Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal the Association for Computing Machinery, July 1985.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....approach that provides better support for persistent use represents an array by a pair consisting of a unique version stamp and a pointer to a master array, whose elements are finite maps from version stamps to associated values. The implementation by O Neill and Burton [18] uses splay trees [20] for the finite maps and a version stamp scheme based on the list order problem [5] Their implementation exhibits Q(1) bounds for single threaded use and Q(logn) amortized bounds for persistent use. A common characteristic of the two approaches above is the internal use of destructive updates. ....

Daniel D Sleator and Robert E Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652--686, July 1985.

....Consequently, there is a large body of work on optimizing search trees for nonuniform distributions in a variety of contexts: 1. Known distribution on a RAM optimal binary search trees [1, 18] and variations [15] and Hu man codes [16] 2. Unknown distribution on a RAM splay trees [17, 21]. 3. Known distribution in external memory optimal binary search trees in the HMM model [22] 4. Unknown distribution in external memory alternatives to splay trees [17] Fixed Tree Topology. Search trees frequently encode decision trees that cannot be rebalanced because the operations lack ....

Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652-686, July 1985.

....data structures form a central topic of online algorithms research. The area of Competitive Analysis began with the results of Sleator and Tarjan showing that splay trees achieve static optimality for search trees, and that Move to Front is constant competitive for the list update problem [ST85a, ST85b]. In a parallel development, powerful algorithms have been developed in Machine Learning for problems of online prediction [LW94, FS96] This paper is inspired by the observation made in [BB00] that if computational decision making costs are not considered, then these weighted experts ....

.... of online algorithms research, beginning with the results of Sleator and Tarjan that splay trees perform within a constant factor of the best static search tree for any request sequence, and that Move to Front is constant competitive with respect to the best o line (adaptive) algorithm for lists [ST85a, ST85b]. Online algorithms for lists have subsequently been well studied, with upper and lower bounds of 1.6 and 1.5 respectively, on the competitive ratio of randomized algorithms [AvSW95, Tei93] The case of trees appears much harder: it is not known whether splay trees or any other online algorithm ....

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Daniel D. Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32:652-686, 1985.

....assuming all edge times are between 0 and 1. This is similar to the bounds of Takimoto and Warmuth [16] and their specialized algorithm is also efficient. For the next problem, no efficient near optimal algorithm was previously known. Tree update problem. This problem is a classic online problem [15] introduced by Sleator and Tarjan with Splay Trees, around the same time as they introduced the list update problem [14] In the tree update problem, one maintains a binary search tree over n items in the face of an unknown sequence of accesses to these items. For each access, i.e. lookup, the ....

Daniel Sleator and Robert Tarjan. Self-Adjusting Binary Search Trees. Journal of the ACM 32:652-686, 1985.

....vector c at each coordinate is the number of edges between u and v, and a cut is represented by a 0 1 vector with 1s in the coordinates where u and v are on di erent sides. The feasible set in this case is not convex. ith coordinate is the depth of node i. In the online version of this problem [16], we have to repeatedly choose a tree in anticipation of the next access. 3) Linear programming: Consider the speci c maximization example illustrated in Figure 1a. The o ine version is trivial, but the online version, where we have a sequence of cost vectors c j , is not. This example ....

....v in period j of FLL is identical to the distribution over r c 1: j 1 in FRL, i.e. d j (x) d (x c 1: j 1 ) because FLL uses M(v) while FRL uses M(r c 1: j 1 ) This can be seen by induction. The base case is trivial. Our algorithm is (1 o(1) competitive. In contrast, Splay trees [16] are 3 log 2 3 competitive [16] We are referring to static optimality. We do not address the longstanding conjecture of dynamic optimality of splay trees, and our algorithms are not dynamically optimal. If d j (x) d (x c 1: j 1 ) then we can write d j 1 (x) in terms of the probabilities ....

