Combining Binary Search Trees

@article{Demaine2013CombiningBS,
  title={Combining Binary Search Trees},
  author={Erik D. Demaine and John Iacono and Stefan Langerman and {\"O}zg{\"u}r {\"O}zkan},
  journal={ArXiv},
  year={2013},
  volume={abs/1304.7604}
}
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 "well-behaved" bound on the running time of the given BSTs, for any online access sequence. (A BST has a well-behaved bound with f(n) overhead if it spends at most $\mathcal{O}(f(n))$ time per access and its bound satisfies a weak sense of closure under subsequences.) In particular, we obtain a BST… Expand
The landscape of bounds for binary search trees
TLDR
This paper introduces novel properties that explain the efficiency of sequences not captured by any of the previously known properties, and which provide new barriers to the dynamic optimality conjecture, and establishes connections between various properties. Expand
2 The landscape of bounds for binary search trees
Binary search trees (BSTs) with rotations can adapt to various kinds of structure in search sequences, achieving amortized access times substantially better than the Θ(logn) worst-case guarantee.Expand
Belga B-trees
TLDR
A B-Tree model data structure is introduced, the Belga B-tree, that executes any sequence of searches within a \(O(\log \log N)\) factor of the best offline B- tree model algorithm, provided \(B=\log ^{O(1)}N\). Expand
Multi-finger binary search trees
TLDR
It is shown that BSTs can efficiently serve queries that are close to some recently accessed item, a (restricted) form of the unified property that was previously not known to hold for any BST algorithm, online or offline. Expand
Pattern-Avoiding Access in Binary Search Trees
TLDR
Two different relaxations of the traversal conjecture for GREEDY are proved: (i) GREEDy is almost linear for preorder traversal, and (ii) if a linear-time preprocessing1 is allowed,GREEDY is in fact linear. Expand
Belga B-Trees
TLDR
The B-Tree model is formalized as a natural generalization of the BST model, and a Belga B-tree is introduced, that executes any sequence of searches within a O ( log log N ) $O(\log N)$ factor of the best offline B- tree model algorithm. Expand
Binary search trees, rectangles and patterns
TLDR
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. Expand
Weighted dynamic finger in binary search trees
TLDR
This result is the strongest finger-type bound to be proven for binary search trees, and compared to the previous proof of the dynamic finger bound for Splay trees, it is significantly shorter, stronger, simpler, and has reasonable constants. Expand
New Paths from Splay to Dynamic Optimality
TLDR
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. Expand
The Power and Limitations of Static Binary Search Trees with Lazy Finger
TLDR
A non-entropy based asymptotically-tight expression for the runtime of the optimal lazy finger trees is derived, and a dynamic programming-based method is presented to compute the optimal tree. Expand
...
1
2
...

References

SHOWING 1-10 OF 26 REFERENCES
De-amortizing Binary Search Trees
TLDR
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. Expand
Adaptive binary search trees
TLDR
A framework for generating lower bounds on the cost that any BST algorithm must pay to execute a query sequence is introduced, and it is shown that this framework generalizes previous lower bounds, which suggests that the optimal lower bound in the framework is a good candidate for being tight to within a constant factor of the optimal BST algorithm for each input. Expand
O(log log n)-competitive dynamic binary search trees
TLDR
The multi-splay tree (MST) data structure is introduced, which is the first O(log log n)-competitive BST to simultaneously achieve O( Log n) amortized cost and O( log2 n) worst-case cost per query and proves the sequential access lemma for MSTs, which states that sequentially accessing all keys takes linear time. Expand
Layered Working-Set Trees
TLDR
It is shown how layered working-set trees can be used to achieve the unified bound to within a small additive term in the amortized sense while maintaining in the worst case an access time that is both logarithmic and within asmall multiplicative factor of the working- set bound. Expand
Alternatives to splay trees with O(log n) worst-case access times
  • J. Iacono
  • Mathematics, Computer Science
  • SODA '01
  • 2001
TLDR
The unified conjecture is presented, which subsumes the working set theorem and dynamic finger theorem, and accurately bounds the performance of splay trees over some classes of sequences where the existing theorems' bounds are not tight. Expand
Self-adjusting binary search trees
TLDR
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. Expand
Confluently Persistent Tries for Efficient Version Control
TLDR
This work considers a data-structural problem motivated by version control of a hierarchical directory structure in a system like Subversion, and presents a general technique for global rebuilding of fully persistent data structures, which is nontrivial because amortization and persistence do not usually mix. Expand
On the dynamic finger conjecture for splay trees
  • R. Cole
  • Mathematics, Computer Science
  • STOC '90
  • 1990
TLDR
In this paper, the following bound is shown: the amortized cost of an access is 0 (1 + log d), where the current access is at distance d from the previous access (distance being measured in terms of the number of items straddled by the two successive accesses). Expand
Skip-Splay: Toward Achieving the Unified Bound in the BST Model
TLDR
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. Expand
Improved Upper Bounds for Pairing Heaps
  • J. Iacono
  • Computer Science, Mathematics
  • SWAT
  • 2000
TLDR
It is shown that pairing heaps have a distribution sensitive behavior whereby the cost to perform an extract-min on an element x is O(log min(n, k) where k is the number of heap operations performed since x's insertion. Expand
...
1
2
3
...