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Basic Concepts in Probability
Published in X. Rong Li, Probability, Random Signals, and Statistics, 2017
If every element of set A is also an element of set B, then A is said to be a subset of B, denoted as A ⊂ B or B ⊃ A. Set A is said to be equal to set B if A⊂ B and A ⊃ B, denoted as A = B. In this case, A and B have exactly the same elements. Two sets are said to be disjoint if they do not have any element in common.
Findset, Find Min, and Find Word
Published in Suman Saha, Shailendra Shukla, Advanced Data Structures, 2019
The disjoint-set data structure is characterized by its unique operations and provides the way to understand the data which is always arranged as a disjoint collection of sets. The operations are described below. Separate three operations below by vertical spaces so they’re easier to read. makeset(x): The makeset(x) operation uses an element x as input and returns a new singleton set represented by itself.union(x,y): The union(x,y) operation uses two sets represented by x and y elements as inputs and unites those sets to return a new set consisting of elements from both the sets.findset(x): The findset(x) operation uses an element as input and returns the representative of the set to which it belongs.Find connected components of a graph G(V, E)In this example, we present an algorithm to compute connected components of a given graph. The algorithm is very straightforward and given below.Algorithm to determine connected components of a graph G(V,E)1: procedure CONNECTED-COMPONENTS G(V,E) ⊳ computes connected components of a graph2: for ∀ v ∈ Vdo3: makeset(v) ⊳ creates |V| number of singleton sets4: remember representatives5: for ∀ (u, v) ∈ Edo6: if findset(u) ≠ findset(v) then7: union(u,v) ⊳ unite if different set8: reduce representative9: return representative ⊳ one representative for each component
Synthesis of regular expression problems and solutions
Published in International Journal of Computers and Applications, 2020
A context-free grammar (or CFG) G is a four-tuple: where: N is a set of nonterminals, also known as ‘syntactic variables .' Nonterminal represent phrases/clauses in a sentence. Hence, nonterminals are sometimes referred to as syntactic categories, with every nonterminal defining a sub-language of the language G. Σ is a finite set of terminal symbols, disjoint from N, from which the actual content of a sentence is composed. Σ is referred to as the alphabet of the language defined by the grammar G. P is the set of productions, each production consisting of a nonterminal, called the left hand side of the production, a forward arrow, and a sequence of terminal and/or nonterminal symbols, called the right hand side of the production. S is the start nonterminal (or start symbol), used to denote the entire sentence. The relation must always hold.
Group of L-homeomorphisms and L f -representability of Permutation Groups
Published in Fuzzy Information and Engineering, 2020
[18] Letbe any set andbe a subgroup ofgenerated bywherebe an indexed family of disjoint cycles with equal length. Thenis t-representable ifor.is not t-representable ifand.
Automatic data matching for geospatial models: a new paradigm for geospatial data and models sharing
Published in Annals of GIS, 2019
As with spatial similarity, temporal similarity calculations also include temporal topology and distance calculations. The biggest difference between temporal topology and spatial topology is that the former needs to consider the time direction (before or after) (Zhu et al. 2017a). Generally, it is assumed that the new data are better than the old. Therefore, when the temporal topology is disjoint or adjacent but the time of is later than , the similarity is not set to 0. The key to temporal distance calculation is to unify the time granularity of and . For example, the temporal granularity of is day, and the counterpart of should be day rather than year or hour; otherwise, the temporal granularity of must be transformed to day.