scholarly journals On Ensuring Multilayer Wirability by Stretching Layouts

VLSI Design ◽  
1998 ◽  
Vol 7 (4) ◽  
pp. 365-383
Author(s):  
Teofilo F. Gonzalez ◽  
Si-Qing Zheng
Keyword(s):  
Dynamic Programming ◽  
Lower Bound ◽  
Upper Bound ◽  
Area Increase ◽  
Different Types ◽  
Np Complete ◽  
Layout Area ◽  
The One

Every knock-knee layout is four-layer wirable. However, there are knock-knee layouts that cannot be wired in less than four layers. While it is easy to determine whether a knock-knee layout is one-layer wirable or two-layer wirable, the problem of determining three-layer wirability of knock-knee layouts is NP-complete. A knock-knee layout may be stretched vertically (horizontally) by introducing empty rows (columns) so that it can be wired in fewer than four layers. In this paper we discuss two different types of stretching schemes. It is known that under these two stretching schemes, any knock-knee layout is three-layer wirable by stretching it up to (4/3) of the knock-knee layout area (upper bound). We show that there are knock-knee layouts that when stretched and wired in three layers under scheme I (II) require at least 1.2 (1.07563) of the original layout area. Our lower bound for the area increase factor can be used to guide the search for effective stretching-based dynamic programming three-layer wiring algorithms similar to the one presented in [8].

scholarly journals The number of olfactory stimuli that humans can discriminate is still unknown

eLife ◽  
2015 ◽  
Vol 4 ◽  
Author(s):  
Richard C Gerkin ◽  
Jason B Castro
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Olfactory Stimuli ◽  
The One ◽  
Reported Data

It was recently proposed (<xref ref-type="bibr" rid="bib2">Bushdid et al., 2014</xref>) that humans can discriminate between at least a trillion olfactory stimuli. Here we show that this claim is the result of a fragile estimation framework capable of producing nearly any result from the reported data, including values tens of orders of magnitude larger or smaller than the one originally reported in (<xref ref-type="bibr" rid="bib2">Bushdid et al., 2014</xref>). Additionally, the formula used to derive this estimate is well-known to provide an upper bound, not a lower bound as reported. That is to say, the actual claim supported by the calculation is in fact that humans can discriminate at most one trillion olfactory stimuli. We conclude that there is no evidence for the original claim.

ESTIMATE OF THE HAUSDORFF MEASURE OF THE SIERPINSKI TRIANGLE

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 137-148
Author(s):  
PÉTER MÓRA
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Upper Bound ◽  
Open Problem ◽  
Dimensional Hausdorff Measure ◽  
Sierpinski Triangle ◽  
The One

It is well-known that the Hausdorff dimension of the Sierpinski triangle Λ is s = log 3/ log 2. However, it is a long standing open problem to compute the s-dimensional Hausdorff measure of Λ denoted by [Formula: see text]. In the literature the best existing estimate is [Formula: see text] In this paper we improve significantly the lower bound. We also give an upper bound which is weaker than the one above but everybody can check it easily. Namely, we prove that [Formula: see text] holds.

scholarly journals Sur les solutions friables de l'équation a+b=c

2013 ◽  
Vol 154 (3) ◽  
pp. 439-463 ◽  
Author(s):  
SARY DRAPPEAU
Keyword(s):  
Lower Bound ◽  
Asymptotic Behaviour ◽  
Recent Work ◽  
Upper Bound ◽  
Smooth Function ◽  
Riemann Hypothesis ◽  
Smooth Solutions ◽  
Compactly Supported ◽  
The One

AbstractIn a recent paper [5], Lagarias and Soundararajan study the y-smooth solutions to the equation a+b=c. Conditionally under the Generalised Riemann Hypothesis, they obtain an estimate for the number of those solutions weighted by a compactly supported smooth function, as well as a lower bound for the number of bounded unweighted solutions. In this paper, we prove a more precise conditional estimate for the number of weighted solutions that is valid when y is relatively large with respect to x, so as to connect our estimate with the one obtained by La Bretèche and Granville in a recent work [2]. We also prove, conditionally under the Generalised Riemann Hypothesis, the conjectured upper bound for the number of bounded unweighted solutions, thus obtaining its exact asymptotic behaviour.

