J. Nievergelt and E. M. Reingold. Binary search trees of bounded balance. SIComp, 2:33-43, 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....guarantees a relative bound between subtrees with a common root, so the size di erence between subtrees of the same height may be large. In contrast, strong weight balance requires an absolute constraint that relates the sizes of all subtrees at the same level. For example, BB[ trees [NR73] are weight balanced binary search trees based on rotations, but they are not strongly weight balanced. Search trees that satisfy Properties 1 and 2 include weight balanced B trees [AV96] deterministic skip lists [MPS92] and skip lists [Pug89] in the expected sense. We choose to use ....

J. Nievergelt and E. M. Reingold. Binary search trees of bounded balance. SIAM Journal on Computing, 2:33-43, 1973.

....basic structure is a balanced tree having a fan out proportional to the disk block size B. The B tree uses linear space and its query and update performance are O(log B N ) memory transfers. This is a O(logB) factor improvement over the O(log 2 N ) bound obtained by the RAM model structures (e.g. [1, 21, 30, 33, 37, 38]) This improvement translates to approximately an order of magnitude speedup, depending on the application. Although B trees are in widespread use, they do have several limitations. They depend critically on Department of Computer Science, State University of New York, Stony Brook, NY ....

J. Nievergelt and E. M. Reingold. Binary search trees of bounded balance. SIAM Journal on Computing, 2:33{ 43, 1973.

....average shift cost per node, total insertion cost, and average successful search cost for ISA[k] trees. We also present an insertion algorithm with associated predecessor and successor functions for ISA[k] trees. For large binary search trees (over 160 nodes) our results suggest the use of ISA[2] or ISA[3] trees for best performance. key words : Binary search trees Inorder shifting Internal path length Optimal searching INTRODUCTION Binary search trees are very popular and useful data structures for efficiently searching, inserting and deleting data items. Unfortunately, binary search ....

....ISA[k] trees. An ISA[k] tree with n nodes (n.0) is a binary search tree that is complete to level L, and levels L11toL1k(k.0) are all either empty or are the last nonempty levels of the tree. Notice that all trees satisfying Definition 1 are ISA[1] trees. Figure 4(a) illustrates an example ISA[2] tree. The tree is complete to level L=1, and levels L11 and L12 are not complete. Let T be an ISA[k] tree complete to level L. Then the number of possible nodes, n,inTis bounded by 2 1 n 2 . The total number of nodes in a binary tree complete to level L is 2 1. Recall that ....

[Article contains additional citation context not shown here]

J. Nievergelt and E. M. Reingold, `Binary search trees of bounded balance', Siam J. Comput, 2, (1), 33--43 (1973).

....height denotes the maximum height of the tree, and node pointer is a pointer to a node of the tree. The actual value of height depends on which kind of tree is used, e.g. standard binary trees or AVL trees. 6.2.2. Weight Balanced Trees. So called weight balanced trees have been introduced in [18] and are treated in detail in [19] and in [20] Definition 6.4. We define: 1) Let T be a binary tree with left subtree Tt and right subtree Tr. Then is called the root balance of T. Here IT] denotes the number of leaves of tree T. 2) Tree T is of bounded balance c if for every subtree T of ....

I. Nievergelt and E. Reingold. Binary search trees of bounded balance. SIAM Journal of Computing, 2(1):33-43, 1973.

....The traditional approach to elude these problems is to impose additional constraints on the heights, sizes, etc. of the subtrees; many kinds of balanced search trees have been proposed, like AVLs [1] red black trees [10] weight balanced trees (also known as BB[ff] trees) [19], height ratiobalanced trees [8] All balanced search trees guarantee logarithmic performance of the search and update operations in the worst case. The insertion and deletion algorithms must guarantee that the resulting BST does not violate any of the constraints; typically, this is ....

J. Nievergelt and E. Reingold. Binary search trees of bounded balance. SIAM J. Comp., 2(1):33-- 43, 1973.

....interval tree [21, 22] In Section 2 we present the basic structure where the endpoints of the intervals stored in the structure belong to a xed set of N points. In Section 3 we then remove the xed endpoint set assumption. In internal memory, the assumption is normally removed using a BB[ tree [33] as the base search tree structure [32] and this leads to amortized update bounds. However, as BB[ trees are unsuitable for implementation in external memory, we develop a new weight balanced B tree for use in external memory. Like in internal memory, this results in amortized update bounds. In ....