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Daniel Sleator and Robert Tarjan. Self-Adjusting Binary Search Trees. Journal of the ACM 32:652686, 1985. 8 Appendix: L 2 guarantees Here we assume jc j j 2  1 and D 2 is an upper bound on the diameter of S. 12 Theorem 4. Let  be uniform on the ball of radius  . Then for FEL(; s) (any s  1) and FLL(), E[FLL's cost t ] = E[FEL's cost t ]  min-cost t +D 2

....tree, which requires O(log n) time for searching and O(log n) amortized time for updating, with just O(1) RAM registers needed to operate dynamically. An interesting feature of our data structure is that it achieves the best space saving possible. In this context self adjusting search trees [25] and random search trees [17] are often mentioned for their ability, among others, to avoid balancing information for saving space while requiring O(log n) amortized or expected time. However, they still need n n) words of memory (for the pointers) Compared to previous work in the eld, the ....

Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652-686, July 1985.

....be viewed as a linear minimization problem where the sequence of accesses is represented by an objective vector c whose ith coordinate is the number of accesses to node i, and each tree is represented by a vector whose ith coordinate is the depth of node i. In the online version of this problem [16], we Department of Mathematics and Laboratory for Computer Science, MIT. Email: fakalai, vempalag math.mit.edu This may seem like an unfair comparison, because we allow online algorithms the exibility to change each period. However, o ine algorithms are given the c j s in advance, and in ....

....given the c j s in advance, and in general, no online algorithm can compare to minx2S c j x. have to repeatedly choose a tree in anticipation of the next access. For this problem our algorithm (lazy trees, Section 4) is (1 ) competitive. In contrast, Splay trees are 3 log 2 3 competitive [16]. We are referring to static optimality. We do not address the longstanding conjecture of dynamic optimality of splay trees, and our algorithms are not dynamically optimal. The feasible sets in both these examples are discrete. However, we can consider S to be the convex hull of the feasible ....

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Daniel Sleator and Robert Tarjan. Self-Adjusting Binary Search Trees. Journal of the ACM 32:652686, 1985. 7 Appendix: L 2 guarantees Here we assume jc j j 2  1 and D 2 is an upper bound on the L 2 diameter of S. 11 Theorem 5. Let  be uniform on the ball of radius  . Then for FEL(; s) (any s  1) and FLL(), E[FLL's cost t ] = E[FEL's cost t ]  min-cost t +D 2

.... I Adam Kalai I Abstract Adaptive data structures form a central topic of online algorithms research, beginning with the results of Sleator and Tarjan showing that splay trees achieve static optimality for search trees, and that Move toFront is constant competitive for the list update prob lem [ST85a, ST85b]. This paper is inspired by the observation that one can in fact achieve a 1 e ra tio against the best static object in hindsight for a wide range of data structure problems via weighted experts techniques from Machine Learning, if computational decision making costs are not considered. In ....

.... of online algorithms research, beginning with the results of Sleator and Tarjan that splay trees perform within a constant factor of the best static search tree for any request sequence, and that Move to Front is constant competitive with respect to the best off line (adaptive) algorithm for lists [ST85a, ST85b]. Online algorithms for lists have subsequently been well studied, with upper and lower bounds of 1.6 and 1.5 respectively, on the competitive ratio of randomized algorithms for this problem [AvW95, Tei93] The case of trees appears much harder: it is not known whether splay trees or il: ....

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Daniel D. Sleator and Robert Endre Tarjan. Selfadjusting binary search trees. Journal of the ACM, 32:652-686, 1985.

....is enforced. Nievergelt and Reingold [53] introduce such trees and present algorithms for implementing the basic dictionary operations in O(log n) worst case time. The first published data structure that does not store any extra information at each node are Splay trees due to Sleator and Tarjan [64]. They achieve O(log n) amortized complexity per operation. However, splay trees do not guarantee a logarithmic worst case bound on the cost of a Search, and require restructuring even during searches (unlike scapegoat trees, which do have a logarithmic worst case cost of a Search and do not ....

