Could we potentially place A in a smaller complexity class if we consider other computational models?
|
|
- Alexis Tate
- 5 years ago
- Views:
Transcription
1 Introduction to Complexity Theory Big O Notation Review Linear function: r(n) =O(n). Polynomial function: r(n) =2 O(1) Exponential function: r(n) =2 no(1) Logarithmic function: r(n) = O(log n) Poly-log function: r(n) =log O(1) n Definition 1 (TIME) Let t : ℵ ℵ. Define the time complexity class, TIME(t(n)) to be TIME(t(n)) = {L DTM M which decides L in time O(t(n))}. Definition 2 (NTIME) Let t : ℵ ℵ. Define the time complexity class, NTIME(t(n)) to be NTIME(t(n)) = {L NDTM M which decides L in time O(t(n))}. Example 1 Consider the language A = {0 k 1 k k 0}. Is A TIME(n 2 )? Is A TIME(n log n)? Is A TIME(n)? Could we potentially place A in a smaller complexity class if we consider other computational models? Theorem 1 If t(n) n, then every t(n) time multitape Turing machine has an equivalent O(t 2 (n)) time single-tape turing machine. see Theorem 7.8 in Sipser (pg. 232) Theorem 2 If t(n) n, then every t(n) time RAM machine has an equivalent O(t 3 (n)) time multi-tape turing machine. optional exercise Conclusion: Linear time is model specific; polynomical time is model indepedent. Definition 3 (The Class P ) P = k TIME(n k ) Definition 4 (The Class NP)
2 NP = k NTIME(n k ) Equivalent Definition of NP NP = {L L has a polynomial time verifier }. A polynomial time verifier for a language A is an algorithm, V,whereA = {w V acepts <w,c> for some string c}. Example 2 Let RELP RIME = {< x,y > gcd(x, y) = 1}. Is RELP RIME NP? Is RELP RIME P? Example 3 PATH = {<G,s,t > G has a directed path from s to t}. IsPATH P? Construct a deterministic polynomial time Turing machine, M, that decides P AT H. M = On Input <G,s,t > where G is a directed graph with nodes s and t. 1. Place a mark on node s. 2. While there exists an unmarked node do 3. Search for all edges (a, b) wherea is marked and b unmarked, and mark such b. 4. If t is marked, accept; otherwise, reject. To show an algorithm runs in polynomial, one must show that each step is executed only a polynomial number of steps as well as each steps executes in polynomial time. Example 4 SAT = {< φ> φ is a satisfiable boolean formula }. IsSAT NP? Construct a nondeterministic polynomial time Turing machine, M, as follows: M = On Input <φ>: 1. Nondeterministically select an assignment of the variables, x. 2. Teset where x satisfies φ. 3. If thest test passes, accept; otherwise, reject. Example 5 VALUE = {< φ,x > φ(x) where φ is a boolean formula and x an assignment of variables }. IsVALUE P? Example 6 RATIONALROOT = {< p> q Q such that p(q) =0where p(x) =a 0 + a 1 x a k x k is a polynomial of degree k where i =1...k, a i ℵ. Is RATIONALROOT P? Construct a nondeterministic polynomial time Turing machine, M, for RATIONALROOT: M = On Input <p>where p(x) =a 0 + a 1 x a k x k, a i ℵ.
