Sharp Bounds on Davenport-Schinzel Sequences of Every Order

Pettie, Seth

doi:10.1145/2794075

Cited by 10 publications

(42 citation statements)

References 96 publications

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“…The function λs(y) is monotonically increasing, and slightly super linear for s ≥ 3, for example λs(y) = O y · 2 O((α(y)) s ) , where α is the inverse Ackermann function (for the currently best bounds known, see [Pet13]). The conditions on the functions in F give us the following.…”

Section: Complexity Of the Overlay Of Lowerenvelopes Of Functions In Ricmentioning

confidence: 99%

On the Complexity of Randomly Weighted Voronoi Diagrams

Har-Peled

Raichel

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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In this paper, we provide an O(n polylog n) bound on the expected complexity of the randomly weighted Voronoi diagram of a set of n sites in the plane, where the sites can be either points, interior-disjoint convex sets, or other more general objects. Here the randomness is on the weight of the sites, not their location. This compares favorably with the worst case complexity of these diagrams, which is quadratic. As a consequence we get an alternative proof to that of Agarwal et al.[AHKS13] of the near linear complexity of the union of randomly expanded disjoint segments or convex sets (with an improved bound on the latter). The technique we develop is elegant and should be applicable to other problems.

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Section: Complexity Of the Overlay Of Lowerenvelopes Of Functions In Ricmentioning

confidence: 99%

On the Complexity of Randomly Weighted Voronoi Diagrams

Har-Peled

Raichel

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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“…The function Ex(u, n, m) has been used to find bounds on Ex(u, n). Bounds on Ex(u, n) are known for several families of sequences such as alternations [1,20,22] and more generally the sequences up(r, t) = (a 1 . .…”

Section: Introductionmentioning

confidence: 99%

“…a r ) t [10]. Let a s denote the alternation of length s. It is known that that Ex(a 3 , n) = n, Ex(a 4 , n) = 2n − 1, Ex(a 5 , n) = 2nα(n) + O(n), Ex(a 6 , n) = Θ(n2 α(n) ), Ex(a 7 , n) = Θ(nα(n)2 α(n) ), and Ex(a s+2 , n) = n2 1,20,22]. Relatively little about Ex(u, n) is known for arbitrary forbidden sequences u.…”

Section: Introductionmentioning

confidence: 99%

“…Nivasch [20] and Pettie [22] found tight bounds on F r,s (n) for all fixed r, s > 0. In particular, F r,5 (n) = Θ(n2 α(n) ) and F r,s+1 (n) = n2 α t (n) t!…”

Section: Introductionmentioning

confidence: 99%

“…Although the known bounds on F r,s (n) are tight, they are not exact, even for s = 2. Klazar [15], Nivasch [20], and Pettie [22] showed that F r,2 (n) < rn, F r,3 (n) < 2rn, F r,2 (n, m) < n+(r−1)m, and F r,3 (n, m) < 2n+(r−1)m. In this paper, we prove that F r,2 (n, m) = n + (r − 1)(m − 1), F r,3 (n, m) = 2n + (r − 1)(m − 2), F r,2 (n) = (n − r)r + 2r−1, and F r,3 (n) = 2(n−r)r+3r−1. We also prove a conjecture from [10] that fw(abc(acb) t abc) = 2t+3 for all t ≥ 0.…”

Section: Introductionmentioning

confidence: 99%

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A Relationship Between Generalized Davenport-Schinzel Sequences and Interval Chains

Geneson

2015

Electron. J. Combin.

