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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 ...
Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible along each branch before backtracking.
Any vertex that is not on a directed cycle forms a strongly connected component all by itself: for example, a vertex whose in-degree or out-degree is 0, or any vertex of an acyclic graph. The basic idea of the algorithm is this: a depth-first search (DFS) begins from an arbitrary start node (and subsequent depth-first searches are conducted on ...
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).
In the example on the left, there are two arrays, C and R. Array C stores the adjacency lists of all nodes. Array R stored the index in C, the entry R[i] points to the beginning index of adjacency lists of vertex i in array C. The CSR is extremely fast because it costs only constant time to access vertex adjacency.
For each vertex we store the list of adjacencies (out-edges) in order of the planarity of the graph (for example, clockwise with respect to the graph's embedding). We then initialize a counter = + and begin a Depth-First Traversal from . During this traversal, the adjacency list of each vertex is visited from left-to-right as needed.
In the analysis of algorithms, the input to breadth-first search is assumed to be a finite graph, represented as an adjacency list, adjacency matrix, or similar representation. However, in the application of graph traversal methods in artificial intelligence the input may be an implicit representation of an infinite graph. In this context, a ...
The adjacency matrix distributed between multiple processors for parallel Prim's algorithm. In each iteration of the algorithm, every processor updates its part of C by inspecting the row of the newly inserted vertex in its set of columns in the adjacency matrix. The results are then collected and the next vertex to include in the MST is ...