scholarly journals Prefix Block-Interchanges on Binary and Ternary Strings

2019 ◽  
Author(s):  
Md. Khaledur Rahman ◽  
M. Sohel Rahman
Keyword(s):  
Upper Bound ◽  
Genome Rearrangement ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Linear Time Algorithm ◽  
Minimum Number ◽  
Binary Strings ◽  
Better Than

AbstractThe genome rearrangement problem computes the minimum number of operations that are required to sort all elements of a permutation. A block-interchange operation exchanges two blocks of a permutation which are not necessarily adjacent and in a prefix block-interchange, one block is always the prefix of that permutation. In this paper, we focus on applying prefix block-interchanges on binary and ternary strings. We present upper bounds to group and sort a given binary/ternary string. We also provide upper bounds for a different version of the block-interchange operation which we refer to as the ‘restricted prefix block-interchange’. We observe that our obtained upper bound for restricted prefix block-interchange operations on binary strings is better than that of other genome rearrangement operations to group fully normalized binary strings. Consequently, we provide a linear-time algorithm to solve the problem of grouping binary normalized strings by restricted prefix block-interchanges. We also provide a polynomial time algorithm to group normalized ternary strings by prefix block-interchange operations. Finally, we provide a classification for ternary strings based on the required number of prefix block-interchange operations.

scholarly journals Acyclic Coloring of Graphs of Maximum Degree $\Delta$

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Guillaume Fertin ◽  
André Raspaud
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Acyclic Coloring ◽  
International Audience ◽  
Better Than

International audience An acyclic coloring of a graph $G$ is a coloring of its vertices such that: (i) no two neighbors in $G$ are assigned the same color and (ii) no bicolored cycle can exist in $G$. The acyclic chromatic number of $G$ is the least number of colors necessary to acyclically color $G$, and is denoted by $a(G)$. We show that any graph of maximum degree $\Delta$ has acyclic chromatic number at most $\frac{\Delta (\Delta -1) }{ 2}$ for any $\Delta \geq 5$, and we give an $O(n \Delta^2)$ algorithm to acyclically color any graph of maximum degree $\Delta$ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding $a(G) \leq \Delta (\Delta -1) +2$. By a deeper study of the case $\Delta =5$, we also show that any graph of maximum degree $5$ can be acyclically colored with at most $9$ colors, and give a linear time algorithm to achieve this bound.

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.

How to Keep an Eye on Small Things

2021 ◽  
pp. 1-24
Author(s):  
Bengt J. Nilsson ◽  
Paweł Żyliński
Keyword(s):  
Linear Time ◽  
Simple Polygon ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Approximation Factor ◽  
Fpt Algorithm ◽  
Optimal Linear ◽  
Small Things ◽  
Set Of Points ◽  
Better Than

We present new results on two types of guarding problems for polygons. For the first problem, we present an optimal linear time algorithm for computing a smallest set of points that guard a given shortest path in a simple polygon having [Formula: see text] edges. We also prove that in polygons with holes, there is a constant [Formula: see text] such that no polynomial-time algorithm can solve the problem within an approximation factor of [Formula: see text], unless P=NP. For the second problem, we present a [Formula: see text]-FPT algorithm for computing a shortest tour that sees [Formula: see text] specified points in a polygon with [Formula: see text] holes. We also present a [Formula: see text]-FPT approximation algorithm for this problem having approximation factor [Formula: see text]. In addition, we prove that the general problem cannot be polynomially approximated better than by a factor of [Formula: see text], for some constant [Formula: see text], unless P [Formula: see text]NP.

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 Obtaining a Proportional Allocation by Deleting Items

Algorithmica ◽  
2021 ◽  
Author(s):  
Britta Dorn ◽  
Ronald de Haan ◽  
Ildikó Schlotter
Keyword(s):  
Control Problem ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Lower And Upper Bounds ◽  
Proportional Allocation ◽  
Minimum Number

AbstractWe consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small—we show that the problem is $${\mathsf {W}}[3]$$ W [ 3 ] -hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k. Considering the possibilities for approximation, we prove a strong inapproximability result for PID. Finally, we also study a variant of the problem where we are given an allocation $$\pi $$ π in advance as part of the input, and our aim is to delete a minimum number of items such that $$\pi $$ π is proportional in the remainder; this variant turns out to be $${{\mathsf {N}}}{{\mathsf {P}}}$$ N P -hard for six agents, but polynomial-time solvable for two agents, and we show that it is $$\mathsf {W[2]}$$ W [ 2 ] -hard when parameterized by the number k of

scholarly journals Computing large and small stable models

2001 ◽  
Vol 2 (1) ◽  
pp. 1-23 ◽  
Author(s):  
MIROSŁAW TRUSZCZYŃSKI
Keyword(s):  
Upper Bound ◽  
Linear Time ◽  
Logic Program ◽  
Time Algorithm ◽  
Stable Model ◽  
Linear Time Algorithm ◽  
Constant Independent ◽  
Fixed Integer ◽  
Fixed Parameter ◽  
Stable Models

In this paper, we focus on the problem of existence and computing of small and large stable models. We show that for every fixed integer k, there is a linear-time algorithm to decide the problem LSM (large stable models problem): does a logic program P have a stable model of size at least [mid ]P[mid ]−k? In contrast, we show that the problem SSM (small stable models problem) to decide whether a logic program P has a stable model of size at most k is much harder. We present two algorithms for this problem but their running time is given by polynomials of order depending on k. We show that the problem SSM is fixed-parameter intractable by demonstrating that it is W[2]-hard. This result implies that it is unlikely an algorithm exists to compute stable models of size at most k that would run in time O(mc), where m is the size of the program and c is a constant independent of k. We also provide an upper bound on the fixed-parameter complexity of the problem SSM by showing that it belongs to the class W[3].

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