CSE 105 Theory of Computation

Transcription

1 CSE 105 Theory of Computation Professor Jeanne Ferrante 1

2 Today s Agenda P and NP (7.2, 7.3) Next class: Review Reminders and announcements: CAPE & TA evals are open: Please participate! Final Exam Sat Jun 4, 11:30 am 2:29 pm in WLH 2001 Seat Assignments: Will be out soon. Last chance to request Left-handed seat 11:59 pm today You can bring your own a 5 in by 8 in index card (both sides) to the exam (no magnifying aids!!) 2

3 Time Complexity Def. Let M be a deterministic TM that always halts. The running time or time complexity of M is the function f: N N where f(n) is the maximum number of steps of the TM M on any Input of length n. Let M be a nondeterministic TM that always halts. The running time or time complexity of M is the function f: N N where f(n) is the maximum number of steps of TM M on any branch of the computation on any Input of length n. 3

4 Running times of decider TM s Deterministic q0 Nondeterministic q0 f(n)= MAX number of steps with input length n q rej f(n) = MAX number of steps on any branch with input length n q rej q rej q acc q acc 4

5 Polynomial vs Exponential DTIME Polynomial running time: n c for some constant c Exponential running time: 2 nδ for some real number δ > 0 If we double the input size from n to 2n: Polynomial time: n c (2n) c = 2 c n c Exponential time: 2 nδ 2 (2n)δ = 2 2δ n δ» Quickly explodes in size = ( 2 nδ ) 2δ 5

6 Review: Big-O Notation f(n) = 4nlog(n) + 3n 2 + 3n + 10 is equal to which of the following in Big-O notation? A. O(n 3 ) B. O(nlog(n)) C. O(n 2 ) D. None or more than one of the above 6

7 Time Complexity Classes Def. Let t: N R + be a function. The deterministic time complexity class DTIME(t(n)) is the collection of languages L that are decidable by a O(t(n)) deterministic, single tape TM. The nondeterministic time complexity class NTIME(t(n)) is the collection of languages L that are decidable by a O(t(n)) nondeterministic, single tape TM. DTIME(n 2 ) DTIME(n 3 )? A. TRUE B. FALSE C. Don t Know 7

8 Here, model resources matter! t(n) time deterministic multitape TM O(t(n) 2 )time deterministic single tape TM t(n) time nondeterministic single tape TM 2 O(t(n)) time deterministic single tape TM (Where t(n) >= n) 8

9 Example TM: M1 M1 = On input w: 1. Scan the input from left to right to check whether it is of form {0,1}*#{0,1}*. If not, reject. If the input consists of only #, accept. 2. Return the head to the left hand end of tape. 3. Zig-zag across the tape, checking that the first unmarked symbol to the left of the # is the same as the corresponding unmarked symbol following #. If the corresponding symbols do not match, or there is no unmarked symbol left after the #, reject. Otherwise if the symbols match, mark them and continue. 4. If all symbols to the left of # have been marked, check for unmarked symbols after the #. If any unmarked symbols remain to the right of the #, reject; if none are found, accept. L(M1) is in A.DTIME(n) B. DTIME(n 2 ) C. DTIME(nlogn) D.None or more than 1 of the above 9

10 Most useful! CLASS P (DETERMINISTIC POLYNOMIAL TIME) 10

11 P = U DTIME(n k ) k P is the class of languages that can be decided in polynomial time on a deterministic, single-tape TM Encodings in P Need polynomial time encodings Graph encodings Analyze graph problems in number n of nodes; still polynomial in size of graph Can t use Nondeterminism Can use multiple tapes Brute force approach may not be enough 11

12 Examples in Class P PATH = {<G,s,t> G is a directed graph with n nodes and a directed path from node s to node t} RELPRIME = { <x,y> x and y are relatively prime} Use Euclidean Algorithm to show in P Every CFL {w w is generated by CFG G} Use Dynamic Programming to show in P Most sorting algorithms To be in P, avoid brute-force searches! 12

13 May be exponential time deterministically! CLASS NP (NONDETERMINISTIC POLYNOMIAL TIME) 13

14 Class NP = U NTIME(n k ) For many problems, only solution is equivalent to brute force search NP is the class of languages that can be decided in polynomial time on a non-deterministic TM For problems in NP, best deterministic algorithms are exponential time P NP! Outstanding open problem in CS: P = NP? Prevailing opinion: P NP k 14

