balter , Tom Leighton , Daniel Lewin, Resource discovery in distributed networks, Proceedings of the eighteenth annual ACM symposium on Principles of distributed computing, p.229-237, May 04-06, 1999, Atlanta, Georgia, United States [doi>10.1145/301308.301362]
• SV82: Yossi Shiloach , Uzi Vishkin, An O(n2 log n) parallel max-flow algorithm, Journal of Algorithms, v.3 n.2, p.128-146, Feb. 1982 [doi>10.1016/0196-6774(82)90013-X]

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.2 Graph Theory
          Subjects: Network problems

General Terms:
Algorithms, Design