....a GB 0 (c) tree gets rebuilt arbitrarily many times. 108 To sum up a scapegoat trees require less total rebuilding to assure the same balance criterion as GB 0 (c) trees. The usage of these schemes for multiway trees as well as their use for upgrading existing code are novel. Sleator and Tarjan [64] as well as Andersson [3] do not report experimental measurements of their suggested structures performance. In the next section we present a practical study of tree based dictionary solutions that do not require storage of balance enforcing information at the nodes. 3.10 Experimental Results ....

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Daniel D. Sleator and Robert E. Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652--686, 1985. 118

.... O(n log n) by a divideand conquer algorithm, by applying the Ham Sandwich Theorem and using a linear time algorithm for nding ham sandwich cuts from [14] 4: Self Adjusting Trees The class of self adjusting trees considered in the exercise are often denoted splay trees, and were described in [20]. Homework 4 1: Disk intersection Exercise 35.2 6 in [6] 2: Edit distance Computing the edit distance of two strings is one of the classical applications of dynamic programming. Hirshberg developed in [11] a general technique that, among others, allows the computation of the edit sequence in ....

Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652-686, 1985.

....an alloc actually be followed by an insert, or those allocated blocks will be lost. 3.2.4 Support data structures Several data structures support the operation of the above components. These are described here brie y. 6 Splay tree. Based on the algorithm in Sleator and Tarjan s original paper,[8] the class SplayTree is a container with parameters for the key, value, and an asymmetric and transitive key comparison relation class. SplayTree provides the expected basic functionality: insertion (both exclusive values for a key and multiple values associated with the same key) removal, ....

Daniel Sleator and Robert Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652-686, July 1985. 37

.... y Adam Kalai z Abstract Adaptive data structures form a central topic of online algorithms research, beginning with the results of Sleator and Tarjan showing that splay trees achieve static optimality for search trees, and that Move toFront is constant competitive for the list update problem [ST85a, ST85b]. This paper is inspired by the observation that one can in fact achieve a 1 ratio against the best static object in hindsight for a wide range of data structure problems via weighted experts techniques from Machine Learning, if computational decision making costs are not considered. In ....

.... of online algorithms research, beginning with the results of Sleator and Tarjan that splay trees perform within a constant factor of the best static search tree for any request sequence, and that Move to Front is constant competitive with respect to the best o line (adaptive) algorithm for lists [ST85a, ST85b]. Online algorithms for lists have subsequently been well studied, with upper and lower bounds of 1.6 and 1.5 respectively, on the competitive ratio of randomized algorithms for this problem [AvW95, Tei93] The case of trees appears much harder: it is not known whether splay trees or email: ....

[Article contains additional citation context not shown here]

Daniel D. Sleator and Robert Endre Tarjan. Selfadjusting binary search trees. Journal of the ACM, 32:652-686, 1985.

.... following restriction on our tree: if the nodes u and v have the the same parent p, then p is the parent of all w # [l u , l v ] We will keep all children of a node p in a structure called List Of Trees (LOT) implemented as splay trees(splay trees are self adjusting binary search trees, see [6]) We will enforce the above restriction for any LOT the algorithm will manipulate: if u and v are in a LOT and w # [l u , l v ] then w is in that same LOT. Furthermore, w will be placed between u and v in the LOT, if possible (the only case where this will not be possible is when the LOT ....

....have to add the two functions ptotcost(path p) computing the sum of the cost of all edges in p, and plength(path p) returning the number of nodes in p. Both can be trivially implemented as a variation of pmincost( As mentioned before, LOTs will be implemented as splay trees. A splay tree [6] is a self adjusting binary search structure based on the splay(tree T, node v) operation, which brings the node v to the root of the tree T , while maintaining the inorder of the nodes. We will store in the root of the splay tree, the name of the LOT it corresponds to. The operation lotof(vertex ....

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Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652--686, July 1985.

