admin
	ps1 back
	ps2 due 10/2
	ps3 out later today
today
	expander mixing lemma
	random walks on expanders
next time
	random walks on expanders
	explicit constructions
expander mixing lemma
	draw graph
		subsets S, T
	Q. how many edges between them? |cut(S,T)|
		count directed edges
		u\ne v\in S\cap T then (u,v) counts *twice*
	A. if random graph, each edge w/p d/n
		\E[\cut(S,T)]=|S|*|T|*d/n
		[[how close is an expander?]]
	lem: A adjacency matrix, |\cut(S,T)|= 1_S^\tr A 1_T = d * 1_S^\tr M 1_T, M random walk matrix
		1_S is indicator vector for S
		draw matrix equation
	pf
		# edges = \sum_{(i,j)\in E, i\in S, j\in T} 1
		=\sum_{(i,j)\in E} 1_S(i) A_{i,j} 1_T (j)
		=1_S^\tr A 1_T
	lem[expander mixing lemma]
		|S|=\alpha n
		|T|=\beta n
		| |\cut(S,T)|/d*n - |S|*|T|/n^2|
			=| |\cut(S,T)|/d*n - \alpha*\beta|
			\le \lambda \sqrt{\alpha\beta(1-\alpha)(1-\beta)}
		[[non-trivial if \lambda <<\sqrt{\alpha\beta}
			as sqrt increases things here as less than one]]
	pf
		v=1_S
			=v^\par+v^\perp
		||v||=\sqrt{\alpha *n}
		||v^\par||=<v,1/\sqrt{n}>=\alpha n /\sqrt{n}=\alpha\sqrt{n}
		||v^\perp||^2=|v|^2-||v^\par||^2=\alpha n -\alpha^2 n = \alpha(1-\alpha)n
			||v^\perp||=\sqrt{\alpha(1-\alpha) n}
		w=1_T
			norms are similar
		|\cut(S,T)|
			= d * 1_S^\tr M 1_T
			= d (v^\par + v^\perp)^\tr M (w^\par+w^\perp)
			= d (v^\par M w^\par+v^\par M w^\perp + v^\perp M w^\par + v^\per M w^\perp)
				Mw^\par=w^\par
				v^\par M=v^\par
				v^\par w^\perp =0
			= d (<v^\par w^\par> +< v^\per, M w^\perp>)
				v^\par = ||v^\par|| * \vec{1}/\sqrt{n}
				<v^\par w^\par>= ||v^\par||*||w^\par||
				|< v^\per, M w^\perp>|\le ||v^\per||*||M w^\perp|| (cauchy schwartz)
					\le \lambda ||v^\per|| ||w^\per||
			= d (\alpha\sqrt{n}\beta\sqrt{n} \pm \lambda \sqrt{\alpha(1-\alpha)\beta(1-\beta)} n)
		divide by dn to get the result
	Cor: \gamma spectral expansion => (N/2,\gamma/2) edge expander
		ie |\cut(S,\bar{S})|\ge \gamma/2 *D*|S| for S with |S|\le N/2
		only use expander mixing lemma for disjoint sets here
	pf
		|\cut(S,\bar{S})|/DN\ge |S||\bar{S}|/N^2-\gamma \sqrt{|S|/n (1-|S|/n) (1-|S|/n) |S|/n}
				= |S|/N (1-|S|/N)-\gamma |S|/n (1-|S|/n)
				= |S|/n((1-|S|/N)(1-\lambda)
		|\cut|\ge D|S| (1-|S|/N) * (1-\lambda)
			(1-|S|/N)\ge 1/2
	Thm: (N/2,\eps) edge expander, and at least \alpha fraction of edges leaving each vertex are self loops then G has spectral expansion \alpha\eps^2/2
		eg, D-regular and 1 self loop per vertex then \alpha\ge 1/D
		Q. why self-loops?
		A. to avoid bipartite graphs
random walks on expanders
	draw graph
		start with random node
		then randomly walk
	Q. what is the random walk of a complete graph (w/ self loops)
	A. independent samples
	Q. if expanders are pseudorandom, can we replace independent samples w/ walk on expander?
	def: G D-regular graph
		random walk is function from [N]\times [D]^{t-1} to [N]^t
		for v\in[N] i\in [D] say E(v,i)=i-th neighbor of v
		RandWalk_j=E(...E(v,i_1),i_2),...i_{j-1})
	mixing results: sectral gap \gamma
		for any prob dist \pi, t-th output of RandWalk(\pi,(\rU_{[D]})^{t-1}) is \approx U_{[N]}
			\ellinfty error \eps in O(\log(N/\eps)/\gamma) steps
		=> invest \log D*O(\log(N/\eps)/\gamma) random bits to get \log N nearly random output
			[[this is basically optimal for constant D, \eps, \gamma]]
		=> hitting times by taking many fresh random walks
	Q. can we get more from this?
		[[look at the whole output of the walk, not just the end?]]
