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26
On the Evolution of Random Graphs
 PUBLICATION OF THE MATHEMATICAL INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES
, 1960
"... his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_ ..."
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Cited by 1849 (7 self)
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his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_
The Watershed Transform: Definitions, Algorithms and Parallelization Strategies
, 2001
"... The watershed transform is the method of choice for image segmentation in the field of mathematical morphology. We present a critical review of several definitions of the watershed transform and the associated sequential algorithms, and discuss various issues which often cause confusion in the li ..."
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Cited by 135 (3 self)
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The watershed transform is the method of choice for image segmentation in the field of mathematical morphology. We present a critical review of several definitions of the watershed transform and the associated sequential algorithms, and discuss various issues which often cause confusion in the literature. The need to distinguish between definition, algorithm specification and algorithm implementation is pointed out. Various examples are given which illustrate differences between watershed transforms based on different definitions and/or implementations. The second part of the paper surveys approaches for parallel implementation of sequential watershed algorithms.
On the complexity of a concentrator
 7th International Teletraffic Conference
, 1973
"... In this paper a swi tcbing network with n inputs and m outputs is considered. The network satisfies the following condition: any k ~ m inputs can be simultaneously connected to some k outputs. Such networks are r eferred to as (n, m) concentrators. The problem of constructing a concentrator with a m ..."
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Cited by 61 (0 self)
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In this paper a swi tcbing network with n inputs and m outputs is considered. The network satisfies the following condition: any k ~ m inputs can be simultaneously connected to some k outputs. Such networks are r eferred to as (n, m) concentrators. The problem of constructing a concentrator with a mini mum possible number of crosspoints is investigated. A c oncentrator with less t han 29n c IQ sspoints is constructed. Two cases are considered: a) ~O for n oO, b) ~1 for nOG • n n The constructed concentrator has asymptotically no more than 3n c rosspoints in the case (a) and 4n c rosspoints in the case (b). The paper is concerned with a switching network having n inputs and m outputs (m <n) and satisfying the following condition: any k ~ m inputs can be simultaneously connected to some k outputs. Such networks are referred to as (n, m) concentrators. The paper deals with the problem of constructing concentrators with a minimum possible number of c rosspoint s. As in [1J, the problem is formulated and solved in terms of the theory of graphs. It is necessary to construct an oriented n inputs m out puts graph f in which any m of n inputs can be connected to m outputs by nonintersecting paths. The number Q(f) of crosspoints of a concentrator is determined as the number of edges in this graph f. Let Fn,m be a set of all ~riented graphs f which are (n, m) concentrators and Q(n, m) = min Q(f) (1) fE Fn,m The main results of the work can be summarized in the following theorem. Theorem: (a) 2n2 ~ Q(n, m) < cn, m~2 (2) where the constant c <29 is independent of n and m; (b) Q(n, an) ~3n(1+o(1)), 0(1)0 forv<. ~ 0, n.:. 00
Matching: a wellsolved class of integer linear programs
 in: Combinatorial structures and their applications (Gordon and Breach
, 1970
"... A main purpose of this work is to give a good algorithm for a certain welldescribed class of integer linear programming problems, called matching problems (or the matching problem). Methods developed for simple matching [2,3], a special case to which these problems can be reduced [4], are applied di ..."