15 Travelling Salesperson Problem in NP Input: Encodings of collection {C1, Cn} of cities, Distance k, Distance Matrix D(i,j): distance between Ci and Cj Problem: Is there a tour of all cities with total distance less than k? Deterministically enumerating all tours to check distance is exponential Best known algorithms are exponential e.g. dynamic programming In NP: Nondeterministically guess a tour including all cities Checking if a given tour has distance less than k can be done in deterministic polynomial time 15

16 Another Example in NP SAT = {<E> E is a satisfiable Boolean expression} A Boolean expression has operators AND, OR and NOT, and Boolean values which take the value 0 (false) or 1 (true) A Boolean expression is satisfiable if there is some assignment to its variables which makes the expression evaluate to 1 (true) x AND (y OR z) is satisfiable when x = 1, y = 1, z = 0 x AND (NOT x) is never satisfiable 16

17 Given <E> of size n: Decider for SAT 1. Nondeterministically guess an assignment of values to the variables in E 2. Check whether the assignment satisfies E; if so accept SAT is in A. DTIME(n c ), c a constant B. DTIME (2 nc ), c a constant C. NTIME(n c ), c a constant D. None or more than one of the above 17

18 Another Example In NP CLIQUE = {<G,k> G is an undirected graph with a clique of size k} A clique is a subgraph of G where every 2 nodes are connected by an edge Deterministically enumerating all subsets of k nodes and checking whether they have edges between each node is exponential Checking if a given set of edges forms a clique of size k can be done in deterministic polynomial time A. True B. False C. Don t Know 18

19 Summary Problems in P PATH Any CFL RELPRIME ADDITION MULTIPLICATION Problems in NP Any problem in P Traveling Salesperson SAT CLIQUE 19

20 DOES P = NP or P NP??? L? CF L Regular P NP?? Decidable P = Class of languages for which membership can be decided in deterministic polynomial time NP = Class of languages for which membership can be checked in deterministic polynomial time 20

21 Advances in P = NP? Stephen Cook won the Turing Award (1982) for his 1971 paper that laid the foundations for the theory of NP- Completeness. Intuitively: A problem X in NP is NP-complete if showing the problem X is in P guarantees all problems in NP are in P Cook-Levin TH: SAT is NP-complete So if SAT is in P, then P = NP! 21

22 Polynomial Time Computable Functions A function f : Σ* Σ* is polynomial time computable if some deterministic polynomial time TM M exists that given a string w in Σ* as input, halts with just f(w) on its tape. Note that f need not be 1-1 or onto M must always halt Can be any function a TM M can compute in deterministic polynomial time! TM M computes a polynomial time computable function: M = On input w, make a second copy of w immediately after the input, and halt A. True B. False C. Don t Know 22

23 Polynomial Time Reducibility: A p B Language A is polynomial time reducible to language B, written A P B, if there is a polynomial time computable function f : Σ* Σ* such that for any w in Σ* w є A IFF f(w) є B Turns question of whether a in A into question about whether f(a) in B in polynomial time Note that if w NOT in A, then there must be b = f(w) NOT in B If A p B and B p C THEN A p C IS A. TRUE B. FALSE C. DON T KNOW 23

24 Th. 7.31: If A p B AND B IS IN P, THEN A IS IN P. Proof: Let M be the polynomial time decider for B, and let f be the polynomial time reduction from A to B. We define a polynomial time decider N for A: N = On input w: 1. Compute f(w) 2. Run M on input f(w) and if M accepts, accept; if M rejects, reject Correctness: w in A IFF M accepts f(w) IFF N accepts w because f is a reduction and by construction of N. N runs in polynomial time because steps 1 and 2 both run in polynomial time (this follows in step 2 because f(w) is polynomial in the length of w, and then M runs in polynomial time on f(w)) 24

25 NP-Completeness A language B is NP-Complete if 1. B is in NP 2. Every A in NP is polynomial time reducible to B (A p B ) TH 7.35: If B is NP-complete and in P, then P = NP Proof: If B is NP-complete, then by definition, for every A in NP, A p B. But B is also in P, so by TH A is in P. 25

26 Many NP-Complete Problems SAT: Determining if Boolean Formula is satisfiable CLIQUE: Determining if undirected graph has clique of size k HAMPATH SUBSET-SUM But NONE yet shown to be in P! $1M Prize NEXT TIME: REVIEW 26