N. Blum and K. Mehlhorn. On the average number of rebalancing operations in weightbalanced trees. Theoretical Computer Science, 11:303-320, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 T: c p(T ) 1 c (3) BB[c] is the set of ....

....take time O(logN) in BB[c] trees. Here N is the number of leaves in the BB[c] tree. Some of the above operations can move the root balance of some nodes on the path of search outside the permissible range [c, 1 c] This can be repaired by single and double rotations (for details see [19] and [20]) BB[c] trees are binary trees with bounded height. In fact it is proved in [19] that ldN 1 height(T) ld(1 c) 1, where N is the number of leaves in the BB[c] tree T. A template for the above operations is shown in Figure 7, where floor(x) is ne node pointer : node pointer.left [ ....

N. Blum and K. Mehlhorn. On the average number of rebalancing operations in weightbalanced trees. Theoretical Computer Science, 11:303-320, 1980.

.... B(p) 1 c for every node p in T. By definition, the empty tree is in WB(c 0 for all c. Lemma I ( The maximum height, hmax(n) of an n node tree in WB(a) is log (n q 1) NIEV73] 2) Inserts and deletes can be performed in an n node tree in WB( in O(logn) time for 2 11 c 1 v 2 [BLUM80]. 3) Each search operation in an n node tree in WB( takes O(logn) time [NIEV73] In the case of weight balanced trees WB(oz) an LL rotation is performed when B(gp) 1 oz and B(p) oz (1 oz) see Figure l(a) NIEV73] So, s(p) 1 s(p) 1 s(gp) 1 s(p) s(d) 2 a 2a 1 s(d) s(p) 1 ....

....1 2oz oz s(q) s(p) i i oz For oz 1 3, s(q) s(d) and LR (RL) rotations do not increase search cost. Thus, in the case of WB(oz) trees, the rebalancing rotations do not increase search cost. This statement remains true if the conditions for LL and LR rotation are changed to those in [BLUM80]. While rotations do not increase the search cost of WB(oz) trees, these trees miss per forming some rotations that would reduce search cost. For example, it is possible to have oz B(gp) 1 oz, B(p) 5, and s(q) s(d) Since B(gp) isn t high enough, an LL rotation isn t performed. Yet, ....

N. Blum and K. Mehlhorn, On the Average Number of Rebalancing Operations in Weight-balanced Trees, Theoretical Computer Science, vol 11, pp.303-320, 1980.

....by modifying the balance conditions somewhat, we get a class of d dimensional range trees, for which an update changes only O( logn) nodes. We define range trees as follows. For the definitions of BB[a] trees and aBBtrees, we refer the reader to Nievergelt and Reingold [6] Blum and Mehlhorn [3], and Olivi [7,8] Definition 3 Let S be a set of points in the d dimensional euclidean space. A d dimensional range tree T, representing the set S, is defined as follows. 23 1. I d = 1, then T is an aBB tree, containing the points o in sorted order in its leaves. I d 1, then T ....

....If the root of the main tree gets out of balance, we perfoi;nx a rotation and, hence, we have to rebuild the associated structures of the sons of the root. Since these associated structures are (d 1) dimensional range trees, this changes O(n(log n) nodes. It was shown by Blum and Mehlhorn [3] that for a proper choice of the root of the main tree gets out of balance at most once every (n) updates. Hence the average number of nodes that are changed due to our visit of the root of the main tree is bounded by O( log n)a a) It follows that C(n, d) satisfies the following recurrence: ....

N. Blum and K. Mehlhorn. On the Average Number of Rebalancing Operations in Weight-Balanced Trees. Theor. Comp. Sci. 11 (1980), pp. 303-320.

....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 reorganization of the secondary structures as a result of a rebalance operation on v can be performed in O(w) time, we get an O(1) amortized bound on performing a rebalancing operation. Unfortunately BB[ trees are not suitable for implementation in external memory; ....

N. Blum and K. Mehlhorn. On the average number of rebalancing operations in weightbalanced trees. Theoretical Computer Science, 11:303-320, 1980.

.... Mucke Zhu proved that the expected running time of their method is O(k p n=k) which reaches its optimum (O(n 1=3 ) when k is Theta(n 1=3 ) If we now assume that the sites are organized as a dynamically maintained weight balanced binary search tree (bounded balanced trees, cf. [1]) based on xy order, BinSearch Walk proceeds as follows: start from the root of the tree, search for q; while doing so, determine, among all sites visited, the one, i, closest to q. Finally walk from i to q, as for Jump Walk. We now prove the following Theorem: Theorem 2.1 It takes procedure ....

