Dijkstra’s Algorithm

1. Dijkstra’s Algorithm

2. Announcements • Assignment #2 Due Tonight • Exams Graded • Assignment #3 Posted

3. Announcements

4. Last Time • Breadth First Search, Depth First Search • Different algorithms for searching all nodes in a graph. • Topological Sort • Produces an ordering of vertices in a graph, given certain constrains (A before B, B before C, etc.) • Dijkstra’s Algorithm • Finds the shortest paths from a source vertex to all other vertices.

5. Today • Finish up Dijkstra’s • Path reconstruction • MSTs – Minimum Spanning Tree algorithms • Prim’s • Kruskal’s

6. Dijkstra’s Algorithm • This algorithm finds the shortest path from a source vertex to all other vertices in a weighted directed graph without negative edge weights. 5 10 2 a b d 2 3 8 4 3 4 12 e c 16 13

7. Dijkstra’s Path Reconstruction • Example shown on the Board

8. Minimum Spanning Trees • In a weighted, undirected graph, it is a tree formed by connecting all of the vertices with minimal cost. • The MST is a tree because it’s acyclic. • It’s spanning because it covers every vertex. • And it’s minimum because it has minimum cost. 2 v1 v2 4 1 3 10 2 7 v3 v4 v5 4 8 5 6 v6 v7 1 2 v1 v2 1 2 v3 v4 v5 4 6 v6 v7 1

9. MSTs – proof that greedy works • Let G be a graph with vertices in the set V partitioned into two sets V1 and V2. Then the minimum weight edge, e, that connects a vertex from V1 to V2 is part of a minimum spanning tree of G. • Proof: Consider a MST T of G that does NOT contain the minimum weight edge e. • This MUST have at least one edge in between a vertex from V1 to V2. (Otherwise, no vertices between those two sets would be connected.) • Let G contain edge f that connects V1 to V2. • Now, add in edge e to T. • This creates a cycle. In particular, there was already one path from every vertex in V1 to V2 and with the addition of e, there are two. • Thus, we can form a cycle involving both e and f. Now, imagine removing f from this cycle. • This new graph, T' is also a spanning tree, but it's total weight is less than or equal to T because we replaced e with f, and e was the minimum weight edge.

10. MSTs – proof that greedy works • As a spanning tree is created • If the edge that is added is the one of minimum cost that avoids creation of a cycle. • Then the cost of the resulting spanning tree cannot be improved • Because any replacement edge would have cost at least as much as an edge already in the spanning tree. • This is why greedy works!

11. Kruskal’s Algorithm Let V =  For i=1 to n-1, (where there are n vertices in a graph) V = V  e, where e is the edge with the minimum edge weight not already in V, and that does NOT form a cycle when added to V. Return V • Basically, you build the MST of the graph by continually adding in the smallest weighted edge into the MST that doesn't form a cycle. • When you are done, you'll have an MST. • You HAVE to make sure you never add an edge the forms a cycle and that you always add the minimum of ALL the edges left that don't.

12. Kruskal’s Given Graph G: Edges in sorted order: 2 v1 v2 v4 4 1 3 10 v1 4 2 7 1 v3 v4 v5 v7 4 8 v4 v3 5 6 v6 v7 1 v6 v7 5 1 v6 v5 Determine the MST: 2 v1 v2 6 v7 2 v1 v2 4 2 3 v3 v4 7 1 v4 v5 2 CYCLE! v3 v4 v2 v5 v4 3 4 8 6 v4 v6 v6 v7 v1 v2 1 4 10 All Vertices, we’re done!! v3 v5

13. Kruskal’s • The reason this works: • is that each added edge is connecting between two sets of vertices, • and since we select the edges in order by weight, • we are always selecting the minimum edge weight that connects the two sets of vertices. • Cycle detection: • Keep track of disjoint sets. • Initially, each vertex is in its own disjoint set. • When you add an edge you are unioning two sets. • A union cannot happen if the two vertices are already in the same set. • This would create a cycle.

14. Prim’s Algorithm • This is quite similar to Kruskal's with one big difference: • The tree that you are "growing" ALWAYS stays connected. • Whereas in Kruskal's you could add an edge to your growing tree that wasn't connected to the rest of it, here you can NOT do it. Here is the algorithm: 1) Set S = . 1) Pick any vertex in the graph. 2) Add the minimum edge incident to that vertex to S. 3) Continue to add edges into S (n-2 more times) using the following rule: Add the minimum edge weight to S that is incident to S but that doesn't form a cycle when added to S.

15. Given Graph G: Prim’s 2 v1 v2 Edges in sorted order: 4 1 3 10 v4 2 7 v1 v3 v4 v5 4 4 8 1 5 6 v7 v6 v7 v4 v3 1 v6 v7 5 Determine the MST, using Prim’s starting with vertexV1: 1 v6 v5 2 v1 v2 6 v7 2 v1 v2 4 2 3 v3 v4 7 1 v4 v5 2 CYCLE! v3 v4 v2 v5 v4 3 4 8 6 v4 v6 v6 v7 v1 v2 1 4 10 All Vertices, we’re done!! v3 v5

16. Example on the board

17. Huffman Encoding • Compress the storage of data using variable length codes. • For example, each character in a text file is stored using 8 bits. • Nice and easy because we always read in 8 bits for a single character. • Not the most efficient… • What if ‘e’ is used 10 times more frequently than ‘q’. • It would be more advantageous for us to use a 7 bit code for e and a 9 bit code for q.

18. Huffman Coding • Finds the optimal way to take advantage of varying character frequencies in a particular file. • On average, standard files can shrink them anywhere from 10% to 30% depending to the character distribution. • The idea behind the coding is to give less frequent characters and groups of characters longer codes. • Also, the coding is constructed in such a way that no two constructed codes are prefixes of each other. • This property about the code is crucial with respect to easily deciphering the code.

19. Building a Huffman Tree • Example on the board

20. References • Slides adapted from Arup Guha’s Computer Science II Lecture notes: http://www.cs.ucf.edu/~dmarino/ucf/cop3503/lectures/ • Additional material from the textbook: Data Structures and Algorithm Analysis in Java (Second Edition) by Mark Allen Weiss • Additional images: www.wikipedia.com xkcd.com