....B N T=B) I Os worst case and such that updates can be performed in O(log B N) I Os amortized. 7 3 General external interval tree In order to remove the xed endpoint assumption from our external interval tree, we need to use a dynamic search tree as base tree. In internal memory BB[ trees [33] are often used as base tree for structures with secondary structures. In such trees, a node v with weight w (i.e. with w elements below it) can only be involved in a rebalancing operation once for every w) updates that access (i.e. pass through) v [16, 32] Since the necessary ....

J. Nievergelt and E. M. Reingold. Binary search tree of bounded balance. SIAM Journal of Computing, 2(1):33-43, 1973.

....result in the height of the tree decreasing by one. B tree variants and extensions. Recently, several important variants and extensions of B trees have been considered. Arge and Vitter [30] developed the weight balanced B trees, which can be viewed as an external version of BB[ff] trees [120]. Weight balanced B trees are very similar to B trees but with a weight constraint imposed on each node in the tree instead of a degree constraint. The weight of a node v is defined as the number of elements in the leaves of the subtree rooted in v. Like B trees, weight balanced B trees are ....

J. Nievergelt and E. M. Reingold. Binary search tree of bounded balance. SIAM Journal of Computing, 2(1):33--43, 1973.

....structures for manipulating arbitrary data is a balanced search tree. The basic functionality that we ask of a balanced search tree is that it maintain an ordered collection of elements subject to insertions, deletions, and searches. Thus, balanced search trees such as AVL trees [1] BB[ trees [22], red black trees [17] randomized search trees [29] skip lists [26] and splay trees [33] are appropriate for main memory, whereas B trees [10] are more appropriate for external memory. In this paper we develop search trees that are memoryefficient at all levels of the memory hierarchy. A ....

....its descendants. More formally, we require the following property: Property 1 Suppose that whenever we rebalance a node v we also touch all of v s descendents. Then the amortized number of elements that are touched per insertion is only O(log N) Many weight balanced trees, such as BB[ trees [22], have this property [21] A tree is weight balanced if the left subtree (including the root) and the right subtree (including the root) have sizes that differ by at most a constant factor. This is stronger than merely requiring that most modifications are near the leaves. We require a stronger ....

J. Nievergelt and E. M. Reingold. Binary search trees of bounded balance. SIAM J. Comput., 2:33--43, 1973.

....secondary structures it turns out that a rebalancing operation requires work proportional to the number of elements in the subtree rooted at the node being rebalanced this is normally called the weight of the node. In internal memory a natural choice of dynamic base tree is 28 the BB[ff] tree [96], because in this structure a node with weight w can only be involved in a rebalancing operation for every Omega Gamma w) updates that access (go through) the node [30, 90] This leads to an O(1) amortized bound on performing a rebalancing operation and thus to an O(log B n) amortized rebalance ....

....tree [53, 54] In Section 2 we present the basic structure where the endpoints of the intervals stored in the structure belong to a fixed set of points. In Section 3 we then remove the fixed endpoint set assumption. In internal memory the assumption is normally removed by using a BB[ff] tree [96] as base search tree structure [90] and this leads to amortized update bounds. But as BB[ff] trees are not very suitable for implementation in external memory, we develop a special weight balanced B tree for use in our external interval tree structure. Like in internal memory this results in ....

[Article contains additional citation context not shown here]

J. Nievergelt and E. M. Reingold. Binary search tree of bounded balance. SIAM Journal of Computing, 2(1), 1973.

....are abundant in computational geometry. Examples of augmented trees are range trees [3, 21] segment trees [4] interval trees [8] and dynamization of order decomposable searching problems [17] Asymptotically efficient realizations of augmented trees are usually based on weight balanced trees [16]; in the case of range and segment trees implementations are available in LEDA. Skip lists [18, 19] were recently proposed as an alternative to balanced trees for one dimensional dictionaries. We argue in section 5 that skip lists can also serve as the basis for augmented tree schemes. They lead ....

....secondary structures along a path of the primary tree. If C(m) bounds the cost of an operation on the secondary structure of size m then total cost is C(n) C(an) C(a 2 n) For many functions C, this sums to O(C(n) The geometric size decrease property holds for weight balanced trees [16] [11, section III.5.1. and their derivatives, and for randomized search trees [2] it does not hold for any kind of height balanced tree scheme. Consider for example a (2; 3) tree of height 3h where a subtree T of height 2h consists solely for nodes of degree 3 and all 8 nodes outside the ....

