LOWER BOUNDS FOR STREETS AND GENERALIZED STREETS

2001 ◽  
Vol 11 (04) ◽  
pp. 401-421 ◽  
Author(s):  
ALEJANDRO LÓPEZ-ORTIZ ◽  
SVEN SCHUIERER
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real Line ◽  
Upper Bound ◽  
Competitive Ratio ◽  
Search Strategy ◽  
The Real ◽  
Simple Polygons ◽  
On Line ◽  
Special Classes

We present lower bounds for on-line searching problems in two special classes of simple polygons called streets and generalized streets. In streets we assume that the location of the target is known to the robot in advance and prove a lower bound of [Formula: see text] on the competitive ratio of any deterministic search strategy—which can be shown to be tight. For generalized streets we show that if the location of the target is not known, then there is a class of orthogonal generalized streets for which the competitive ratio of any search strategy is at least [Formula: see text] in the L2-metric—again matching the competitive ratio of the best known algorithm. We also show that if the location of the target is known, then the competitive ratio for searching in generalized streets in the L1-metric is at least 9 which is tight as well. The former result is based on a lower bound on the average competitive ratio of searching on the real line if an upper bound of D to the target is given. We show that in this case the average competitive ratio is at least 9-O(1/ log D).

scholarly journals On-line Search in Two-Dimensional Environment

2019 ◽  
Vol 63 (8) ◽  
pp. 1819-1848
Author(s):  
Dariusz Dereniowski ◽  
Dorota Osula
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Mobile Agents ◽  
Line Search ◽  
Search Strategy ◽  
A Priori ◽  
Two Dimensional ◽  
Pursuit Evasion ◽  
On Line ◽  
Evasion Problem

Abstract We consider the following on-line pursuit-evasion problem. A team of mobile agents called searchers starts at an arbitrary node of an unknown network. Their goal is to execute a search strategy that guarantees capturing a fast and invisible intruder regardless of its movements using as few searchers as possible. We require that the strategy is connected and monotone, that is, at each point of the execution the part of the graph that is guaranteed to be free of the fugitive is connected and whenever some node gains a property that it cannot be occupied by the fugitive, the strategy must operate in such a way to keep this property till its end. As a way of modeling two-dimensional shapes, we restrict our attention to networks that are embedded into partial grids: nodes are placed on the plane at integer coordinates and only nodes at distance one can be adjacent. Agents do not have any knowledge about the graph a priori, but they recognize the direction of the incident edge (up, down, left or right). We give an on-line algorithm for the searchers that allows them to compute a connected and monotone strategy that guarantees searching any unknown partial grid with the use of $O(\sqrt {n})$ O ( n ) searchers, where n is the number of nodes in the grid. As for a lower bound, there exist partial grids that require ${\varOmega }(\sqrt {n})$ Ω ( n ) searchers. Moreover, we prove that for each on-line searching algorithm there is a partial grid that forces the algorithm to use ${\varOmega }(\sqrt {n})$ Ω ( n ) searchers but $O(\log n)$ O ( log n ) searchers are sufficient in the off-line scenario. This gives a lower bound on ${\varOmega }(\sqrt {n}/\log n)$ Ω ( n / log n ) in terms of achievable competitive ratio of any on-line algorithm.

Bounds for the best constant in Landau/s inequality on the line

1983 ◽  
Vol 95 (3-4) ◽  
pp. 257-262 ◽  
Author(s):  
Z. M. Franco ◽  
Hans G. Kaper ◽  
Man Kam Kwong ◽  
A. Zettl
Keyword(s):  
Lower Bounds ◽  
Real Line ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Analyse Math ◽  
Best Constant ◽  
The Real ◽  
Explicit Formulae

SynopsisExplicit formulae and numerical values for upper and lower bounds for the best constant in Landau/s inequality on the real line are given. For p > 3, the value of the upper bound is less than the value of the best constant conjectured by Gindler and Goldstein (J. Analyse Math. 28 (1975), 213–238).

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.

scholarly journals On the Rank of $p$-Schemes

10.37236/3097 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Fateme Raei Barandagh ◽  
Amir Rahnamai Barghi
Keyword(s):  
Lower Bound ◽  
Prime Number ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Minimum Rank

Let $n>1$ be an integer and $p$ be a prime number. Denote by $\mathfrak{C}_{p^n}$ the class of non-thin association $p$-schemes of degree $p^n$. A sharp upper and lower bounds on the rank of schemes in $\mathfrak{C}_{p^n}$ with a certain order of thin radical are obtained. Moreover, all schemes in this class whose rank are equal to the lower bound are characterized and some schemes in this class whose rank are equal to the upper bound are constructed. Finally, it is shown that the scheme with minimum rank in $\mathfrak{C}_{p^n}$ is unique up to isomorphism, and it is a fusion of any association $p$-schemes with degree $p^n$.

scholarly journals Lattice self-similar sets on the real line are not Minkowski measurable

