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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.
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
Origins of nonlinearity in DavenportSchinzel sequences. Manuscript in submission. Avalabile from http://www.eecs.umich.edu/~pettie
, 2009
"... A generalized DavenportSchinzel sequence is one over a finite alphabet that excludes subsequences isomorphic to a fixed forbidden subsequence. The fundamental problem in this area is bounding the maximum length of such sequences. Following Klazar, we let Expσ, nq be the maximum length of a sequence ..."
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Cited by 6 (5 self)
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A generalized DavenportSchinzel sequence is one over a finite alphabet that excludes subsequences isomorphic to a fixed forbidden subsequence. The fundamental problem in this area is bounding the maximum length of such sequences. Following Klazar, we let Expσ, nq be the maximum length of a sequence over an alphabet of size n excluding subsequences isomorphic to σ. It has been proved that for every σ, Expσ, nq is either linear or very close to linear. In particular it is Opn2 αpnqOp1q q, where α is the inverseAckermann function and Op1q depends on σ. In much the same way that the complete graphs K5 and K3,3 represent the minimal causes of nonplanarity, there must exist a set ΦNonlin of minimal nonlinear forbidden subsequences. Very little is known about the size or membership of ΦNonlin. In this paper we construct an infinite antichain of nonlinear forbidden subsequences which, we argue, strongly supports the conjecture that ΦNonlin is itself infinite. Perhaps the most novel contribution of this paper is a succinct, humanly readable code for expressing the structure of forbidden subsequences. 1
Generalized DavenportSchinzel Sequences and Their 01 Matrix Counterparts
, 2010
"... A generalized DavenportSchinzel sequence is one over a finite alphabet whose subsequences are not isomorphic to a forbidden subsequence σ. What is the maximum length of such a σfree sequence, as a function of its alphabet size n? is the extremal function linear or nonlinear? and what characteristi ..."
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Cited by 5 (3 self)
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A generalized DavenportSchinzel sequence is one over a finite alphabet whose subsequences are not isomorphic to a forbidden subsequence σ. What is the maximum length of such a σfree sequence, as a function of its alphabet size n? is the extremal function linear or nonlinear? and what characteristics of σ determine the answers to these questions? It is known that such sequences have length at most n · 2 (α(n))O(1), where α is the inverseAckermann function and the O(1) depends on σ. We resolve a number of open problems on the extremal properties of generalized DavenportSchinzel sequences. Among our results: 1. We give a nearly complete characterization of linear and nonlinear σ ∈ {a, b, c} ∗ over a threeletter alphabet. Specifically, the only repetitionfree minimally nonlinear forbidden sequences are ababa and abcacbc. 2. We prove there are at least four minimally nonlinear forbidden sequences. 3. We prove that in many cases, doubling a forbidden sequence has no significant affect its extremal function. For example, Nivasch’s upper bounds on alternating sequences of the form (ab) t and (ab) t a, for t ≥ 3, can be extended to forbidden sequences of the form (aabb) t and (aabb) t a. 4. Finally, we show that the absence of simple subsequences in σ tells us nothing about σ’s extremal function. For example, for any t, there exists a σt avoiding ababa whose extremal function is Ω(n·2 αt (n) Most of our results are obtained by translating questions about generalized DavenportSchinzel sequences into questions about the density of 01 matrices avoiding certain forbidden submatrices. We give new and often tight bounds on the extremal functions of numerous forbidden 01 matrices.
On the Structure and Composition of Forbidden Sequences, with Geometric Applications
, 2010
"... Forbidden substructure theorems have proved to be among of the most versatile tools in bounding the complexity of geometric objects and the running time of geometric algorithms. To apply them one typically transcribes an algorithm execution or geometric object as a sequence over some alphabet or a 0 ..."
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Cited by 3 (1 self)
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Forbidden substructure theorems have proved to be among of the most versatile tools in bounding the complexity of geometric objects and the running time of geometric algorithms. To apply them one typically transcribes an algorithm execution or geometric object as a sequence over some alphabet or a 01 matrix, proves that this object avoids some subsequence or submatrix σ, then uses an off the shelf bound on the maximum size of such a σfree object. As a historical trend, expanding our library of forbidden substructure theorems has led to better bounds and simpler analyses of the complexity of geometric objects. We establish new and tight bounds on the maximum length of generalized DavenportSchinzel sequences, which are those whose subsequences are not isomorphic to some fixed sequence σ. (The standard DavenportSchinzel sequences restrict σ to be of the form abab · · ·.) 1. We prove that Nshaped forbidden subsequences (of the form abc · · · xyzyx · · · cbabc · · · xyz) have a linear extremal function. Our proof dramatically improves an earlier one of Klazar and Valtr in the leading constants and overall simplicity. This result tightens the (astronomical) leading constants in Valtr’s O(n log n) bound on geometric graphs without
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,
CAPTAIN: TAKE OFF EVERY ’ZIG’!! CAPTAIN: YOU KNOW WHAT YOU DOING. CAPTAIN: MOVE ’ZIG’. CAPTAIN: FOR GREAT JUSTICE.