		[[don't ask for independent random vars, but show there is some property of independence that is preserved?]]
		ie, use outut of walk as a sampler
	strategy
		vector decomposition [[expander mixing lemma]]
			v=v^\par+v^\perp
			Mv^\par=v^\par
			||Mv^\perp||\le \lambda ||v^\perp||
			track how parallel and perp components mix
			[[can get messy]]
		matrix decomposition [[simpler, but less optimal]]
			def
				spectral norm (\elltwo operator norm)
				||M||=\max_{|x|\le 1} ||Mx||
				can define for any matrix
			lem: M random walk matrix, J=all ones matrix /n
				M has spectral gap \gamma => M=\gamma J+\lambda E, for ||E||\le 1
			Pf
				=>
					define E=(M-\gamma J)/\lambda
						E symmetric so spectral norm captured by eigenvalues
						suffices to bound |<v,Ev>| for any eigenvector v
					E v^\par=
						1/\lambda (Mv^\par-\gamma Jv^\par)
						=1/\lambda (v^\par-\gamma v^\par)
						=1/\lambda (1-\gamma) v^\par)
						=v^\par
						[[ie, \vec{1} is eigenvector w eigenvalue 1]]
					E v^\perp=1/\lambda (Mv^\perp-\gamma Jv^\perp)
						=1/\lambda (Mv^\perp)
					|<perp,Eperp>|\le 1*\lambda*1/\lambda=1
			Rmk: J is random walk matrix of complete graph
				[[intuition then is that w/p \gamma we do completely random walk, w/p 1-\gamma=\lambda we do arbitrary walk]]
			Rmk: w/ matrix decomposition get \lambda \sqrt{\alpha\beta} error term in expander mixing lemma
				[[getting the better term important for getting into edge expansion]]
hitting property of expander walks
	RP algo, want to amplify probability of success
		[[x \notin L is ok]]
		[[x \in L: want to see witness w/ better prob]]
		bad random strings B\subset V
		want to find v\in V\B
	algo
		do t-step random walk starting with uniform vertex
		acc x if ever acc
	analysis
		Pr[failure]=\Pr[ walk stays inside B^t]
	Q. how to analyze
		[[t=2 can use expander mixing lemma]]
	thm: S\subset V, \mu=|S|/N
		expansion 1-\lambda
		Pr[random t-step walk, starting from uniform, stays in S^t]\le (\mu+\lambda(1-\mu))^t
			=(\gamma \mu+(1-\gamma))^t
			\le 1
	intuition
		lambda=0
			complete graph, aka independent samples
			get \mu^t as usual
		\lambda>0
			w/p \gamma do complete graph, pickup factor of \mu
			w/p 1-\gamma do arbitrary step, get nothing
	Pf
		\pi prob dist
		P\in \bits^n projection onto S
			P_{i,j}=0 for i\ne j
			P_i 	= 0 for i\in S
				=1 for i\notin S
		lem: (P\pi)_i=Pr[i\from \pi is in B]
		lem: ((PM)^{t-1}P\pi)_i= Pr[draw from \pi, are in B, do t-1 step random walk, still in B, land at i]
		lem: <1,(PM)^{t-1}P\pi>=Pr[t-step random walk starting at \pi is in B^t]
		some tricks
			<1,(PM)^{t-1}P\pi>
			=<P1,(PMP)^{t-1}P\pi>
		cauchy schwartz
			|<P1,(PMP)^{t-1}P\pi>|
				\le |P1| |(PMP)^{t-1}P\pi>|
				\le |P1| |(PMP)^{t-1}||P\pi|
				=\sqrt{\mu n} |(PMP)^{t-1}| \sqrt{\mu/n}
				=\sqrt{\mu n} |PMP|^{t-1} \sqrt{\mu/n}
				=\mu |(PMP)^{t-1}|
		lem: |PMP|\le \gamma\mu+\lambda
		Pf
			PMP=\gamma PJP+\lambda PEP
			|P|\le 1 => |PEP|\le 1
			PJP:
				y=Px
				Jy=|y|_1*1
				y has support on S
				|y|_1=|<1_S,x>|\le \sqrt{\mu N} ||x||=\sqrt{\mu N}
				PJPx=|y|*P1/n
				|PJPx|
					\le |y|_1 ||P1/n||_2
					\le \sqrt{\mu N} ||1_S/n||_2
					= \sqrt{\mu N}* \sqrt{\mu/N}
					=\mu
	Cor: RP algo with error \le 1/2, 
		G expander with spectral gap \lambda \le 1/4
		t-step random walk
			Pr[error]\le (3/4)^t
			randomness m+\log D*t=m+O(t)
	Rmk: but we need an *explicit* expander
admin
	ps1 back
	ps2 due 10/2
	ps3 out later today
today
	expander mixing lemma
	random walks on expanders
next time
	random walks on expanders
	explicit constructions