Results 1  10
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15
Improved bounds and new techniques for DavenportSchinzel sequences and their generalizations
 In Proceedings 20th ACMSIAM Symposium on Discrete Algorithms (SODA
, 2009
"... We present several new results regarding λs(n), the maximum length of a Davenport–Schinzel sequence of order s on n distinct symbols. First, we prove that λs(n) ≤ n · 2 (1/t!)α(n)t +O(α(n) t−1), n · 2 (1/t!)α(n)t log 2 α(n)+O(α(n) t), s ≥ 4 even; s ≥ 3 odd; where t = ⌊(s − 2)/2⌋, and α(n) denotes th ..."
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Cited by 18 (1 self)
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We present several new results regarding λs(n), the maximum length of a Davenport–Schinzel sequence of order s on n distinct symbols. First, we prove that λs(n) ≤ n · 2 (1/t!)α(n)t +O(α(n) t−1), n · 2 (1/t!)α(n)t log 2 α(n)+O(α(n) t), s ≥ 4 even; s ≥ 3 odd; where t = ⌊(s − 2)/2⌋, and α(n) denotes the inverse Ackermann function. The previous upper bounds, by Agarwal, Sharir, and Shor (1989), had a leading coefficient of 1 instead of 1/t! in the exponent. The bounds for even s are now tight up to lowerorder terms in the exponent. These new bounds result from a small improvement on the technique of Agarwal et al. More importantly, we also present a new technique for deriving upper bounds for λs(n). This new technique is based on some recurrences very similar to those used by the author, together with Alon, Kaplan, Sharir, and Smorodinsky (SODA 2008), for the problem of stabbing interval chains with jtuples. With this new technique we: (1) rederive the upper bound of λ3(n) ≤ 2nα(n)+O ( n √ α(n) ) (first shown by Klazar, 1999); (2) rederive our own new upper bounds for general s; and (3) obtain improved upper bounds for the generalized Davenport–Schinzel sequences considered by Adamec, Klazar, and Valtr (1992). Regarding lower bounds, we show that λ3(n) ≥ 2nα(n) − O(n) (the previous lower bound (Sharir and Agarwal, 1995) had a coefficient of 1 2), so the coefficient 2 is tight. We also present a simpler variant of the construction of Agarwal, Sharir, and Shor that achieves the known lower bounds of λs(n) ≥ n·2 (1/t!)α(n)t−O(α(n) t−1) for s ≥ 4 even.
Splay trees, DavenportSchinzel sequences, and the deque conjecture
, 2007
"... We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a DavenportSchinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct ..."
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Cited by 15 (5 self)
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We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a DavenportSchinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct this technique towards Tarjan’s deque conjecture and prove that n deque operations require O(nα ∗ (n)) time, where α ∗ (n) is the minimum number of applications of the inverseAckermann function mapping n to a constant. We are optimistic that this approach could be directed towards other open conjectures on splay trees such as the traversal and split conjectures.
A unified access bound on comparisonbased dynamic dictionaries
 Theoretical Computer Science
"... We present a dynamic comparisonbased search structure that supports insertions, deletions, and searches within the unified bound. The unified bound specifies that it is quick to access an element that is near a recently accessed element. More precisely, if w(y) distinct elements have been accessed ..."
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Cited by 12 (1 self)
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We present a dynamic comparisonbased search structure that supports insertions, deletions, and searches within the unified bound. The unified bound specifies that it is quick to access an element that is near a recently accessed element. More precisely, if w(y) distinct elements have been accessed since the last access to element y, and d(x, y) denotes the rank distance between x and y among the current set of elements, then the amortized cost to access element x is O(miny log[w(y) + d(x, y) + 2]). This property generalizes the workingset and dynamicfinger properties of splay trees. Preprint submitted to Elsevier Science 31 January 2007 1
Weak ɛnets and interval chains
, 2007
"... We construct weak ɛnets of almost linear size for certain types of point sets. Specifically, for planar point sets in convex position we construct weak 1 rnets of size O(rα(r)), where α(r) denotes the inverse Ackermann function. For point sets along the moment curve in Rd we construct weak 1 rnet ..."