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Cited by 60 (1 self)
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A main purpose of this work is to give a good algorithm for a certain welldescribed class of integer linear programming problems, called matching problems (or the matching problem). Methods developed for simple matching [2,3], a special case to which these problems can be reduced [4], are applied directly to the larger class. In the process, we derive a description of a system of linear inequalities whose polyhedron is the convex hull of the admissible solution vectors to the given matching problem. At the same time, various combinatorial results about matchings are derived and discussed in terms of graphs. The general integer linear programming problem can be stated as: (1) Minimize z = ∑ j∈E cjxj, where cj is a given real number, subject to (2) xj an integer for each j ∈ E; (3) 0 ≤ xj ≤ αj, j ∈ E, where αj is a given positive integer or +∞; (4) ∑ j∈E aijxj = bi, i ∈ V, where aij and bi are given integers; V and E are index sets having cardinalities V  and E. (5) The integer program (1) is called a matching problem whenever i∈V aij  ≤ 2 holds for all j ∈ E. (6) A solution to the integer program (1) is a vector [xj], j ∈ E, satisfying (2), (3), and (4), and an optimum solution is a solution which minimizes z among all solutions. When the integer program is a matching problem, a solution is called a matching and an optimum solution is an optimum matching. If the integer restriction (2) is omitted, the problem becomes a linear program. An optimum solution to that linear program will typically have fractional values. There is an important class of linear programs, called transportation or network flow problems, which have the property that for any integer righthand side bi, i ∈ V, and any cost vector cj, j ∈ E, there is an optimum solution which has all integer xj, j ∈ E. The class of matching probems includes that class of linear programs, but, in addition, includes problems for which omitting
Characterization of Glushkov automata
 Theoretical Computer Science
, 1996
"... Glushkov algorithm computes a nondeterministic finite automaton without ffltransition and with n + 1 states from a simple regular expression having n occurrences of letters. The aim of this paper is to give a set of necessary and sufficient conditions characterizing this automaton. Our characteriza ..."
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Cited by 23 (1 self)
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Glushkov algorithm computes a nondeterministic finite automaton without ffltransition and with n + 1 states from a simple regular expression having n occurrences of letters. The aim of this paper is to give a set of necessary and sufficient conditions characterizing this automaton. Our characterization theorem is formulated in terms of directed graphs. Moreover these conditions allow us to produce an algorithm of conversion of a Glushkov automaton into a regular expression of small size. 1 Introduction During the last forty years, the synthesis of automaton 1 has stimulated a good deal of research. The taxonomy that B.W. Watson [Wa95] has dedicated to this topic enlights two wide categories of algorithms: the first ones yield an automaton with ffltransitions [Tho68] [HU79] [Sed83], the second ones, a nondeterministic automaton without ffltransition [MNY60] [Gl61] [Mir65]. The second approach provides in a "natural" way [BK93] an automaton of "small" size [Bst87] currently called...
A Log (N) Distributed Mutual Exclusion Algorithm Based on the Path Reversal
 Journal of Parallel and Distributed Computing
, 1996
"... In this paper, we present a distributed algorithm for mutual exclusion based on path reversal. The algorithm does not use logical clocks to serialize the concurrent events, and all the variables are bounded. When a process invokes a critical section, it sends a request to the tail of a queue. A dyna ..."
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Cited by 12 (1 self)
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In this paper, we present a distributed algorithm for mutual exclusion based on path reversal. The algorithm does not use logical clocks to serialize the concurrent events, and all the variables are bounded. When a process invokes a critical section, it sends a request to the tail of a queue. A dynamical rooted tree gives the path to this tail. The algorithm requires only O(Log (n)) messages on average, where n is the number of processes in the network. The performances analysis of the algorithm is based on generating formal power series. Support This work was financed in part by the CNRS: Unité associée 040822 and the C 3 coordinated research program. Index Terms Distributed algorithm, mutual exclusion, logical rooted tree, distributed variables, Dyck words, Path reversal. * LIB, Faculté des sciences, Route de Gray 25030 BESANCONCEDEX, FRANCE ** LaBRI , Université BORDEAUX I, 33405 TALENCECEDEX, FRANCE 2 I INTRODUCTION Algorithms for mutual exclusion may vary from centralized s...
On the History of Combinatorial Optimization (till 1960)
"... Introduction As a coherent mathematical discipline, combinatorial optimization is relatively young. When studying the history of the field, one observes a number of independent lines of research, separately considering problems like optimum assignment, shortest spanning tree, transportation, and the ..."
