Graphs

1. Graphs

2. Definitions • A graph is two sets. • A set of nodes or vertices V • A set of edges E • Edges connect nodes. • The number of vertices |V| • The number of edges |E| • A sparse graph is one with relatively few edges • A dense graph is one with relatively many edges • A graph with all possible edges is complete

3. More Definitions • A graph with edges directed from one vertex to another is a direct graph or a diagraph • A graph whose edges are not directed is called an undirected graph • A graph with labels associated with its vertices is called a labeled graph • Two vertices that share an edge are called adjacent. They are also called neighbors. • An edge connecting vertices u and v is written (u,v). The edge is said to be incident on u and v. • Associated with each edge may be a cost or a weight.

4. Still More Definitions • A sequence of vertices v1, v2,…,vn forms a path of length n-1 if there exist edges from vi to vi+1 for 1 <= i < n. • A path is simple if all vertices on the path are unique • A cycle is a path of length 3 or more that connects some vertex to itself. • A cycle is simple if the path is simple except for the first and last vertices.

5. Yes More Definitions • A subgraph S is formed from Graph G by selecting a subset of Vs of G’s vertices and a subset Es of G’s edges such that for every edge E in Es, both whose vertices are in Vs. • An undirected graph is connected if there is at least one path from any vertex to any other. • A graph without cycles is called acyclic. • A directed graph without cycles is called directed acyclic graph or DAG

6. Graph Representations • Adjacency Matrix • If a pair of vertices have an edge between them, then there is a 1 in the matrix. • Adjacency List • If a vertex has an edge, then a node is added to its list with the other node as the data

7. Graph Implementations • A common activity that graphs must support is traversals • This usually requires finding the node that is closest or the first node • Then you usually need to find the next node after some given node. • Another common member needed for a graph implementation is a way to mark each of the vertices

8. Graph Traversals • To visit the vertices of a graph in some specific order based on the graph’s topology. • There are some troublesome issues • It may not be possible to reach all of the vertices from each other • The graph may contain cycles and we need to make sure that the cycles do not cause an infinite loop

9. The Mark • Graphs will typically maintain a mark for each vertex in the graph. • They will clear the marks before a traversal begins and set the mark as they go. • When the traversal is done we can check the marks to see if all the vertices have been visited.

10. Depth-first Search • Whenever a vertex is visited, a DFS will recursively visit all of its unvisited neighbors. void DFS(Graph* G, int v) { Previsit(G, v); G->setMark(v, VISITED); for(int w=G->first(v); w<G->n(); w=G->next(v, w) ) if (G->getMark(v) == UNVISITED) DFS(G, w); PostVisit(G,v); }

11. Breadth-First Search • Examines all vertices connected to the start vertex before visiting vertices further away • Similar to DFS except a queue replaces the recursion stack. • If the graph is a tree, this is equivalent to traversing level by level.

12. Shortest Paths Problem • Single-source shortest-path • Given a vertex v in a graph G, what is the shortest path from v to all other vertices in G • This is usually solved by a classic algorithm. • Dijkstra’s Algorithm

13. Dijkstra’s Algorithm • The algorithm maintains a distance estimate from each vertex to every other vertex. • Initially, every distance is set to infinity. • Vertices are process in order of distance. • Whenever a vertex is processed, its distance is updated for all of its neighbors.

14. Minimum-Cost Spanning Trees • MST problem takes a connected, undirected, weighted graph. • The MST is the graph containing the vertices of G along with the subset of G’s edges that • Has minimum total cost as measured by summing the values for all of the edges in the subset • Keeps the vertices connected.

15. Minimum-Cost Spanning Tree • There are two main algorithms for solving this problem • Prim’s algorithm • Kruskal’s algorithm