Randomized Algorithm pick a uniform random rES; SCF check whether fr)=0; S1=2d f∫丰0 d Prf)=o】≤ 1 三 -2 Fundamental Theorem of Algebra: A degree d polynomial has at most d roots
A degree d polynomial has at most d roots. Fundamental Theorem of Algebra: Randomized Algorithm pick a uniform random r ; check whether f(r) = 0 ; ∈S S F if f 0 Pr[f(r) = 0] |S| d |S| = 2d = 1 2
Checking ldentity 北京 database 1 Are they identical? 上海 database 2
Checking Identity database 1 database 2 Are they identical? 北京 上海
Communication Complexity (Yao1979) f(a,b) #of bits communicated a b Han Meimei Li Lei EQ:{0,1}m×{0,1}m→{0,1} aa={ 1 a=b a夫b
Communication Complexity (Yao 1979) Han Meimei Li Lei EQ : {0, 1}n × {0, 1}n → {0, 1} # of bits communicated a b f(a, b) EQ(a, b) = ! 1 a = b 0 a ̸= b
Communication Complexity (Yao 1979) f(a,b) #of bits communicated a b Han Meimei Li Lei EQ:{0,1}m×{0,1}m→{0,1} Theorem(Yao,1979) There is no deterministic communication protocol solving EQ with less than n bits in the worst-case
Communication Complexity (Yao 1979) Han Meimei Li Lei EQ : {0, 1}n × {0, 1}n → {0, 1} # of bits communicated a b f(a, b) EQ(a, b) = ! 1 a = b There is no deterministic communication protocol 0 a ̸= b solving EQ with less than n bits in the worst-case. Theorem (Yao, 1979)
Communication Complexity m-1 m-1 f=】 r)=8(r)? 2=0 2=0 r,8(r) a∈{0,1}m b∈{0,1}n Han Meimei Li Lei by PIT: pick uniform 1 random r∈2n one-sided error≤ -2 of bit communicated: too large!
Communication Complexity Han Meimei Li Lei a ∈{0, 1} b n ∈{0, 1}n f = n 1 i=0 aixi pick uniform random r ∈[2n] r, g(r) f(r)=g(r) ? one-sided error 1 2 by PIT: # of bit communicated: too large! g = n X1 i=0 bixi