Randomized Complexity Classes; RP ² Let N be a polynomial-time precise NTM that runs in time p(n) and has 2 nondeterministic choices at each step. ² N is a polynomial Monte Carlo Turing machine for a language L if the following conditions hold: { If x 2 L, then at least half of the 2p(n) computation paths of N on x halt with \yes" where n = j x j. { If x 62 L, then all computation paths halt with \no." ² The class of all languages with polynomial Monte Carlo TMs is denoted RP (randomized polynomial time).a aAdleman and Manders (1977). °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 512 Comments on RP ² Nondeterministic steps can be seen as fair coin flips. ² There are no false positive answers. ² The probability of false negatives, 1 ¡ ², is at most 0.5. ² But any constant between 0 and 1 can replace 0.5. { By repeating the algorithm k = d¡ 1 e times, the log2 1¡² probability of false negatives becomes (1 ¡ ²)k · 0:5. ² In fact, ² can be arbitrarily close to 0 as long as it is of the order 1=q(n) for some polynomial q(n). { ¡ 1 = O( 1 ) = O(q(n)). log2 1¡² ² °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 513 Where RP Fits ² P ⊆ RP ⊆ NP. { A deterministic TM is like a Monte Carlo TM except that all the coin flips are ignored. { A Monte Carlo TM is an NTM with extra demands on the number of accepting paths. ² compositeness 2 RP; primes 2 coRP; primes 2 RP.a { In fact, primes 2 P.b ² RP [ coRP is an alternative \plausible" notion of e±cient computation. aAdleman and Huang (1987). bAgrawal, Kayal, and Saxena (2002). °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 514 ZPPa (Zero Probabilistic Polynomial) ² The class ZPP is de¯ned as RP \ coRP. ² A language in ZPP has two Monte Carlo algorithms, one with no false positives and the other with no false negatives. ² If we repeatedly run both Monte Carlo algorithms, eventually one de¯nite answer will come (unlike RP). { A positive answer from the one without false positives. { A negative answer from the one without false negatives. aGill (1977). °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 515 The ZPP Algorithm (Las Vegas) 1: fSuppose L 2 ZPP.g 2: fN1 has no false positives, and N2 has no false negatives.g 3: while true do 4: if N1(x) = \yes" then 5: return \yes"; 6: end if 7: if N2(x) = \no" then 8: return \no"; 9: end if 10: end while °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 516 ZPP (concluded) ² The expected running time for the correct answer to emerge is polynomial. { The probability that a run of the 2 algorithms does not generate a de¯nite answer is 0.5 (why?). { Let p(n) be the running time of each run of the while-loop. { The expected running time for a de¯nite answer is X1 0:5iip(n) = 2p(n): i=1 ² Essentially, ZPP is the class of problems that can be solved without errors in expected polynomial time. °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 517 Large Deviations ² Suppose you have a biased coin. ² One side has probability 0:5 + ² to appear and the other 0:5 ¡ ², for some 0 < ² < 0:5. ² But you do not know which is which. ² How to decide which side is the more likely side|with high con¯dence? ² Answer: Flip the coin many times and pick the side that appeared the most times. ² Question: Can you quantify the con¯dence? °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 518 The Cherno® Bounda Theorem 69 (Cherno® (1952)) Suppose x1; x2; : : : ; xn are independent random variables taking the values 1 and 0 Pn with probabilities p and 1 ¡ p, respectively. Let X = i=1 xi. Then for all 0 · θ · 1, 2 prob[ X ¸ (1 + θ) pn ] · e¡θ pn=3: ² The probability that the deviate of a binomial random variable from its expected value Xn E[ X ] = E[ xi ] = pn i=1 decreases exponentially with the deviation. aHerman Cherno® (1923{). The Cherno® bound is asymptotically optimal. °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 519 The Proof ² Let t be any positive real number. ² Then prob[ X ¸ (1 + θ) pn ] = prob[ etX ¸ et(1+θ) pn ]: ² Markov's inequality (p. 460) generalized to real-valued random variables says that £ ¤ prob etX ¸ kE[ etX ] · 1=k: ² With k = et(1+θ) pn=E[ etX ], we have prob[ X ¸ (1 + θ) pn ] · e¡t(1+θ) pnE[ etX ]: °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 520 The Proof (continued) Pn ² Because X = i=1 xi and xi's are independent, E[ etX ] = (E[ etx1 ])n = [ 1 + p(et ¡ 1) ]n: ² Substituting, we obtain prob[ X ¸ (1 + θ) pn ] · e¡t(1+θ) pn[ 1 + p(et ¡ 1) ]n t · e¡t(1+θ) pnepn(e ¡1) as (1 + a)n · ean for all a > 0. °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 521 The Proof (concluded) ² With the choice of t = ln(1 + θ), the above becomes prob[ X ¸ (1 + θ) pn ] · epn[ θ¡(1+θ) ln(1+θ)]: θ2 θ3 θ4 ² The exponent expands to ¡ 2 + 6 ¡ 12 + ¢ ¢ ¢ for 0 · θ · 1, which is less than µ ¶ µ ¶ θ2 θ3 1 θ 1 1 θ2 ¡ + · θ2 ¡ + · θ2 ¡ + = ¡ : 2 6 2 6 2 6 3 °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 522 Power of the Majority Rule 2 ¡ θ pn From prob[ X · (1 ¡ θ) pn ] · e 2 (prove it): Corollary 70 If p = (1=2) + ² for some 0 · ² · 1=2, then " # Xn ¡²2n=2 prob xi · n=2 · e : i=1 ² The textbook's corollary to Lemma 11.9 seems incorrect. ² Our original problem (p. 518) hence demands ¼ 1:4k=²2 independent coin flips to guarantee making an error with probability at most 2¡k with the majority rule. °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 523 BPPa (Bounded Probabilistic Polynomial) ² The class BPP contains all languages for which there is a precise polynomial-time NTM N such that: { If x 2 L, then at least 3=4 of the computation paths of N on x lead to \yes." { If x 62 L, then at least 3=4 of the computation paths of N on x lead to \no." ² N accepts or rejects by a clear majority. aGill (1977). °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 524 Magic 3=4? ² The number 3=4 bounds the probability of a right answer away from 1=2. ² Any constant strictly between 1=2 and 1 can be used without a®ecting the class BPP. ² In fact, 0.5 plus any inverse polynomial between 1=2 and 1, 1 0:5 + ; p(n) can be used. °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 525 The Majority Vote Algorithm Suppose L is decided by N by majority (1=2) + ². 1: for i = 1; 2;:::; 2k + 1 do 2: Run N on input x; 3: end for 4: if \yes" is the majority answer then 5: \yes"; 6: else 7: \no"; 8: end if °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 526 Analysis ² The running time remains polynomial, being 2k + 1 times N's running time. ² By Corollary 70 (p. 523), the probability of a false 2 answer is at most e¡² k. ² By taking k = d 2=²2 e, the error probability is at most 1=4. ² As with the RP case, ² can be any inverse polynomial, because k remains polynomial in n. °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 527 Probability Ampli¯cation for BPP ² Let m be the number of random bits used by a BPP algorithm. { By de¯nition, m is polynomial in n. ² With k = £(log m) in the majority vote algorithm, we can lower the error probability to, say, · (3m)¡1: °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 528 Aspects of BPP ² BPP is the most comprehensive yet plausible notion of e±cient computation. { If a problem is in BPP, we take it to mean that the problem can be solved e±ciently. { In this aspect, BPP has e®ectively replaced P. ² (RP [ coRP) ⊆ (NP [ coNP). ² (RP [ coRP) ⊆ BPP. ² Whether BPP ⊆ (NP [ coNP) is unknown. ² But it is unlikely that NP ⊆ BPP (p. 544). °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 529 coBPP ² The de¯nition of BPP is symmetric: acceptance by clear majority and rejection by clear majority. ² An algorithm for L 2 BPP becomes one for L¹ by reversing the answer. ² So L¹ 2 BPP and BPP ⊆ coBPP. ² Similarly coBPP ⊆ BPP. ² Hence BPP = coBPP. ² This approach does not work for RP. ² It did not work for NP either. °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 530 BPP and coBPP Ø\HVÙ ØQRÙ ØQRÙ Ø\HVÙ °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 531 \The Good, the Bad, and the Ugly" coNP NP ZPP coRP P RP BPP °c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 532 Circuit Complexity ² Circuit complexity is based on boolean circuits instead of Turing machines. ² A boolean circuit with n inputs computes a boolean function of n variables. ² By identifying true/1 with \yes" and false/0 with \no," a boolean circuit with n inputs accepts certain strings in f 0; 1 gn.