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The Best Algorithms are Randomized Algorithms

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 A A A A A A A. Why do we randomize?. Reasons for Randomness. Fooling Adversaries Symmetry Breaking Searching a Haystack

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The Best Algorithms are Randomized Algorithms

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  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.: AAAAAAAA

  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 17 5 23 9 12 4 19 10 5 9 12 4 23 17 19 10 • 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”

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

  8. Another Adversary Example IP • 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

  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 • Consider the polynomial f(x,y)=(x-y2)(x2+2y+1) • 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

  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 verticesand 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? • Connectivityis 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? • Connectivityis 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 S • 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

  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 randomizedapproach

  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 w 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

  20. Edge Contraction v u w • 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

  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/n2. (n = # vertices) • Fairly low success probability. Is this useful? • Yes! Run the algorithm n2 times. • Pr[ fails to find min cut ]  (1-1/n2)n2 1/e • So algorithm succeeds with probability > 1/2

  24. Proof of Main Theorem • 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 • 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?

  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. Final Algorithm • Input: Graph G • Output: min cut, with probability  1/2 • For i=1,..,n2 • Start from G • While graph has  2 vertices • Pick an edge uv at random • Contract it • End While • Let Ei = { remaining edges } • End For • Output smallest Ei • Running time: O(m¢n2) (m = # edges, n = # vertices) • With more beautiful ideas, improves to  O(n2)

  29. How Many Min Cuts? • Our analysis: for any particular min cut, Pr[ algorithm finds that min cut ]  1/n2 • So suppose C1, C2, ..., Ct are all the min cuts • 1/n2 fraction of the time the algorithm finds C1 • 1/n2 fraction of the time the algorithm finds C2 • ... • 1/n2 fraction of the time the algorithm finds Ct • This is only possible if t ·n2 • Corollary: Any connected graph on n verticeshas · n2 min cuts. (Actually, min cuts)

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

  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 }  n2. • No other proof of this theorem is known!

  32. Classes • CS 466, CS 761 (still active?), C&O 738 Books Mitzenmacher-Upfal Motwani-Raghavan Alon-Spencer

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