Q. F. Stout and B. L. Warren, `Tree rebalancing in optimal time and space', Commun. ACM, 29, (9), 902--908 (1986).  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....section 3.6.2 which inserts all the nodes at the deepest level of the newly built subtree at the leftmost possible positions, instead of spreading them evenly. This simplified the code somewhat and yielded a 6 9 percent speedup over the version described by the pseudo code. Stout and Warren [68] call these route balanced trees. This issue is discussed further in Section 3.11. It is natural to expect that the optimal value for parameter ff should depend on the ratio between the number of searches and the number of modifications in a given sequence of requests to the scapegoat tree. The ....

....distributed, or when the application is search intensive. For the splay trees we used top down splaying as suggested by Sleator and Tarjan [64] The implementation of red black trees follows Chapter 14 in Cormen, Leiserson and Rivest [20] 3. 11 Discussion and Conclusions Stout and Warren [68] present an algorithm which takes an arbitrary binary search tree and rebalances it to form what they call a route balanced tree using linear time and only constant space. This improves upon the logarithmic space required to output a perfectly balanced tree. A route balanced tree is one containing ....

Q. F. Stout and B. L. Warren. Tree rebalancing in optimal time and space. Communications of the ACM, 29(9):902--908, September 1986.

....even though insertions and deletions may be distributed unevenly the tree will not get too far from being AVL balanced. This expectation is achieved by concurrent rebalancers that search continually for places in the tree at which it can perform local rebalancing. There is a different algorithm [30] that takes an arbitrary binary tree and transforms it into a complete binary tree (in which the levels of the leaves differ by at most one) The algorithm always rebuilds the whole tree even when the tree has only some or no balance violations; it cannot be used efficiently in a concurrent ....

Q. F. Stout and B. L. Warren. Tree rebalancing in optimal time and space. Communications of the ACM, 29:902--908, 1986. 25

....trees are widely used index structures in applications where the whole index can be stored in main memory. To avoid the bad worst case behaviour of these trees, various strategies have been developed to maintain them in balance. Global balancing algorithms periodically rebuild the whole tree [3, 9, 15]. Balanced trees, e.g. AVL trees and red black trees, perform rebalancing coupled with each update operation [1, 5, 6, 8, 11, 13] These data structures and algorithms were developed and analyzed assuming the Random Access Memory (RAM) model [2] This was a reasonable assumption when memory was ....

....the value pointed to by ptr and addr of item gives the address of item. The syntax ptr field is equivalent to (ref x) field. We assume the tree is perfectly balanced so that all leaf nodes are either on level height or height Gamma 1. If this is not the case, some global rebalancing algorithm [3, 9, 15] should be employed prior to relocation. We also assume we have a routine Allocate which allocates memory for a given amount of nodes so that the allocated area does not cross a page boundary. After the relocation call the old tree can be freed. 4 Count Nodes(tree : addr of Node) 1 if tree = ....

Q.F. Stout, B.L. Warren, Tree Rebalancing in Optimal Time and Space. Communications of the ACM 29:9 (1986), pp. 902--908.

....trees are widely used index structures in applications where the whole index can be stored in main memory. To avoid the bad worst case behaviour of these trees, various strategies have been developed to maintain them in balance. Global balancing algorithms periodically rebuild the whole tree [3, 9, 15]. Balanced trees, e.g. AVL trees and red black trees, perform rebalancing coupled with each update operation [1, 5, 6, 8, 11, 13] These data structures and algorithms were developed and analyzed assuming the Random Access Memory (RAM) model [2] This was a reasonable assumption when memory was ....

....the value pointed to by ptr and addr of item gives the address of item. The syntax ptr field is equivalent to (ref x) field. We assume the tree is perfectly balanced so that all leaf nodes are either on level height or height Gamma 1. If this is not the case, some global rebalancing algorithm [3, 9, 15] should be employed prior to relocation. We also assume we have a routine Allocate which allocates memory for a given amount of nodes so that the allocated area does not cross a page boundary. After the relocation call the old tree can be freed. Count Nodes(tree : addr of Node) 1 if tree = nil ....

Q.F. Stout, B.L. Warren, Tree Rebalancing in Optimal Time and Space. Communications of the ACM 29:9 (1986), pp. 902--908.

....a tree is the path length between the root and a leaf node, averaged over all the leaf nodes. This measure is also used in the experiments in Chapter 5. Although little if any work has been done on balancing decision trees, a great deal of research has considered balanced search trees (e.g. see [16, 79, 461, 93, 357]) Roughly speaking, this literature deals with techniques to restructure search trees when elements are inserted or deleted, in order to restrict the depth of these trees to a logarithmic function of the number of search keys. Examples of balanced search trees include AVL trees, B trees, and ....

Quentin F. Stout and Bette L. Warren. Tree rebalancing in optimal time and space. Communications of the ACM, 29(9):902--908, September 1986.

