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

Why do we randomize?

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

Picking Passwords Encryption keys are randomly chosen

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

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”

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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

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.

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

Proof of Main Theorem Fix some min cut. Say it has k edges. 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?

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

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)

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 ]

Final Algorithm Running time:  O(m¢n2) (m = # edges, n = # vertices) 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)

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 vertices has · n2 min cuts. (Actually, min cuts)

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

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!

Resources http://www.cs.ubc.ca/~nickhar/W15 Books Mitzenmacher-Upfal Motwani-Raghavan Alon-Spencer