Graphs

1. Graphs A graphs is an abstract representation of a set of objects, called vertices or nodes, where some pairs of the objects are connected by links, called edges or arcs. undirected graph directed graph (digraph) v u v u e e w w e = (v, w) (v, w) = (w, v) v: origin of e w: destination of e

2. Applications of Graphs Java has a sophisticated Map interface, which has an extensive collection of methods and “views”. Here we present a stripped-down version.

3. Some Terminologies • The vertices are adjacent to one another and that the edge is incident to both vertices. • The degree of a vertex is the number of edges incident to it. • A path is a sequence of vertices such that each adjacent pair of vertices is connected by an edge. • A cycle is a path with at least one edge whose first and last vertices are the same. • The length of a path or a cycle is its number of edges it traverses. • The distance from one vertex to another is the length of the shortest path from one to the other.

4. Some Terminologies • An undirected graph is connected if there is a path from every vertex to every other vertex in the graph. A graph that is not connected consists of a set of connected components, which are maximal connected subgraphs. • A directed graph is strongly connected if for every pair of nodes u and v there is a path from u to v and a path from v to u.

5. A few common graphs problems

6. Representing Graphs We need to represent the vertices and the edges of a graph so that for any given vertex we can easily identify its neighbors An adjacency matrix is a V-by-V array of boolean values. Each row and column represents a vertex of the graph. Set the value at row i, column j to true if (i, j) is an edge of the graph. If the graph is undirected, the adjacency matrix is symmetric: row i, column j has the same value as row j, column i. Space Requirement?

7. Representing Graphs An adjacency list is actually a collection of lists. Each vertex v maintains a list of the edges directed out from v. Space Requirement?

8. Representing Graphs An adjacency list is actually a collection of lists. Each vertex v maintains a list of the edges directed out from v. The standard list representations all work — arrays, linked lists, even binary search trees (because you can traverse one in linear time). The total memory used by all the lists is O(V + E).

9. Adjacency Matrix vs. Adjacency List An adjacency list is more space- and time-efficient than an adjacency matrix for a sparse graph, but less efficient for a complete graph. A complete graph is a graph having every possible edge; that is, for every vertex u and every vertex v, (u, v) is an edge of the graph.

10. Depth-First Search DFS starts at an arbitrary vertex and searches a graph as “deeply” as possible and as early as possible. For example, if your graph is an undirected tree rooted at the starting vertex, DFS performs a postorder tree traversal.