....in a subtree, we denote the subtree v immediately after the latest time it was rebuilt as v o . The notation h(v o ) jv o j and so forth follows from above. In our analysis we assume that rebuilding of a subtree v takes linear time, examples of linear algorithms for balancing trees are found in [5, 6, 7, 14]. 3 Analysis of Partial Rebuilding The analysis of ff balanced trees maintained by partial rebuilding is somewhat more complicated than the analysis of weight balanced trees. Our analysis of the update cost is based on a computation of how many updates are required to make an initially ....

Q. F. Stout and B. L. Warren. Tree rebalancing in optimal time and space. Communications of the ACM, 29(9):902--908, 1986.

....Even if the amount of restructuring work is constant it might be of some interest to determine how much is actually required. The exact amount depends on the algorithm used to balance pseudo nodes. There are several algorithms for balancing binary search trees which may be used for this purpose [5, 6, 10, 19]. As an example we can study the algorithm given by Stout and Warren [19] This algorithm is not the most efficient one but it uses rotations which makes the balancing cost comparable with the cost required for other classes of search trees. The Stout Warren algorithm works in two steps: 1. A ....

....to determine how much is actually required. The exact amount depends on the algorithm used to balance pseudo nodes. There are several algorithms for balancing binary search trees which may be used for this purpose [5, 6, 10, 19] As an example we can study the algorithm given by Stout and Warren [19]. This algorithm is not the most efficient one but it uses rotations which makes the balancing cost comparable with the cost required for other classes of search trees. The Stout Warren algorithm works in two steps: 1. A right degenerated tree is produced by repeated right rotations. 2. A balanced ....

Q. F. Stout and B. L. Warren. Tree rebalancing in optimal time and space. Communications of the ACM, 29(9):902--908, 1986.

....D) The average depth of a tree is the path length between the root and a leaf node, averaged over all the leaf nodes. Although little if any work has been done on balancing decision trees, a great deal of research has considered balanced search trees (Baer Schwab, 1977; Chang Iyengar, 1984; Stout Warren, 1986; Cormen, Leiserson, Rivest, 1990; Nakamura, Abe, Ohsawa, Sakauchi, 1993) Roughly speaking, this literature deals with techniques to restructure search trees when elements are inserted or deleted, in order to restrict the depth of these trees to a logarithmic function of the number of search ....

Stout, Q. F., & Warren, B. L. (1986). Tree rebalancing in optimal time and space. Communications of the ACM, 29 (9), 902--908.

....on the right, which was built using lookahead, accuracy = 99.10 , size = 10, maximum depth = 4 and the number of candidate tests was 1,455,901. on TRAIN, but is shallower. Although little work has been done on balancing decision trees, a great deal of research has considered balanced search trees [23, 17]. Roughly speaking, this literature deals with techniques to restructure search trees when elements are inserted or deleted, in order to restrict the depth of these trees to a logarithmic function of the number of search keys. An axis parallel decision tree in a continuous space can be interpreted ....

Q. F. Stout and B. L. Warren. Tree rebalancing in optimal time and space. Communications of the ACM, 29(9):902--908, September 1986.

....Definition 1 A binary tree v is perfectly balanced if and only if and ffi(v) 0 and v s both subtrees are perfectly balanced trees. In the following we assume that rebuilding a (sub )tree v takes O(jvj) time. Examples of linear algorithms for balancing trees can be found in the literature [10, 11, 13, 17, 25]. We also assume that updates are performed by adding or removing nodes at the lowest level of the tree. For each node v, an update below v changes the value of ffi(v) by at most one. Hence, at least ffi(v) updates have been made in the subtree v since the last time it was perfectly balanced. 3 ....

Q. F. Stout and B. L. Warren. Tree rebalancing in optimal time and space. Communications of the ACM, 29(9):902--908, 1986.

....2 A binary tree is optimally balanced if and only if the difference in length between the longest and shortest paths is at most one. In the literature, there are a number of algorithms presented for maintenance of optimally balanced trees, all having an amortized cost of Theta(n) per update [7, 8, 9, 15]. Although this cost is high, it is regarded as affordable when updates are rare and when a low search cost is essential. In this paper we improve the tradeoff between balance and maintenance cost for binary search trees. Starting with optimal balance, that is, at most one incomplete level and a ....

....at node v is called the size of v, denoted jvj. this implies that the size of a tree containing n elements is n 1. We assume that rebuilding the subtree rooted at v to perfect balance requires time proportional to the size of v. Linear algorithms for balancing a binary tree can be found in [7, 8, 15]. 2 Maintaining Near Optimal Balance If we allow the height of a binary search tree to be an additive constant larger than the optimal height, we have a tree of near optimal height. In this section we show that a near optimal height of dlog(n 1) ffle may be maintained at an amortized cost of ....

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Q. F. Stout and B. L. Warren. Tree rebalancing in optimal time and space. Communications of the ACM, 29:902--908, 1986.

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Q. F. Stout and B. L. Warren, `Tree rebalancing in optimal time and space', Commun. ACM, 29, (9), 902--908 (1986).

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Q. F. Stout and B. L. Warren, `Tree rebalancing in optimal time and space', Comm. ACM, 29, 902-- 908 (1986).

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