1 / 12

Space Complexity in Theory of Computation CS464

Discussing NP-complete problems from an encyclopedia, efficient algorithms, space complexity classes, and the impact of Space analysis on algorithms led by Sofya Raskhodnikova.

bsharpe
Download Presentation

Space Complexity in Theory of Computation CS464

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related