scholarly journals Channel Density Minimization by Pin Permutation

VLSI Design ◽  
1994 ◽  
Vol 2 (2) ◽  
pp. 171-183
Author(s):  
Yang Cai ◽  
D. F. Wong ◽  
Jason Cong
Keyword(s):  
Optimal Algorithm ◽  
Linear Time ◽  
Layout Design ◽  
Strong Sense ◽  
Np Hard ◽  
Wire Length ◽  
Channel Density ◽  
Time Optimal ◽  
Two Sides ◽  
Intergrated Circuits

We present in this paper a linear time optimal algorithm for minimizing the density of a channel (with exits) by permuting the terminals on the two sides of the channel. This compares favorably with the previously known near-optimal algorithm presented in [6] that runs in superlinear time. Our algorithm has important applications in hierarchical layout design of intergrated circuits. We also show that the problem of minimizing wire length by permuting terminals is NP-hard in the strong sense.

scholarly journals A TIME-OPTIMAL ALGORITHM FOR THE ESTIMATION OF CONTACT DISTRIBUTION FUNCTIONS OF RANDOM SETS

2011 ◽  
Vol 23 (3) ◽  
pp. 177 ◽  
Author(s):  
Johannes Mayer
Keyword(s):  
Distribution Function ◽  
Optimal Algorithm ◽  
Linear Time ◽  
Distribution Functions ◽  
Distance Distribution ◽  
Random Sets ◽  
Distance Distribution Function ◽  
Structuring Element ◽  
Time Optimal ◽  
Contact Distribution

This paper presents a linear-time and therefore time-optimal algorithm for the estimation of distance distribution functions and contact distribution functions of random sets. The distance distribution function is the area fraction of a dilated set, where this function depends on the size of the structuring element used for the dilation. Furthermore, contact distribution functions are related to distance distribution functions. Minussampling estimators are used for the estimation.

scholarly journals Flipping Modules to Minimize Maximum Wire Length

VLSI Design ◽  
1995 ◽  
Vol 3 (1) ◽  
pp. 37-41
Author(s):  
Kyunrak Chong ◽  
Sartaj Sahni
Keyword(s):  
Linear Time ◽  
Np Hard ◽  
Wire Length

We show that obtaining the optimal orientations of modules to minimize the length of the longest wire is NP-hard. If each module is permitted only two possible orientations, this can be done in linear time. When all four orientations are permissible and wires are restricted to connect modules whose separation is bounded by some constant, the problem also can be solved in linear time.

scholarly journals An Optimum Channel Routing Algorithm in the Knock-knee Diagonal Model

VLSI Design ◽  
1994 ◽  
Vol 1 (3) ◽  
pp. 233-242 ◽  
Author(s):  
Xiaoyu Song
Keyword(s):  
Routing Algorithm ◽  
Difficult Problem ◽  
Layout Design ◽  
Channel Routing ◽  
Routing Problem ◽  
Vlsi Layout ◽  
Channel Density ◽  
Optimum Algorithm ◽  
Important Time ◽  
Optimum Channel

Channel routing problem is an important, time consuming and difficult problem in VLSI layout design. In this paper, we consider the two-terminal channel routing problem in a new routing model, called knock-knee diagonal model, where the grid consists of right and left tracks displayed at +45° and –45°. An optimum algorithm is presented, which obtains d + 1 as an upper bound to the channel width, where d is the channel density.

scholarly journals Efficient Algorithms for Max-Weighted Point Sweep Coverage on Lines

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1457
Author(s):  
Dieyan Liang ◽  
Hong Shen
Keyword(s):  
Optimal Algorithm ◽  
Approximate Solutions ◽  
Approximation Ratio ◽  
Mobile Sensors ◽  
Np Hard ◽  
Coverage Problem ◽  
Polynomial Time Approximation Algorithm ◽  
Weighted Point ◽  
Eulerian Graph ◽  
Set Of Points