I. Nievergelt, E.M. Reingold, Binary Search Trees of Bounded Balance, SIAM Journal of Computing, Vol. 2, 1973, 33-43

....unpredictably, ensuring small depth of all the nodes in the changing search tree is less straightforward. Nonetheless, a fair number of strategies have been developed for maintaining approximate balance in such changing search trees. Examples are AVL trees [1] a; b) trees [4] BB(ff) trees [25], red black trees [13] and many others. All these classes of trees guarantee that accesses and updates can be performed in O(log n) worst case time. Some sort of balance information stored with the nodes is used for the restructuring during updates. All these trees can be implemented so that the ....

J. Nievergelt and E.M. Reingold, Binary search trees of bounded balance. SIAM J. Comput. 2 (1973) 33--43.

....tree [14, 15] In Section 2 we present the basic structure where the endpoints of the intervals stored in the structure belong to a fixed set of points. In Section 3 we then remove the fixed endpoint set assumption. In internal memory the assumption is normally removed by using a BB[ff] tree [30] as base search tree structure [28] and this leads to amortized update bounds. But as BB[ff] trees are not very suitable for implementation in external memory, we develop a special weight balanced B tree for use in our external interval tree structure. Like in internal memory this results in ....

....case) when search trees are augmented with secondary structures it turns out that a rebalancing operation requires work proportional to the number of items in the subtree with the node being rebalanced as root. In internal memory a natural choice of dynamic base tree is therefore the BB[ff] tree [30], because in this structure a node with w items below it (with weight w) can only be involved in a rebalancing operation for every Omega Gamma w) updates that access (pass through) the node [10, 28] This leads to an O(1) amortized bound on performing a rebalancing operation. Unfortunately ....

J. Nievergelt and E. M. Reingold. Binary search tree of bounded balance. SIAM Journal of Computing, 2(1), 1973.

....of insert, delete, and member. The binary search tree is a well known and well studied data structure that can support these operations in O(log n) time in the expected case. Many kinds of balanced trees have been proposed that can support all dictionary operations in O(log n) worst case time [1, 11, 16]; however, these structures cannot adapt to nonuniform or skewed access patterns. We consider the problem of maintaining an adaptive binary search tree (also known as a self organizing search tree [2] and a self adjusting search tree [21] Compared to balanced trees, adaptive trees have the ....

J. Nievergelt and E. M. Reingold. Binary search trees of bounded balance. SIAM Journal on Computing, 2:33--43, 1973.

....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 slow and processors even slower. However, a long term trend has been that processor speed has doubled every two years and memory ....

J. Nievergelt, E.M. Reingold, Binary Search Trees of Bounded Balance. SIAM Journal of Computing 2:1 (1973), pp. 33--43.

....Binary Trees denotes the maximum height of the tree, and node pointer is a pointer to a node of the tree. The actual value of height depends on which kind of tree is used, e.g. standard binary trees or AVL trees. Weight Balanced Trees So called weight balanced trees have been introduced in [NR73] and are treated in detail in [Meh84b] Definition 5.4. We define: 1. Let T be a binary tree with left subtree T and right subtree T r . Then ae(T ) jT j=jT j = 1 Gamma jT r j=jT j is called the root balance of T . Here jT j denotes the number of leaves of tree T . 2. Tree T is of ....

I. Nievergelt and E. Reingold. Binary search trees of bounded balance. SIAM Journal of Computing, 2(1):33--43, 1973.

....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 slow and processors even slower. However, a long term trend has been that processor speed has doubled every two years and memory ....

J. Nievergelt, E.M. Reingold, Binary Search Trees of Bounded Balance. SIAM Journal of Computing 2:1 (1973), pp. 33--43.

....the tree, and node pointer is a pointer to a node of the tree. The actual value of height depends on which kind of tree is used, e.g. standard binary trees or AVL trees. TU Vienna Project WOOP 22 J. Blieberger 6.2.2. Weight Balanced Trees. So called weight balanced trees have been introduced in [18] and are treated in detail in [19] and in [20] Definition 6.4. We define: 1) Let T be a binary tree with left subtree T and right subtree T r . Then ae(T ) jT j=jT j = 1 Gamma jT r j=jT j is called the root balance of T . Here jT j denotes the number of leaves of tree T . 2) Tree T ....

