Balanced allocations for tree-like inputs

Abstract

We consider the following procedure for constructing a directed tree on n vertices: The underlying undirected tree is fixed in advance but the edges of the tree are presented in a random order (all orders are equally likely); each edge is oriented towards its endpoint that has the lower indegree at the time of its insertion. The question is what is E( M( n)), the expected maximum indegree? As we shall explain, this problem has connections with balanced allocations and with on-line load balancing. Previous results by Azar, Naor, and Rom imply that if the insertion order is arbitrary, for any tree, M( n) = O( log n) and that there are trees and insertion orders for which M(n) = Ω(log n). On the other hand, results by Azar, Broder, Karlin, and Upfal imply that if both the underlying tree and the insertion order are random, then E( M( n)) = Θ( log log n). Here we show an intermediate result: for any tree if the insertion order is random, then E(M(n)) = O( log n log log n ) and there are trees for which E(M(n)) = Ω( log n log log n ) .