3 1. Nondeterministically select q Q. 2. Evaluate p(q). 3. If p(q) = 0, accept; otherwise, reject. Does this run in polynomial time? Cearly, to evaulate p(q) is polynomial in the length of q since multipication can be computed in polynomial time. How big is q though? From the Rational Root Theorem in algebra, any rational root, x, ofp(x) =0isoftheformx = s t where s a 0 and t a k. Since a 0 =log(a 0 ) <nand a k =log(a k ) <n, q = O(n). Thus, step 1 and 2 can be computed in polynomial steps. Example 7 Let SUBSETSUM = {< S,t> {y 1,y 2,...,y n } = Y S = {s 1,s 2,...,s n } such that i y i = t} Construct a nondeterministic polynomial time Turing machine, M, as follows: M = On Input <S,t>: 1. Nondeterministically select a subset c of the numbers in S. 2. Test whether c is a collection of numbers that sum to t. 3. If thest test passes, accept; otherwise, reject. Theorem 3 P NP A O(t(n)) DTM has an equivalent O(t(n)) NDTM. Definition 5 (The Class EXPTIME) EXPTIME = k TIME(2 nk ) Definition 6 (The Class NEXPTIME) NEXPTIME = k NTIME(2 nk ) Relations: P NP EXPTIME NEXPTIME Definition 7 (Space Complexity Classes) SPACE(t(n)) = {L DTM M which decides L in space O(t(n))}. NSPACE(t(n)) = {L NDTM M which decides L in space O(t(n))}. PSPACE = k SPACE(n k ) NPSPACE = k NSPACE(n k )
4 Definition 8 (Logarithmic Space Classes) L = SPACE(log n) NL = NSPACE(log n) How can our definition of a turing machine use only logarithmic space since the input tape uses linear space? We introduce a new turing machine with two tapes: a read-only input tape and a read/write tape. Example 8 PATH NL Relations between time and space: Theorem 4 TIME(f(n)) SPACE(f(n)) A deterministic turing machine that decides membership in O(f(n)) steps can use at most O(f(n)) space since one TM step requires can only write to one memory cell. Theorem 5 If f(n) log(n), SPACE(f(n)) TIME(k f(n) ) Define a configuration as a snap shot of a turing machine including the position of the heads, the state of the control unit, and the contents of all cells on tape. If our turing machine is restricted to O(f(n)) space, the work tape head can be in only O(f(n)) locations and the input tape head canbeinonlyo(n). The control unit can be in any of c positions were c is a constant representing the number of states in the fsm. If we have k characters that can be written to a cell, then there are O(k f(n) ) possibilities for the content of the tape. Thus, there are O(cnf(n) k f(n) )=O(k f(n) ) configurations. If the turing machine is to halt, there will be no duplicate configuration and thus it must run in O(k f(n) ). Corollary 1 L P Corollary 2 P PSPACE Corollary 3 PSPACE EXPTIME Complements of complexity classes Definition 9 (coc) The complement of a complexity class of decision problems C, denoted coc, is the set of decision problems that are complemnt of decision problems of C. Example 9 cosat = {< φ> φ is NOT a satisfiable boolean formula }. Is cosat NP? Is cosat EXPTIME? IscoSAT PSPACE? Theorem 6 If C is a deterministic time or space complexity class, then C=coC.
5 Deterministic turing machines are closed under complementation. Is NP = conp? Theorem 7 NP conp {} Let PRIMALITY = {< n> n is prime}. WeshowPRIMALITY NP conp. not quite done. Lemma 1 An integer p>2 is prime iff there is an integer 1 <r<psuch that r p 1 1(modp) and q such that q p 1, r p 1 q 1(modp). not quite done
Complexity Theory 112. Space Complexity
Complexity Theory 112 Space Complexity We ve already seen the definition SPACE(f(n)): the languages accepted by a machine which uses O(f(n)) tape cells on inputs of length n. Counting only work space NSPACE(f(n))
More informationTime Complexity. CS60001: Foundations of Computing Science
Time Complexity CS60001: Foundations of Computing Science Professor, Dept. of Computer Sc. & Engg., Measuring Complexity Definition Let M be a deterministic Turing machine that halts on all inputs. The
More informationLecture 21: Space Complexity (The Final Exam Frontier?)
6.045 Lecture 21: Space Complexity (The Final Exam Frontier?) 1 conp NP MIN-FORMULA conp P NP FIRST-SAT TAUT P FACTORING SAT NP NP NP 2 VOTE VOTE VOTE For your favorite course on automata and complexity
More informationChapter 7: Time Complexity
Chapter 7: Time Complexity 1 Time complexity Let M be a deterministic Turing machine that halts on all inputs. The running time or time complexity of M is the function f: N N, where f(n) is the maximum
More informationComputational Complexity
CS311 Computational Structures Computational Complexity Lecture 16 Andrew P. Black Andrew Tolmach 1 So, itʼs computable! But at what cost? Some things that are computable in principle are in practice intractable
More informationComputability and Complexity CISC462, Fall 2018, Space complexity 1
