Random Cut for each vertex v∈V uniform independent Ye0,1} Y,=1 v∈S Y,=0> v∈T for each edge uv∈E Yu卡Yw IC(S,T)=>Yu )Yu=Yo uU∈D OPT EIC(S,T)川=>PY≠Y] 2 2 uw∈E
for each vertex v 2 V uniform & independent v 2 S v 2 T Yv 2 {0, 1} Yv = 1 Yv = 0 Yuv = ( 1 Yu 6= Yv 0 Yu = Yv for each edge uv 2 E |C(S, T)| = X uv2E Yuv E[|C(S, T)|] = X uv2E Pr[Yu 6= Yv] = |E| 2 OPT 2 Random Cut
Random Cut for each vertex v∈V uniform 2-wise independent YE 10,1} Yo=1> v∈S Y=0> v∈T for each edge uv∈E Yu卡Yw IC(S,T)=>Yu )Yu=Yo uU∈D E OPT EIC(S,T)川=>PY≠Y] 2 2 uw∈E
Random Cut for each vertex v 2 V uniform & 2-wise independent v 2 S v 2 T Yv 2 {0, 1} Yv = 1 Yv = 0 Yuv = ( 1 Yu 6= Yv 0 Yu = Yv for each edge uv 2 E |C(S, T)| = X uv2E Yuv E[|C(S, T)|] = X uv2E Pr[Yu 6= Yv] = |E| 2 OPT 2
Derandomization for each vertex v∈V uniform&2-wise independent Yo∈{0，ly Y=1 v∈S Y=0> v∈T for each edge uv∈E EIC(S,T)川=>Pr[Yu≠Y= E OPT 2 2 uU∈E V={1,2,,vn} Y,Yv2,...,Yvm constructed from log2(n+1)]bits try all 2()=O(n2)possibilities!
Derandomization for each vertex v 2 V uniform & 2-wise independent v 2 S v 2 T Yv 2 {0, 1} Yv = 1 Yv = 0 for each edge uv 2 E E[|C(S, T)|] = X uv2E Pr[Yu 6= Yv] = |E| 2 OPT 2 V = {v1, v2,...,vn} Yv1 , Yv2 ,...,Yvn constructed from dlog bits 2(n + 1)e try all 2dlog2(n+1)e = O(n2) possibilities!
2-wise Independent Variables random source:uniform and independent Xo,X1∈[p Goal:uniform and 2-wise independent Yo,i,.,Yp-1∈p prime p fori∈[pl Y=(Xo+i·X1)modp uniformity:i,a∈[p] Pr[Y:a] 2-wise independence: i卡j,a,b∈[p 1 Pr[Y=a∧Y)=b= p2
2-wise Independent Variables Goal: uniform and 2-wise independent 2 [p] prime p random source: uniform and independent X0, X1 2 [p] for i 2 [p] Yi = (X0 + i · X1) mod p Y0, Y1,...,Yp1 uniformity: 8i, a 2 [p] Pr[Yi = a] = 1 p 2-wise independence: 8i 6= j, a, b 2 [p] Pr[Yi = a ^ Yj = b] = 1 p2
uniform and independent Xo,XIE[p] fori∈[p] Y=(Xo+i·X1)modp uniformity:Y i,a∈pl PrYi=a =Pr[(Xo+i·X1)modp=a =>Pr[X1=j]Pr [(Xo+ij)modp=a] j∈[p] ∑Pr[Xo=(a-i)(mod p)] j∈[p] 1 p
for i 2 [p] Yi = (X0 + i · X1) mod p uniformity: 8i, a 2 [p] uniform and independent X0, X1 2 [p] Pr[Yi = a] = Pr [(X0 + i · X1) mod p = a] = X j2[p] Pr[X1 = j] · Pr [(X0 + ij) mod p = a] = 1 p X j2[p] Pr [X0 ⌘ (a ij) (mod p)] = 1 p