We propose a family of constant-degree routing networks of logarithmic diameter, with the additional property that the addition or removal of a node to the network requires no global coordination, only a constant number of linkage changes in expectation, and a logarithmic number with high probability. Our randomized construction improves upon existing solutions, such as balanced search trees, by ensuring that the congestion of the network is always within a logarithmic factor of the optimum with high probability. Our construction derives from recent advances in the study of peer-to-peer lookup networks, where rapid changes require e#cient and distributed maintenance, and where the lookup e#ciency is impacted both by the lengths of paths to requested data and the presence or elimination of bottlenecks in the network.

Abstract. Relaxed balancing means that, in a dictionary stored as a balanced tree, the necessary rebalancing after updates may be delayed. This is in contrast to strict balancing meaning that rebalancing is performed immediately after the update. Relaxed balancing is important for efficiency in highly dynamic applications where updates can occur in bursts. The rebalancing tasks can be performed gradually after all urgent updates, allowing the concurrent use of the dictionary even though the underlying tree structure is not completely in balance. In this paper we propose a new scheme of how to make known rebalancing techniques relaxed in an efficient way. The idea is applied to the red-black trees, but can be applied to any class of balanced trees. The key idea is to accumulate insertions and deletions such that they can be settled in arbitrary order using the same rebalancing operations as for standard balanced search trees. As a result it can be shown that the number of needed rebalancing operations known from the strict balancing scheme carry over to relaxed balancing. 1

First we present a generalization of symmetric binary B-trees, SBB(k)- trees. The obtained structure has a height of only \Sigma (1 + 1k) log(n + 1)\Upsilon, where k may be chosen to be any positive integer. The maintenance algorithms require only a constant number of rotations per updating operation in the worst case. These properties together with the fact that the structure is relatively simple to implement makes it a useful alternative to other search trees in practical applications. Then, by using an SBB(k)-tree with a varying k we achieve a structure with a logarithmic amortized cost per update and a height of log n + o(log n). This result is an improvement of the upper bound on the height of a dynamic binary search tree. By maintaining two trees simultaneously the amortized cost is transformed into a worstcase cost. Thus, we have improved the worst-case complexity of the dictionary problem.

We present lower and upper bounds on adaptive heuristics for maintaining binary search trees using a constant number of link or pointer changes for each operation (constant linkage cost (CLC)). We show that no adaptive heuristic with an amortized linkage cost of o(log n) can be competitive. In particular, we show that any heuristic that performs f(n) = o(log n) promotions (rotations) amortized over each access has a competitive ratio of at least \Omega\Gammaast n=f(n)) against an oblivious adversary, and any heuristic that performs f(n) = o(log n) pointer changes amortized over each access has a competitive ratio of at least\Omega\Gamma log n f(n) log(log n=f(n)) ) against an adaptive online adversary. In our investigation of upper bounds we present four adaptive heuristics: ffl A randomized, worst-case-CLC heuristic (R2P) whose expected search time is within a constant factor of the search time using an optimal tree; that is, it is statically competitive ffl A randomized, expecte...

Relaxed balancing means that, in a dictionary stored as a balanced tree, the necessary rebalancing after updates may be delayed. This is in contrast to strict balancing meaning that rebalancing is performed immediately after the update. Relaxed balancing is important for efficiency in highly dynamic applications where updates can occur in bursts. The rebalancing tasks can be performed gradually after all urgent updates, allowing the concurrent use of the dictionary even though the underlying tree structure is not completely in balance. The contribution of the present paper is that we introduce a new scheme for relaxed balancing, which is obtained by a simple generalization of strict balancing. Our approach implies a simple proof of the fact that the number of the needed rebalancing operations (to put the tree in balance) for relaxed balancing is the same as for strict balancing. 1 Introduction A dictionary is a scheme for storing a set of data such that the operations sea...

We propose and analyze a randomized variant of the Butterfly network that is amenable to decentralized construction and modification as nodes join and leave a network. The proposed networks have constant out-degree, and maintain a logarithmic diameter and nearly optimal congestion with high probability. In expectation, only a constant number of links are changed during a modification, and a logarithmic number with high probability. These properties make our construction particularly suitable for distributed hash tables (DHTs), which have been heavily promoted as a building block for internet-scale distributed services.

The balance criterion defining the class of ff-balanced trees states that the ratio between the shortest and longest paths from a node to a leaf be at least ff. We show that a straight-forward use of partial rebuilding for maintenance of ff-balanced trees requires an amortized cost of \Omega\Gamma p n) per update. By slight modifications of the maintenance algorithms the cost can be reduced to O(log n) for any value of ff, 0 ! ff ! 1. KEY WORDS ff-balanced trees, partial rebuilding, search trees. CR CATEGORIES: E.1, F.2, I.1.2. 1 Introduction In his thesis Olivie [9] introduced a class of binary search trees, which he calls ff-balanced trees, or ffBB-trees. Let h(v) denote the length for the longest path from a node v to a leaf and let s(v) denote the length of the shortest path. We give a formal definition of ff-balanced trees below. Definition 1 A binary tree is ff-balanced if the following is true for each node v in the tree: s(v) h(v) ff; h(v) 1 1 \Gamma ff (1) h(v) \...

Consider a dynamic set of points in the plane having different colors. Let ]C and 1C2 be two keys according to which points may be sorted, e.g. the x- and the y-coordinate. A point p C S is called proper with respect to a query point po C S iff ]C(po) < ]C(p), ]C2(po ) < ]C2(p) , and if p has a different color than Po.