As an important application of wireless sensor networks (WSNs), deployment of mobile sensors to periodically monitor (sweep cover) a set of points of interest (PoIs) arises in various applications, such as environmental monitoring and data collection. For a set of PoIs in an Eulerian graph, the point sweep coverage problem of deploying the fewest sensors to periodically cover a set of PoIs is known to be Non-deterministic Polynomial Hard (NP-hard), even if all sensors have the same velocity. In this paper, we consider the problem of finding the set of PoIs on a line periodically covered by a given set of mobile sensors that has the maximum sum of weight. The problem is first proven NP-hard when sensors are with different velocities in this paper. Optimal and approximate solutions are also presented for sensors with the same and different velocities, respectively. For M sensors and N PoIs, the optimal algorithm for the case when sensors are with the same velocity runs in O(MN) time; our polynomial-time approximation algorithm for the case when sensors have a constant number of velocities achieves approximation ratio 12; for the general case of arbitrary velocities, 12α and 12(1−1/e) approximation algorithms are presented, respectively, where integer α≥2 is the tradeoff factor between time complexity and approximation ratio.

scholarly journals Ranking top-k trees in tree-based phylogenetic networks

2019 ◽  
Author(s):  
Momoko Hayamizu ◽  
Kazuhisa Makino
Keyword(s):  
Optimal Algorithm ◽  
Linear Time ◽  
Fundamental Problem ◽  
Phylogenetic Network ◽  
Reticulate Evolution ◽  
Interesting Property ◽  
Biological Data ◽  
Phylogenetic Networks ◽  
Linear Delay ◽  
Algorithmic Problems

Abstract 'Tree-based' phylogenetic networks provide a mathematically-tractable model for representing reticulate evolution in biology. Such networks consist of an underlying 'support tree' together with arcs between the edges of this tree. However, a tree-based network can have several such support trees, and this leads to a variety of algorithmic problems that are relevant to the analysis of biological data. Recently, Hayamizu (arXiv:1811.05849 [math.CO]) proved a structure theorem for tree-based phylogenetic networks and obtained linear-time and linear-delay algorithms for many basic problems on support trees, such as counting, optimisation, and enumeration. In the present paper, we consider the following fundamental problem in statistical data analysis: given a tree-based phylogenetic network $N$ whose arcs are associated with probability, create the top-$k$ support tree ranking for $N$ by their likelihood values. We provide a linear-delay (and hence optimal) algorithm for the problem and thus reveal the interesting property of tree-based phylogenetic networks that ranking top-$k$ support trees is as computationally easy as picking $k$ arbitrary support trees.

A Generalization of the Bang-Bang Principle of Linear Control Theory*

1965 ◽  
Vol 8 (6) ◽  
pp. 783-789
Author(s):  
Richard Datko
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Matrix Function ◽  
Linear Time ◽  
Time Optimal Control ◽  
Linear Control ◽  
Dimensional Vector ◽  
R Matrix ◽  
Linear Control Theory ◽  
Time Optimal

In a paper by LaSalle [l] on linear time optimal control the following lemma is proved:Let Ω be the set of all r-dimensional vector functions U(τ) measurable on [ 0, t] with |ui(τ)≦1. Let Ωo be the subset of functions uo(τ) with |uoi(τ) = 1. Let Y(τ) be any (n × r ) matrix function in L1([ 0, t]).

scholarly journals Oriented Coloring on Recursively Defined Digraphs

Algorithms ◽  
2019 ◽  
Vol 12 (4) ◽  
pp. 87 ◽  
Author(s):  
Frank Gurski ◽  
Dominique Komander ◽  
Carolin Rehs
Keyword(s):  
Chromatic Number ◽  
Linear Time ◽  
Directed Graphs ◽  
Graph Isomorphism ◽  
Oriented Graph ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Np Hard ◽  
Graph Isomorphism Problem ◽  
Acyclic Digraph

Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G = ( V , A ) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows that finding the chromatic number of an oriented graph is an NP-hard problem, too. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.

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