Computability and Complexity CISC462, Fall 2018, Space complexity 1 SPACE COMPLEXITY This material is covered in Chapter 8 of the textbook. For simplicity, we define the space used by a Turing machine
More informationCS154, Lecture 17: conp, Oracles again, Space Complexity
CS154, Lecture 17: conp, Oracles again, Space Complexity Definition: conp = { L L NP } What does a conp computation look like? In NP algorithms, we can use a guess instruction in pseudocode: Guess string
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Fall 2016 http://cseweb.ucsd.edu/classes/fa16/cse105-abc/ Logistics HW7 due tonight Thursday's class: REVIEW Final exam on Thursday Dec 8, 8am-11am, LEDDN AUD Note card allowed
More informationCISC 4090 Theory of Computation
CISC 4090 Theory of Computation Complexity Professor Daniel Leeds dleeds@fordham.edu JMH 332 Computability Are we guaranteed to get an answer? Complexity How long do we have to wait for an answer? (Ch7)
More informationLecture 16: Time Complexity and P vs NP
6.045 Lecture 16: Time Complexity and P vs NP 1 Time-Bounded Complexity Classes Definition: TIME(t(n)) = { L there is a Turing machine M with time complexity O(t(n)) so that L = L(M) } = { L L is a language
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION "Winter" 2018 http://cseweb.ucsd.edu/classes/wi18/cse105-ab/ Today's learning goals Sipser Ch 7 Distinguish between computability and complexity Articulate motivation questions
More informationReview of Basic Computational Complexity
Lecture 1 Review of Basic Computational Complexity March 30, 2004 Lecturer: Paul Beame Notes: Daniel Lowd 1.1 Preliminaries 1.1.1 Texts There is no one textbook that covers everything in this course. Some
More informationCSCI 1590 Intro to Computational Complexity
CSCI 1590 Intro to Computational Complexity Space Complexity John E. Savage Brown University February 11, 2008 John E. Savage (Brown University) CSCI 1590 Intro to Computational Complexity February 11,
More informationTheory of Computation Time Complexity
Theory of Computation Time Complexity Bow-Yaw Wang Academia Sinica Spring 2012 Bow-Yaw Wang (Academia Sinica) Time Complexity Spring 2012 1 / 59 Time for Deciding a Language Let us consider A = {0 n 1
More informationDefinition: conp = { L L NP } What does a conp computation look like?
Space Complexity 28 Definition: conp = { L L NP } What does a conp computation look like? In NP algorithms, we can use a guess instruction in pseudocode: Guess string y of x k length and the machine accepts
More informationComplexity. Complexity Theory Lecture 3. Decidability and Complexity. Complexity Classes
Complexity Theory 1 Complexity Theory 2 Complexity Theory Lecture 3 Complexity For any function f : IN IN, we say that a language L is in TIME(f(n)) if there is a machine M = (Q, Σ, s, δ), such that: L
More informationCS5371 Theory of Computation. Lecture 23: Complexity VIII (Space Complexity)
CS5371 Theory of Computation Lecture 23: Complexity VIII (Space Complexity) Objectives Introduce Space Complexity Savitch s Theorem The class PSPACE Space Complexity Definition [for DTM]: Let M be a DTM
More informationP vs. NP Classes. Prof. (Dr.) K.R. Chowdhary.
P vs. NP Classes Prof. (Dr.) K.R. Chowdhary Email: kr.chowdhary@iitj.ac.in Formerly at department of Computer Science and Engineering MBM Engineering College, Jodhpur Monday 10 th April, 2017 kr chowdhary
More informationconp, Oracles, Space Complexity
conp, Oracles, Space Complexity 1 What s next? A few possibilities CS161 Design and Analysis of Algorithms CS254 Complexity Theory (next year) CS354 Topics in Circuit Complexity For your favorite course
More informationTime Complexity. Definition. Let t : n n be a function. NTIME(t(n)) = {L L is a language decidable by a O(t(n)) deterministic TM}
Time Complexity Definition Let t : n n be a function. TIME(t(n)) = {L L is a language decidable by a O(t(n)) deterministic TM} NTIME(t(n)) = {L L is a language decidable by a O(t(n)) non-deterministic
More informationCOMPLEXITY THEORY. PSPACE = SPACE(n k ) k N. NPSPACE = NSPACE(n k ) 10/30/2012. Space Complexity: Savitch's Theorem and PSPACE- Completeness
15-455 COMPLEXITY THEORY Space Complexity: Savitch's Theorem and PSPACE- Completeness October 30,2012 MEASURING SPACE COMPLEXITY FINITE STATE CONTROL I N P U T 1 2 3 4 5 6 7 8 9 10 We measure space complexity
More informationTime Complexity (1) CSCI Spring Original Slides were written by Dr. Frederick W Maier. CSCI 2670 Time Complexity (1)
Time Complexity (1) CSCI 2670 Original Slides were written by Dr. Frederick W Maier Spring 2014 Time Complexity So far we ve dealt with determining whether or not a problem is decidable. But even if it
More informationCSCI 1590 Intro to Computational Complexity
CSCI 1590 Intro to Computational Complexity Complement Classes and the Polynomial Time Hierarchy John E. Savage Brown University February 9, 2009 John E. Savage (Brown University) CSCI 1590 Intro to Computational
