Linear-time algorithm

Let A = (Q, Σ, δ, q 0 , F) and B = (Q, Σ, δ ′ , q 0 , F) be deterministic finite automata on the same state set and with the same input alphabet. For each word w ∈ Σ * , let δ w , δ ′ w ∈ Q Q be the induced transformations. We introduce the differential transition semigroup D(A, B) = {(δ w , δ ′ w) : w ∈ Σ * } ≤ Q Q × Q Q , and the defect set Def(f, g) = {q ∈ Q : f (q) ̸ = g(q)}. A composition law for defects is established and used to prove a backward localization theorem: the disagreement of every word-induced pair is contained in the backward saturation of the set of locally modified states. Consequently, if the initial state lies outside the backward saturation, then the recognized language is invariant and the syntactic semigroups coincide. A local update algorithm is then obtained whose running time depends on the saturation size rather than on the full transition semigroup. The construction yields a rigorous differential framework for dynamic automata and incremental syntactic reconstruction.

IRMS (Belief, Ritual, Space, Symbol) is an interdisciplinary analytical
framework that maps how humans construct meaning through four fundamental axes.
The system does not evaluate ontological truth or aesthetic superiority.
Instead, it asks: "How is something constructed?" Operating along the axes of
density/void, continuity/rupture, and universal/local, IRMS provides a
reproducible and indexable coding system for comparative analysis across
cultures and historical periods. This document presents the complete
typological structure of IRMS including I-Layer (Belief Typology I0-I9),
R-Layer (Ritual Typology R1-R9), M-Layer (Space Typology M1-M9), and S-Layer
(Symbol Typology with morphological, structural, semantic, civilizational, and
chronological codes).

Given a series-parallel network network, for short N, its dual network N is given by interchanging the series connection and the parallel connection of network N. We usually use a series-parallel graph to represent a network. Let w x w X x X G N and G N be graph representations of N and N , respectively. A sequence Ž w x w X x. of edges e , e , . . . , e is said to form a common trail on G N , G N if it is a 1 2 k

Given a series-parallel network network, for short N, its dual network N is given by interchanging the series connection and the parallel connection of network N. We usually use a series-parallel graph to represent a network. Let w x w X x X G N and G N be graph representations of N and N , respectively. A sequence Ž w x w X x. of edges e , e , . . . , e is said to form a common trail on G N , G N if it is a 1 2 k

A vertex in a graph is simplicial if its neighborhood forms a clique. We consider three generalizations of the concept of simplicial vertices: avoidable vertices (also known as OCF -vertices), simplicial paths, and their common generalization avoidable paths, introduced here. We present a general conjecture on the existence of avoidable paths. If true, the conjecture would imply a result due to Ohtsuki, Cheung, and Fujisawa from 1976 on the existence of avoidable vertices, and a result due to Chvátal, Sritharan, and Rusu from 2002 the existence of simplicial paths. In turn, both of these results generalize Dirac's classical result on the existence of simplicial vertices in chordal graphs. We prove that every graph with an edge has an avoidable edge, which settles the first open case of the conjecture. We point out a close relationship between avoidable vertices in a graph and its minimal triangulations, and identify new algorithmic uses of avoidable vertices, leading to new polynomially solvable cases of the maximum weight clique problem in classes of graphs simultaneously generalizing chordal graphs and circular-arc graphs. Finally, we observe that the proved cases of the conjecture have interesting consequences for highly symmetric graphs: in a vertex-transitive graph every induced two-edge path closes to an induced cycle, while in an edge-transitive graph every three-edge path closes to a cycle and every induced three-edge path closes to an induced cycle.

In sequencing by hybridization (SBH), one has to reconstruct a sequence from its k-long substrings. SBH was proposed as a promising alternative to gel-based DNA sequencing approaches, but in its original form the method is not competitive. Positional SBH is a recently proposed enhancement of SBH in which one has additional information about the possible positions of each substring along the target sequence. We give a linear time algorithm for solving the positional SBH problem when each substring has at most two possible positions. On the other hand, we prove that the problem is NP-complete if each substring has at most three possible positions.

Blurred (previously named fuzzy) segments were introduced by Debled-Rennesson et al as an extension of the arithmetical approach of Reveillès [11] on discrete lines, to take into account noise in digital images. An incremental linear-time algorithm was presented to decompose a discrete curve into blurred segments with order bounded by a parameter d. However, that algorithm fails to segment discrete curves into a minimal number of blurred segments. We show in this paper, that this characteristic is intrinsic to the whole class of blurred segments. We thus introduce a subclass of blurred segments, based on a geometric measure of thickness. We provide a new convex hull based incremental linear time algorithm for segmenting discrete curves into a minimal number of thin blurred segments.

