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.

scholarly journals Walking in a Planar Poisson–Delaunay Triangulation: Shortcuts in the Voronoi Path

2018 ◽  
Vol 28 (03) ◽  
pp. 255-269 ◽  
Author(s):  
Olivier Devillers ◽  
Louis Noizet
Keyword(s):  
Point Process ◽  
Delaunay Triangulation ◽  
Poisson Point Process ◽  
Expected Length

Let [Formula: see text] be a planar Poisson point process of intensity [Formula: see text]. We give a new proof that the expected length of the Voronoi path between [Formula: see text] and [Formula: see text] in the Delaunay triangulation associated with [Formula: see text] is [Formula: see text] when [Formula: see text] goes to infinity; and we also prove that the variance of this length is [Formula: see text]. We investigate the length of possible shortcuts in this path, and define a shortened Voronoi path whose expected length can be expressed as an integral that is numerically evaluated to [Formula: see text]. The shortened Voronoi path has the property to be locally defined; and is shorter than the previously known locally defined paths in Delaunay triangulation such as the upper path whose expected length is [Formula: see text].

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 The Provable Virtue of Laziness in Motion Planning

2019 ◽  
Author(s):  
Nika Haghtalab ◽  
Simon Mackenzie ◽  
Ariel D. Procaccia ◽  
Oren Salzman ◽  
Siddhartha Srinivasa
Keyword(s):  
Lower Bound ◽  
Motion Planning ◽  
Shortest Path ◽  
Upper Bound ◽  
Probabilistic Model ◽  
Shortest Paths ◽  
Worst Case ◽  
Asymptotically Optimal ◽  
Cluttered Environments ◽  
Running Time

The Lazy Shortest Path (LazySP) class consists of motion-planning algorithms that only evaluate edges along candidate shortest paths between the source and target. These algorithms were designed to minimize the number of edge evaluations in settings where edge evaluation dominates the running time of the algorithm such as manipulation in cluttered environments and planning for robots in surgical settings; but how close to optimal are LazySP algorithms in terms of this objective? Our main result is an analytical upper bound, in a probabilistic model, on the number of edge evaluations required by LazySP algorithms; a matching lower bound shows that these algorithms are asymptotically optimal in the worst case.

scholarly journals Multiple closed orbits for N-body-type problems

1998 ◽  
Vol 58 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Shiqing Zhang
Keyword(s):  
Lower Bound ◽  
Periodic Solutions ◽  
Upper Bound ◽  
Critical Value ◽  
Body Type ◽  
Fixed Energy ◽  
Closed Orbits

Using the equivariant Ljusternik-Schnirelmann theory and the estimate of the upper bound of the critical value and lower bound for the collision solutions, we obtain some new results in the large concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planar N-body type problems.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

Isolated minima of the product of n linear forms

1953 ◽  
Vol 49 (1) ◽  
pp. 59-62 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Forms ◽  
Image Position

Letbe n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

scholarly journals Encoding Two-Dimensional Range Top-k Queries

Algorithmica ◽  
2021 ◽  
Author(s):  
Seungbum Jo ◽  
Rahul Lingala ◽  
Srinivasa Rao Satti
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Total Order ◽  
Two Dimensional ◽  
Information Theoretic ◽  
Cartesian Tree ◽  
Dimensional Range

AbstractWe consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering $${\text{Top-}}{k}$$ Top- k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an $$m \times n$$ m × n array, with $$m \le n$$ m ≤ n , we first propose an encoding for answering 1-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, whose query range is restricted to $$[1 \dots m][1 \dots a]$$ [ 1 ⋯ m ] [ 1 ⋯ a ] , for $$1 \le a \le n$$ 1 ≤ a ≤ n . Next, we propose an encoding for answering for the general (4-sided) $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries that takes $$(m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))$$ ( m lg ( k + 1 ) n n + 2 n m ( m - 1 ) + o ( n ) ) bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial $$O(nm\lg {n})$$ O ( n m lg n ) -bit encoding, our encoding takes less space when $$m = o(\lg {n})$$ m = o ( lg n ) . In addition to the upper bound results for the encodings, we also give lower bounds on encodings for answering 1 and 4-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, which show that our upper bound results are almost optimal.

On the Modal μ-Calculus Over Finite Symmetric Graphs

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
Giovanna D’Agostino ◽  
Giacomo Lenzi
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Satisfiability Problem ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Symmetric Graphs ◽  
Trivial Consequence

AbstractIn this paper we consider the alternation hierarchy of the modal μ-calculus over finite symmetric graphs and show that in this class the hierarchy is infinite. The μ-calculus over the symmetric class does not enjoy the finite model property, hence this result is not a trivial consequence of the strictness of the hierarchy over symmetric graphs. We also find a lower bound and an upper bound for the satisfiability problem of the μ-calculus over finite symmetric graphs.

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