More informationComputational complexity
COMS11700 Computational complexity Department of Computer Science, University of Bristol Bristol, UK 2 May 2014 COMS11700: Computational complexity Slide 1/23 Introduction If we can prove that a language
More informationComputability Theory
CS:4330 Theory of Computation Spring 2018 Computability Theory The class NP Haniel Barbosa Readings for this lecture Chapter 7 of [Sipser 1996], 3rd edition. Section 7.3. Question Why are we unsuccessful
More informationsatisfiability (sat) Satisfiability unsatisfiability (unsat or sat complement) and validity Any Expression φ Can Be Converted into CNFs and DNFs
Any Expression φ Can Be Converted into CNFs and DNFs φ = x j : This is trivially true. φ = φ 1 and a CNF is sought: Turn φ 1 into a DNF and apply de Morgan s laws to make a CNF for φ. φ = φ 1 and a DNF
More informationComputational Models Lecture 11, Spring 2009
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 1 Computational Models Lecture 11, Spring 2009 Deterministic Time Classes NonDeterministic Time Classes
More informationCS154, Lecture 13: P vs NP
CS154, Lecture 13: P vs NP The EXTENDED Church-Turing Thesis Everyone s Intuitive Notion of Efficient Algorithms Polynomial-Time Turing Machines More generally: TM can simulate every reasonable model of
More informationThe space complexity of a standard Turing machine. The space complexity of a nondeterministic Turing machine
298 8. Space Complexity The space complexity of a standard Turing machine M = (Q,,,, q 0, accept, reject) on input w is space M (w) = max{ uav : q 0 w M u q av, q Q, u, a, v * } The space complexity of
More informationCS154, Lecture 13: P vs NP
CS154, Lecture 13: P vs NP The EXTENDED Church-Turing Thesis Everyone s Intuitive Notion of Efficient Algorithms Polynomial-Time Turing Machines More generally: TM can simulate every reasonable model of
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2016 http://cseweb.ucsd.edu/classes/sp16/cse105-ab/ Today's learning goals Sipser Ch 7.2, 7.3 Distinguish between polynomial and exponential DTIME Define nondeterministic
More informationLecture 19: Finish NP-Completeness, conp and Friends
6.045 Lecture 19: Finish NP-Completeness, conp and Friends 1 Polynomial Time Reducibility f : Σ* Σ* is a polynomial time computable function if there is a poly-time Turing machine M that on every input
More informationFinish K-Complexity, Start Time Complexity
6.045 Finish K-Complexity, Start Time Complexity 1 Kolmogorov Complexity Definition: The shortest description of x, denoted as d(x), is the lexicographically shortest string such that M(w) halts
More informationCSE 105 Theory of Computation
CSE 105 Theory of Computation http://www.jflap.org/jflaptmp/ Professor Jeanne Ferrante 1 Today s Agenda P and NP (7.2, 7.3) Next class: Review Reminders and announcements: CAPE & TA evals are open: Please
More informationDe Morgan s a Laws. De Morgan s laws say that. (φ 1 φ 2 ) = φ 1 φ 2, (φ 1 φ 2 ) = φ 1 φ 2.
De Morgan s a Laws De Morgan s laws say that (φ 1 φ 2 ) = φ 1 φ 2, (φ 1 φ 2 ) = φ 1 φ 2. Here is a proof for the first law: φ 1 φ 2 (φ 1 φ 2 ) φ 1 φ 2 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 a Augustus DeMorgan
More informationTIME COMPLEXITY AND POLYNOMIAL TIME; NON DETERMINISTIC TURING MACHINES AND NP. THURSDAY Mar 20
TIME COMPLEXITY AND POLYNOMIAL TIME; NON DETERMINISTIC TURING MACHINES AND NP THURSDAY Mar 20 COMPLEXITY THEORY Studies what can and can t be computed under limited resources such as time, space, etc Today:
More informationThe Polynomial Hierarchy
The Polynomial Hierarchy Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach. Ahto Buldas Ahto.Buldas@ut.ee Motivation..synthesizing circuits is exceedingly difficulty. It is even
More informationCS151 Complexity Theory. Lecture 1 April 3, 2017
CS151 Complexity Theory Lecture 1 April 3, 2017 Complexity Theory Classify problems according to the computational resources required running time storage space parallelism randomness rounds of interaction,
More informationCS5371 Theory of Computation. Lecture 23: Complexity VIII (Space Complexity)
CS5371 Theory of Computation Lecture 23: Complexity VIII (Space Complexity) Objectives Introduce Space Complexity Savitch s Theorem The class PSPACE Space Complexity Definition [for DTM]: Let M be a DTM
More informationFORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY. FLAC (15-453) Spring l. Blum TIME COMPLEXITY AND POLYNOMIAL TIME;
15-453 TIME COMPLEXITY AND POLYNOMIAL TIME; FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY NON DETERMINISTIC TURING MACHINES AND NP THURSDAY Mar 20 COMPLEXITY THEORY Studies what can and can t be computed
More informationPolynomial Time Computation. Topics in Logic and Complexity Handout 2. Nondeterministic Polynomial Time. Succinct Certificates.