2018 ◽  
Vol 40 (1) ◽  
pp. 221-232
Author(s):  
SABRINA KOMBRINK ◽  
STEFFEN WINTER
Keyword(s):  
Real Line ◽  
Iterated Function System ◽  
Function System ◽  
Open Set Condition ◽  
The Real ◽  
Open Set ◽  
Puzzle Piece ◽  
Iterated Function ◽  
Self Similar ◽  
Special Classes

We show that any non-trivial self-similar subset of the real line that is invariant under a lattice iterated function system (IFS) satisfying the open set condition (OSC) is not Minkowski measurable. So far, this has only been known for special classes of such sets. Thus, we provide the last puzzle-piece in proving that under the OSC a non-trivial self-similar subset of the real line is Minkowski measurable if and only if it is invariant under a non-lattice IFS, a 25-year-old conjecture.

scholarly journals Matrix Games with Interval-Valued 2-Tuple Linguistic Information

Games ◽  
2018 ◽  
Vol 9 (3) ◽  
pp. 62 ◽  
Author(s):  
Anjali Singh ◽  
Anjana Gupta
Keyword(s):  
Linear Programming ◽  
Lower Bound ◽  
Real World ◽  
Upper Bound ◽  
Duality Theorem ◽  
Matrix Game ◽  
Matrix Games ◽  
Linguistic Information ◽  
The Real ◽  
Interval Valued

In this paper, a two-player constant-sum interval-valued 2-tuple linguistic matrix game is construed. The value of a linguistic matrix game is proven as a non-decreasing function of the linguistic values in the payoffs, and, hence, a pair of auxiliary linguistic linear programming (LLP) problems is formulated to obtain the linguistic lower bound and the linguistic upper bound of the interval-valued linguistic value of such class of games. The duality theorem of LLP is also adopted to establish the equality of values of the interval linguistic matrix game for players I and II. A flowchart to summarize the proposed algorithm is also given. The methodology is then illustrated via a hypothetical example to demonstrate the applicability of the proposed theory in the real world. The designed algorithm demonstrates acceptable results in the two-player constant-sum game problems with interval-valued 2-tuple linguistic payoffs.

scholarly journals When is non-trivial estimation possible for graphons and stochastic block models?‡

2017 ◽  
Vol 7 (2) ◽  
pp. 169-181
Author(s):  
Audra McMillan ◽  
Adam Smith
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Random Networks ◽  
Block Models

Abstract Block graphons (also called stochastic block models) are an important and widely studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $k$ of blocks and an upper bound $\rho$ on the values (connection probabilities) of the graphon, every estimator incurs error ${\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)\right)$ in the $\delta_2$ metric with constant probability for at least some graphons. In particular, our bound rules out any non-trivial estimation (that is, with $\delta_2$ error substantially less than $\rho$) when $k\geq n\sqrt{\rho}$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the accuracy of graphon estimation in the $\delta_2$ metric. A similar lower bound to ours was obtained independently by Klopp et al.

The Heine-Borel theorem in extended basic logic

1949 ◽  
Vol 14 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Fundamental Theorem ◽  
Upper Bounds ◽  
Basic Logic ◽  
Real Numbers ◽  
Consistent Theory

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U-reals and is completely represented in Κ′ and if some U-real is an upper bound of С, then there is a U-real which is a least upper bound of С. If D is a class of (U-reals and is completely represented in Κ′, then there is a U-real which is a greatest lower bound of D.

A SEMI-ON-LINE SCHEDULING PROBLEM OF TWO PARALLEL MACHINES WITH COMMON MAINTENANCE TIME

2012 ◽  
Vol 29 (04) ◽  
pp. 1250020 ◽  
Author(s):  
YUHUA CAI ◽  
QI FENG ◽  
WENJIE LI
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Parallel Machines ◽  
Time Interval ◽  
Scheduling Problem ◽  
Maintenance Time ◽  
Identical Machines ◽  
On Line

In this paper, we consider a semi-on-line scheduling problem of two identical machines with common maintenance time interval and nonresumable availability. We prove a lower bound of 2.79129 on the competitive ratio and give an on-line algorithm with competitive ratio 2.79633 for this problem.

Some lower bounds for the distribution of the supremum of the Yeh-Wiener process over a rectangular region

1975 ◽  
Vol 12 (04) ◽  
pp. 824-830
Author(s):  
Arthur H. C. Chan
Keyword(s):  
Lower Bound ◽  
Gaussian Process ◽  
Wiener Process ◽  
Lower Bounds ◽  
Upper Bound ◽  
Exact Distribution ◽  
Rectangular Region ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
Two Parameter

Let W (s, t), s, t ≧ 0, be the two-parameter Yeh–Wiener process defined on the first quadrant of the plane, that is, a Gaussian process with independent increments in both directions. In this paper, a lower bound for the distribution of the supremum of W (s, t) over a rectangular region [0, S]×[0, T], for S, T > 0, is given. An upper bound for the same was known earlier, while its exact distribution is still unknown.

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