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An (r, s)-formation is a concatenation of s permutations of r distinct letters. We define the function F r,s (n) to be the maximum possible length of a sequence with n distinct letters that avoids all (r, s)formations and has every r consecutive letters distinct, and we define the function F r,s (n, m) to be the maximum possible length of a sequence with n distinct letters that avoids all (r, s)-formations and can be partitioned into m blocks of distinct letters. (Nivasch, 2010) and found bounds on F r,s (n) for all fixed r, s > 0, but no exact values were known, even for s = 2. We prove that F r,2 (n, m) = n + (r − 1)(m − 1), F r,3 (n, m) = 2n + (r − 1)(m − 2), F r,2 (n) = (n − r)r + 2r − 1, and F r,3 (n) = 2(n − r)r + 3r − 1, improving on bounds of (Klazar, 1992), (Nivasch, 2010), and.In addition, we improve an upper bound of (Klazar, 2002). For any sequence u, define Ex(u, n) to be the maximum possible length of a sequence with n distinct letters that avoids u and has every r consecutive letters distinct, where r is the number of distinct letters in u. Klazar proved that Ex((a 1 . . . a r ) 2 , n) < (2n+1)L, where L = Ex((a 1 . . . a r ) 2 , K −1)+1 and K = (r−1) 4 +1. Here we prove that K = (r−1) 4 +1 in Klazar's bound can be replaced with K = (r−1) 3 +1. We also prove a conjecture from (Geneson et al., 2014) by showing for t ≥ 1 that Ex(abc(acb) t abc, n) = n2 1 t! α(n) t ±O(α(n) t−1 ) . In addition, we prove that Ex(abcacb(abc) t acb, n) = n2 1 (t+1)! α(n) t+1 ±O(α(n) t ) for t ≥ 1. Furthermore, we extend the equalities F r,2 (n, m) = n + (r − 1)(m − 1) and F r,3 (n, m) = 2n + (r − 1)(m − 2) to formations in d-dimensional 0-1 matrices, sharpening a bound from (Geneson, 2019).

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On the Complexity of Randomly Weighted Multiplicative Voronoi Diagrams

Har-Peled

Raichel

2015

Discrete Comput Geom

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We provide an O(n polylog n) bound on the expected complexity of the randomly weighted multiplicative Voronoi diagram of a set of n sites in the plane, where the sites can be either points, interior-disjoint convex sets, or other more general objects. Here the randomness is on the weight of the sites, not their location. This compares favorably with the worst case complexity of these diagrams, which is quadratic. As a consequence we get an alternative proof to that of Agarwal et al.[AHKS14] of the near linear complexity of the union of randomly expanded disjoint segments or convex sets (with an improved bound on the latter). The technique we develop is elegant and should be applicable to other problems.Generalizations of Voronoi diagrams. In the additive weighted Voronoi diagram, the distance to a Voronoi site is the regular Euclidean distance plus some constant (which depends on the site). Additive Voronoi diagrams have linear descriptive complexity in the plane, as their cells are star shaped (and thus simply connected), as can be easily verified. This holds even if the sites are arbitrary convex sets. In the multiplicative weighted Voronoi diagram, for each site one multiplies the Euclidean distance by a constant (again, that depends on the site). However, unlike the additive case, the worst case complexity for multiplicative weighted Voronoi diagrams is Θ(n 2 ) [AE84], even in the plane. In the weighted case, the cells are not necessarily connected, and a bisector of two sites is either a line or an (Apollonius) circle.In the Power diagram, each site s i has an associated radius r i , and the distance of a point p to this site is s i − p 2 − r 2 i ; that is, the squared length of the tangent from p to the disk of radius r i centered at s i . As such, Power diagrams allow including weight in the distance function, while still having bisectors that are straight lines and having linear combinatorial complexity overall.Klein [Kle88] introduced (and this was further refined by Klein et al. [KLN09]) the notion of abstract Voronoi diagrams to help unify the ever growing list of variants of Voronoi diagrams which have been considered. Specifically, a simple set of axioms was identified, focusing on the bisectors and the regions they define, which classifies a large class of Voronoi diagrams with linear complexity (hence such axioms are not intended to model, for example, multiplicative diagrams).Randomization and Expected Complexity. In many cases, there is a big discrepancy between the worst case analysis of a structure (or an algorithm) and its average case behavior. This suggests that in practice, the worst case is seldom encountered. For example, recently, Agarwal et al. [AKS13, AHKS14], showed that the expected union complexity of a set of randomly expanded disjoint segments is O(n log n), while in the worst case the union complexity can be quadratic. In other words, Agarwal et al. bounded the expected complexity of a level set of the randomly weighted Voronoi diagram of disjoint segments.If the sites are placed...

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Sharp Bounds on Davenport-Schinzel Sequences of Every Order

Cited by 10 publications

References 96 publications

On the Complexity of Randomly Weighted Voronoi Diagrams

On the Complexity of Randomly Weighted Voronoi Diagrams

A Relationship Between Generalized Davenport-Schinzel Sequences and Interval Chains

On the Complexity of Randomly Weighted Multiplicative Voronoi Diagrams

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