Graphs, Dioids and Semirings: New Models and Algorithms by Michel Gondran

By Michel Gondran

The origins of Graph conception date again to Euler (1736) with the answer of the prestigious 'Koenigsberg Bridges Problem'; and to Hamilton with the well-known 'Trip around the globe' video game (1859), declaring for the 1st time an issue which, in its newest model – the 'Traveling Salesman challenge' -, remains to be the topic of lively examine. but, it's been over the past fifty years or so—with the increase of the digital computers—that Graph thought has develop into an quintessential self-discipline when it comes to the quantity and significance of its functions around the technologies. Graph thought has been specially significant to Theoretical and Algorithmic laptop technology, and automated regulate, platforms Optimization, financial system and Operations learn, facts research within the Engineering Sciences. shut connections among graphs and algebraic constructions were popular within the research and implementation of effective algorithms for plenty of difficulties, for instance: transportation community optimization, telecommunication community optimization and making plans, optimization in scheduling and construction structures, etc.

The fundamental pursuits of GRAPHS, DIOÏDS AND SEMIRINGS: New types and Algorithms are to stress the deep relatives current among the semiring and dioïd buildings with graphs and their combinatorial homes, whereas demonstrating the modeling and problem-solving potential and adaptability of those buildings. furthermore the ebook offers an in depth evaluate of the mathematical homes hired through "nonclassical" algebraic buildings, which both expand traditional algebra (i.e., semirings), or correspond to a brand new department of algebra (i.e., dioïds), except the classical buildings of teams, earrings, and fields.

Show description

Read or Download Graphs, Dioids and Semirings: New Models and Algorithms PDF

Similar graph theory books

Managing and Mining Graph Data

Managing and Mining Graph information is a entire survey booklet in graph administration and mining. It comprises huge surveys on various very important graph subject matters similar to graph languages, indexing, clustering, information new release, trend mining, type, key-phrase seek, trend matching, and privateness. It additionally reports a couple of domain-specific situations similar to circulate mining, internet graphs, social networks, chemical and organic info. The chapters are written via popular researchers within the box, and supply a wide viewpoint of the world. this is often the 1st complete survey publication within the rising subject of graph info processing.

Managing and Mining Graph info is designed for a various viewers composed of professors, researchers and practitioners in undefined. This quantity can also be appropriate as a reference ebook for advanced-level database scholars in machine technology and engineering.

Tree lattices

Crew activities on timber provide a unified geometric method of recasting the bankruptcy of combinatorial team thought facing unfastened teams, amalgams, and HNN extensions. a few of the relevant examples come up from rank one uncomplicated Lie teams over a non-archimedean neighborhood box performing on their Bruhat--Tits timber.

Genetic Theory for Cubic Graphs

This e-book used to be inspired via the idea that the various underlying hassle in hard circumstances of graph-based difficulties (e. g. , the touring Salesman challenge) might be “inherited” from less complicated graphs which – in a suitable feel – will be noticeable as “ancestors” of the given graph example. The authors suggest a partitioning of the set of unlabeled, hooked up cubic graphs into disjoint subsets named genes and descendants, the place the cardinality of the descendants dominates that of the genes.

Additional info for Graphs, Dioids and Semirings: New Models and Algorithms

Sample text

G. Salomaa 1969, Eilenberg, 1974). (Let us observe however that the axioms of regular languages include, in addition to the above, the closure operation denoted ∗ ). || An interesting special case of an idempotent-cancellative dioid is one where the operation ⊕ is not only idempotent but also selective. 3. We call selective-cancellative dioid a dioid which has a selective monoid structure for ⊕ and a cancellative monoid structure for ⊗. 4. Let us take for E the set of nonnegative reals R+ ∪ {+∞} and let us define the operations ⊕ and ⊗ as: ∀a, b ∈ E: a ⊕ b = Min{a, b} ∀a, b ∈ E: a ⊗ b = a + b (addition of reals) (E, ⊕) is a selective monoid with neutral element ε = +∞, and (E, ⊗) is a cancellative monoid with neutral element e = 0.

It is therefore not a hemi-group. 2. A cancellative commutative monoid (E, ⊕) is a hemi-group if it satisfies the so-called zero-sum-free condition: a ⊕ b = ε ⇒ a = ε and b = ε. Proof. It suffices to show that (E, ⊕) is canonically ordered. Let us then assume that: a ≤ b and b ≤ a, So: ∃ c such that: b = a ⊕ c ∃ d such that: a = b ⊕ d hence: a ⊕ b = a ⊕ b ⊕ ε = a ⊕ b ⊕ c ⊕ d. Since a ⊕ b is a cancellative element, we deduce c ⊕ d = ε The condition of positivity then implies c=d=ε hence a = b. The canonical preorder relation is therefore clearly an order relation.

In the case where ε, the neutral element added for ⊕, is absorbing for ⊗, we have a semiring structure, see Sect. 5. 2. There exist many cases where there is neither right distributivity nor left distributivity. As an example, the structure (E, ⊕, ⊗) with E = [0, 1], a ⊕ b = a + b − ab and a ⊗ b = ab does not enjoy distributivity and is therefore not a pre-semiring. The same applies to the structure (E, ⊕, ⊗) with E = [0, 1], a ⊕ b = Min(1, a + b), a ⊗ b = Max(0, a + b − 1). || 4 Pre-Semirings and Pre-Dioids 21 The reason why it is of interest not to assume both right and left distributivity in the most basic structure (the pre-semiring structure) is that there exist interesting applications which do not enjoy both properties.

Download PDF sample

Rated 4.49 of 5 – based on 15 votes