....expected running time of the procedure is O Gamma p n(1= p n 1=k) Delta = O( p n= log n) since k = Theta(log n) This concludes the proof of Theorem 2.1. 2 Since it is possible to insert or delete an element in worst case O(log n) time in a bounded balanced tree with n nodes ([1]) the location algorithm may be used in a fully dynamic context. However and for obvious reasons, its worst case performance is linear in the size of the data. Oversampled BinSearch Walk Observe that the optimum of O(k p n=k) is O(n 1=4 ) and is reached when k is Theta(n 1=4 ) ....

N. Blum and K. Mehlhorn. On the average number of rebalancing operations in weight-balanced trees. Theoretical Computer Science, 11:303-- 320, 1980.

....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 bound. Unfortunately BB[ff] trees are not very suitable for implementation in external memory. The natural candidate however, the B tree, does not have the property that a ....

....as root. In internal memory a natural choice of dynamic base tree is therefore the BB[ff] tree [96] 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 [30, 90]. This leads to an O(1) amortized bound on performing a rebalancing operation. Unfortunately BB[ff] trees are not suitable for implementation in external memory. The main problem is of course that BB[ff] trees are binary and that there does not seem to be an easy way of grouping nodes together in ....

N. Blum and K. Mehlhorn. On the average number of rebalancing operations in weightbalanced trees. Theoretical Computer Science, 11:303--320, 1980. 162

....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 BB[ff] trees are not suitable for implementation in external memory. The main problem is of course that BB[ff] trees are binary and that there does not seem to be an easy way of grouping nodes together in ....

N. Blum and K. Mehlhorn. On the average number of rebalancing operations in weightbalanced trees. Theoretical Computer Science, 11:303--320, 1980.

....the same node which is the bottom most node of the two internal nodes in the rotation. It is easy to show that after it has lost at most two leaf children, its children are going to be internal nodes from that point on. Deletions can be viewed as reverse insertions. AVL trees [1] and BB[a] trees [9, 4, 8] also have limited leaf action. As soon as one gets just some fixed constant distance up towards the root from the place of an update, then the relationship between heights and sizes, respectively, of subtrees implies that none of the involved subtrees can be leaves. Other Complexity ....

N. Blum and K. Mehlhorn, On the Average Number of Rebalancing Operations in Weight-Balanced Trees, Theoretical Computer Science, 11, 303--320, 1980.

....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 is of bounded balance ff if for every subtree ....

....O(logN ) in BB[ff] trees. Here N is the number of leaves in the BB[ff] tree. Some of the above operations can move the root balance of some nodes on the path of search outside the permissible range [ff; 1 Gamma ff] This can be repaired by single and double rotations (for details see [19] and [20]) BB[ff] trees are binary trees with bounded height. In fact it is proved in [19] that height(T ) ld N Gamma 1 Gamma ld(1 Gamma ff) 1; where N is the number of leaves in the BB[ff] tree T . A template for the above operations is shown in Figure 7, where floor(x) is 1 discrete ....

N. Blum and K. Mehlhorn. On the average number of rebalancing operations in weightbalanced trees. Theoretical Computer Science, 11:303--320, 1980.

....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. Unfortunately BB[#] trees are not very suitable for implementation in external memory. The natural candidate however, the B tree, does not have the property that a ....

....being rebalanced as root. In internal memory a natural choice of dynamic base tree is therefore the BB[#] tree [96] because in this structure a node with w items below it (with weight w) can only be involved in a rebalancing operation for every## w) updates that access (pass through) the node [30, 90]. This leads to an O(1) amortized bound on performing a rebalancing operation. Unfortunately BB[#] trees are not suitable for implementation in external memory. The main problem is of course that BB[#] trees are binary and that there does not seem to be an easy way of grouping nodes together in ....

N. Blum and K. Mehlhorn. On the average number of rebalancing operations in weightbalanced trees. Theoretical Computer Science, 11:303--320, 1980. 162

....rebalanced as root. In internal memory a natural choice of dynamic base tree is therefore the BB[#] 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## w) updates that access (pass through) the node [10, 28]. This leads to an O(1) amortized bound on performing a rebalancing operation. Unfortunately BB[#] trees are not suitable for implementation in external memory. The main problem is of course that BB[#] trees are binary and that there does not seem to be an easy way of grouping nodes together in ....

