scholarly journals VERTEX SPLITTING IN DAGS AND APPLICATIONS TO PARTIAL SCAN DESIGNS AND LOSSY CIRCUITS

1998 ◽  
Vol 09 (04) ◽  
pp. 377-398 ◽  
Author(s):  
DOOWON PAIK ◽  
SUDHAKAR REDDY ◽  
SARTAJ SAHNI
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Directed Acyclic Graphs ◽  
Partial Scan ◽  
Acyclic Graphs ◽  
Vertex Splitting ◽  
Minimum Number ◽  
Path Lengths ◽  
Circuit Delays ◽  
Pipelined Circuits

Directed acyclic graphs (dags) are often used to model circuits. Path lengths in such dags represent circuit delays. In the vertex splitting problem, the objective is to determine a minimum number of vertices to split so that the resulting dag has no path of length > δ. This problem has application to the placement of flip-flops in partial scan designs, placement of latches in pipelined circuits, placement of signal boosters in lossy circuits and networks, etc. Several simplified versions of this problem are shown to be NP-hard. A linear time algorithm is obtained for the case when the dag is a tree. A backtracking algorithm and heuristics are developed for general dags and experimental results using dags obtained from ISCAS benchmark circuits are obtained.

scholarly journals Minimum Cell Connection in Line Segment Arrangements

2017 ◽  
Vol 27 (03) ◽  
pp. 159-176
Author(s):  
Helmut Alt ◽  
Sergio Cabello ◽  
Panos Giannopoulos ◽  
Christian Knauer
Keyword(s):  
Linear Time ◽  
Optimal Solution ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Constant Number ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Number Of Segments ◽  
Connection Problems

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points [Formula: see text] and [Formula: see text] in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting [Formula: see text] to [Formula: see text] that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting [Formula: see text] to [Formula: see text] must stay inside a given polygon [Formula: see text] with a constant number of holes, the segments are contained in [Formula: see text], and the endpoints of the segments are on the boundary of [Formula: see text]. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution.

scholarly journals Embedding Sun Graphs in a Single Page

2013 ◽  
Vol 5 ◽  
pp. 31-36
Author(s):  
Mahavir Banukumar
Keyword(s):  
Page Number ◽  
Linear Time ◽  
Dimensional Space ◽  
Time Algorithm ◽  
Linear Ordering ◽  
Linear Time Algorithm ◽  
The Sun ◽  
3 Dimensional ◽  
Book Embedding ◽  
Minimum Number

A book consists of a line in the 3-dimensional space, called the spine, and a number of pages, each a half-plane with the spine as boundary. A book embedding (p, r) of a graph consists of a linear ordering of p, of vertices, called the spine ordering, along the spine of a book and an assignment r, of edges to pages so that edges assigned to the same page can be drawn on that page without crossing. That is, we cannot find vertices u, v, x, y with p(u) < p(x) < p(v) < p(y), yet the edges uv and xy are assigned to the same page, that is r(uv) = r(xy). The book thickness or page number of a graph G is the minimum number of pages in required to embed G in a book. In this paper we consider the Sun Graph or the Trampoline graph and obtain the printing cycle for embedding the Sun Graph in a single page. We also give a linear time algorithm for such an embedding.

Search Numbers in Networks with Special Topologies

2019 ◽  
Vol 19 (01) ◽  
pp. 1940004
Author(s):  
BOTING YANG ◽  
RUNTAO ZHANG ◽  
YI CAO ◽  
FARONG ZHONG
Keyword(s):  
Approximation Algorithms ◽  
Search Strategy ◽  
Linear Time ◽  
Time Algorithm ◽  
Time Transformation ◽  
Linear Time Algorithm ◽  
Optimal Layout ◽  
Explicit Formulas ◽  
Minimum Number ◽  
Search Numbers

In this paper, we consider the problem of finding the minimum number of searchers to sweep networks/graphs with special topological structures. Such a number is called the search number. We first study graphs, which contain only one cycle, and present a linear time algorithm to compute the vertex separation and the optimal layout of such graphs; by a linear-time transformation, we can find the search number of this kind of graphs in linear time. We also investigate graphs, in which every vertex lies on at most one cycle and each cycle contains at most three vertices of degree more than two, and we propose a linear time algorithm to compute their search number and optimal search strategy. We prove explicit formulas for the search number of the graphs obtained from complete k-ary trees by replacing vertices by cycles. We also present some results on approximation algorithms.

