Review Big O and Algorithms Master Theorem Complexity P vs NP NP G. Carl Evans University of Illinois Summer 2013 NP Review Big O and Algorithms Master Theorem Complexity P vs NP Outline 1 Review Big O and Algorithms 2 Master Theorem 3 Complexity 4 P vs NP NP Review Big O and Algorithms Master Theorem Complexity P vs NP Review Algorithms Model algorithm complexity in terms of how much the cost increases as the input/parameter size increases In characterizing algorithm computational complexity, we care about Large inputs Dominant terms p 1 log n n n n log n n2 n3 2n 3n n! NP Review Big O and Algorithms Master Theorem Complexity P vs NP Example: multiplying large numbers Multiplying small numbers in binary 101 ×011 Complexity: m m Multiplying large numbers x = x12 + x0 and y = y12 + y0 m m xy = (x12 + x0)(y12 + y0) 2m m = x1y12 + (x0y1 + y0x1)2 + x0y0 Complexity: NP Review Big O and Algorithms Master Theorem Complexity P vs NP Example: multiplying large numbers m m Multiplying large numbers x = x12 + x0 and y = y12 + y0 m m xy = (x12 + x0)(y12 + y0) 2m m = x1y12 + (x0y1 + y0x1)2 + x0y0 Trick by Anatolii Karatsuba (x0y0 + y0x1) = (x1 + x0)(y1 + yo) − x1y1 − x0y0 Complexity: NP Review Big O and Algorithms Master Theorem Complexity P vs NP Review Big O f (n) is O(g(n)) if the dominant terms in f (n) are equivalent or dominated by the dominant terms in g(n) f (n) is Ω(g(n)) if the dominant terms in f (n) are equivalent or dominate the dominant terms in g(n) f (n) is Θ(g(n)) if the dominant terms in f (n) are equivalent the dominant terms in g(n) NP Review Big O and Algorithms Master Theorem Complexity P vs NP Master Theorem n T (n) = aT + f (n) where a ≥ 1; b > 1 b NP Review Big O and Algorithms Master Theorem Complexity P vs NP Leaf term dominates n T (n) = aT + f (n) where a ≥ 1; b > 1 b c logba If f (n) = Θ(n ) with c < logb a, then T (n) = Θ(n ) n Example: T (n) = 4T 2 + O(n) NP Review Big O and Algorithms Master Theorem Complexity P vs NP Balanced expansion n T (n) = aT + f (n) where a ≥ 1; b > 1 b c k If f (n) = Θ(n log n) with k ≥ 0; c = logb a, then T (n) = Θ(nc logk+1 n) n Example: T (n) = 2T 2 + O(n) NP Review Big O and Algorithms Master Theorem Complexity P vs NP Slow expansion n T (n) = aT + f (n) where a ≥ 1; b > 1 b c c If f (n) = Θ(n ) with c > logb a, then T (n) = Θ(n ) n 2 Example: T (n) = 2T 2 + O(n ) NP Review Big O and Algorithms Master Theorem Complexity P vs NP Computational Complexity ALL AH RE R PR ELEMENTARY NEEE P-Sel NEEXP EEE EEXP EESPACE MIP_{EXP} EXPSPACE IP_{EXP} PEXP EXPH SEH NEXP^{NP} NEE EXP^{NP} AM_{EXP} NEXP/poly MA_{EXP} BPEE EXP/poly NEXP BPEXP EE +EXP EXP QRG ESPACE RG Almost-PSPACE QPSPACE PSPACE Coh PL_{infty} CH MP^{#P} AvgE P^{PP} EH PP/poly P^{#P[1]} BP.PP MP PH SF_4 Sigma_3P AmpMP QMIP QMIP_{le} QMIP_{ne} SQG Delta_3P BQP/qpoly MIP* QIP MIP RG[1] Sigma_2P BPP^{NP} Complexity Classes BQP/mpoly NE/poly XOR-MIP*[2,1] QMA(2) QIP[2] IP RP^{NP} frIP BQP/poly NP/poly QSZK QAM AM[polylog] compIP ZPP^{NP} QS_2P PP SF_3 (NP-cap-coNP)/poly BQP/qlog DQP CZK AM S_2P