Graphs

1. Graphs Trees Without Rules

2. Graph • A data structure that consists of a set of nodes (called vertices) and a set of edges that relate the nodes to each other

3. Facebook Graph Vertices: people Edges: friends June Katie Susan Sarah Judy Lila Robert Rebecca John

4. Airline Routes Graph

5. Course Prerequisites Graph

6. Graphs • Another recursive data structure • Node with any number of pointers to other nodes • No restrictions on connectivity • Allows disconnected nodes, multiple paths, and cycles • Terminology • Node • Arc • Directed • Connected • Path • Cycle

7. Implementation Strategies • Create a set of all arcs • (a->b, a->c, b->d, c->d, d->a, d->b) • Adjacency List • a: {b, c} • b: {d} • c:{d} • d:{a, b} • Adjacency matrix • Consider the data • Sparse vs. Dense • Space vs. Time a c b d

8. Coding Graphs • Graph itself is set or vector of node * • Why not just pointer to a root, like a tree? • Could you designate an arbitrary node as root? Struct nodeT { //data for node goes here vector <nodeT *> connected; };

9. Coding Graphs • Often graphs have data associated with the arc itself. • Unlike trees and lists where links are only for connecting nodes • Arcs may have information like distance or cost. • Add struct to hold arc info • Arc has pointers to start/end node and a node has a collection of arcs • Circular reference

10. Circular Reference

11. Arcs have Nodes and Nodes have Arcs struct arcT { //arc fields go here nodeT *start, *end; //error! nodeT not defined }; struct nodeT { //node fields go here vector <arcT *> outgoing; //arcT already defined };

12. Making the Gears Work struct nodeT; //forward reference struct arcT { //arc fields go here nodeT *start, *end; }; struct nodeT { //node fields go here vector <arcT *> outgoing; };

13. Graph Traversals • Traverse reachable nodes • Start from a node and follow arcs to other nodes • Some graphs not fully connected, so not all nodes are reachable • Depth-first and Breadth-first • Similar to tree’s post-order, pre-order and in-order • Both visit all reachable nodes, but in a different order • Possibility of cycles means you must track “visited” nodes to avoid infinite loop • Could maintain a visited flag per node or set of visited nodes

14. Depth-First Traversal • Choose a starting node • No root node in graph, may have specific start in mind or just choose one randomly • Go Deep • Pick a neighbor, explore all reachable from there • Backtrack • After fully exploring everything reachable from first neighbor, choose another neighbor and go again • Continue until all neighbors are exhausted • Base case?

15. Depth-First Pseudocode void DepthFirstSeach(node *cur, set<nodeT *> & visited) { //if visited contains cur, do nothing. //otherwise, add cur to visited set //for all arcT’s in the graph vector named outgoing DepthFirstSearch(cur->outgoing[i]->end, visited); }

16. Trace A H B D G C E A B C D E F G H F

17. Breadth-First Traversal • Choose starting node • Visit all immediate neighbors • Those directly connected to start node • Branch out to all neighbors 2 hops away • Again to 3 hops and so on • Until all reachable nodes are visited • How to manage nodes to vist • Perfect for the queue • What about cycles/multiple paths? • Need to track visited status

18. Breadth-First Pseudocode void BreadthFirstSearch(nodeT *start) { queue<nodeT *> q; set<nodeT *> visited; q.enqueue(start); //while q is not empty nodeT *cur = q.dequeue(); //if visited does not contain cur, add cur to visited and //for all arcTs in vector named outgoing... q.enqueue(cur->outgoing[i]->end); }

19. Graph Search Algorithms • Many interesting questions are really just graph searches • Which nodes are reachable from this node? • Does the graph have a cycle? • Is the graph fully connected? • Longest path without a cycle? • Is there a continuous path that visits all nodes once and exactly once?