....and Interprocess Communication 6.7.3 Persistence 7 Algorithmic Analyses for Data Dependent Algorithms 7.1 Tree Algorithms [Blah blah. Perhaps an example is in order here. One well known recent invention is the SleatorTarjan splay tree, a kind of self adjusting binary trees presented in [ST85]. Using an amortized analysis, Sleator and Tarjan have shown that splay trees worst case time performance is at least competitive (within a constant factor) with that of any conventional binary tree, for a sufficiently long sequence of inputs. This result is interesting in itself, and has ....

Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3), 1985.

.... O(n log n) by a divideand conquer algorithm, by applying the Ham Sandwich Theorem and using a linear time algorithm for nding ham sandwich cuts from [14] 4: Self Adjusting Trees The class of self adjusting trees considered in the exercise are often denoted splay trees, and were described in [20]. Homework 4 1: Disk intersection Exercise 35.2 6 in [6] 2: Edit distance Computing the edit distance of two strings is one of the classical applications of dynamic programming. Hirshberg developed in [11] a general technique that, among others, allows the computation of the edit sequence in ....

Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652-686, 1985.

....As each pass runs in time proportional to the combined length of the right spines of the arguments of merge, the operation runs in O(log n) time. The remaining operations are straightforward. 2.4. SUMMARY 41 2.3. 7 Splay Heaps Okasaki [38] presents an implementation of heaps using splay trees [49]. A splay tree is a binary tree that does not maintain any balance information but consistently re structures itself in a manner that tends to balance the tree. For example, as the elements are stored in symmetric order, the deleteMin operation must remove the leftmost node. After this node is ....

Daniel D. K. Sleator and Robert E. Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652--686, July 1985. (p 41)

....See also Page s critique in [Pag82] Section 4. 1) 64 Splay trees are particularly interesting for this application, since they have an adaptive characteristic that may adjust well to the patterns in allocator requests, as well as having amortized complexity within a constant factor of optimal [ST85] 65 We suspect that earlier researchers often simply didn t worry about this because memory sizes were quite small (and block sizes were often rather large) Since this point was not generally made explicit, however, the obvious applicability of scalable data structures was simply left out of ....

....They were interested in both time and space costs, and in scalability to large heaps. Five of their allocators were variants of next fit. 124 The others included simple segregated storage (with powers of two size classes) 125 address ordered first fit (using a self adjusting splay tree [ST85] segregated fits (using Fibonacci spaced size classes) better fit (using Stephenson s Cartesian tree scheme) and two best fit algorithms (one using a balanced binary tree, and the other a splay tree) It may be significant that Korn and Vo modified most of their allocators to include a ....

Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3), 1985.

....decreases by the size of the allocated object, a running total of volume allocated increases by the same amount, and a data structure is created to represent the object. The simulator maintains a global set of data structures representing all objects present in the heap. A splaytree data structure [Sleator and Tarjan, 1983] is used for the purpose, because of its good temporal locality properties. For each D record, the simulator marks the corresponding data structure as known dead, but does not discard it immediately. For each U record, the simulator first identifies the object in which the updated field is and the ....

Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. In Proceedings of the ACM SIGACT Symposium on Theory, pages 235--245, Boston, Massachusetts, April 1983.

....Fortune s code to use a heap to store events, they obtained O(n log n) running time and better performance on large point sets (having more than 50,000 points) However, bucketing outperforms a heap on small point sets. Triangle s implementation uses a heap as well, and also uses a splay tree [17] to store mesh boundary edges, so that an O(n log n) running time is attained, regardless of the distribution of points. Not all boundary edges are stored in the splay tree; when a new edge is created, it is inserted into the tree with probability 0:1. The value 0:1 was chosen empirically to ....

Daniel Dominic Sleator and Robert Endre Tarjan. SelfAdjusting Binary Search Trees. Journal of the Association for Computing Machinery 32(3):652--686, July 1985.