Energy of m-Polar Fuzzy Digraphs

2020 ◽  
pp. 469-491 ◽  
Author(s):  
Muhammad Akram ◽  
Danish Saleem ◽  
Ganesh Ghorai
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Fuzzy Graph ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Different Types ◽  
Matrix Eigenvalues

In this chapter, firstly some basic definitions like fuzzy graph, its adjacency matrix, eigenvalues, and its different types of energies are presented. Some upper bound and lower bound for the energy of this graph are also obtained. Then certain notions, including energy of m-polar fuzzy digraphs, Laplacian energy of m-polar fuzzy digraphs and signless Laplacian energy of m-polar fuzzy digraphs are presented. These concepts are illustrated with several example, and some of their properties are investigated.

scholarly journals The Degree/Diameter Problem in Maximal Planar Bipartite graphs

10.37236/4468 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Cristina Dalfó ◽  
Clemens Huemer ◽  
Julián Salas
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Upper Bounds ◽  
The One

The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.

scholarly journals Edge Forcing in Butterfly Networks

2021 ◽  
Vol 182 (3) ◽  
pp. 285-299
Author(s):  
G. Jessy Sujana ◽  
T.M. Rajalaxmi ◽  
Indra Rajasingh ◽  
R. Sundara Rajan
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Higher Dimensions ◽  
Zero Forcing ◽  
Minimum Cardinality ◽  
Forcing Number ◽  
Np Complete ◽  
Forcing Set ◽  
Zero Forcing Set

A zero forcing set is a set S of vertices of a graph G, called forced vertices of G, which are able to force the entire graph by applying the following process iteratively: At any particular instance of time, if any forced vertex has a unique unforced neighbor, it forces that neighbor. In this paper, we introduce a variant of zero forcing set that induces independent edges and name it as edge-forcing set. The minimum cardinality of an edge-forcing set is called the edge-forcing number. We prove that the edge-forcing problem of determining the edge-forcing number is NP-complete. Further, we study the edge-forcing number of butterfly networks. We obtain a lower bound on the edge-forcing number of butterfly networks and prove that this bound is tight for butterfly networks of dimensions 2, 3, 4 and 5 and obtain an upper bound for the higher dimensions.

scholarly journals Number of Directions Determined by a Set in $$\mathbb {F}_{q}^{2}$$ and Growth in $$\mathrm {Aff}(\mathbb {F}_{q})$$

2021 ◽  
Author(s):  
Daniele Dona
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Upper Bound ◽  
Finite Plane ◽  
Structural Theorem ◽  
Prime Fields ◽  
The One

AbstractWe prove that a set A of at most q non-collinear points in the finite plane $$\mathbb {F}_{q}^{2}$$ F q 2 spans more than $${|A|}/\!{\sqrt{q}}$$ | A | / q directions: this is based on a lower bound by Fancsali et al. which we prove again together with a different upper bound than the one given therein. Then, following the procedure used by Rudnev and Shkredov, we prove a new structural theorem about slowly growing sets in $$\mathrm {Aff}(\mathbb {F}_{q})$$ Aff ( F q ) for any finite field $$\mathbb {F}_{q}$$ F q , generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.

