Parallel Load Balancing on constrained client-server topologies

We study parallel Load Balancing protocols for the client-server distributed model defined as follows. There is a set C of n clients and a set S of n servers where each client has (at most) a constant number d ⩾ 1 of requests that must be assigned to some server. The client set and the server one are connected to each other via a fixed bipartite graph: the requests of client v can only be sent to the servers in its neighborhood N ( v ). The goal is to assign every client request so as to minimize the maximum load of the servers. In this setting, efficient parallel protocols are available only for dense topologies. In particular, a simple protocol, named raes, has been recently introduced by Becchetti et al. [1] for regular dense bipartite graphs. They show that this symmetric, non-adaptive protocol achieves constant maximum load with parallel completion time O ( log ⁡ n ) and overall work O ( n ), w. h. p. Motivated by proximity constraints arising in some client-server systems, we analyze raes over almost-regular bipartite graphs where nodes may have neighborhoods of small size. In detail, we prove that, w. h. p. , the raes protocol keeps the same performances as above (in terms of maximum load, completion time, and work complexity, respectively) on any almost-regular bipartite graph with degree Ω ( log 2 ⁡ n ). Our analysis significantly departs from that in [1] since it requires to cope with non-trivial stochastic-dependence issues on the random choices of the algorithmic process which are due to the worst-case, sparse topology of the underlying graph.

Authors

• Andrea Clementi
• Emanuele Natale
• Isabella Ziccardi

Keywords

• Parallel balanced allocations
• Balls-into-bins processes
• Randomized algorithms