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An adjacency list representation for a graph associates each vertex in the graph with the collection of its neighbouring vertices or edges. There are many variations of this basic idea, differing in the details of how they implement the association between vertices and collections, in how they implement the collections, in whether they include both vertices and edges or only vertices as first ...
The time complexity of operations in the adjacency list representation can be improved by storing the sets of adjacent vertices in more efficient data structures, such as hash tables or balanced binary search trees (the latter representation requires that vertices are identified by elements of a linearly ordered set, such as integers or ...
An example of bag structure with 23 elements. There are some special data structures that parallel BFS can benefit from, such as CSR (Compressed Sparse Row), bag-structure, bitmap and so on. In the CSR, all adjacencies of a vertex is sorted and compactly stored in a contiguous chunk of memory, with adjacency of vertex i+1 next to the adjacency ...
In the context of efficient representations of graphs, J. H. Muller defined a local structure or adjacency labeling scheme for a graph G in a given family F of graphs to be an assignment of an O(log n)-bit identifier to each vertex of G, together with an algorithm (that may depend on F but is independent of the individual graph G) that takes as input two vertex identifiers and determines ...
Provided the graph is described using an adjacency list, Kosaraju's algorithm performs two complete traversals of the graph and so runs in Θ(V+E) (linear) time, which is asymptotically optimal because there is a matching lower bound (any algorithm must examine all vertices and edges).
This simple model is commonly known as the adjacency list model and was introduced by Dr. Edgar F. Codd after initial criticisms surfaced that the relational model could not model hierarchical data. [citation needed] However, the model is only a special case of a general adjacency list for a graph.
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As Itai & Rodeh (1978) observe, the graph contains a triangle if and only if its adjacency matrix and the square of the adjacency matrix contain nonzero entries in the same cell. Therefore, fast matrix multiplication techniques can be applied to find triangles in time O ( n 2.376 ) .