On Updating and Balancing Relaxed Balanced Search Trees in Main Memory

In this thesis, various methods for maintaining balanced search trees in main memory are considered. The work concentrates on algorithms applying relaxed balancing in which updates and rebalancing are uncoupled from each other. A new relaxed balanced tree, the rank-valued tree, which is a relaxed version of symmetric binary B-trees, also known as red-black trees, is introduced and analyzed. Rank-valued trees have the following benefits compared with chromatic trees, an older relaxed version of redblack trees. First, no rebalancing transformation will make the tree less balanced. Second, no rotations are needed if the tree is initially in balance. New algorithms for rebalancing relaxed balanced trees are given. The algorithm R1 rebalances a chromatic tree after M updates in O(M log(N+M)) time and using O(N+M) extra space, where N is the size of the initial tree. The algorithm R2 rebalances a heightvalued tree that is a relaxed AVL tree, in O(M logN) time, and it requires O(logN) extra space. A group-update operation means that a large number of records is inserted or deleted at one time. In this thesis, a new method of implementing group updates is discussed. Contrary to previous methods, in which balancing was carried out during the updates, in the new method balancing is postponed. This is important because in this way the updates are immediately accessible and there is no delay due to the rebalancing of the tree. The results imply that if M records are inserted below p leaves, denoted by r1, r2, . . . , rp, in an AVL tree T , the total time for performing the updates and rebalancing the tree is O(M + logN + ∑p−1 i=1 log di) in which di denotes the distance of the nodes ri and ri+1 in T , i.e., the number of leaves between them. A similar bound is obtained when deletions are allowed. Previously, the same bounds have been obtained for the method in which updates and rebalancing are tightly coupled. However, the bounds presented in the thesis also hold in the case when an arbitrary combination of insertions and deletions is performed, a result not obtained previously. Experimental tests have been performed using the presented algorithms. The results imply that group operations can speed up the performance considerably compared with standard search and update operations. An observation was made that the memory layout of the tree is very important for the behavior of balanced search trees. Thus, the efficiency of a simple tree searching algorithm can be increased by a factor of two when nodes that are logically close in the tree are physically close in the memory. The main reason for gaining efficiency is the reduced thrashing of cache memories. This implies that many tree algorithms should be redesigned to maintain the locality of the data structure.