A discrete chain of degrees of index sets

1972 ◽  
Vol 37 (1) ◽  
pp. 139-149 ◽  
Author(s):  
Louise Hay
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Local Information ◽  
Truth Table ◽  
Partial Ordering ◽  
Index Set ◽  
Recursively Enumerable ◽  
Index Sets ◽  
The One ◽  
Standard Enumeration

Let {Wi} be a standard enumeration of all recursively enumerable (r.e.) sets, and for any class A of r.e. sets, let θA denote the index set of A = {n ∣ Wn ∈ A}. (Clearly, .) In [1], the index sets of nonempty finite classes of finite sets were classified under one-one reducibility into an increasing sequence {Ym}, 0 ≤ m < ∞. In this paper we examine further properties of this sequence within the partial ordering of one-one degrees of index sets. The main results are as follows: (1) For each m, Ym < Ym + 1 and < Ym + 1; (2) Ym is incomparable to ; (3) Ym + 1 and ; are immediate successors (among index sets) of Ym and m; (4) the pair (Ym + 1, ) is a “least upper bound” for the pair (Ym, ) in the sense that any successor of both Ym and is ≥ Ym + 1or; (5) the pair (Ym, ) is a “greatest lower bound” for the pair (Ym + 1, ) in the sense that any predecessor of both Ym + 1 and is ≤ Ym or . Since and all Ym are in the bounded truth-table degree of K, this yields some local information about the one-one degrees of index sets which are “at the bottom” in the one-one ordering of index sets.

scholarly journals An Efficient Algorithm to Count Tree-Like Graphs with a Given Number of Vertices and Self-Loops

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 923 ◽  
Author(s):  
Naveed Ahmed Azam ◽  
Aleksandar Shurbevski ◽  
Hiroshi Nagamochi
Keyword(s):  
Dynamic Programming ◽  
Graph Theory ◽  
Lower Bound ◽  
Upper Bound ◽  
Efficient Algorithm ◽  
Rooted Tree ◽  
Chemical Compounds ◽  
Natural Sciences ◽  
Interesting Problem ◽  
Cycle Rank

Graph enumeration with given constraints is an interesting problem considered to be one of the fundamental problems in graph theory, with many applications in natural sciences and engineering such as bio-informatics and computational chemistry. For any two integers n≥1 and Δ≥0, we propose a method to count all non-isomorphic trees with n vertices, Δ self-loops, and no multi-edges based on dynamic programming. To achieve this goal, we count the number of non-isomorphic rooted trees with n vertices, Δ self-loops and no multi-edges, in O(n2(n+Δ(n+Δ·min{n,Δ}))) time and O(n2(Δ2+1)) space, since every tree can be uniquely viewed as a rooted tree by either regarding its unicentroid as the root, or in the case of bicentroid, by introducing a virtual vertex on the bicentroid and assuming the virtual vertex to be the root. By this result, we get a lower bound and an upper bound on the number of tree-like polymer topologies of chemical compounds with any “cycle rank”.

Heuristic Algorithms for Waste Load Allocation in a River Basin

1989 ◽  
Vol 21 (8-9) ◽  
pp. 1057-1064 ◽  
Author(s):  
Vijay Joshi ◽  
Prasad Modak
Keyword(s):  
Dynamic Programming ◽  
Heuristic Algorithms ◽  
Water Quality Management ◽  
User Preferences ◽  
Theory And Practice ◽  
Waste Load Allocation ◽  
Practical Applications ◽  
Waste Load ◽  
On Line ◽  
The One

Waste load allocation for rivers has been a topic of growing interest. Dynamic programming based algorithms are particularly attractive in this context and are widely reported in the literature. Codes developed for dynamic programming are however complex, require substantial computer resources and importantly do not allow interactions of the user. Further, there is always resistance to utilizing mathematical programming based algorithms for practical applications. There has been therefore always a gap between theory and practice in systems analysis in water quality management. This paper presents various heuristic algorithms to bridge this gap with supporting comparisons with dynamic programming based algorithms. These heuristics make a good use of the insight gained in the system's behaviour through experience, a process akin to the one adopted by field personnel and therefore can readily be understood by a user familiar with the system. Also they allow user preferences in decision making via on-line interaction. Experience has shown that these heuristics are indeed well founded and compare very favourably with the sophisticated dynamic programming algorithms. Two examples have been included which demonstrate such a success of the heuristic algorithms.

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