Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences

P. K. Agarwal; M. Sharir; P. Shor

doi:10.1016/0097-3165(89)90032-0

Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences

P. K. Agarwal^*, M. Sharir, P. Shor

^*Corresponding author for this work

School of Computer Science

Research output: Contribution to journal › Article › peer-review

105 Scopus citations

Abstract

We obtain sharp upper and lower bounds on the maximal length λ_s(n) of (n, s)-Davenport-Schinzel sequences, i.e., sequences composed of n symbols, having no two adjacent equal elements and containing no alternating subsequence of length s + 2. We show that (i) λ₄(n) = Θ(n·2^α(n)); (ii) for s > 4, λ_s(n) ≤ n·2^{(α(n))^{(s - 2) 2} + C_s(n)} if s is even and λ_s(n) ≤ n·2^{(α(n))^{(s - 3) 2}log(α(n)) + C_s(n)} if s is odd, where C_s(n) is a function of α(n) and s, asymptotically smaller than the main term; and finally (iii) for even values of s > 4, λ_s(n) = Ω(n·2^{K_s(α(n))^{(s - 2) 2} + Q_s(n)}), where K_s = (( (s - 2) 2)!)^-1 and Q_s is a polynomial in α(n) of degree at most (s - 4) 2.

Original language	English
Pages (from-to)	228-274
Number of pages	47
Journal	Journal of Combinatorial Theory. Series A
Volume	52
Issue number	2
DOIs	https://doi.org/10.1016/0097-3165(89)90032-0
State	Published - Nov 1989

Funding

Funders	Funder number
National Council for Forest Research and Development
International Business Machines Corporation
Digital Equipment Corporation
National Center for Research and Development
National Science Foundation	NSF-DCR-83-20085, 8320085
Office of Naval Research	NOOO14-82-K-0381

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title = "Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences",

abstract = "We obtain sharp upper and lower bounds on the maximal length λs(n) of (n, s)-Davenport-Schinzel sequences, i.e., sequences composed of n symbols, having no two adjacent equal elements and containing no alternating subsequence of length s + 2. We show that (i) λ4(n) = Θ(n·2α(n)); (ii) for s > 4, λs(n) ≤ n·2(α(n)) (s - 2) 2 + Cs(n) if s is even and λs(n) ≤ n·2(α(n)) (s - 3) 2log(α(n)) + Cs(n) if s is odd, where Cs(n) is a function of α(n) and s, asymptotically smaller than the main term; and finally (iii) for even values of s > 4, λs(n) = Ω(n·2Ks(α(n)) (s - 2) 2 + Qs(n)), where Ks = (( (s - 2) 2)!)-1 and Qs is a polynomial in α(n) of degree at most (s - 4) 2.",

author = "Agarwal, {P. K.} and M. Sharir and P. Shor",

note = "Funding Information: *Work on this paper by the first two authors has been supported by Office of Naval Research Grant NOOO14-82-K-0381, by National Science Foundation Grant NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation. Work by the second author has also been supported by a research grant from the NCRD -the Israeli National Council for Research and Development.",

year = "1989",

month = nov,

doi = "10.1016/0097-3165(89)90032-0",

language = "אנגלית",

volume = "52",

pages = "228--274",

journal = "Journal of Combinatorial Theory. Series A",

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number = "2",

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TY - JOUR

T1 - Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences

AU - Agarwal, P. K.

AU - Sharir, M.

AU - Shor, P.

N1 - Funding Information: *Work on this paper by the first two authors has been supported by Office of Naval Research Grant NOOO14-82-K-0381, by National Science Foundation Grant NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation. Work by the second author has also been supported by a research grant from the NCRD -the Israeli National Council for Research and Development.

PY - 1989/11

Y1 - 1989/11

N2 - We obtain sharp upper and lower bounds on the maximal length λs(n) of (n, s)-Davenport-Schinzel sequences, i.e., sequences composed of n symbols, having no two adjacent equal elements and containing no alternating subsequence of length s + 2. We show that (i) λ4(n) = Θ(n·2α(n)); (ii) for s > 4, λs(n) ≤ n·2(α(n)) (s - 2) 2 + Cs(n) if s is even and λs(n) ≤ n·2(α(n)) (s - 3) 2log(α(n)) + Cs(n) if s is odd, where Cs(n) is a function of α(n) and s, asymptotically smaller than the main term; and finally (iii) for even values of s > 4, λs(n) = Ω(n·2Ks(α(n)) (s - 2) 2 + Qs(n)), where Ks = (( (s - 2) 2)!)-1 and Qs is a polynomial in α(n) of degree at most (s - 4) 2.

AB - We obtain sharp upper and lower bounds on the maximal length λs(n) of (n, s)-Davenport-Schinzel sequences, i.e., sequences composed of n symbols, having no two adjacent equal elements and containing no alternating subsequence of length s + 2. We show that (i) λ4(n) = Θ(n·2α(n)); (ii) for s > 4, λs(n) ≤ n·2(α(n)) (s - 2) 2 + Cs(n) if s is even and λs(n) ≤ n·2(α(n)) (s - 3) 2log(α(n)) + Cs(n) if s is odd, where Cs(n) is a function of α(n) and s, asymptotically smaller than the main term; and finally (iii) for even values of s > 4, λs(n) = Ω(n·2Ks(α(n)) (s - 2) 2 + Qs(n)), where Ks = (( (s - 2) 2)!)-1 and Qs is a polynomial in α(n) of degree at most (s - 4) 2.

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Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences

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