N. Blum and K. Mehlhorn. On the average number of rebalancing operations in weight-balanced trees. Theoretical Computer Science, 11:303--320, 1980.

....which is the bottom most node of the two internal nodes in the rotation. It is easy to show that after it has lost at most two leaf children, its children are going to be internal nodes from that point on. Furthermore, deletions can be viewed as reverse insertions. On the other hand, BB[ff] trees [9, 4, 8] are not neighborhood balanced. It is possible to construct a sequence of double rotations where one leaf gets a different parent a logarithmic number of times. However, the number of structural changes after an update is amortized constant. This means that printing the whole history of a key ....

N. Blum and K. Mehlhorn. On the Average Number of Rebalancing Operations in Weight-Balanced Trees. Theoretical Computer Science, 11:303--320, 1980.

....one balances the height of the subtrees of each node, while in the latter, one balances the number of nodes in the subtrees of each node. Examples of height balanced trees are AVL trees [1] and red black trees [75] Weight balanced trees were first presented in [119] and include BB[ff] trees [25]. All of these trees use O(n) space and support each operation of the above repertory in time O(log n) An important variation of balanced trees stores weighted items, where the weight of an item is usually associated with its access frequency in membership queries. Several schemes have been ....

N. Blum and K. Mehlhorn, "On the Average Number of Rebalancing Operations in WeightBalanced Trees," Theoretical Computer Science 11 (1980), 303--320.

....node p in T. By definition, the empty tree is in WB(ff) for all ff. Lemma 1 (1) The maximum height, hmax(n) of an n node tree in WB(ff) is log 1 1 Gammaff (n 1) NIEV73] 2) Inserts and deletes can be performed in an n node tree in WB(ff) in O(log n) time for 2=11 ff 1 Gamma p 2=2 [BLUM80]. 3) Each search operation in an n node tree in WB(ff) takes O(log n) time [NIEV73] In the case of weight balanced trees WB(ff) an LL rotation is performed when B(gp) 1 Gamma ff and B(p) ff= 1 Gamma ff) see Figure 1(a) NIEV73] So, 1 Gamma ff s(p) 1 s(gp) 1 = s(p) 1 s(p) ....

....1 Gamma ff Gamma ff 1 Gamma ff : For ff 1=3, s(q) s(d) and LR (RL) rotations do not increase search cost. Thus, in the case of WB(ff) trees, the rebalancing rotations do not increase search cost. This statement remains true if the conditions for LL and LR rotation are changed to those in [BLUM80]. While rotations do not increase the search cost of WB(ff) trees, these trees miss performing some rotations that would reduce search cost. For example, it is possible to have ff B(gp) 1 Gamma ff, B(p) ff 1 Gammaff , and s(q) s(d) Since B(gp) isn t high enough, an LL rotation isn t ....

N. Blum and K. Mehlhorn, On the Average Number of Rebalancing Operations in Weight-balanced Trees, Theoretical Computer Science, vol 11, pp.303-320, 1980.

.... low height at a reasonable cost during insertions and deletions, and a multitude of schemes exist for keeping a height of c Delta log(n) 1 for some constant c 1, while updating in time O(log n) Examples include AVL trees [1] with c = 1:44, red black trees [9] with c = 2, BB(ff) trees [7, 13] with 2 c 3:45 (depending on ff) and many more. The trivial lower bound on the height of a binary tree of n nodes is dlog(n 1)e. While being within a constant factor of this lower bound may be sufficient for practical purposes, a theoretically fundamental (but surprisingly overlooked) ....

N. Blum and K. Mehlhorn. On the average number of rebalancing operations in weight-balanced trees. Theoretical Computer Science, 11:303-- 320, 1980.

....the cost of a local change at v is proportional to size(v) A tree scheme is said to have a strong bound for the rebalancing frequency if the amortized rebalancing cost is logarithmic even if the cost of a local change is proportional to the size of the subtrees involved. Weight balanced trees [21, 5] and randomized search trees [2] have strong bounds but no height balanced scheme has. Also, at least for weight balanced trees the known bounds involve large constant factors. For example, 21] states that every additional dimension in a range tree multiplies the bound for the update cost by 100 ....

N. Blum, K. Mehlhorn, On the Average Number of Rebalancing Operations in Weight-Balanced Trees, Theoretical Computer Science 11, 1980, 303-320

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N. Blum and K. Mehlhorn, "On the Average Number of Rebalancing Operations in Weight-Balanced Trees," Theoretical Computer Science, 11, 1980, 303-- 320.

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