In Pursuit of the Dynamic Optimality Conjecture

Iacono, John

doi:10.1007/978-3-642-40273-9_16

Cited by 18 publications

(13 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Related work. For an overview of results related to dynamic optimality, we refer to the survey of Iacono [19]. Besides Tango trees, other O(log log n)-competitive BST algorithms have been proposed, similarly using Wilber's first bound (multi-splay [38] and chain-splay [17]).…”

Section: Overview Of Techniquesmentioning

confidence: 99%

Pattern-Avoiding Access in Binary Search Trees

Chalermsook

Goswami

Kozma

et al. 2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

View full text Add to dashboard Cite

The dynamic optimality conjecture is perhaps the most fundamental open question about binary search trees (BST). It postulates the existence of an asymptotically optimal online BST, i.e. one that is constant factor competitive with any BST on any input access sequence. The two main candidates for dynamic optimality in the literature are splay trees [Sleator and Tarjan, 1985], and Greedy [Lucas, 1988;Munro, 2000;Demaine et al. 2009]. Despite BSTs being among the simplest data structures in computer science, and despite extensive effort over the past three decades, the conjecture remains elusive. Dynamic optimality is trivial for almost all sequences: the optimum access cost of most length-n sequences is Θ(n log n), achievable by any balanced BST. Thus, the obvious missing step towards the conjecture is an understanding of the "easy" access sequences, and indeed the most fruitful research direction so far has been the study of specific sequences, whose "easiness" is captured by a parameter of interest. For instance, splay provably achieves the bound of O(nd) when d roughly measures the distances between consecutive accesses (dynamic finger), the average entropy (static optimality), or the delays between multiple accesses of an element (working set). The difficulty of proving dynamic optimality is witnessed by other highly restricted special cases that remain unresolved; one prominent example is the traversal conjecture [Sleator and Tarjan, 1985], which states that preorder sequences (whose optimum is linear) are linear-time accessed by splay trees; no online BST is known to satisfy this conjecture.In this paper, we prove two different relaxations of the traversal conjecture for Greedy: (i) Greedy is almost linear for preorder traversal, (ii) if a linear-time preprocessing 1 is allowed, Greedy is in fact linear. These statements are corollaries of our more general results that express the complexity of access sequences in terms of a pattern avoidance parameter k. Pattern avoidance is a well-established concept in combinatorics, and the classes of input sequences thus defined are rich, e.g. the k = 3 case includes preorder sequences. For any sequence X with parameter k, our most general result shows that Greedy achieves the cost n2 α(n) O(k) where α is the inverse Ackermann function. Furthermore, a broad subclass of parameter-k sequences has a natural combinatorial interpretation as k-decomposable sequences. For this class of inputs, we obtain an n2 O(k 2 ) bound for Greedy when preprocessing is allowed. For k = 3, these results imply (i) and (ii). To our knowledge, these are the first upper bounds for Greedy that are not known to hold for any other online BST. To obtain these results we identify an input-revealing property of Greedy. Informally, this means that the execution log partially reveals the structure of the access sequence. This property facilitates the use of rich technical tools from forbidden submatrix theory.Further studying the intrinsic complexity of k-decomposable sequences, we make several observ...

show abstract

Section: Overview Of Techniquesmentioning

confidence: 99%

Pattern-Avoiding Access in Binary Search Trees

Chalermsook

Goswami

Kozma

et al. 2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

View full text Add to dashboard Cite

show abstract

“…Here we consider the total number of elementary operations, finger moves and rotations, required to search a given sequence of nodes in the tree, and wish this number to lie within a constant factor of that of any other algorithm, even when the la er has access to the sequence of queries in advance. We refer to Iacono for a survey on this question [Iac13].…”

Section: Dynamic Optimality Of Binary Search Treesmentioning

confidence: 99%

Competitive Online Search Trees on Trees

Bose

Cardinal

Iacono

et al. 2020

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

Self Cite

View full text Add to dashboard Cite

We consider the design of adaptive data structures for searching elements of a tree-structured space. We use 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.is model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. e model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online O(log log n)-competitive search tree data structure in this model, matching the best known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one which we call Steiner-closed search trees, which may be of independent interest. Moreover our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees, and secondly from these trees into paths.

show abstract

“…There has been much work on this conjecture, as well as on the more fundamental question of whether there exists any dynamically optimal BST algorithm (see [9] for a recent review). In the past decade progress was made on this latter question and BST algorithms with better competitive ratio were discovered: Tango trees [6] was the first Oplog log nq-competitive BST; Multi-Splay trees [15] and Zipper trees [1] also have the same competitive ratio along with some additional properties.…”

Section: Introductionmentioning

confidence: 99%