In Pursuit of the Dynamic Optimality Conjecture

  title={In Pursuit of the Dynamic Optimality Conjecture},
  author={John Iacono},
  • J. Iacono
  • Published 2 June 2013
  • Computer Science
  • ArXiv
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… 
Competitive Dynamic Bsts .. Two Cost Models
This work describes a conceptually simple but computationally expensive dynamic binary search tree that is-competitive if the cost of rotations is ignored, and proposes a deterministic weakly bounded skip list that is dynamically optimal.
New Paths from Splay to Dynamic Optimality
This work attempts to lay the foundations for a proof of the dynamic optimality conjecture, which is that the cost of splaying is always within a constant factor of the optimal algorithm for performing searches.
Better Analysis of GREEDY Binary Search Tree on Decomposable Sequences
A Foundation for Proving Splay is Dynamically Optimal
It is proved that a lower bound on optimal execution cost is approximately monotone and details how to adapt this proof from the lower bound to Splay, and how to overcome the remaining barriers to establishing dynamic optimality.
Pinning Down the Strong Wilber 1 Bound for Binary Search Trees
It is shown that the gap between the stronger WB-1 bound and $OPT$ may be as large as $\Omega(\log\log n/\log\ log n)$.
Binary search trees, rectangles and patterns
Splay, a popular BST algorithm that has several proven efficiency properties, is generalized, and a set of sufficient (and, in a limited sense, necessary) criteria that guarantee the efficient behavior of a BST algorithm is defined.
Greedy Is an Almost Optimal Deque
This paper extends the geometric binary search tree model of Demaine, Harmon, Iacono, Kane, and Pǎtrascu (DHIKP) to accommodate for insertions and deletions and studies the online Greedy BST algorithm introduced by DHIKP.
Splay trees on trees
It is shown that a $(1 + \frac{1}{t})-approximation of an optimal size-$n STT for a given search distribution can be computed in time, and a broad family of STTs with linear rotation-distance is identified, allowing the generalization of Splay trees to the STT setting.
Competitive Online Search Trees on Trees
This work uses a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree, and describes an online $O(\log \log n)-competitive search tree data structure in this model, matching the best known competitive ratio of binary search trees.
What Does Dynamic Optimality Mean in External Memory?
This paper revisits the question of how dynamic optimality should be defined in external memory and gives an elegant data structure, called the Belga B-tree, that is within an O (log log N )-factor of being dynamically optimal over this class of external-memory search trees.


The geometry of binary search trees
It is shown that there exists an equal-cost online algorithm, transforming the conjecture of Lucas and Munro into the conjecture that the greedy algorithm is dynamically optimal, and achieving a new lower bound for searching in the BST model.
Upper Bounds for Maximally Greedy Binary Search Trees
  • K. Fox
  • Computer Science
  • 2011
The first non-trivial upper bounds for the cost of search operations using GREEDYFUTURE are proved, including giving an access lemma similar to that found in Sleator and Tarjan's classic paper on splay trees.
Self-organizing binary search trees
  • Brian Allen, J. I. Munro
  • Computer Science
    17th Annual Symposium on Foundations of Computer Science (sfcs 1976)
  • 1976
A "move to root" heuristic is shown to yield an expected search time within a constant factor of that of an optimal static binary search tree.
Self-adjusting binary search trees
The splay tree, a self-adjusting form of binary search tree, is developed and analyzed and is found to be as efficient as balanced trees when total running time is the measure of interest.
Static Optimality and Dynamic Search-Optimality in Lists and Trees
A (computationally in efficient) algorithm is shown that can achieve a 1+ε ratio with respect to the best static list in hindsight, by a simple efficient algorithm, and what is called ``dynamic search optimality'': dynamic optimality if the on-line algorithm is allowed to make free rotations after each request.
De-amortizing Binary Search Trees
This paper presents a general method for de-amortizing essentially any Binary Search Tree (BST) algorithm and proves that if there is an O(1)-competitive online BST algorithm, then there is also one that performs every search in O(logn) operations worst case.
Dynamic Optimality - Almost
This is the first major progress on Sleator and Tarjan's dynamic optimality conjecture of 1985 that O(1)-competitive binary search trees exist and presents an O(lg lg n)-competitive online binary search tree.
Combining Binary Search Trees
We present a general transformation for combining a constant number of binary search tree data structures (BSTs) into a single BST whose running time is within a constant factor of the minimum of any
On the Dynamic Finger Conjecture for Splay Trees. Part II: The Proof
  • R. Cole
  • Computer Science
    SIAM J. Comput.
  • 2000
On an n-node splay tree, the amortized cost of an access at distance d from the preceding access is O(log (d+1)) and there is an O(n) initialization cost.
Skip-Splay: Toward Achieving the Unified Bound in the BST Model
The skip-splay algorithm is simple and similar to the splay algorithm, and is the first binary search tree algorithm known to have a running time that nearly achieves the unified bound.