scholarly journals A Note on the Expected Length of the Longest Common Subsequences of two i.i.d. Random Permutations

10.37236/6974 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Christian Houdré ◽  
Chen Xu
Keyword(s):  
Lower Bound ◽  
Random Permutations ◽  
Longest Common Subsequences ◽  
Expected Length ◽  
Uniform Case

We address a question and a conjecture on the expected length of the longest common subsequences of two i.i.d. random permutations of $[n]:=\{1,2,...,n\}$. The question is resolved by showing that the minimal expectation is not attained in the uniform case. The conjecture asserts that $\sqrt{n}$ is a lower bound on this expectation, but we only obtain $\sqrt[3]{n}$ for it.

scholarly journals Stretch factor in a planar Poisson–Delaunay triangulation with a large intensity

2018 ◽  
Vol 50 (01) ◽  
pp. 35-56 ◽  
Author(s):  
Nicolas Chenavier ◽  
Olivier Devillers
Keyword(s):  
Lower Bound ◽  
Point Process ◽  
Delaunay Triangulation ◽  
Shortest Path ◽  
Upper Bound ◽  
Poisson Point Process ◽  
Large Intensity ◽  
Stretch Factor ◽  
Expected Length

Abstract Let X := X n ∪ {(0, 0), (1, 0)}, where X n is a planar Poisson point process of intensity n. We provide a first nontrivial lower bound for the distance between the expected length of the shortest path between (0, 0) and (1, 0) in the Delaunay triangulation associated with X when the intensity of X n goes to ∞. Simulations indicate that the correct value is about 1.04. We also prove that the expected length of the so-called upper path converges to 35 / 3π2, yielding an upper bound for the expected length of the smallest path.

Longest common subsequences of two random sequences

1975 ◽  
Vol 12 (02) ◽  
pp. 306-315 ◽  
Author(s):  
Vacláv Chvátal ◽  
David Sankoff
Keyword(s):  
Monte Carlo ◽  
Molecular Evolution ◽  
Lower Bounds ◽  
Monte Carlo Methods ◽  
Longest Common Subsequence ◽  
Upper And Lower Bounds ◽  
Longest Common Subsequences ◽  
Expected Length ◽  
Common Subsequence ◽  
Limiting Behaviour

Summary Given two random k-ary sequences of length n, what is f(n,k), the expected length of their longest common subsequence? This problem arises in the study of molecular evolution. We calculate f(n,k) for all k, where n ≦ 5, and f(n,2) where n ≦ 10. We study the limiting behaviour of n –1 f(n,k) and derive upper and lower bounds on these limits for all k. Finally we estimate by Monte-Carlo methods f(100,k), f(1000,2) and f(5000,2).

Longest common subsequences of two random sequences

1975 ◽  
Vol 12 (2) ◽  
pp. 306-315 ◽  
Author(s):  
Vacláv Chvátal ◽  
David Sankoff
Keyword(s):  
Monte Carlo ◽  
Molecular Evolution ◽  
Lower Bounds ◽  
Monte Carlo Methods ◽  
Longest Common Subsequence ◽  
Upper And Lower Bounds ◽  
Longest Common Subsequences ◽  
Expected Length ◽  
Common Subsequence ◽  
Limiting Behaviour

SummaryGiven two random k-ary sequences of length n, what is f(n,k), the expected length of their longest common subsequence? This problem arises in the study of molecular evolution. We calculate f(n,k) for all k, where n ≦ 5, and f(n,2) where n ≦ 10. We study the limiting behaviour of n–1f(n,k) and derive upper and lower bounds on these limits for all k. Finally we estimate by Monte-Carlo methods f(100,k), f(1000,2) and f(5000,2).

On the Difference of Expected Lengths of Minimum Spanning Trees

2009 ◽  
Vol 18 (3) ◽  
pp. 423-434 ◽  
Author(s):  
WENBO V. LI ◽  
XINYI ZHANG
Keyword(s):  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Exact Formula ◽  
Expected Length ◽  
Wheel Graphs ◽  
Edge Distribution ◽  
Tutte Polynomials ◽  
Uniform Case ◽  
The Difference ◽  
Identical Edge

An exact formula for the expected length of the minimum spanning tree of a connected graph, with independent and identical edge distribution, is given, which generalizes Steele's formula in the uniform case. For a complete graph, the difference of expected lengths between exponential distribution, with rate one, and uniform distribution on the interval (0, 1) is shown to be positive and of rate ζ(3)/n. For wheel graphs, precise values of expected lengths are given via calculations of the associated Tutte polynomials.

