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 ExampleIP
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

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

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 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 ExampleS
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 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 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 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?

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 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)

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

30. 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!

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