Upper Bounds for Maximally Greedy Binary Search Trees

Fox, Kyle

doi:10.1007/978-3-642-22300-6_35

Cited by 16 publications

(28 citation statements)

References 14 publications

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“…, and hence splay is approximately monotone on all such executions. 21 Exactly the same logic applies to executions also containing Remark 2.5. The proof of the dynamic finger theorem given in [13,12] is notoriously complicated.…”

Section: Sequences Of Increments and Decrementsmentioning

confidence: 89%

A New Path from Splay to Dynamic Optimality

Levy¹,

Tarjan²

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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Consider the task of performing a sequence of searches in a binary search tree. After each search, an algorithm is allowed to arbitrarily restructure the tree, at a cost proportional to the amount of restructuring performed. The cost of an execution is the sum of the time spent searching and the time spent optimizing those searches with restructuring operations. This notion was introduced by Sleator and Tarjan in (JACM, 1985), along with an algorithm and a conjecture.The algorithm, Splay, is an elegant procedure for performing adjustments while moving searched items to the top of the tree. The conjecture, called dynamic optimality, is that the cost of splaying is always within a constant factor of the optimal algorithm for performing searches. The conjecture stands to this day. In this work, we attempt to lay the foundations for a proof of the dynamic optimality conjecture.Central to our methods are simulation embeddings and approximate monotonicity. A simulation embedding maps each execution to a list of keys that induces a target algorithm to simulate the execution. Approximately monotone algorithms are those whose cost does not increase by more than a constant factor when keys are removed from the list. As we shall see, approximately monotone algorithms with simulation embeddings are dynamically optimal. Building on these ideas:• We construct a simulation embedding for Splay by inducing Splay to perform arbitrary subtree transformations. Thus, if Splay is approximately monotone then it is dynamically optimal. We also show that approximate monotonicity is a necessary condition for dynamic optimality. (Section 2)• We show that if Splay is dynamically optimal, then with respect to optimal cost, its additive overhead is at most linear in the sum of initial tree size and the number of requests. (Section 3)• We prove that a known lower bound on optimal execution cost by Wilber [58] is approximately monotone. (Section 4 and Appendix C)• We speculate about how one might establish dynamic optimality by adapting the proof of approximate monotonicity from the lower bound to Splay. (Section 5)• We demonstrate that two related conjectures, traversal [48] and deque [55], also follow if Splay is approximately monotone, and that many results in this paper extend to a broad class of "path-based" algorithms. (Section 6) Appendix A generalizes the tree transformations used to build simulation embeddings, and Appendix B includes proofs of selected pieces of "folklore" that have appeared throughout the literature.

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Section: Sequences Of Increments and Decrementsmentioning

confidence: 89%

A New Path from Splay to Dynamic Optimality

Levy¹,

Tarjan²

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The left-depth of every sub-root decreases from smallest to largest. 8 3. The splay operation takes constant amortized time.…”

Section: Postordersmentioning

confidence: 99%

“…Balanced TreesLet |x| denote the size of the subtree rooted at x. Following[18], we say T is α weight balanced for α ∈ (0, 1/2] if min{| left(x)|, | right(x)|} + 1 ≥ α · (|x| + 1)8 The first two invariants dictate that the ancestors of sub-roots form a right-toothed comb.…”

mentioning

confidence: 99%

Splaying Preorders and Postorders

Levy

Tarjan

2019

Lecture Notes in Computer Science

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Let T be a binary search tree of n nodes with root r, left subtree L = left(r), and right subtree R = right(r). The preorder and postorder of T are defined as follows: the preorder and postorder of the empty tree is the empty sequence, andwhere ⊕ denotes sequence concatenation. 1 We prove the following results about the behavior of splaying [21] preorders and postorders: 1. Inserting the nodes of preorder(T ) into an empty tree via splaying costs O(n). (Theorem 2.) 2. Inserting the nodes of postorder(T ) into an empty tree via splaying costs O(n). (Theorem 3.) 3. If T has the same keys as T and T is weight-balanced [18] then splaying either preorder(T ) or postorder(T ) starting from T costs O(n). (Theorem 4.) * Research at Princeton University partially supported by an innovation research grant from Princeton and a gift from Microsoft.

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“…However, the conversion in the other direction does not preserve online-ness. The method used by Fox [Fox11] to turn a particular online ASS superset algorithm into an online BST algorithm could probably be adapted to turn any online ASS superset algorithm into an online BST algorithm.…”

Section: Geometric Viewmentioning

confidence: 99%

“…While this method seems intuitively to be a good idea, basic facts like O(log n) amortized time per search were not known until the work of Fox [Fox11], who also showed that there is an equivalent online BST to this method. Showing that this method which greedily looks into the future has an equivalent online method provides some support for the belief that the best online and offline BST algorithms have asymptotically the same runtime.…”

Section: Greedymentioning

confidence: 99%

In Pursuit of the Dynamic Optimality Conjecture

Iacono

2013

Lecture Notes in Computer Science

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In 1985, Sleator and Tarjan introduced the splay tree, a self-adjusting binary search tree algorithm. Splay trees were conjectured to perform within a constant factor as any offline rotation-based search tree algorithm on every sufficiently long sequence-any binary search tree algorithm that has this property is said to be dynamically optimal. However, currently neither splay trees nor any other tree algorithm is known to be dynamically optimal. Here we survey the progress that has been made in the almost thirty years since the conjecture was first formulated, and present a binary search tree algorithm that is dynamically optimal if any binary search tree algorithm is dynamically optimal.

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Upper Bounds for Maximally Greedy Binary Search Trees

Cited by 16 publications

References 14 publications

A New Path from Splay to Dynamic Optimality

A New Path from Splay to Dynamic Optimality

Splaying Preorders and Postorders

In Pursuit of the Dynamic Optimality Conjecture

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