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

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 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 Linux 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

Another Symmetry Breaking Example Ethernet Media Access Control Try to send packet If collision detected – Wait for a random delay – Retry sending packet

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)

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 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

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 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

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

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/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

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

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 ]

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 )

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)

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

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!

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