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The order-4 shuffle-exchange network, with its vertices arranged in numerical order. In graph theory, the shuffle-exchange network is an undirected cubic multigraph, whose vertices represent binary sequences of a given length and whose edges represent two operations on these sequence, circular shifts and flipping the lowest-order bit.
Example of shuffling five letters using Durstenfeld's in-place version of the Fisher–Yates shuffle The Fisher–Yates shuffle is an algorithm for shuffling a finite sequence . The algorithm takes a list of all the elements of the sequence, and continually determines the next element in the shuffled sequence by randomly drawing an element from ...
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G).
A simple algorithm to generate a permutation of n items uniformly at random without retries, known as the Fisher–Yates shuffle, is to start with any permutation (for example, the identity permutation), and then go through the positions 0 through n − 2 (we use a convention where the first element has index 0, and the last element has index n − 1), and for each position i swap the element ...
Mashaghi A. et al., for example, demonstrated that a transformation which converts random graphs to their edge-dual graphs (or line graphs) produces an ensemble of graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient. [9]
These graphs are partial cubes, graphs in which the nodes can be labeled by bitvectors in such a way that the graph distance equals the Hamming distance between labels. In the case of a line arrangement, each coordinate of the labeling assigns 0 to nodes on one side of one of the lines and 1 to nodes on the other side. [26]
Examples of edge dominating sets. In graph theory, an edge dominating set for a graph G = (V, E) is a subset D ⊆ E such that every edge not in D is adjacent to at least one edge in D. An edge dominating set is also known as a line dominating set. Figures (a)–(d) are examples of edge dominating sets (thick red lines).
A natural approach would be to shuffle the deck and then take the top k cards. In the general case, the shuffle also needs to work even if the number of cards in the deck is not known in advance, a condition which is satisfied by the inside-out version of the Fisher–Yates shuffle : [ 9 ]