Construction of two-dimensional complementary orthogonal sequences with ZCZ and their lower bound

2005 ◽  
Author(s):  
Fanxin Zeng ◽  
Zhenyu Zhang ◽  
Lijia Ge
Keyword(s):  
Lower Bound ◽  
Two Dimensional ◽  
Orthogonal Sequences

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.

A new lower bound for the non-oriented two-dimensional bin-packing problem

2007 ◽  
Vol 35 (3) ◽  
pp. 365-373 ◽  
Author(s):  
François Clautiaux ◽  
Antoine Jouglet ◽  
Joseph El Hayek
Keyword(s):  
Lower Bound ◽  
Bin Packing ◽  
Packing Problem ◽  
Two Dimensional ◽  
Bin Packing Problem

scholarly journals Hausdorff dimension and uniform exponents in dimension two

2018 ◽  
Vol 167 (02) ◽  
pp. 249-284 ◽  
Author(s):  
YANN BUGEAUD ◽  
YITWAH CHEUNG ◽  
NICOLAS CHEVALLIER
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Packing Dimension ◽  
Two Dimensional ◽  
Joint Work ◽  
Unpublished Work

AbstractIn this paper we prove that the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponentμin (1/2, 1) is equal to 2(1 −μ) forμ⩾$\sqrt2/2$, whereas forμ<$\sqrt2/2$it is greater than 2(1 −μ) and at most equal to (3 − 2μ)(1 − μ)/(1 −μ+μ2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) whenμtends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. Moreover, we prove a lower bound for the packing dimension, which appears to be strictly greater than the Hausdorff dimension for μ ⩾ 0.565. . . .

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.

TWO FOR ONE: TIGHT APPROXIMATION OF 2D BIN PACKING

2013 ◽  
Vol 24 (08) ◽  
pp. 1299-1327 ◽  
Author(s):  
ROLF HARREN ◽  
KLAUS JANSEN ◽  
LARS PRÄDEL ◽  
ULRICH M. SCHWARZ ◽  
ROB VAN STEE
Keyword(s):  
Lower Bound ◽  
Knapsack Problem ◽  
Bin Packing ◽  
Packing Problem ◽  
Approximate Algorithm ◽  
Two Dimensional ◽  
Chip Design ◽  
Practical Applications ◽  
Vlsi Chip ◽  
Bin Packing Problem

In this paper, we study the two-dimensional geometrical bin packing problem (2DBP): given a list of rectangles, provide a packing of all these into the smallest possible number of unit bins without rotating the rectangles. Beyond its theoretical appeal, this problem has many practical applications, for example in print layout and VLSI chip design. We present a 2-approximate algorithm, which improves over the previous best known ratio of 3, matches the best results for the problem where rotations are allowed and also matches the known lower bound of approximability. Our approach makes strong use of a PTAS for a related 2D knapsack problem and a new algorithm that can pack instances into two bins if OPT = 1.

DISASSEMBLING TWO-DIMENSIONAL COMPOSITE PARTS VIA TRANSLATIONS

1993 ◽  
Vol 03 (01) ◽  
pp. 71-84 ◽  
Author(s):  
DORON NUSSBAUM ◽  
JÖRG-RÜDIGER SACK
Keyword(s):  
Computational Complexity ◽  
Lower Bound ◽  
Two Dimensional ◽  
Composite Part ◽  
Disassembly Sequence

This paper deals with the computational complexity of disassembling 2-dimensional composite parts (comprised of M simple n-vertex polygons) via collision-free translations. The first result of this paper is an O(Mn+M log M) algorithm for computing a sequence of translations (performed in a common direction) to disassemble composite parts. The algorithm improves on the O(Mn log Mn) bound previously established for this problem and is easily seen to be optimal. The problem had been posed by Nurmi and by Toussaint. The second result of this paper is an Ω(Mn+M log M) lower bound proof for the problem of detecting whether a composite part can be disassembled, or contains interlocking subparts. Thus, detecting the existence of a disassembly sequence is as hard as computing one. As a consequence, the algorithm for computing a disassembly sequence is optimal also for the detecting problem.

scholarly journals Remarks on the boundary set of spectral equipartitions

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120492 ◽  
Author(s):  
P. Bérard ◽  
B. Helffer
Keyword(s):  
Riemannian Manifold ◽  
Lower Bound ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Two Dimensional ◽  
Nodal Sets ◽  
Nodal Domains ◽  
Open Set ◽  
Ground State Energies

Given a bounded open set in (or in a Riemannian manifold), and a partition of Ω by k open sets ω j , we consider the quantity , where λ ( ω j ) is the ground state energy of the Dirichlet realization of the Laplacian in ω j . We denote by ℒ k ( Ω ) the infimum of over all k -partitions. A minimal k -partition is a partition that realizes the infimum. Although the analysis of minimal k -partitions is rather standard when k =2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of k becomes non-trivial and quite interesting. Minimal partitions are in particular spectral equipartitions, i.e. the ground state energies λ ( ω j ) are all equal. The purpose of this paper is to revisit various properties of nodal sets, and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We prove a lower bound for the length of the boundary set of a partition in the two-dimensional situation. We consider estimates involving the cardinality of the partition.

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