A min-cut that seperates vertices s and t in a network is an edge set of minimum weight whose removal will disconnect s and t. This problem is the dual of the well known s-t max-flow problem. Several algorithms for the min-cut problem are based on max-flow computation although the fastest known min-cut algorithms are not flow based. The well known Karger's randomized algorithm for min-cut is a non-flow based method for solving the (global) min-cut problem of finding the min s-t cut over all pair of vertices s,t in a weighted undirected graph. This paper presents an adaptation of Karger's algorithm for a synchronous distributed setting where each node is allowed to perform only local computations. The paper essentially addresses the technicalities involved in circumventing the limitations imposed by a distributed setting to the working of Karger's algorithm. While the correctness proof follows directly from Karger's algorithm, the complexity analysis differs significa...

69 On the Wiener Polynomials of Some Trees Ali A. Ali Ahmed M. Ali [email protected] [email protected] College of Computer Sciences and Mathematics University of Mosul, Iraq Received on: 02/05/2006 Accepted on: 16/08/2006 ABSTRACT The Wiener index is a graphical invariant which has found many applications in chemistry. The Wiener Polynomial of a connected graph G is the generating function of the sequence (C(G,k)) whose derivative at x=1 is the Wiener index W(G) of G, in which C(G,k) is the number of pairs of vertices distance k apart. The Wiener Polynomials of star-like trees and other special trees are found in this paper; and hence a formula of the Wiener index for each such trees is obtained .

69 On the Wiener Polynomials of Some Trees Ali A. Ali Ahmed M. Ali [email protected] [email protected] College of Computer Sciences and Mathematics University of Mosul, Iraq Received on: 02/05/2006 Accepted on: 16/08/2006 ABSTRACT The Wiener index is a graphical invariant which has found many applications in chemistry. The Wiener Polynomial of a connected graph G is the generating function of the sequence (C(G,k)) whose derivative at x=1 is the Wiener index W(G) of G, in which C(G,k) is the number of pairs of vertices distance k apart. The Wiener Polynomials of star-like trees and other special trees are found in this paper; and hence a formula of the Wiener index for each such trees is obtained .

We prove that [n/2J vertex guards are always sufficient and sometimes necessary to guard the surface of an n-vertex polyhedral terrain. We also show that l(4n -4)/13J edge guards are sometimes necessary to guard the surface of an n-vertex polyhedral terrain. The upper bound on the number of edge guards is ln/3J (Everett and Rivera-Campo, 1994). Since both upper bounds are based on the four color theorem, no practical polynomial time algorithm achieving these bounds seems to exist, but we present a linear time algorithm for placing L3n/5] vertex guards for covering a polyhedral terrain and a linear time algorithm for placing L2n/5J edge guards.

For a connected graph G and any non-empty S ⊆ V (G), S is called a weakly connected dominating set of G if the subgraph obtained from G by removing all edges each joining any two vertices in V (G) \ S is connected. The weakly connected domination number γ w (G) is defined to be the minimum integer k with |S| = k for some weakly connected dominating set S of G. In this note, we extend a result on the lower bound for the weakly connected domination number γ w (G) on trees to cycle-e-disjoint graphs, i.e., graphs in which no cycles share a common edge. More specifically, we show that if G is a connected cycle-e-disjoint graph, then , where n c (G) is the number of cycles in G, n oc (G) is the number of odd cycles in G and v 1 (G) is the number of vertices of degree 1 in G. The graphs for which equality holds are also characterised.

We consider offline scheduling algorithms that incorporate speed scaling to address the bicriteria problem of minimizing energy consumption and a scheduling metric. For makespan, we give a linear-time algorithm to compute all non-dominated solutions for the general uniprocessor problem and a fast arbitrarily-good approximation for multiprocessor problems when every job requires the same amount of work. We also show that the multiprocessor problem becomes NP-hard when jobs can require different amounts of work. For total flow, we show that the optimal flow corresponding to a particular energy budget cannot be exactly computed on a machine supporting exact real arithmetic, including the extraction of roots. This hardness result holds even when scheduling equal-work jobs on a uniprocessor. We do, however, extend previous work by Pruhs et al. to give an arbitrarily-good approximation for scheduling equal-work jobs on a multiprocessor.

Given a graph G = (V, E), k natural numbers n 1 , n 2 , ..., n k such that The problem of finding a k-partition of a graph G is NP-hard in general. It is known that every k-connected graph has a k-partition. But there is no polynomial time algorithm for finding a k-partition of a k-connected graph. In this paper we give a simple linear-time algorithm for finding a k-partition of a "doughnut graph" G.

A graph is planar if and only if it does not contain a Kuratowski subdivision. Hence such a subdivision can be used as a witness for non-planarity. Modern planarity testing algorithms allow to extract a single such witness in linear time. We present the first linear time algorithm which is able to extract multiple Kuratowski subdivisions at once. This is of particular interest for, e.g., Branch-and-Cut algorithms which require multiple such subdivisions to generate cut constraints. The algorithm is not only described theoretically, but we also present an experimental study of its implementation.