Graphs - II CS 2110, Spring 2016

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Graphs - II CS, Spring Where did David leave that book? http://www.geahvet.com Where did David leave that book?

com Go as far down a path as possible before backtracking http://www.geahvet.

1 Graphs - II CS, Spring Where did David leave that book? Where did David leave that book? Where did David leave that book? Go as far down a path as possible before backtracking Graph Algorithms Search Depth-first search Breadth-first search Shortest paths Dijkstra's algorithm Minimum spanning trees Prim's algorithm Kruskal's algorithm Node v is node u if there is a path from u to v. node?

2 Node v is node u if there is a path from u to v. Node v is node u if there is a path from u to v. node?,,,, node? 7 8 Node v is node u if there is a path from u to v. We need an invariant! node?,,, How to determine reachability efficiently? 9 Node v is node u without green nodes if there is a path from u to v without green nodes. node without Node v is node u without green nodes if there is a path from u to v without green nodes. node without

3 Node v is node u without green nodes if there is a path from u to v without green nodes. node without Node v is node u without green nodes if there is a path from u to v without green nodes. node without None! Node is green, so all paths from node contain a green node! node? Start at node node? Extend path to some child node? Extend path to some child node? 7 8

4 node? No new way to extend path, so backtrack Extend path to a different child node? 9 Extend path to some child node? Already visited, so backtrack node? node? No new way to extend path, so backtrack node? No new way to extend path, so backtrack

5 Extend path to a different child node? Extend path to some child node? Already visited, so backtrack node? node? No new way to extend path, so backtrack 7 8 using Recursion /** Visit all nodes u without visited nodes */ Nothing to backtrack, so all done! node? 9 node without None!

6 using Recursion /** Visit all nodes u without visited nodes */ using Recursion /** Visit all nodes u without visited nodes */ for (Node v with edge from u to v) dfs(v); using Recursion /** Visit all nodes u without visited nodes */ for (Node v with edge from u to v) dfs(v); using Recursion /** Visit all nodes u without visited nodes */ for (Node v with edge from u to v) dfs(v); OO-style Recursive class Node { final List<Node> targets; // edges go from this to targets boolean visited= false; // has this node been visited? Node(Node targets) { this.targets= Arrays.asList(targets); /*Visit all nodes this without visited nodes*/ void dfs() { if (visited) return; visited= true; for (Node v : targets) v.dfs(); using Iteration /** Visit all nodes u without visited nodes */ Collection<Node> work= new Stack<Node>(); work.add(u); // inv: all nodes that have to be visited are // reachable (without visited nodes) from some node in work while (!work.isempty()) { Node u= work.pop(); // Remove first node and put it in u if (!u.hasbeenvisited() ) { for (Node v with edge from u to v) work.add(v); // Stack adds nodes to front

7 Breadth-First Search Breadth-First Search node? node? Visit nodes distance from node 7 8 Breadth-First Search Breadth-First Search node? node? Visit nodes distance from node Visit nodes distance from node 9 Breadth-First Search No nodes at distance, so all done! node? using Iteration /** Visit all nodes u without visited nodes */ Collection<Node> work= new Stack<Node>(); work.add(u); // inv: all nodes that have to be visited are // reachable (without visited nodes) from some node in work while (!work.isempty()) { Node u= work.pop(); // Remove first node and put it in u if (!u.hasbeenvisited()) { for (Node v with edge from u to v) work.add(v); // Stack adds nodes to front 7

8 Breadth-First Search using Iteration /** Visit all nodes u without visited nodes */ void bfs(node u) { Collection<Node> work= new Queue<Node>(); work.add(u); // inv: all nodes that have to be visited are // reachable (without visited nodes) from some node in work while (!work.isempty()) { Node u= work.pop(); // Remove first node and put it in u if (!u.hasbeenvisited()) { for (Node v with edge from u to v) work.add(v); // Queue adds nodes to back 8

Graphs - II CS 2110, Spring 2016

Graphs - II CS 2110, Spring 2016 Graphs - II CS, Spring Where did David leave that book? http://www.geahvet.com Where did David leave that book? http://www.geahvet.com Where did David leave that book? Go as far down a path as possible