I. Nievergelt and E. Reingold. Binary search trees of bounded balance. SIAM Journal of Computing, 2(1):33--43, 1973.

....secondary structures it turns out that a rebalancing operation requires work proportional to the number of elements in the subtree rooted at the node being rebalanced this is normally called the weight of the node. In internal memory a natural choice of dynamic base tree is 28 the BB[#] tree [96], because in this structure a node with weight w can only be involved in a rebalancing operation for every## w) updates that access (go through) the node [30, 90] This leads to an O(1) amortized bound on performing a rebalancing operation and thus to an O(log B n) amortized rebalance bound. ....

....tree [53, 54] In Section 2 we present the basic structure where the endpoints of the intervals stored in the structure belong to a fixed set of points. In Section 3 we then remove the fixed endpoint set assumption. In internal memory the assumption is normally removed by using a BB[#] tree [96] as base search tree structure [90] and this leads to amortized update bounds. But as BB[#] trees are not very suitable for implementation in external memory, we develop a special weight balanced B tree for use in our external interval tree structure. Like in internal memory this results in ....

[Article contains additional citation context not shown here]

J. Nievergelt and E. M. Reingold. Binary search tree of bounded balance. SIAM Journal of Computing, 2(1), 1973.

.... t) Updates. If updates are allowed, the binary tree should be replaced by a dynamic search tree, for which efficient rebalancing methods are known. To maintain insertions and deletions of line based segments in the data structure described above, we replace the binary tree with a BB[ff] tree [6, 17], 0 ff 1 Gamma 1= p 2. We store balance values in internal nodes of the BB[ff] tree and maintain the optimal O(log n) height of the tree by performing O(log n) single or double rotations during an update. The update cost consists of O(log n) operations for the search and balance maintenance ....

....n(log B n log 2 B IL (B) t) Insertions. The 2LDS proposed above has been designed for the static case. The extension of the proposed schema to the semi dynamic case is based on: i) the use of a weighted balanced B tree [3] as first level data structure; 8 (ii) the use of a BB[ff] tree [6, 17], 0 ff 1 Gamma 1= p 2 as the second level structure G for long fragments; iii) the definition of some additional operations on multislab lists (similar to those presented in [12] guaranteeing the O(1) I O amortized complexity of bridge navigation. Such extensions allow insertions to be ....

J. Nievergelt and E. M. Reingold. Binary Search Tree of Bounded Balance. SIAM J. Computing, 2(1):33--43, 1973.

....knowledge of the representation of pointers in the machine so it is inherently nonportable. In high level languages like SML the pointers are hidden completely from the programmer so this technique is not available. 6 CSTR 92 10 BALANCING MAINTENANCE ALGORITHMS 6. Nievergelt and Reingold [4] use a slightly different criterion that compares the size of a subtree to the size of the whole tree. For their purposes, which included producing analytical results on the behaviour of the tree, this criterion is cleaner. For our purposes it is simpler to compare the sizes of the left and right ....

....a tree is logarithmic in its size, one tree should never be more than a fixed amount higher than its sibling. The rotations (figure 1) must be applied to lift the larger (hence taller) trees at the expense of the smaller trees. This is exactly what T does. Both analytical and empirical evidence ([4], 3] suggests that there is little to choose between various balancing schemes, and that balancing is unnecessary on random data. Balancing is only necessary to prevent poor worst case behaviour and the evidence suggests that almost any scheme to avoid poor behaviour in the pathological cases ....

Nievergelt, J. and E. M. Reingold, 1973. "Binary search trees of bounded balance", SIAM J. Computing 2(1), March 1973. 32 CSTR 92-10

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J. Nievergelt and E. M. Reingold. Binary search trees of bounded balance. SIComp, 2:33-43, 1973.

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J. Nievergelt and E. M. Reingold. Binary search trees of bounded balance. SIAM J. Comput., 2:33--43, 1973.

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J. Nievergelt and E. M. Reingold. Binary search trees of bounded balance. SIAM Journal on Computing, 2:33-43, 1973.

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J. Nievergelt and E. M. Reingold, `Binary search trees of bounded balance', SIAM J. Computing, 2,33-- 43 (1973).

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J. Nievergelt and E. M. Reingold, Binary Search Trees of Bounded Balance, SIAM J. Computing, Vol. 2, No. 2, pp. 33-43, March 1973.

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