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**CSCI 3160 Design and Analysis of Algorithms Tutorial 4**

Chengyu Lin

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**Outline More examples about 2-SAT More randomized algorithms**

Verifying Matrix Multiplication String Equality Test Design Patterns

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**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?

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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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

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**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

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**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

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**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

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**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

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**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

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**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

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**2-SAT We always move to the right with probability at least 1/2 1 2 3**

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 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

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**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

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**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)

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**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

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**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?

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**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

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**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

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**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?

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**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

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**Design Patterns Las Vegas algorithm Always produces the correct answer**

Usually only expected running time is guaranteed Example: randomized quick sort

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**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

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End Questions?

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