Complexity.

What is a rand. alg?

• What is an alg?
• Turing Machines. RAM with large ints. log-cost RAM as TM.
• language as decision problem (vs optimization problems) ``graphs with small min-cut.'' algos accept/reject
• complexity class as set of languages
• P. polynomial time in input size
• NP as P with good advice string. witnesses
• polytime reductions. hardness, completeness.
• Randomized algorithms have advice string, but it is random
• measure probs over space of advice strings
• equivalence to fliping unbiased random bits
• ZPP (zero error probabilistic polytime)
• Polynomial expected time
• A(x) accepts iff x $\in$L.
• Las Vegas algorithms
• RP (randomized polytime) (MC with one-sided error).
• polytime (always)
• x $\not\in$L $\Rightarrow$rejects (always).
• x $\in$L $\Rightarrow$accepts with probability > 1/2.
• Monte Carlo algorithm
• one sided error
• precise numbers unimportant: amplification.
• min-cut example
• coRP.
• What if NOT worst case polytime? stop when passes time bound and accept.
• ZPP = RP $\cap$coRP
• PP (probabilistic polytime) (two-sided MC)
• Worst case polytime (can force)
• x $\in$L $\Rightarrow$accepts prob > 1/2
• x$\not\in$L $\Rightarrow$accepts prob < 1/2
• weakness: NP $\subseteq$PP
• BPP (bounded probabilistic polytime)
• worst case polytime (can force)
• x $\in$L $\Rightarrow$accepts prob > 3/4
• x$\not\in$L $\Rightarrow$accepts prob < 1/4
• precise numbers unimportant.

·       Clearly P $\subseteq$RP $\subseteq$NP. Open questions:

o      RP = coRP? (equiv RP = ZPP)

o      BPP $\subseteq$NP?

Tree evaluation.

Moving LOE through a (linear) recurrence.

• define. algo cost is number of leaves. n = 2^h
• NOR model

deterministic model: must examine all leaves. time 2^h = 4^h/2 = n

• by induction: on any tree of height h, as questions are asked, can answer such that root is not determined until all leaves checked.
• Note: bad instance being constructed on the fly as algorithm runs.
• But, since algorithm deterministic, bad instance can be built in advance by simulating algorithm.

nondeterministic/checking

• T(0) = 1
• winning position can guess move. W(h) = L(h - 1)
• losing must check both. L(h) = 2W(h - 1)
• follows W(h) = 2*W(h - 2) = 2^h/2 = n^1/2

randomized-guess which leaf wins.

• T(0) = 1
• W(T) is a random variable: time it takes to verify T is a win. Ditto L(T).
• W(h) = max over all height-h winning trees of E[W(T)]
• explain W(T) = L(T₁) + ${\frac{1}{2}}$L(T₂)
• Suppose T₁ wins.

$$\le$$

$$\approx$$

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