1. CSCI 3160 Design and Analysis of Algorithms Tutorial 4
Chengyu Lin

2. Outline More examples about 2-SAT More randomized algorithms
Verifying Matrix Multiplication
String Equality Test
Design Patterns

3. 2-SAT Given (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
(x1, x2, x3, x4, x5) = (T, T, T, T, T) is a satisfying assignment.
Suppose you do not know about this solution
You do not even know if there exists a solution for this formula
How to decide if there is one using randomness?

4. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Start with a random assignment,
say (x1, x2, x3, x4, x5) = (F, T, F, F, T)
Number of xis that agrees with the solution (i.e. number of i such that xi = T) 1 2 3 4 5

5. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Find an unsatisfied clause: (x4∨x3)
flip one of the value of x3 and x4 randomly
If we flip x3, then we jump from 2 to 3
If we flip x4, then we jump from 2 to 3
clauses x1 x2 x3 x4 x5
current F T

6. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Find an unsatisfied clause: (x1∨¬x2)
flip one of the value of x1 and x2 randomly
If we flip x1, then we jump from 3 to 4
If we flip x2, then we jump from 3 to 2
clauses x1 x2 x3 x4 x5
current F T

7. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Find an unsatisfied clause: (x2∨¬x3)
flip one of the value of x2 and x3 randomly
If we flip x2, then we jump from 2 to 3
If we flip x3, then we jump from 2 to 1
clauses x1 x2 x3 x4 x5
current F T

8. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Find an unsatisfied clause: (x1∨¬x2)
flip one of the value of x1 and x2 randomly
If we flip x1, then we jump from 3 to 4
If we flip x2, then we jump from 3 to 2
clauses x1 x2 x3 x4 x5
current F T

9. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Find an unsatisfied clause: (x4∨¬x5)
flip one of the value of x4 and x5 randomly
If we flip x4, then we jump from 4 to 5
If we flip x5, then we jump from 4 to 3
clauses x1 x2 x3 x4 x5
current T F

10. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Find an unsatisfied clause: none
We have a satisfying assignment! =)
clauses x1 x2 x3 x4 x5
current T

11. Analysis
To do the analysis, assume we have a satisfying assignment in mind, call it solution
Of course, we do not know if a satisfying assignment exists when we run the algorithm
we do this for the analysis only
We compare the current assignment with the solution
And record the number of variables that are assigned to the same T/F value in both (current and solution) assignments

12. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Find an unsatisfied clause: (x4∨x3)
flip one of the value of x3 and x4 randomly
If we flip x3, then we jump from 2 to 3
If we flip x4, then we jump from 2 to 3
clauses x1 x2 x3 x4 x5
Solution T
current F
Number of xis that agrees with the solution (i.e. xi = T)
with prob. 1 1 2 3 4 5

13. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Find an unsatisfied clause: (x1∨¬x2)
flip one of the value of x1 and x2 randomly
If we flip x1, then we jump from 3 to 4
If we flip x2, then we jump from 3 to 2
clauses x1 x2 x3 x4 x5
Solution T
current F
Number of xis that agrees with the solution (i.e. xi = T)
with prob. 1/2
with prob. 1/2 1 2 3 4 5

14. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Find an unsatisfied clause: (x2∨¬x3)
flip one of the value of x2 and x3 randomly
If we flip x2, then we jump from 2 to 3
If we flip x3, then we jump from 2 to 1
clauses x1 x2 x3 x4 x5
Solution T
current F
Number of xis that agrees with the solution (i.e. xi = T)
with prob. 1/2
with prob. 1/2 1 2 3 4 5

15. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Find an unsatisfied clause: (x1∨¬x2)
flip one of the value of x1 and x2 randomly
If we flip x1, then we jump from 3 to 4
If we flip x2, then we jump from 3 to 2
clauses x1 x2 x3 x4 x5
Solution T
current F
Number of xis that agrees with the solution (i.e. xi = T)
with prob. 1/2
with prob. 1/2 1 2 3 4 5

16. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Find an unsatisfied clause: (x4∨¬x5)
flip one of the value of x4 and x5 randomly
If we flip x4, then we jump from 4 to 5
If we flip x5, then we jump from 4 to 3
clauses x1 x2 x3 x4 x5
Solution T
current F
Number of xis that agrees with the solution (i.e. xi = T)
with prob. 1/2
with prob. 1/2 1 2 3 4 5

17. 2-SAT (x1∨¬x2)∧(x2∨¬x3)∧(x4∨x3)∧(x5∨x1)∧(x4∨¬x5)
Find an unsatisfied clause: none
We have a satisfying assignment! =)
clauses x1 x2 x3 x4 x5
Solution T
current
Number of xis that agrees with the solution (i.e. xi = T) 1 2 3 4 5

18. 2-SAT We always move to the right with probability at least 1/2 1 2 31 2 3 4 5
with prob. 1/2
with prob. 1/2 1 2 3 4 5
with prob. 1/2
with prob. 1/2 1 2 3 4 5
with prob. 1/2
with prob. 1/2 1 2 3 4 5
with prob. 1/2
with prob. 1/2 1 2 3 4 5 1 2 3 4 5
We always move to the right with probability at least 1/2

19. 2-SAT We always move forward with probably at least 1/2
a satisfying assignment must satisfy every clause
When we find an unsatisfied clause, it must be bad
The values of the literals in a satisfying assignment must be one of the good ones
Hence, can always flip one to move to the right
e.g. (x1∨¬x2) x1 x2
good T F
bad

20. 2-SAT We always move forward with probably at least 1/2
When we reach n (i.e. 5 in the example), we have a satisfying assignment.
Sufficient condition, not necessary.
There may be other satisfying assignments
But that will only improve the running time
This is random walk on line
and so we expect O(n2) jumps to hit n (refer to lecture notes)

21. 2-SAT Randomized algorithm: Repeat O(n2) times,
if we hit 5 in the middle of these O(n2) flips, return there is an assignment
otherwise, return there is NO satisfying assignment

22. Verifying Matrix Multiplication
Given 3 n x n integer matrices A, B and C, verify if AB = C
Naïve algorithm
Compute AB, check if it is equal to C
Runs in time O(n3), dominated by computing AB
How to do faster with high probability?

23. Verifying Matrix Multiplication
Given 3 n x n integer matrices A, B and C, verify if AB = C
Consider AB – C
Check if AB – C is a zero matrix
Lecture: how to check if two polynomials are identical?
Evaluate the polynomials at different points

24. Verifying Matrix Multiplication
Given 3 n x n integer matrices A, B and C, verify if AB = C
Now, multiply AB – C by different random vectors
To avoid computing AB, we compute A(Bx) – Cx
We have
Working over modulo 2,
If AB ≠ C, then Pr[(AB - C)x ≠ 0] ≥ 1/2

25. String equality Alice has an n-bit string (a1, a2, …, an)
Bob has an n-bit string (b1, b2, …, bn)
Alice can communicate with Bob
How to send as few bits as possible so that they know the two strings are equal?

26. String equality Alice has an n-bit string (a1, a2, …, an)
Bob has an n-bit string (b1, b2, …, bn)
Represent
(a1, a2, …, an) by a1 + a2x + … + anxn-1
(b1, b2, …, bn) by b1 + b2x + … + bnxn-1
Polynomial Identity Testing!
This idea can be extended to pattern matching

27. Design Patterns Las Vegas algorithm Always produces the correct answer
Usually only expected running time is guaranteed
Example: randomized quick sort

28. Design Patterns Monte Carlo algorithm Running time is deterministic
Small probability of error
Usually repeats a simple procedure to get a large success probability
Example: min-cut

29. End
Questions?