1. The Best Algorithms are Randomized Algorithms N. Harvey C&O Dept TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AAAA A A A

2. WHY DO WE RANDOMIZE?

3. Reasons for Randomness Fooling Adversaries Symmetry Breaking Searching a Haystack Sampling huge sets...

4. Picking Passwords Encryption keys are randomly chosen

5. The Hallway Dance Randomly go left or right After a few twists you won’t collide (usually!)

6. QuickSort 1752391241910 If I knew an approximate median, I could partition into 2 groups & recurse A random element is probably close to the median Like “searching for hay in a haystack” 1752391241910

7. Reasons for Randomness Fooling Adversaries Symmetry Breaking Searching a Haystack...

8. Another Adversary Example Linux 2.4.20 denial-of-service attack Kernel uses a hash table to cache info about IP traffic flows Adversary sends many IP packets that all have same hash value Hash table becomes linked list; Kernel very slow Solution: Randomized hash function IP

9. Another Symmetry Breaking Example Ethernet Media Access Control Try to send packet If collision detected – Wait for a random delay – Retry sending packet http://picasaweb.google.com/tomatobasil

10. Another Haystack Example Problem: Find a point (x,y) that is not a zero of f If f  0 then, for randomly chosen (x,y), Pr[ f(x,y)=0 ] = 0 Even over finite fields, Pr[ f(x,y)=0 ] is small Consider the polynomial f(x,y)=(x-y 2 )(x 2 +2y+1)

11. Reasons for Randomness Fooling Adversaries Symmetry Breaking Searching a Haystack... Beauty: randomization often gives really cool algorithms!

12. Graph Basics A graph is a collection of vertices and edges Every edge joins two vertices The degree of a vertex is # edges attached to it

13. Graph Connectivity A connected graph A disconnected graph Problem: How many edges do you need to remove to make a graph disconnected?

14. Graph Connectivity Problem: How many edges do you need to remove to make a graph disconnected? Connectivity is minimum # edges whose removal disconnects the graph Observation: for every vertex v, degree(v)  connectivity Because removing all edges attached to vertex v disconnects the graph

15. Graph Connectivity Problem: How many edges do you need to remove to make a graph disconnected? Connectivity is minimum # edges whose removal disconnects the graph Useful in many applications: How many MFCF network cables must be cut to disrupt MC’s internet connectivity? How many bombs can my oil pipelines withstand?

16. Graph Connectivity Example Removing red edges disconnects graph Definition: A cut is a set of edges of the form { all edges with exactly one endpoint in S } where S is some set of vertices. Fact: min set of edges that disconnects graph is always a cut S

17. Algorithms for Graph Connectivity Given a graph, how to compute connectivity? Standard Approach: Network Flow Theory – Ford-Fulkerson Algorithm computes an st min cut (minimum # edges to disconnect vertices s & t) – Compute this value for all vertices s and t, then take the minimum – Get min # edges to disconnect any two vertices CO 355 Approach: – Compute st min cuts by ellipsoid method instead Next: An amazing randomized approach

18. Algorithm Overview Input: A haystack Output: A needle (maybe) While haystack not too small – Pick a random handful – Throw it away End While Output whatever is left

19. Edge Contraction u v Key operation: contracting an edge uv Delete the edge uv Combine u & v into a single vertex w Any edge with endpoint u or v now ends at w w

20. Edge Contraction u v Key operation: contracting an edge uv Delete the edge uv Combine u & v into a single vertex w Any edge with endpoint u or v now ends at w This can create “parallel” edges w

21. Randomized Algorithm for Connectivity Input: A graph Output: Minimum cut (maybe) While graph has  2 vertices “Not too small” – Pick an edge uv at random “Random Handful” – Contract it “Throw it away” End While Output remaining edges

22. Graph Connectivity Example We were lucky: Remaining edges are the min cut How lucky must we be to find the min cut? Theorem (Karger ‘93): The probability that this algorithm finds a min cut is  1/(# vertices) 2.

23. But does the algorithm work? How lucky must we be to find the min cut? Theorem (Karger ‘93): The probability that this algorithm finds a min cut is  1/n 2. (n = # vertices) Fairly low success probability. Is this useful? Yes! Run the algorithm n 2 times. Pr[ fails to find min cut ]  (1-1/n 2 ) n 2  1/e So algorithm succeeds with probability > 1/2

24. Fix some min cut. Say it has k edges. If algorithm doesn’t contract any edge in this cut, then the algorithm outputs this cut – When contracting edge uv, both u & v are on same side of cut So what is probability that this happens? While graph has  2 vertices “Not too small” – Pick an edge uv at random “Random Handful” – Contract it “Throw it away” End While Output remaining edges Proof of Main Theorem

25. Initially there are n vertices. Claim 1: # edges in min cut=k  every vertex has degree  k  total # edges  nk/2 Pr[random edge is in min cut] = # edges in min cut / total # edges  k / (nk/2) = 2/n

26. Now there are n-1 vertices. Claim 2: min cut in remaining graph is  k Why? Every cut in remaining graph is also a cut in original graph. So, Pr[ random edge is in min cut ]  2/(n-1)

27. In general, when there are i vertices left Pr[ random edge is in min cut ]  2/i So Pr[ alg never contracts an edge in min cut ]

28. Input: Graph G Output: min cut, with probability  1/2 For i=1,..,n 2 Start from G While graph has  2 vertices – Pick an edge uv at random – Contract it End While Let E i = { remaining edges } End For Output smallest E i Final Algorithm Running time:  O(m ¢ n 2 ) (m = # edges, n = # vertices) With more beautiful ideas, improves to  O(n 2 )

29. How Many Min Cuts? Our analysis: for any particular min cut, Pr[ algorithm finds that min cut ]  1/n 2 So suppose C 1, C 2,..., C t are all the min cuts  1/n 2 fraction of the time the algorithm finds C 1  1/n 2 fraction of the time the algorithm finds C 2...  1/n 2 fraction of the time the algorithm finds C t This is only possible if t · n 2 Corollary: Any connected graph on n vertices has · n 2 min cuts. (Actually, min cuts)

30. An n-cycle has exactly min cuts! How Many Min Cuts? Corollary: Any connected graph on n vertices has min cuts. Is this optimal?

31. How Many Approximate Min Cuts? A similar analysis gives a nice generalization: Theorem (Karger-Stein ‘96): Let G be a graph with min cut size k. Then # { cuts with  k edges }  n 2 . No other proof of this theorem is known!

32. Books Motwani-RaghavanMitzenmacher-UpfalAlon-Spencer Classes CS 466, CS 761 (still active?), C&O 738