Error locating: degree constraints

10.32920/ryerson.14661957 ◽

2021 ◽

Author(s):

Christopher Dennis

Keyword(s):

Maximum Degree ◽

Linear Time ◽

Time Algorithm ◽

Mathematical Tool ◽

Linear Time Algorithm ◽

Total Size ◽

Special Cases ◽

Locating Array ◽

General Error

Error graphs are a useful mathematical tool for representing failing interactions in a system. This representation is used as the basis for constructing an error locating array (ELA). However, if too many errors are present in a given error graph, it may not be possible to locate all interactions. We say that a graph is locatable if an ELA can be built. Bounds on the total size of an error graph are known, bounds on the degree an error graph can have have not been considered. In this thesis we explore the maximum degree an error graph may have while still guaranteeing its locatability. We consider special cases for 3 and 4 partite error graphs as well as developing bounds on the degree of a general error graph. We describe a linear time algorithm which can be used to generate tests which have at most one failing interaction.

Download Full-text

Acyclic Coloring of Graphs of Maximum Degree $\Delta$

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3450 ◽

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.

Download Full-text

Comparing Degenerate Strings

A Mosaic of Computational Topics: from Classical to Novel ◽

10.3233/stal200005 ◽

2020 ◽

Author(s):

Mai Alzamel ◽

Lorraine A.K. Ayad ◽

Giulia Bernardini ◽

Roberto Grossi ◽

Costas S. Iliopoulos ◽

...

Keyword(s):

Lower Bound ◽

Linear Time ◽

Simple Algorithm ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Total Size ◽

Empty Intersection ◽

Compact Representations ◽

String Comparison ◽

Combinatorial Result

Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string Ŝ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1, …, kn-1. Our main result is an O(N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in Ŝ in O(min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in Ŝ. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach.

Download Full-text

COLORING ALGORITHMS ON SUBCUBIC GRAPHS

International Journal of Foundations of Computer Science ◽

10.1142/s0129054104002285 ◽

2004 ◽

Vol 15 (01) ◽

pp. 21-40 ◽

Cited By ~ 2

Author(s):

SAN SKULRATTANAKULCHAI ◽

HAROLD N. GABOW

Keyword(s):

Maximum Degree ◽

Linear Time ◽

Edge Coloring ◽

Time Algorithm ◽

Linear Time Algorithm ◽

The Third ◽

Erew Pram ◽

List Edge Coloring ◽

Subcubic Graphs ◽

Decomposition Principle

We present efficient algorithms for three coloring problems on subcubic graphs. (A subcubic graph has maximum degree at most three.) The first algorithm is for 4-edge coloring, or more generally, 4-list-edge coloring. Our algorithm runs in linear time, and appears to be simpler than previous ones. The second algorithm is the first randomized EREW PRAM algorithm for the same problem. It uses O(n/ log n) processors and runs in O( log n) time with high probability, where n is the number of vertices of the graph. The third algorithm is the first linear-time algorithm to 5-total-color subcubic graphs. The fourth algorithm generalizes this to get the first linear-time algorithm to 5-list-total-color subcubic graphs. Our sequential algorithms are based on a method of ordering the vertices and edges by traversing a spanning tree of a graph in a bottom-up fashion. Our parallel algorithm is based on a simple decomposition principle for subcubic graphs.

Download Full-text

Comparing Degenerate Strings

Fundamenta Informaticae ◽

10.3233/fi-2020-1947 ◽

2020 ◽

Vol 175 (1-4) ◽

pp. 41-58

Author(s):

Mai Alzamel ◽

Lorraine A.K. Ayad ◽

Giulia Bernardini ◽

Roberto Grossi ◽

Costas S. Iliopoulos ◽

...

Keyword(s):

Lower Bound ◽

Linear Time ◽

Simple Algorithm ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Total Size ◽

Empty Intersection ◽

Compact Representations ◽

String Comparison ◽

Combinatorial Result

Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string Ŝ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1, . . . , kn-1. Our main result is an 𝒪(N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in Ŝ in 𝒪(min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in Ŝ. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach.

Download Full-text

Locating Eigenvalues of Perturbed Laplacian Matrices of Trees

TEMA (São Carlos) ◽

10.5540/tema.2017.018.03.479 ◽

2018 ◽

Vol 18 (3) ◽

pp. 479

Author(s):

Rodrigo Orsini Braga ◽

Virgínia Maria Rodrigues

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Real Interval ◽

Linear Time Algorithm ◽

Laplacian Matrices ◽

Special Cases

We give a linear time algorithm to compute the number of eigenvalues of any perturbedLaplacian matrix of a tree in a given real interval. The algorithm can be applied to weightedor unweighted trees. Using our method we characterize the trees that have up to $5$ distincteigenvalues with respect to a family of perturbed Laplacian matrices that includes the adjacencyand normalized Laplacian matrices as special cases, among others.

Download Full-text

Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽

10.3390/math9030293 ◽

2021 ◽

Vol 9 (3) ◽

pp. 293

Author(s):

Xinyue Liu ◽

Huiqin Jiang ◽

Pu Wu ◽

Zehui Shao

Keyword(s):

Linear Time ◽

Minimum Weight ◽

Domination Number ◽

Time Algorithm ◽

Simple Graph ◽

Linear Time Algorithm ◽

Domination Problem ◽

Np Complete ◽

Chordal Bipartite Graphs ◽

Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

Download Full-text