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Cited by 6 (1 self)
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We construct weak ɛnets of almost linear size for certain types of point sets. Specifically, for planar point sets in convex position we construct weak 1 rnets of size O(rα(r)), where α(r) denotes the inverse Ackermann function. For point sets along the moment curve in Rd we construct weak 1 rnets of size r · 2poly(α(r)) , where the degree of the polynomial in the exponent depends (quadratically) on d. Our constructions result from a reduction to a new problem, which we call stabbing interval chains with jtuples. Given the range of integers N = [1, n], an interval chain of length k is a sequence of k consecutive, disjoint, nonempty intervals contained in N. A jtuple p = (p1,..., pj) is said to stab an interval chain C = I1 · · · Ik if each pi falls on a different interval of C. The problem is to construct a smallsize family Z of jtuples that stabs all kinterval chains in N. Let z (j)
Applications of forbidden 01 matrices to search tree and path compression based data structures
, 2009
"... In this paper we improve, reprove, and simplify a variety of theorems concerning the performance of data structures based on path compression and search trees. We apply a technique very familiar to computational geometers but still foreign to many researchers in (nongeometric) algorithms and data s ..."
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Cited by 6 (5 self)
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In this paper we improve, reprove, and simplify a variety of theorems concerning the performance of data structures based on path compression and search trees. We apply a technique very familiar to computational geometers but still foreign to many researchers in (nongeometric) algorithms and data structures, namely, to bound the complexity of an object via its forbidden substructures. To analyze an algorithm or data structure in the forbidden substructure framework one proceeds in three discrete steps. First, one transcribes the behavior of the algorithm as some combinatorial object M; for example, M may be a graph, sequence, permutation, matrix, set system, or tree. (The size of M should ideally be linear in the running time.) Second, one shows that M excludes some forbidden substructure P, and third, one bounds the size of any object avoiding this substructure. The power of this framework derives from the fact that M lies in a more pristine environment and that upper bounds on the size of a Pfree object M may be reused in different contexts. All of our proofs begin by transcribing the individual operations of a dynamic data structure
Dynamic Optimality for Skip Lists and BTrees
, 2008
"... Sleator and Tarjan [39] conjectured that splay trees are dynamically optimal binary search trees (BST). In this context, we study the skip list data structure introduced by Pugh [35]. We prove that for a class of skip lists that satisfy a weak balancing property, the workingset bound is a lower bou ..."
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Cited by 5 (1 self)
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Sleator and Tarjan [39] conjectured that splay trees are dynamically optimal binary search trees (BST). In this context, we study the skip list data structure introduced by Pugh [35]. We prove that for a class of skip lists that satisfy a weak balancing property, the workingset bound is a lower bound on the time to access any sequence. Furthermore, we develop a deterministic selfadjusting skip list whose running time matches the workingset bound, thereby achieving dynamic optimality in this class. Finally, we highlight the implications our bounds for skip lists have on multiway branching search trees such as Btrees, (ab)trees, and other variants as well as their binary tree representations. In particular, we show a selfadjusting Btree that is dynamically optimal both in internal and external memory.
Dynamic optimality and multisplay trees
, 2004
"... The Dynamic Optimality Conjecture [ST85] states that splay trees are competitive (with a constant competitive factor) among the class of all binary search tree (BST) algorithms. Despite 20 years of research this conjecture is still unresolved. Recently Demaine et al. [DHIP04] suggested searching for ..."
