goals
	review administrivia
		ps1
		typos
	computational models
		motivate via PIT
	\eps-approx counting
		motivate via PIT
today
	perfect matching
	polynomial identity testing
	randomized algorithms
	tail bounds
	sampling
administrivia
	CS598: Pseudorandomness
	miforbes.cs.illinois.edu/teaching/
	ps1 will be out later today
		will email about it
		due 9/13
	survey form
		fill it out
perfect matching
	G=(L\cup R, E) bipartite graph	
		draw it
	M\subseteq E matching if \deg_M (v)\le 1
	perfect	if deg = 1
	Q. given graph, does it have a perfect matching?
		basic computational question
		in practice
			kidney exchanges
			assigning medical residents to hospitals
		in theory
			matroids
			max flow
			linear programming
	A. many efficient deterministic algorithms
		many *sequential*
	thm: algo for perf matching in polylog(n)-parallel-time w/ poly(n) processors
		won't define model of computation here
	Pf
		write down symbol adj matrix A
		A_{i,j}= 0, or x_{i,j}
		det A = \sum_{\sigma\in S_n} \sgn(\sigma) \prod_{i=1}^n A_{i,\sigma(i)}
		lem \det A =0 iff no perfect matching
polynomial identity testing
	Q. "given" polynomial f, is it =0?
		test if polynomial is identically zero
		"simplifiy" in mathematica (which is based in champaign)
	what is a polynomial
		field F
		many variables
		list of coefficients
		equal to zero if all coeffs are zero
			compare to x^2-x over F_2
	"given"
		for bday
		in the mail
		list of coefficients
		oracle
		algebraic circuit
			draw example
	randomized algo
		lem(SZ)
			intuition from univariate case
		=> randomized polytime algo, choose |S|=2d
		Rmk:
			need to construct field extensions
			randomness is over coin tosses of algo
	Pf of perf-match
		do PIT for \det A
		\det A can be computed efficiently in parallel
	derandomization?
		this algorithm uses poly(n) random bits
		FGT16: reduced to O(\lg^2 n) random bits
			doing better is open!
		Fact:
			derandomizing PIT in general implies ckt lbs
			derandomizing PIT in special cases implies ckt lbs 
				chasm at depth-3
			derandomizing PIT can sometimes be done
randomized algorithms
	randomized TM
		draw it, with random state
		equiv: rand() function returns random bit
	defn of P	
		language L\subseteq {0,1}^\star is in P if poly(n)-time TM M
			why we study languages
		x\in L => M acc x =~ Pr[M acc x]=1
		x\not L => M rej x =~ Pr[M acc x]=0
		rmk:
			randomness only over coins, not over x
	defn of RP
		"randomized polytime"
		like NP
		one-sided error
		defn is robust to change in constant 1/2
		coRP
		Cor: PIT is in coRP
			f =0 => f always reported as zero
			f \ne 0 => w/p 1/2 get non-zero
	defn of BPP
		two-sided error
		bounded-error probabilistic polytime
		defn is also robust
	Open P vs RP vs BPP
tail bounds
	thm[chernoff]
	lem: 1/2-1/n error BPP => 1/2^n error BPP
sampling
	sampling oracle problem
		ie, polling for an electio
	lem: solve via randomness
	lem: query lb
	rmk: doesn't show P\ne BPP
		approximation problems, not languages
		is an *oracle* problem
today
	ps1 out tonight
	fill out survey form
	perfect matching
	poly ident testing
	randomized algorithms
	sampling
next time
	no class Monday (labor day)
	approximate counting
	random walks