A POLYNOMIAL-TIME ALGORITHM FOR COMPUTING THE RESILIENCE OF ARRANGEMENTS OF RAY SENSORS

2014 ◽  
Vol 24 (03) ◽  
pp. 225-236 ◽  
Author(s):  
DAVID KIRKPATRICK ◽  
BOTING YANG ◽  
SANDRA ZILLES
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Np Hard ◽  
Sensor Coverage ◽  
Entire Plane ◽  
Line Segments ◽  
Minimum Number

Given an arrangement A of n sensors and two points s and t in the plane, the barrier resilience of A with respect to s and t is the minimum number of sensors whose removal permits a path from s to t such that the path does not intersect the coverage region of any sensor in A. When the surveillance domain is the entire plane and sensor coverage regions are unit line segments, even with restricted orientations, the problem of determining the barrier resilience is known to be NP-hard. On the other hand, if sensor coverage regions are arbitrary lines, the problem has a trivial linear time solution. In this paper, we study the case where each sensor coverage region is an arbitrary ray, and give an O(n2m) time algorithm for computing the barrier resilience when there are m ⩾ 1 sensor intersections.

scholarly journals Algorithms for Computing Wiener Indices of Acyclic and Unicyclic Graphs

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Bo Bi ◽  
Muhammad Kamran Jamil ◽  
Khawaja Muhammad Fahd ◽  
Tian-Le Sun ◽  
Imran Ahmad ◽  
...  
Keyword(s):  
Physical Properties ◽  
Topological Index ◽  
Linear Time ◽  
Molecular Graph ◽  
Wiener Index ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Unicyclic Graphs ◽  
Polarity Index ◽  
Acyclic Graphs

Let G = V G , E G be a molecular graph, where V G and E G are the sets of vertices (atoms) and edges (bonds). A topological index of a molecular graph is a numerical quantity which helps to predict the chemical/physical properties of the molecules. The Wiener, Wiener polarity, and the terminal Wiener indices are the distance-based topological indices. In this paper, we described a linear time algorithm (LTA) that computes the Wiener index for acyclic graphs and extended this algorithm for unicyclic graphs. The same algorithms are modified to compute the terminal Wiener index and the Wiener polarity index. All these algorithms compute the indices in time O n .

OPTIMUM GUARD COVERS AND m-WATCHMEN ROUTES FOR RESTRICTED POLYGONS

1993 ◽  
Vol 03 (01) ◽  
pp. 85-105 ◽  
Author(s):  
SVANTE CARLSSON ◽  
BENGT J. NILSSON ◽  
SIMEON NTAFOS
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Np Hard ◽  
Art Gallery ◽  
Art Galleries ◽  
Art Gallery Problem ◽  
Natural Connection ◽  
Route Problem ◽  
Minimum Number

A watchman, in the terminology of art galleries, is a mobile guard. We consider several watchman and guard problems for different classes of polygons. We introduce the notion of vision spans along a path or route which provide a natural connection between the art gallery problem, the m-watchmen routes problem and the watchman route problem. We prove that finding the minimum number of vision points, i.e., static guards, along a shortest watchman route is NP-hard. We provide a linear time algorithm to compute the best set of static guards in a histogram polygon. The m-watchmen routes problem, minimize total length of routes for m watchmen, is NP-hard for simple polygons. We give a Θ(n3+n2m2)-time algorithm to compute the best set of m watchmen in a histogram.