1 2 Topics in Logic and Complexity Handout 2 Anuj Dawar MPhil Advanced Computer Science, Lent 2010 Polynomial Time Computation P = TIME(n k ) k=1 The class of languages decidable in polynomial time. The
More informationTheory of Computation. Ch.8 Space Complexity. wherein all branches of its computation halt on all
Definition 8.1 Let M be a deterministic Turing machine, DTM, that halts on all inputs. The space complexity of M is the function f : N N, where f(n) is the maximum number of tape cells that M scans on
More informationCSE 135: Introduction to Theory of Computation NP-completeness
CSE 135: Introduction to Theory of Computation NP-completeness Sungjin Im University of California, Merced 04-15-2014 Significance of the question if P? NP Perhaps you have heard of (some of) the following
More informationChapter 1 - Time and Space Complexity. deterministic and non-deterministic Turing machine time and space complexity classes P, NP, PSPACE, NPSPACE
Chapter 1 - Time and Space Complexity deterministic and non-deterministic Turing machine time and space complexity classes P, NP, PSPACE, NPSPACE 1 / 41 Deterministic Turing machines Definition 1.1 A (deterministic
More informationCS5371 Theory of Computation. Lecture 24: Complexity IX (PSPACE-complete, L, NL, NL-complete)
CS5371 Theory of Computation Lecture 24: Complexity IX (PSPACE-complete, L, NL, NL-complete) Objectives PSPACE-complete languages + Examples The classes L and NL NL-complete languages + Examples PSPACE-complete
More informationTheory of Computation CS3102 Spring 2015 A tale of computers, math, problem solving, life, love and tragic death
Theory of Computation CS3102 Spring 2015 A tale of computers, math, problem solving, life, love and tragic death Robbie Hott www.cs.virginia.edu/~jh2jf Department of Computer Science University of Virginia
More information22c:135 Theory of Computation. Analyzing an Algorithm. Simplifying Conventions. Example computation. How much time does M1 take to decide A?
Example computation Consider the decidable language A = {0 n 1 n n 0} and the following TM M1 deciding A: M1 = "On input string w: 1. Scan across the tape and reject if a 0 appears after a 1 2. Repeat
More informationSpace Complexity. Huan Long. Shanghai Jiao Tong University
Space Complexity Huan Long Shanghai Jiao Tong University Acknowledgements Part of the slides comes from a similar course in Fudan University given by Prof. Yijia Chen. http://basics.sjtu.edu.cn/ chen/
More informationSpace Complexity. The space complexity of a program is how much memory it uses.