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Cited by 9 (0 self)
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Introduction As a coherent mathematical discipline, combinatorial optimization is relatively young. When studying the history of the field, one observes a number of independent lines of research, separately considering problems like optimum assignment, shortest spanning tree, transportation, and the traveling salesman problem. Only in the 1950's, when the unifying tool of linear and integer programming became available and the area of operations research got intensive attention, these problems were put into one framework, and relations between them were laid. Indeed, linear programming forms the hinge in the history of combinatorial optimization. Its initial conception by Kantorovich and Koopmans was motivated by combinatorial applications, in particular in transportation and transshipment. After the formulation of linear programming as generic problem, and the development in 1947 by Dantzig of the simplex method as a tool, one has tried to attack about all combinatorial opti
Duality between the Watershed by Image Foresting Transform and the Fuzzy Connectedness Segmentation Approaches
 IN XIX BRAZILIAN SYMP. ON COMPUTER GRAPH. AND IMAGE PROC. (SIBGRAPI’06
, 2006
"... This paper makes a rereading of two successful image segmentation approaches, the fuzzy connectedness (FC) and the watershed (WS) approaches, by analyzing both by means of the Image Foresting Transform (IFT). This graphbased transform provides a sound framework for analyzing and implementing these m ..."
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Cited by 3 (2 self)
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This paper makes a rereading of two successful image segmentation approaches, the fuzzy connectedness (FC) and the watershed (WS) approaches, by analyzing both by means of the Image Foresting Transform (IFT). This graphbased transform provides a sound framework for analyzing and implementing these methods. This paradigm allows to show the duality existing between the WS by IFT and the FC segmentation approaches. Both can be modeled by an optimal forest computation in a dual form (maximization of the similarities or minimization of the dissimilarities), the main difference being the input parameters: the weights associated to each arc of the graph representing the image. In the WS approach, such weights are based on the (possibly filtered) image gradient values whereas they are based on much more complex affinity values in the FC theory. An efficient algorithm for both FC and IFTWS computation is proposed. Segmentation robustness issue is also discussed.
Bipolar Ranking from Pairwise Fuzzy Outrankings
"... kernels (see Bisdor# & Roubens [3, 4]) to the problem of constructing a global ranking from a pairwise outranking relation defined on a set of decision alternatives as encountered in the fuzzy preference modelling context (see Roy & Bouyssou [11] for instance). Our approach is based on a repet ..."
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Cited by 2 (1 self)
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kernels (see Bisdor# & Roubens [3, 4]) to the problem of constructing a global ranking from a pairwise outranking relation defined on a set of decision alternatives as encountered in the fuzzy preference modelling context (see Roy & Bouyssou [11] for instance). Our approach is based on a repetitive selection of best and worst candidates from sharpest or most credible initial and terminal kernels (see Bisdor# [6]). A practical illustration will concern the global ranking of movies from individual evaluations by a given set of movie critics.
A Hamiltonian Property of Connected Sets in the Alternative Set Theory
, 1994
"... . The representation of indiscernibility phenomena by ßequivalences and of accessibility phenomena by oeequivalences enables a graphtheoretical formulation of topological notions in the alternative set theory. Generalizing the notion of Hamiltonian graph we will introduce the notion of Hamiltoni ..."
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. The representation of indiscernibility phenomena by ßequivalences and of accessibility phenomena by oeequivalences enables a graphtheoretical formulation of topological notions in the alternative set theory. Generalizing the notion of Hamiltonian graph we will introduce the notion of Hamiltonian embedding and prove that for any finite graph without isolated vertices there is a Hamiltonian embedding into any infinite set connected with respect to some ß or oeequivalence. Roughly speaking, in some sense this means that each such an infinite connected set, (in particular, each connected set in a complete metrizable topological space), contains each finite graph inside, and even is exhausted by the images of its edges. Moreover, the main Theorem 3, dealing with the so called deeply connected sets, is in fact a theorem of nonstandard arithmetic. 1. A Brief Outline of Alternative Set Theory The alternative set theory (AST), similarly as nonstandard analysis, enables to treat the in...