A lower bound on the expected length of one-to-one codes

1994 ◽  
Vol 40 (5) ◽  
pp. 1670-1672 ◽  
Author(s):  
N. Alon ◽  
A. Orlitsky
Keyword(s):  
Lower Bound ◽  
Expected Length ◽  
One To One

scholarly journals On the Longest Common Subsequence of Conjugation Invariant Random Permutations

10.37236/8669 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Mohamed Slim Kammoun
Keyword(s):  
Lower Bound ◽  
Good Control ◽  
Universal Constant ◽  
Longest Common Subsequence ◽  
Random Permutations ◽  
Common Subsequence ◽  
The Law

Bukh and Zhou conjectured that the expectation of the length of the longest common subsequence of two i.i.d random permutations of size $n$ is greater than $\sqrt{n}$. We prove in this paper that there exists a universal constant $n_1$ such that their conjecture is satisfied for any pair of i.i.d random permutations of size greater than $n_1$ with distribution invariant under conjugation.  More generally, in the case where the laws of the two permutations are not necessarily the same, we give a lower bound for the expectation. In particular, we prove that if one of the permutations is invariant under conjugation and with a good control of the expectation of the number of its cycles, the limiting fluctuations of the length of the longest common subsequence are of Tracy-Widom type. This result holds independently of the law of the second permutation.

Bounding the Expected Length of Longest Common Subsequences and Forests

1999 ◽  
Vol 32 (4) ◽  
pp. 435-452 ◽  
Author(s):  
R. A. Baeza-Yates ◽  
R. Gavaldà, G. Navarro ◽  
R. Scheihing
Keyword(s):  
Longest Common Subsequences ◽  
Expected Length

scholarly journals Increasing subsequences of random walks

2016 ◽  
Vol 163 (1) ◽  
pp. 173-185 ◽  
Author(s):  
OMER ANGEL ◽  
RICHÁRD BALKA ◽  
YUVAL PERES
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Lower Bound ◽  
Upper Bound ◽  
Simple Random Walk ◽  
Real Numbers ◽  
Record Times ◽  
One Dimensional ◽  
Expected Length ◽  
Longest Increasing Subsequence

AbstractGiven a sequence of n real numbers {Si}i⩽n, we consider the longest weakly increasing subsequence, namely i1 < i2 < . . . < iL with Sik ⩽ Sik+1 and L maximal. When the elements Si are i.i.d. uniform random variables, Vershik and Kerov, and Logan and Shepp proved that ${\mathbb E} L=(2+o(1)) \sqrt{n}$.We consider the case when {Si}i⩽n is a random walk on ℝ with increments of mean zero and finite (positive) variance. In this case, it is well known (e.g., using record times) that the length of the longest increasing subsequence satisfies ${\mathbb E} L\geq c\sqrt{n}$. Our main result is an upper bound ${\mathbb E} L\leq n^{1/2 + o(1)}$, establishing the leading asymptotic behavior. If {Si}i⩽n is a simple random walk on ℤ, we improve the lower bound by showing that ${\mathbb E} L \geq c\sqrt{n} \log{n}$.We also show that if {Si} is a simple random walk in ℤ2, then there is a subsequence of {Si}i⩽n of expected length at least cn1/3 that is increasing in each coordinate. The above one-dimensional result yields an upper bound of n1/2+o(1). The problem of determining the correct exponent remains open.

scholarly journals A quantum lower bound for distinguishing random functions from random permutations

2014 ◽  
Vol 14 (13&14) ◽  
pp. 1089-1097
Author(s):  
Henry Yuen
Keyword(s):  
Lower Bound ◽  
Quantum Algorithm ◽  
Input Function ◽  
Random Permutation ◽  
Query Complexity ◽  
Random Permutations ◽  
Random Functions ◽  
Collision Problem ◽  
Bounded Error ◽  
Quantum Query

The problem of distinguishing between a random function and a random permutation on a domain of size $N$ is important in theoretical cryptography, where the security of many primitives depend on the problem's hardness. We study the quantum query complexity of this problem, and show that any quantum algorithm that solves this problem with bounded error must make $\Omega(N^{1/5}/\polylog N)$ queries to the input function. Our lower bound proof uses a combination of the Collision Problem lower bound and Ambainis's adversary theorem.

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