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Intro to Theory of Computation

L ECTURE 28 Last time Examples of NP-complete problems Today Space complexity. Intro to Theory of Computation. CS 464. Sofya Raskhodnikova. Homework 11 due Homework 12 out Extra credit programming assignment. An encyclopedia of NP-complete problems. From Garey and Jonhson.

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Intro to Theory of Computation

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  1. LECTURE28 • Last time • Examples of NP-complete problems • Today • Space complexity Intro to Theory of Computation CS 464 Sofya Raskhodnikova Homework 11 due Homework 12 out Extra credit programming assignment

  2. An encyclopedia of NP-complete problems

  3. From Garey and Jonhson ``I thought about it for weeks, but I can’t come up with an efficient algorithm. I guess I’m just too dumb.’’ ``We need an efficient algorithm that constructs a design of iThingythat meets the maximum # of requirements at lowest cost.’’

  4. I-clicker question (frequency: AC) • 3SAT • is NP-complete • (1) and (2) are both true • (1) and (2) are both false • (1) is true and (2) is false • (2) is true and (1) is false • At least one of (1) and (2) is an open question

  5. Space analysis If M is a TM and then “M runs in space ” means for every input of length , M on uses at most tape cells. S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson If M is a nondterministicTM that halts on all inputs then is the maximum number of cells M uses on any input of length .

  6. Space complexity classes SPACE() is a class of languages. SPACE()means that some 1-tape TM M that runs in space O() decides A.

  7. Prove: SAT SPACE() M = `` On input , where is a Boolean formula, with variables : For each truth assignment to Evaluate on that truth assignment. Accept if ever evaluates to 1. O.w. reject.’’ If is the input length, M uses space

  8. Space complexity classes NSPACE() is a class of languages. NSPACE()means that some 1-tape nondeterministic TM M that runs in space O() decides A.

  9. Prove: NSPACE() N = `` On input , where is an NFA: Place marker on the start state of M. Repeat times where is the # of states of M: Nondeterministically select . Adjust the markers to simulate M reading Accept if at any point none of the markers are on an accept state. O.w. reject.’’ ={ is an NFA and L If is the input length, N uses nondeterministic space

  10. Savitch’s theorem Theorem. Let be a function, where NSPACE(). Proof: • Let N be an NTM deciding a language A in space. • We give a deterministic TM M deciding A. • More general problem: • Given configurations of N and integer t, decide whether N can go from to in steps on some nondeterministic path. • Procedure CANYIELD

  11. Savitch’s theorem CANYIELD = `` On input : If acceptif or yields in one transition. O.w. reject. If then configs of N with cells: Run CANYIELD(). Run CANYIELD(). If both runs accept, accept Reject.’’ Theorem. Let be a function, where NSPACE(). Proof:

  12. The class PSPACE • PSPACE is the class of languages decidable in polynomial space on a deterministic TM: NSPACE – the same, but for NTMs. By Savitch’s Theorem, P NP PSPACE = NPSPACE EXPTIME

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