admin
	ps1 out, due 2pm 9/13
		I emailed about it, let me know if you didn't get that email
today
	approximate counting
	random walks
approximate counting
	defn
		"given" f, approx \E[f] to \eps error, w/p \ge 1-\delta
	"given"
		as oracle
		as boolean ckt, over AND, OR, NOT
			do example
	oracle: [[last time]]
		easy randomized algorithm
		no deterministic algorithm
			not language problem
			is oracle problem
	defn \pm\eps approx counting, CA^\eps
		input (C,p)
		\E[C]\ge p+\eps => output yes
		\E[C]\le p-\eps => output no
	rmk: promise problem
		\PI_y\sqcup \PI_N\subseteq\bits^\star
		define prP,prBPP
		generalize language problems
		essentially the same complexity as actually estimating \E[C]
			via binary search, won't show here
	rmk: easy randomized algo
	Q. deterministic algo?
	thm: CA^{1/6} is \prBPP-complete
	pf
		in \prBPP
			via random sampling + chernoff
		\prBPP hard
			ie, give reduction
				given (\PI_N,\PI_Y) in \prBPP
				give map  x-> (C_x,1/2)
					x\in \PI_Y -> (C_x,1/2)\in (CA^\eps)_Y
					x\in \PI_N -> (C_x,1/2)\in (CA^\eps)_N
				map is deterministic polytime
				so algo for CA^{1/6} can be used to solve any prBPP problem
			the reduction
				(\PI_N,\PI_Y) has prBPP algo A(x,r)
					x\in \PI_Y -> A(x,r)=1 w/p \ge 2/3 over r
					x\in \PI_N -> A(x,r)=1 w/p \le 1/3 over r
				define C_x(r)\eqdef A(x,r)
					can conver TM's to ckts, see complexity books
	open: BPP-complete problem
multiplicative approx counting
	more natural in applications
	defn (1\pm\eps) approx counting, CA^{\times(1\pm\eps)}
		input (C,p)
		\E[C]\ge (1+\eps)p => output yes
		\E[C]\ge (1-\eps)p => output no
	Rmk:
		\NP-hard even for CNFs
			CNFs=...
			eg p~=0 distinguishes satisfiable from unsatisfiable formulas
	Thm[KarpLuby83]: \times (1\pm\eps) approx for DNFs in poly(|\phi|,1/\eps,\ln 1/\delta) randomized time
	Rmk:
		DNF = ...
			m clauses
			n variables
		SAT is easy for DNFs
	Pf
		randomized sampling doesn't work
			draw picture of small # of samples in large space
		idea: map satisfting assignments to new (smaller) space
			draw picture
			draw actual picture
		key point: sat assign to clause are easy to enumerate
			x_1\AND \not x_3 AND x_5 -> 1*0*1*****
		define
			B={(\alpha,i): \alpha is sat assignment to \phi}
			A'={(\alpha,i): \alpha is sat assignment to \phi, clause i is first satisfied clause}
				canonical explaination
			draw table of sat assign vs clauses
				table = B
				1's = A
		lem: A/B\ge 1/m
			as each row has at least 1 one
			this says A is dense in B, so can randomly sample
		facts:
			|B|=\sum_i 2^{n-n_i}
				n_i=# var in i-th clause
					after removing degenerate clauses eg x AND \not(x)
			=> can efficiently sample from B
			=> can get [\pm\gamma]-approx to A/B in poly(n,m,1/\gamma) rand step
				can decide membership in A
			=> can get [(1\pm\eps]-approx to A/B in poly(n,m,1/\eps) rand step
				|\hat{\mu}-A/B|\le \eps/m \le \eps A/B
					\gamma=\eps/m
	later: techniques for n^{\polylog}-time algo
	best: (nm)^{\lg\lg (nm)} for constant \eps
		can even do slightly subconstant
random walks
	defn of STCONN
		input (G,s,t), G=(V,E), |V|=n
		is there an s-t path
	recall:
		polytime algorithm via depth-first search
		but uses \Omega(n) space
	Q. STCONN in small space?
	model for logspace computation
		draw
			TM
			input tape
			work tape 
			write-only output tape
		space = size of work tape
		randomness = randomized state
			if you want to remember randomness, need to write it down
	defn of L,RL,BPL
	Q. L=RL=BPL?
		seems much easier than P vs BPP
		actually have progress on this
	thm[savitch]	
		STCONN in DSPACE(log^2) (time poly^log)
	thm[AKLLR79]
		USTCONN in RL
	Thm{Reinghold05]
		USTCONN in L
		[[willlater]]
	Pf.
		random walk algo
			v=s
			for poly(n) steps
				pick random neighbor of v, call it u
				v<-u
			if ever saw t, output yes
			else		output no
		complexity
			log(n) bits to store current vertex
			in logspace can compute random neighbor
		analysis
			digraph
				directed graph
				allow parallel edges
				allow self-loops
			defn hitting time, hit(G)
				max_{i,j} min \{t: \Pr[rand walk starting at i hits j in t steps]\ge 1/2]\}
			thm: G connected, undirected => hit(G)\le \poly(n)
			Rmk: exist G directed with hit(G)\ge exp(n^{\Omega(1)})
			Pf[hw]
				via spectral graph theory
				wlog G non-bipartite, d-regular graph
					add self-loops
					will only increase hitting time
					will only do regular graphs in this class
				random walk matrix
					M\in\R^{n\times n}
					M_{i,j}=Pr[j\to i]=(# j->i edges)/degree(j)
						reverse than in book
						will use column vectors in class, row vectors in book
				lem: G regular, undirected => M symmetric
				thm[spectral thm]
					A symmetric. Then A has orthogonal eigenbasis
					[[ask who is comfortable with this]]
				\lem \pi prob dist, M\pi = prob dist after one step of random walk
				obs
					\vu\eqdef uniform dist = 1/n*\vec{1}
					G regular => Mu=u
						eigenvalue 1
				want: M\pi closer to \vu than \pi is