B-trees with relaxed balance

B-trees with relaxed balance have been deened to facilitate fast updating in a concurrent database environment. In that structure, updating and rebalancing are uncoupled such that extensive locking can be avoided in connection with updates. Constraints, weaker than the usual ones, are maintained such that the tree can still be balanced independent of the updating processes. We nd the idea of relaxed balance very promising. However, the original proposal suuers from a number of problems, the worst of which is the lack of a proof of complexity. >From previous work on relaxed data structures, it is known that the choice of constraints and conditions under which the diierent rebalancing operations can be carried out is critical for the overall complexity. Small alterations in the deenitions can change the overall complexity with an order of magnitude. In this paper, we deene a modiied set of rebalancing operations, and prove that each update gives rise to at most blog a (N=2)c + 1 rebalancing operations, where a is the degree of the B-tree, and N is the bound on its maximal size since it was last in balance. In addition to the logarithmic bounds, we also prove that rebalancing can be performed in amortized constant time. So, in the long run, rebalancing is constant time on average, even if any particular update could give rise to logarithmic time rebalancing. We also prove that the amount of rebalancing done at any particular level decreases exponentially going from the leaves towards the root. This is important since locking close to the root can signiicantly reduce the amount of parallelism possible in the system. Though the object of interest to us here is the B-tree, the results are in fact obtained for the more general (a; b)-trees, so we have results for both of the common B-tree versions as well as 2-3 trees and 2-3-4 trees.