Space Complexity The space complexity of a program is how much memory it uses. Measuring Space When we compute the space used by a TM, we do not count the input (think of input as readonly). We say that
More informationComplexity Theory. Knowledge Representation and Reasoning. November 2, 2005
Complexity Theory Knowledge Representation and Reasoning November 2, 2005 (Knowledge Representation and Reasoning) Complexity Theory November 2, 2005 1 / 22 Outline Motivation Reminder: Basic Notions Algorithms
More informationLecture 3: Reductions and Completeness
CS 710: Complexity Theory 9/13/2011 Lecture 3: Reductions and Completeness Instructor: Dieter van Melkebeek Scribe: Brian Nixon Last lecture we introduced the notion of a universal Turing machine for deterministic
More informationComputability and Complexity
Computability and Complexity Lecture 10 More examples of problems in P Closure properties of the class P The class NP given by Jiri Srba Lecture 10 Computability and Complexity 1/12 Example: Relatively
More informationLecture 18: PCP Theorem and Hardness of Approximation I
Lecture 18: and Hardness of Approximation I Arijit Bishnu 26.04.2010 Outline 1 Introduction to Approximation Algorithm 2 Outline 1 Introduction to Approximation Algorithm 2 Approximation Algorithm Approximation
More informationHierarchy theorems. Evgenij Thorstensen V18. Evgenij Thorstensen Hierarchy theorems V18 1 / 18
Hierarchy theorems Evgenij Thorstensen V18 Evgenij Thorstensen Hierarchy theorems V18 1 / 18 Comparing functions To prove results like TIME(f(n)) TIME(g(n)), we need a stronger notion of one function growing
More informationLecture 3: Nondeterminism, NP, and NP-completeness
CSE 531: Computational Complexity I Winter 2016 Lecture 3: Nondeterminism, NP, and NP-completeness January 13, 2016 Lecturer: Paul Beame Scribe: Paul Beame 1 Nondeterminism and NP Recall the definition
More informationCMPT 710/407 - Complexity Theory Lecture 4: Complexity Classes, Completeness, Linear Speedup, and Hierarchy Theorems
CMPT 710/407 - Complexity Theory Lecture 4: Complexity Classes, Completeness, Linear Speedup, and Hierarchy Theorems Valentine Kabanets September 13, 2007 1 Complexity Classes Unless explicitly stated,
More informationAlgorithms & Complexity II Avarikioti Zeta
Algorithms & Complexity II Avarikioti Zeta March 17, 2014 Alternating Computation Alternation: generalizes non-determinism, where each state is either existential or universal : Old: existential states
More informationSpace Complexity. Master Informatique. Université Paris 5 René Descartes. Master Info. Complexity Space 1/26
Space Complexity Master Informatique Université Paris 5 René Descartes 2016 Master Info. Complexity Space 1/26 Outline Basics on Space Complexity Main Space Complexity Classes Deterministic and Non-Deterministic
More information6.045 Final Exam Solutions
6.045J/18.400J: Automata, Computability and Complexity Prof. Nancy Lynch, Nati Srebro 6.045 Final Exam Solutions May 18, 2004 Susan Hohenberger Name: Please write your name on each page. This exam is open
More informationLecture 24: Randomized Complexity, Course Summary
6.045 Lecture 24: Randomized Complexity, Course Summary 1 1/4 1/16 1/4 1/4 1/32 1/16 1/32 Probabilistic TMs 1/16 A probabilistic TM M is a nondeterministic TM where: Each nondeterministic step is called
More informationUndirected ST-Connectivity in Log-Space. Omer Reingold. Presented by: Thang N. Dinh CISE, University of Florida, Fall, 2010
Undirected ST-Connectivity in Log-Space Omer Reingold Presented by: Thang N. Dinh CISE, University of Florida, Fall, 2010 Undirected s-t connectivity(ustcon) Is t reachable from s in a undirected graph
More informationComplexity (Pre Lecture)
Complexity (Pre Lecture) Dr. Neil T. Dantam CSCI-561, Colorado School of Mines Fall 2018 Dantam (Mines CSCI-561) Complexity (Pre Lecture) Fall 2018 1 / 70 Why? What can we always compute efficiently? What
More information1 Non-deterministic Turing Machine
1 Non-deterministic Turing Machine A nondeterministic Turing machine is a generalization of the standard TM for which every configuration may yield none, or one or more than one next configurations. In
More informationComplexity: moving from qualitative to quantitative considerations
Complexity: moving from qualitative to quantitative considerations Textbook chapter 7 Complexity Theory: study of what is computationally feasible (or tractable) with limited resources: running time (main
More informationPrinciples of Knowledge Representation and Reasoning
Principles of Knowledge Representation and Reasoning Complexity Theory Bernhard Nebel, Malte Helmert and Stefan Wölfl Albert-Ludwigs-Universität Freiburg April 29, 2008 Nebel, Helmert, Wölfl (Uni Freiburg)
More informationLecture 22: PSPACE
6.045 Lecture 22: PSPACE 1 VOTE VOTE VOTE For your favorite course on automata and complexity Please complete the online subject evaluation for 6.045 2 Final Exam Information Who: You On What: Everything
More informationMTAT Complexity Theory October 13th-14th, Lecture 6
MTAT.07.004 Complexity Theory October 13th-14th, 2011 Lecturer: Peeter Laud Lecture 6 Scribe(s): Riivo Talviste 1 Logarithmic memory Turing machines working in logarithmic space become interesting when
More informationAdvanced Topics in Theoretical Computer Science
Advanced Topics in Theoretical Computer Science Part 5: Complexity (Part II) 30.01.2014 Viorica Sofronie-Stokkermans Universität Koblenz-Landau e-mail: sofronie@uni-koblenz.de 1 Contents Recall: Turing
More informationComplete problems for classes in PH, The Polynomial-Time Hierarchy (PH) oracle is like a subroutine, or function in
Oracle Turing Machines Nondeterministic OTM defined in the same way (transition relation, rather than function) oracle is like a subroutine, or function in your favorite PL but each call counts as single
More informationLecture 20: PSPACE. November 15, 2016 CS 1010 Theory of Computation
Lecture 20: PSPACE November 15, 2016 CS 1010 Theory of Computation Recall that PSPACE = k=1 SPACE(nk ). We will see that a relationship between time and space complexity is given by: P NP PSPACE = NPSPACE
More informationCS5371 Theory of Computation. Lecture 19: Complexity IV (More on NP, NP-Complete)
CS5371 Theory of Computation Lecture 19: Complexity IV (More on NP, NP-Complete) Objectives More discussion on the class NP Cook-Levin Theorem The Class NP revisited Recall that NP is the class of language
More informationAnnouncements. Problem Set 7 graded; will be returned at end of lecture. Unclaimed problem sets and midterms moved!