P^{QMA} A_0PP SF_2 P/poly BQP/mlog N.NISZK SZK QMA Delta_2P BPP_{path} APP Check BPP//log BQP/log YQP NIQSZK NISZK_h QCMA SBP MA_E P^{NP[log^2]} AWPP C_=P IC[log,poly] BPP/rlog BQP NE NISZK MA WAPP BPE P^{NP[log]} WPP Inherent Complexity BPP/mlog HeurBPP YPP N.BPP PZK AmpP-BQP RPE BPQP UE BH ModP NP/log BPP/log AVBPP FH TreeBQP ZPE BH_2 LWPP Mod_5P +P Mod_3P Nearly-P NP/one P/log BPP E US RP^{PromiseUP} SPP NT* AvgP NP RQP SUBEXP P^{FewP} NT YP compNP RBQP ZQP QP Few EP ZBQP RP EQP betaP QPLIN FewP UAP QNC ZPP Q beta_2P UP P-Close RNC HalfP NLINSPACE P NLIN polyL NC LIN SC QCFL AL NC^2 +L/poly +SAC^1 L^{DET} AC^1 NL/poly +L SAC^1 PL CSL L/poly C_=L 1NAuxPDA^p GCSL NL SPL CFL BPL LFew RL DCFL FewL LogFew FewUL FOLL UL R_HL QNC^1 L TC^0/poly NC^1 PBP QACC^0 TC^0 REG PT_1 QAC^0 MAC^0 ACC^0 (k>=5)-PBP PL_1 AC^0/poly AC^0[2] MAJORITY 4-PBP SPARSE QNC_f^0 AC^0 3-PBP TALLY QNC^0 SAC^0 +SAC^0 2-PBP NC^0 PARITY NONE https://www.math.ucdavis.edu/~greg/zoology/intro.html NP Review Big O and Algorithms Master Theorem Complexity P vs NP P and NP A problem is in the class P if a polynomial-time solution exists A problem is in the class NP (non-deterministic polynomial time) if if a solution can be checked in polynomial time. NP Review Big O and Algorithms Master Theorem Complexity P vs NP Examples 3-SAT Boolean satisfiability: Determine if any assignment of n boolean variables can satisfy a set of logical expressions 3-SAT: The formula must be in CNF with exactly 3 literals per clause. NP Review Big O and Algorithms Master Theorem Complexity P vs NP Examples Sorting Given a array of integers can you sort the set. NP Review Big O and Algorithms Master Theorem Complexity P vs NP Examples Graph Coloring Determining if the graph is n-colorable Determining if the graph is not (n+1)-colorable NP Review Big O and Algorithms Master Theorem Complexity P vs NP CIRCUIT-SAT and Cook-Levin theorem The circuit satisfiability problem (CIRCUIT-SAT) asks the question if there is a set of inputs to a boolean circuit such that the output is true. The Cook-Levin theorem says that this problem is NP-complete. A problem is NP-complete if it is in NP and if there is a polynomial time solution to the problem P=NP NP Review Big O and Algorithms Master Theorem Complexity P vs NP P = NP? Conceptually this is the question \Is it easer to check a problem then to find the solution?" Yes P 6= NP No P = NP Proof is worth $1,000,000 (Millennium Prize Problem) NP Review Big O and Algorithms Master Theorem Complexity P vs NP How it all fits together P NP NP-complete NP-Hard NP Review Big O and Algorithms Master Theorem Complexity P vs NP Things to remember Be able to analyze code for computational cost Tools: finding loops and recursive calls, using recursion trees Sometimes need to know inner-workings of a library to determine Be able to convert to big-O or big-Θ and be familiar with basic complexity terms Problems in NP can be checked in polynomial time but probably not solved in polynomial time P = NP is an open problem but most think P 6= NP. NP.