scholarly journals Computing directed Steiner path covers

2021 ◽  
Author(s):  
Frank Gurski ◽  
Dominique Komander ◽  
Carolin Rehs ◽  
Jochen Rethmann ◽  
Egon Wanke
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Time Algorithm ◽  
Integer Programs ◽  
Path Cover ◽  
Hamiltonian Path Problem ◽  
Directed Paths ◽  
Minimum Number ◽  
Cover Problem ◽  
Vertex Disjoint

AbstractIn this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a directed graph $$G=(V,E)$$ G = ( V , E ) and a set $$T \subseteq V$$ T ⊆ V of so-called terminal vertices, the problem is to find a minimum number of vertex-disjoint simple directed paths, which contain all terminal vertices and a minimum number of non-terminal vertices (Steiner vertices). The primary minimization criteria is the number of paths. We show how to compute in linear time a minimum Steiner path cover for directed co-graphs. This leads to a linear time computation of an optimal directed Steiner path on directed co-graphs, if it exists. Since the Steiner path problem generalizes the Hamiltonian path problem, our results imply the first linear time algorithm for the directed Hamiltonian path problem on directed co-graphs. We also give binary integer programs for the (directed) Hamiltonian path problem, for the (directed) Steiner path problem, and for the (directed) Steiner path cover problem. These integer programs can be used to minimize change-over times in pick-and-place machines used by companies in electronic industry.

scholarly journals Display Sets of Normal and Tree-Child Networks

10.37236/9128 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Janosch Döcker ◽  
Simone Linz ◽  
Charles Semple
Keyword(s):  
Decision Problem ◽  
Phylogenetic Trees ◽  
Phylogenetic Network ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Directed Acyclic Graphs ◽  
Phylogenetic Networks ◽  
Acyclic Graphs ◽  
Normal Network ◽  
Normal Networks

Phylogenetic networks are leaf-labelled directed acyclic graphs that are used in computational biology to analyse and represent the evolutionary relationships of a set of species or viruses. In contrast to phylogenetic trees, phylogenetic networks have vertices of in-degree at least two that represent reticulation events such as hybridisation, lateral gene transfer, or reassortment. By systematically deleting various combinations of arcs in a phylogenetic network $\mathcal N$, one derives a set of phylogenetic trees that are embedded in $\mathcal N$. We recently showed that the problem of deciding if two binary phylogenetic networks embed the same set of phylogenetic trees is computationally hard, in particular, we showed it to be $\Pi^P_2$-complete. In this paper, we establish a polynomial-time algorithm for this decision problem if the initial two networks consist of a normal network and a tree-child network; two well-studied topologically restricted subclasses of phylogenetic networks, with normal networks being more structurally constrained than tree-child networks. The running time of the algorithm is quadratic in the size of the leaf sets.

Linear time algorithm for dominator chromatic number of trestled graphs

2019 ◽  
Vol 11 (06) ◽  
pp. 1950066
Author(s):  
S. Arumugam ◽  
K. Raja Chandrasekar
Keyword(s):  
Chromatic Number ◽  
Linear Time ◽  
Open Neighborhood ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Proper Coloring ◽  
Coloring Problem ◽  
Color Class ◽  
Minimum Number ◽  
Dominator Coloring

A dominator coloring (respectively, total dominator coloring) of a graph [Formula: see text] is a proper coloring [Formula: see text] of [Formula: see text] such that each closed neighborhood (respectively, open neighborhood) of every vertex of [Formula: see text] contains a color class of [Formula: see text] The minimum number of colors required for a dominator coloring (respectively, total dominator coloring) of [Formula: see text] is called the dominator chromatic number (respectively, total dominator chromatic number) of [Formula: see text] and is denoted by [Formula: see text] (respectively, [Formula: see text]). In this paper, we prove that the dominator coloring problem and the total dominator coloring problem are solvable in linear time for trestled graphs.

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