N P NP Announcements Problem Set 7 graded; will be returned at end of lecture. Unclaimed problem sets and midterms moved! Now in cabinets in the Gates open area near the drop-off box. The Complexity Class
More informationResource-Bounded Computation
Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space, other resources? Def: L is decidable within time
More informationLogarithmic space. Evgenij Thorstensen V18. Evgenij Thorstensen Logarithmic space V18 1 / 18
Logarithmic space Evgenij Thorstensen V18 Evgenij Thorstensen Logarithmic space V18 1 / 18 Journey below Unlike for time, it makes sense to talk about sublinear space. This models computations on input
More informationIntroduction to Computational Complexity
Introduction to Computational Complexity George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 400 George Voutsadakis (LSSU) Computational Complexity September
More informationTuring Machines and Time Complexity
Turing Machines and Time Complexity Turing Machines Turing Machines (Infinitely long) Tape of 1 s and 0 s Turing Machines (Infinitely long) Tape of 1 s and 0 s Able to read and write the tape, and move
More informationApplied Computer Science II Chapter 7: Time Complexity. Prof. Dr. Luc De Raedt. Institut für Informatik Albert-Ludwigs Universität Freiburg Germany
Applied Computer Science II Chapter 7: Time Complexity Prof. Dr. Luc De Raedt Institut für Informati Albert-Ludwigs Universität Freiburg Germany Overview Measuring complexity The class P The class NP NP-completeness
More informationTime-Space Tradeoffs for SAT
Lecture 8 Time-Space Tradeoffs for SAT April 22, 2004 Lecturer: Paul Beame Notes: Definition 8.1. TIMESPACE(T (n), S(n)) = {L {0, 1} offline, multitape TM M such that L = L(M) and M uses time O(T (n))
More informationTheory of Computation Space Complexity. (NTU EE) Space Complexity Fall / 1
Theory of Computation Space Complexity (NTU EE) Space Complexity Fall 2016 1 / 1 Space Complexity Definition 1 Let M be a TM that halts on all inputs. The space complexity of M is f : N N where f (n) is
More informationCS151 Complexity Theory. Lecture 4 April 12, 2017
CS151 Complexity Theory Lecture 4 A puzzle A puzzle: two kinds of trees depth n...... cover up nodes with c colors promise: never color arrow same as blank determine which kind of tree in poly(n, c) steps?
More informationFORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY
15-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY THURSDAY APRIL 3 REVIEW for Midterm TUESDAY April 8 Definition: A Turing Machine is a 7-tuple T = (Q, Σ, Γ, δ, q, q accept, q reject ), where: Q is a
More informationChapter 2 : Time complexity
Dr. Abhijit Das, Chapter 2 : Time complexity In this chapter we study some basic results on the time complexities of computational problems. concentrate our attention mostly on polynomial time complexities,
More informationComputer Sciences Department
Computer Sciences Department 1 Reference Book: INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION, by: MICHAEL SIPSER Computer Sciences Department 3 ADVANCED TOPICS IN C O M P U T A B I L I T Y
More informationThe Complexity of Optimization Problems
The Complexity of Optimization Problems Summary Lecture 1 - Complexity of algorithms and problems - Complexity classes: P and NP - Reducibility - Karp reducibility - Turing reducibility Uniform and logarithmic
More informationLecture Notes Each circuit agrees with M on inputs of length equal to its index, i.e. n, x {0, 1} n, C n (x) = M(x).
