Method of Conditional Probabilities and Expectations

Derandomization.

• Theory: is P=RP?
• practice: avoid chance of error, chance of slow.

Conditional Expectation. Max-Cut

• Imagine placing one vertex at a time.
• x_i = 0 or 1 for left or right side
• E[C] = (1/2)E[C| x₁ = 0] + (1/2)E[C| x₁ = 1]
• Thus, either E[C| x₁ = 0] or E[C| X₁ = 1]$\ge$E[C]
• Pick that one, continue
• More general, whole tree of element settings.
• Let C(a) = E[C | a].
• For node a with children b, c, C(b) or C(c)$\ge$C(a).
• By induction, get to leaf with expected value at least E[C]
• But no randomness left, so that is actual cut value.
• Problem: how compute node values? Easy.

Conditional Probabilities. Set balancing. (works for wires too)

• Review set-balancing Chernoff bound
• Think of setting item at a time
• Let Q be bad event (unbalanced set)
• We know Pr[Q] < 1/n.
• Pr[Q] = 1/2 Pr[Q | x_i0] + 1/2 Pr[Q | x_i1]
• Follows that one of conditional probs. less than Pr[Q] < 1/n.
• More general, whole tree of element settings.
• Let P(a) = Pr[Q | a].
• For node a with children b, c, P(b) or P(c) < P(a).
• P(r) < 1 sufficient at root r.
• at leaf l, P(l )= 0 or 1.
• One big problem: need to compute these probabilities!

Pessimistic Estimators.

• Alternative to computing probabilities
• three neceessary conditions:
• $\hat{P}$(r) < 1
• min{$\hat{P}$(b),$\hat{P}$(c)} < $\hat{P}$(a)
• $\hat{P}$computable

Imply can use $\hat{P}$instead of actual.

• Let Q_i = Pr[unbalanced set i]
• Let $\hat{P}$(a) = $\sum$Pr[Q_b | a] at tree node a
• Claim 3 conditions.
• HW
• Result: deterministic O($\sqrt{n\ln n}$) bias.
• more sophisticated pessimistic estimator for wiring.

Oblivious routing

• recall: choose random routing. Only 1/N chance of failure
• Choose N³ random routines.
• whp, for every permutation, at most 2N² bad routes.
• given the N³ routes, pick one at random.
• so for any permutation, prob 2/N of being bad.

Fingerprinting

Basic idea: compare two things from a big universe U

• generally takes log U, could be huge.
• Better: randomly map U to smaller V, compare elements of V.
• Probability(same)= 1/| V|
• intuition: log V bits to compare, error prob. 1/| V|

We work with fields

• add, subtract, mult, divide
• 0 and 1 elements
• eg reals, rats, (not ints)
• talk about Z_p
• which field often won't matter.

Verifying matrix multiplications:

• Claim AB = C
• check by mul: n³, or n^2.376 with deep math
• Freivald's O(n²).
• Good to apply at end of complex algorithm (check answer)

Freivald's technique:

• choose random r $\in${0, 1}ⁿ
• check ABr = Cr
• time O(n²)
• if AB = C, fine.
• What if AB$\ne$C?
• trouble if (AB - C)r = 0 but D = AB - C$\ne$ 0
• find some nonzero row (d₁,..., d_n)
• wlog d₁$\ne$ 0
• trouble if $\sum$d_ir_i = 0
• ie r₁ = ($\sum_{i > 1}^{}$d_ir_i)/d₁
• principle of deferred decisions: choose all i$\ge$2 first
• then have exactly one error value for r₁
• prob. pick it is at most 1/2

How improve detection prob?

• k trials makes 1/2^k failure.
• Or choosing r $\in$[1, s] makes 1/s.
• Doesn't just do matrix mul.
• check any matrix identity claim
• useful when matrices are ``implicit'' (e.g. AB)
• We are mapping matrices (n² entries) to vectors (n entries).

String matching

Checksums:

• Alice and Bob have bit strings of length n
• Think of n bit integers a, b
• take a prime number p, compare a mod p and b mod p with log p bits.
• trouble if a = b(mod p). How avoid? How likely?
• c = a - b is n-bit integer.
• so at most n prime factors.
• How many prime factors less than k? $\Theta$(k/ln k)
• so take 2n²log n limit
• number of primes about n²
• So on random one, 1/n error prob.
• O(log n) bits to send.
• implement by add/sub, no mul or div!

How find prime?

• Well, a randomly chosen number is prime with prob. 1/ln n,
• so just try a few.
• How know its prime? Simple randomized test (later)

Pattern matching in strings

• m-bit pattern
• n-bit string
• work mod prime p of size at most t
• prob. error at particular point most m/(t/log t)
• so pick big t, union bound
• implement by add/sub, no mul or div!