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Then, 8| E | > | V | 2 /8 when | E |/| V | 2 > 1/64, that is the adjacency list representation occupies more space than the adjacency matrix representation when d > 1/64. Thus a graph must be sparse enough to justify an adjacency list representation. Besides the space trade-off, the different data structures also facilitate different operations.
[8] [9] Intersection graphs An interval graph is the intersection graph of a set of line segments in the real line. It may be given an adjacency labeling scheme in which the points that are endpoints of line segments are numbered from 1 to 2n and each vertex of the graph is represented by the numbers of the two endpoints of its corresponding ...
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.
Adjacency matrix [3] A two-dimensional matrix, in which the rows represent source vertices and columns represent destination vertices. Data on edges and vertices must be stored externally. Only the cost for one edge can be stored between each pair of vertices. Incidence matrix [4]
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 of i. In the example on the left, there are two arrays, C and R. Array C stores the adjacency lists of all nodes.
The degree of all nodes is 1, so each node is selected with probability 1/2, and with probability 1/4 both nodes in a component are not chosen. Hence, the number of nodes drops by a factor of 4 each step, and the expected number of steps is log 4 ( n ) {\displaystyle \log _{4}(n)} .
The primitive graph operations that the algorithm uses are to enumerate the vertices of the graph, to store data per vertex (if not in the graph data structure itself, then in some table that can use vertices as indices), to enumerate the out-neighbours of a vertex (traverse edges in the forward direction), and to enumerate the in-neighbours of a vertex (traverse edges in the backward ...
1 S ← empty sequence 2 u ← target 3 if prev[u] is defined or u = source: // Proceed if the vertex is reachable 4 while u is defined: // Construct the shortest path with a stack S 5 insert u at the beginning of S // Push the vertex onto the stack 6 u ← prev[u] // Traverse from target to source