"... Everything was balanced before the computers went off line. Try and adjust something, and you unbalance something else. Try and adjust that, you unbalance two more and before you know what’s happened, the ship is out of control. — Blake, Blake’s 7, “Breakdown ” (March 6, 1978) A good scapegoat is ne ..."
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Everything was balanced before the computers went off line. Try and adjust something, and you unbalance something else. Try and adjust that, you unbalance two more and before you know what’s happened, the ship is out of control. — Blake, Blake’s 7, “Breakdown ” (March 6, 1978) A good scapegoat is nearly as welcome as a solution to the problem. Let’s play.
CAPTAIN: TAKE OFF EVERY ’ZIG’!! CAPTAIN: YOU KNOW WHAT YOU DOING. CAPTAIN: MOVE ’ZIG’. CAPTAIN: FOR GREAT JUSTICE.
"... Everything was balanced before the computers went off line. Try and adjust something, and you unbalance something else. Try and adjust that, you unbalance two more and before you know what’s happened, the ship is out of control. — Blake, Blake’s 7, “Breakdown ” (March 6, 1978) A good scapegoat is ne ..."
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Everything was balanced before the computers went off line. Try and adjust something, and you unbalance something else. Try and adjust that, you unbalance two more and before you know what’s happened, the ship is out of control. — Blake, Blake’s 7, “Breakdown ” (March 6, 1978) A good scapegoat is nearly as welcome as a solution to the problem. Let’s play.
Sharp Bounds on DavenportSchinzel Sequences of Every Order ∗
"... One of the oldest unresolved problems in extremal combinatorics is to determine the maximum length of DavenportSchinzel sequences, where an orders DS sequence is defined to be one over an nletter alphabet that avoids alternating subsequences of the form a · · · b · · · a · · · b · · · wit ..."
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One of the oldest unresolved problems in extremal combinatorics is to determine the maximum length of DavenportSchinzel sequences, where an orders DS sequence is defined to be one over an nletter alphabet that avoids alternating subsequences of the form a · · · b · · · a · · · b · · · with length s + 2. These sequences were introduced by Davenport and Schinzel in 1965 to model a certain problem in differential equations and have since become an indispensable tool in computational geometry and the analysis of discrete geometric structures. Let λs(n) be the extremal function for such sequences. What is λs asymptotically? This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, and Nivasch) when s is even or s ≤ 3. However, since the work of Agarwal, Sharir, and Shor in the 1980s there has been a persistent gap in our understanding of the odd orders, a gap that is just as much qualitative as quantitative. In this paper we establish the following bounds on λs(n) for every order s. n s = 1 2n − 1 s = 2 ⎪ ⎨ 2nα(n) + O(n) s = 3 λs(n) = Θ(n2 α(n) ) s = 4 Θ(nα(n)2 α(n) ) s = 5 n2 (1+o(1))αt (n)/t! s ≥ 6, t = ⌊ s−2 2 ⌋ These results refute a conjecture of Alon, Kaplan, Nivasch, Sharir, and Smorodinsky and run counter to common sense. When s is odd, λs behaves essentially like λs−1.
Chapter 14 DavenportSchinzel Sequences
"... The complexity of a simple arrangement of n lines in R 2 is Θ(n 2) and so every algorithm that uses such an arrangement explicitly needs Ω(n 2) time. However, there are many scenarios in which we do not need the whole arrangement but only some part of it. For instance, to construct a hamsandwich cu ..."
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The complexity of a simple arrangement of n lines in R 2 is Θ(n 2) and so every algorithm that uses such an arrangement explicitly needs Ω(n 2) time. However, there are many scenarios in which we do not need the whole arrangement but only some part of it. For instance, to construct a hamsandwich cut for two sets of points in R 2 one needs the median levels of the two corresponding line arrangements only. As mentioned in the previous section, the relevant information about these levels can actually be obtained in linear time. Similarly, in a motion planning problem where the lines are considered as obstacles we are only interested in the cell of the arrangement we are located in. There is no way to ever reach any other cell, anyway. This chapter is concerned with analyzing the complexity—that is, the number of vertices and edges—of a single cell in an arrangement of n curves in R 2. In case of a line arrangement this is mildly interesting only: Every cell is convex and any line can appear at most once along the cell boundary. On the other hand, it is easy to construct an example in which there is a cell C such that every line appears on the boundary ∂C. But when we consider arrangement of line segments rather than lines, the situation