CS 221: Computational Complexity Prof. Salil Vadhan Lecture Notes 4 February 3, 2010 Scribe: Jonathan Pines 1 Agenda P-/NP- Completeness NP-intermediate problems NP vs. co-np L, NL 2 Recap Last time, we
More informationStudent#: CISC-462 Exam, December XY, 2017 Page 1 of 12
Student#: CISC-462 Exam, December XY, 2017 Page 1 of 12 Queen s University, Faculty of Arts and Science, School of Computing CISC-462 Final Exam, December XY, 2017 (Instructor: Kai Salomaa) INSTRUCTIONS
More informationCSE200: Computability and complexity Space Complexity
CSE200: Computability and complexity Space Complexity Shachar Lovett January 29, 2018 1 Space complexity We would like to discuss languages that may be determined in sub-linear space. Lets first recall
More informationP and NP. Or, how to make $1,000,000.
P and NP Or, how to make $1,000,000. http://www.claymath.org/millennium-problems/p-vs-np-problem Review: Polynomial time difference between single-tape and multi-tape TMs Exponential time difference between
More informationIntro to Theory of Computation
Intro to Theory of Computation LECTURE 24 Last time Relationship between models: deterministic/nondeterministic Class P Today Class NP Sofya Raskhodnikova Homework 9 due Homework 0 out 4/5/206 L24. I-clicker
More informationComplexity: Some examples
Algorithms and Architectures III: Distributed Systems H-P Schwefel, Jens M. Pedersen Mm6 Distributed storage and access (jmp) Mm7 Introduction to security aspects (hps) Mm8 Parallel complexity (hps) Mm9
More informationTechnische Universität München Summer term 2010 Theoretische Informatik August 2, 2010 Dr. J. Kreiker / Dr. M. Luttenberger, J. Kretinsky SOLUTION
Technische Universität München Summer term 2010 Theoretische Informatik August 2, 2010 Dr. J. Kreiker / Dr. M. Luttenberger, J. Kretinsky SOLUTION Complexity Theory Final Exam Please note : If not stated
More informationLecture 20: conp and Friends, Oracles in Complexity Theory
6.045 Lecture 20: conp and Friends, Oracles in Complexity Theory 1 Definition: conp = { L L NP } What does a conp computation look like? In NP algorithms, we can use a guess instruction in pseudocode:
More informationIntractable Problems [HMU06,Chp.10a]
Intractable Problems [HMU06,Chp.10a] Time-Bounded Turing Machines Classes P and NP Polynomial-Time Reductions A 10 Minute Motivation https://www.youtube.com/watch?v=yx40hbahx3s 1 Time-Bounded TM s A Turing
More informationCS256 Applied Theory of Computation
CS256 Applied Theory of Computation Compleity Classes III John E Savage Overview Last lecture on time-bounded compleity classes Today we eamine space-bounded compleity classes We prove Savitch s Theorem,
More informationTheory of Computation
Theory of Computation Dr. Sarmad Abbasi Dr. Sarmad Abbasi () Theory of Computation 1 / 38 Lecture 21: Overview Big-Oh notation. Little-o notation. Time Complexity Classes Non-deterministic TMs The Class
More informationThe purpose here is to classify computational problems according to their complexity. For that purpose we need first to agree on a computational
1 The purpose here is to classify computational problems according to their complexity. For that purpose we need first to agree on a computational model. We'll remind you what a Turing machine is --- you
More informationLecture 17: Cook-Levin Theorem, NP-Complete Problems
6.045 Lecture 17: Cook-Levin Theorem, NP-Complete Problems 1 Is SAT solvable in O(n) time on a multitape TM? Logic circuits of 6n gates for SAT? If yes, then not only is P=NP, but there would be a dream
More informationCSCI 1590 Intro to Computational Complexity
CSCI 59 Intro to Computational Complexity Overview of the Course John E. Savage Brown University January 2, 29 John E. Savage (Brown University) CSCI 59 Intro to Computational Complexity January 2, 29
More informationOutline. Complexity Theory. Example. Sketch of a log-space TM for palindromes. Log-space computations. Example VU , SS 2018
Complexity Theory Complexity Theory Outline Complexity Theory VU 181.142, SS 2018 3. Logarithmic Space Reinhard Pichler Institute of Logic and Computation DBAI Group TU Wien 3. Logarithmic Space 3.1 Computational
More information