....find the data structure best suited for our simulator, we intend to implement the three most promising structures and test their performance empirically. The selection of structures to test is based on the referenced literature and personal communication [26] 27] 28] with the splay tree [29] [30], the calendar queue [31] and Henriksen s algorithm [32] 33] to be evaluated first. C. Modularity In contradiction to the current simulator the new simulator will fully take advantage of hierarchal modelling, i.e. we are fully using the inheritance concept in an object oriented programming ....

Daniel Dominic Sleator and Robert Endre Tarjan, "Selfadjusting binary search trees", Journal of the Association for Computing Machinery, vol. 32, no. 3, pp. 652--686, Jul 1985.

....allow searching for exact key matches. Since we want to support object identification from field addresses whenever the assumption of object contiguity holds, we prefer a data structure that allows inexact matches. Instead of hash tables, we use self adjusting binary search trees (splay trees) [4, 6]. Although the sequential allocation sequence causes the trees to become quite imbalanced, the inherent locality of object accesses is well matched to the self adjusting operation of splay trees, and the average access time (number of splay operations) is remarkably low. We report below some ....

Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. In Proceedings of the ACM SIGACT Symposium on Theory, pages 235--245, Boston, Massachusetts, April 1983.

....of 1 8 of a block. Further, as also discussed in [Ungar and Jackson, 1988] any object that contains few pointers and that exceeds some threshold in size should be stored in LOS to avoid the overhead of copying. LOS uses free list allocation based on splay trees [Sleator and Tarjan, 1983, Sleator and Tarjan, 1985, Jones, 1986] and, once allocated, an LOS object is never moved. However, LOS objects still belong to a step, which is indicated by threading the objects onto a doubly linked list rooted in the step data structure. When a LOS object is promoted, we simply unchain it from one list and chain it ....

Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM 32, 3 (July 1985).

....we used the heuristic threshold of 1 8 of a block. Further, as also discussed in [Ungar and Jackson, 1988] any object that contains few pointers and that exceeds some threshold in size should be stored in LOS to avoid the overhead of copying. LOS uses free list allocation based on splay trees [Sleator and Tarjan, 1983, Sleator and Tarjan, 1985, Jones, 1986] and, once allocated, an LOS object is never moved. However, LOS objects still belong to a step, which is indicated by threading the objects onto a doubly linked list rooted in the step data structure. When a LOS object is promoted, we simply unchain it from ....

Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. In Proceedings of the ACM SIGACT Symposium on Theory (Boston, Massachusetts, April 1983), pp. 235--245.

....decreases by the size of the allocated object, a running total of volume allocated increases by the same amount, and a data structure is created to represent the object. The simulator maintains a global set of data structures representing all objects present in the heap. A splaytree data structure [Sleator and Tarjan, 1983] is used for the purpose, because of its good temporal locality properties. For each D record, the simulator marks the corresponding data structure as known dead, but does not discard it immediately. For each U record, the simulator first identifies the object in which the updated field is and the ....

Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. In Proceedings of the ACM SIGACT Symposium on Theory, pages 235--245, Boston, Massachusetts, April 1983.

....TO space s unprocessed and allocation pointers, we must process each large object added to the step s list of large objects. The algorithm refinements are straightforward so we skip the details. We chose not to compact large object space. Rather, we maintain free list information using splay trees [Sleator and Tarjan, 1985], which offer excellent average and worst case performance for free list maintenance operations. 2.4 Root Set The roots of a generation set G are the pointers from outside G into G. Scavenging must locate all members of the root set, move the objects in G at which the roots point, and update the ....

Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM 32, 3 (July 1985).

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Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal the Association for Computing Machinery, July 1985.

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Daniel Sleator and Robert E. Tarjan (1985). Self-adjusting binary search trees. JACM 32:3, 652-686

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Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3), July 1985.

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Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. In Proceedings of the ACM SIGACT Symposium on Theory, pages 235--245, Boston, Massachusetts, April 1983.

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