1. Randomized Algorithms

2. Deterministic Vs Randomized Algorithms
Deterministic Algorithms
Randomized Algorithms
Algorithm
Algorithm
I/P
O/P
I/P
O/P
Random Bits

3. What are Randomized Algorithms?
Algorithms whose behavior depends on:
Input(s)
A sequence of random bits (these random bits lead to: random steps, choices, decisions,…,etc. during the running of the algorithm)
Can be faster and simpler than a deterministic algorithm
Two types:
Las Vegas
Mont Carlo

4. Las Vegas or Mont Carlo Always returns Correct answer
Running time Fast with a High Probability
Always runs Fast
Returns Correct answer with a High Probability

5. Example 1 Input: array of n chars, half ‘a’s, half ‘b’s
Problem: Find location of any ‘a’
Deterministic Approach:
Search One by One  worst case: O(n/2)
Randomized Approach:
Check entries at random until ‘a’ is found
Probability of finding ‘a’ right away = 0.5
Probability of missing ‘a’ k times in a row = 0.5k
Which type of algorithm is it?

6. Example 2: Microchip Test
We want to know if the manufacturer tested a batch of chips
For an untested batch, P(bad) = 0.1
Deterministic Algorithm: test them all  O(n)
Randomized Algorithm: test chips at random

7. Example 2: Microchip Test
Question to ask: “Has this batch NOT been tested?”
TRUE – if random algorithm finds a bad chip
UNKNOWN – if algorithm finds a good chip
After k chips, algorithm concludes FALSE, therefore the batch is GOOD.
Probability a good chip came from untested batch = 0.9
Probability “good” batch is actually bad = 0.9k

8. Example 2: Microchip Test
0.9k = probability a bad batch is being tested and we keep finding the good chips, k times in a row
For k = 66, chances of a bad batch slipping through is less than 1 in a 1000 => small
Running time = O(k), independent of n, determined by k set by programmer => fast
Monte Carlo: running time fast, correctness probability high

9. Example 3: Min-Cut Input: A connected, undirected graph G = (V,E)
Output: A min-cut, i.e. smallest set of edges to remove that splits graph into two
Possible approaches:
Try them all – exponential since there are 2|E| subsets of edges
Modified max-flow algorithm for all s-t pairs – O(|V|3) x O(|V|2) pairs of vertices = O(|V|5)

10. Example 3: Min-Cut Randomized approach: O((nlogn)2)
Same idea as chip test:
one time through the algorithm yields a relatively low probability of success
repeat algorithm to reduce probability of repeated failure until satisfied with probability of success
Will only show algorithm and that it works with a certain probability of success

11. Example 3: Min-Cut Algorithm:
Randomly select an edge
Merge its end-points
Repeat until two vertices remain
How do we know this algorithm is reliable?

12. Pr[min-cut edge] £ c/(nc/2)
Analysis I
Min-cut is small---few edges
Suppose graph has min-cut c
Then minimum degree at least c
Thus at least nc/2 edges
Random edge is probably safe
Pr[min-cut edge] £ c/(nc/2)
= 2/n 15

13. Analysis II
Algorithm succeeds if never accidentally contracts min-cut edge
Contracts #vertices from n down to 2
When k vertices, chance of error is 2/k
thus, chance of being right is 1-2/k
Pr[always right] is product of probabilities of being right each time 16

14. Analysis III
…not too good! 17

15. Repetition Repetition amplifies success probability
basic failure probability 1 - 2/n2
so repeat 7n2 times 18