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Cited by 3 (2 self)
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The Dynamic Optimality Conjecture [ST85] states that splay trees are competitive (with a constant competitive factor) among the class of all binary search tree (BST) algorithms. Despite 20 years of research this conjecture is still unresolved. Recently Demaine et al. [DHIP04] suggested searching for alternative algorithms which have small, but nonconstant competitive factors. They proposed tango, a BST algorithm which is nearly dynamically optimal – its competitive ratio is £¥¤§¦©¨���¦�¨����� � instead of a constant. Unfortunately, for many access patterns, tango is worse than other BST algorithms by a factor of ¦�¨���¦�¨��� �. In this paper we introduce multisplay trees, which can be viewed as a variant of splay trees. We prove the multisplay access lemma, which resembles the access lemma for splay trees. With different assignment of weights, this lemma allows us to prove various bounds on the performance of multisplay trees. Specifically, we prove that multisplay trees are £¥¤�¦�¨���¦©¨����� �competitive, and amortized £¥¤�¦�¨����� �. This is the first BST data structure to simultaneously achieve these two bounds. In addition, the algorithm is simple enough that we include code for its key parts. This work raises many open questions about the performance of multisplay trees. Does sequential access take linear time? (Our experiments indicate the answer is “yes”.) Are multisplay trees dynamically optimal? How do multisplay trees compare to splay trees? Specifically, are there sequences where one outperformes the other? What can be proved if we allow insertions and deletions in a multisplay tree? 1
Untangling binary trees via rotations
 Comput. J
"... In this paper we present a polynomial time algorithm for finding the shortest sequence of rotations that converts one binary tree into another when both binary trees are of a restricted form. The initial tree must be a degenerate tree, where every node has exactly one child, and the destination bina ..."
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Cited by 2 (0 self)
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In this paper we present a polynomial time algorithm for finding the shortest sequence of rotations that converts one binary tree into another when both binary trees are of a restricted form. The initial tree must be a degenerate tree, where every node has exactly one child, and the destination binary tree must also be degenerate, of a more restricted nature. Previous work on rotation distance has focused on approximation algorithms. Our algorithm is the only known nontrivial polynomial time algorithm for exact rotation distance between special cases of binary trees. 1.
Properties of MultiSplay Trees
, 2009
"... We show that multisplay trees have most of the properties that splay trees have. Specifically, we show that multisplay trees have the following properties: the access lemma, static optimality, the static finger property, the working set property, and keyindependent optimality. Moreover, we prove ..."
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Cited by 1 (1 self)
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We show that multisplay trees have most of the properties that splay trees have. Specifically, we show that multisplay trees have the following properties: the access lemma, static optimality, the static finger property, the working set property, and keyindependent optimality. Moreover, we prove that multisplay trees have the deque property, which was conjectured by Tarjan in 1985 for splay trees, but remains unproven despite a significant amount of research toward proving it. Efficiently maintaining and manipulating sets of elements from a totally ordered universe is a fundamental problem in computer science. Specifically, many algorithms need a data structure that can efficiently support at least the following operations: insert, delete, predecessor, and successor, as well as membership testing. A standard data structure that maintains a totally ordered set and
Adaptive Binary Search Trees
, 2009
"... A ubiquitous problem in the field of algorithms and data structures is that of searching for an element from an ordered universe. The simple yet powerful binary search tree (BST) model provides a rich family of solutions to this problem. Although BSTs require Ω(lg n) time per operation in the wors ..."
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Cited by 1 (0 self)
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A ubiquitous problem in the field of algorithms and data structures is that of searching for an element from an ordered universe. The simple yet powerful binary search tree (BST) model provides a rich family of solutions to this problem. Although BSTs require Ω(lg n) time per operation in the worst case, various adaptive BST algorithms are capable of exploiting patterns in the sequence of queries to achieve tighter, inputsensitive, bounds that can be o(lg n) in many cases. This thesis furthers our understanding of what is achievable in the BST model along two directions. First, we make progress in improving instancespecific lower bounds in the BST model. In particular, we introduce a framework for generating lower bounds on the cost that any BST algorithm must pay to execute a query sequence,