Computational Complexity Theory
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Notes for Lecture 2
Notes on Complexity Theory Last updated: December, 2011 Lecture 2 Jonathan Katz 1 Review The running time of a Turing machine M on input x is the number of \steps" M takes before it halts. Machine M is said to run in time T (¢) if for every input x the running time of M(x) is at most T (jxj). (In particular, this means it halts on all inputs.) The space used by M on input x is the number of cells written to by M on all its work tapes1 (a cell that is written to multiple times is only counted once); M is said to use space T (¢) if for every input x the space used during the computation of M(x) is at most T (jxj). We remark that these time and space measures are worst-case notions; i.e., even if M runs in time T (n) for only a fraction of the inputs of length n (and uses less time for all other inputs of length n), the running time of M is still T . (Average-case notions of complexity have also been considered, but are somewhat more di±cult to reason about. We may cover this later in the semester; or see [1, Chap. 18].) Recall that a Turing machine M computes a function f : f0; 1g¤ ! f0; 1g¤ if M(x) = f(x) for all x. We will focus most of our attention on boolean functions, a context in which it is more convenient to phrase computation in terms of languages. A language is simply a subset of f0; 1g¤. -
Time Complexity
Chapter 3 Time complexity Use of time complexity makes it easy to estimate the running time of a program. Performing an accurate calculation of a program’s operation time is a very labour-intensive process (it depends on the compiler and the type of computer or speed of the processor). Therefore, we will not make an accurate measurement; just a measurement of a certain order of magnitude. Complexity can be viewed as the maximum number of primitive operations that a program may execute. Regular operations are single additions, multiplications, assignments etc. We may leave some operations uncounted and concentrate on those that are performed the largest number of times. Such operations are referred to as dominant. The number of dominant operations depends on the specific input data. We usually want to know how the performance time depends on a particular aspect of the data. This is most frequently the data size, but it can also be the size of a square matrix or the value of some input variable. 3.1: Which is the dominant operation? 1 def dominant(n): 2 result = 0 3 fori in xrange(n): 4 result += 1 5 return result The operation in line 4 is dominant and will be executedn times. The complexity is described in Big-O notation: in this caseO(n)— linear complexity. The complexity specifies the order of magnitude within which the program will perform its operations. More precisely, in the case ofO(n), the program may performc n opera- · tions, wherec is a constant; however, it may not performn 2 operations, since this involves a different order of magnitude of data. -
Quick Sort Algorithm Song Qin Dept
Quick Sort Algorithm Song Qin Dept. of Computer Sciences Florida Institute of Technology Melbourne, FL 32901 ABSTRACT each iteration. Repeat this on the rest of the unsorted region Given an array with n elements, we want to rearrange them in without the first element. ascending order. In this paper, we introduce Quick Sort, a Bubble sort works as follows: keep passing through the list, divide-and-conquer algorithm to sort an N element array. We exchanging adjacent element, if the list is out of order; when no evaluate the O(NlogN) time complexity in best case and O(N2) exchanges are required on some pass, the list is sorted. in worst case theoretically. We also introduce a way to approach the best case. Merge sort [4]has a O(NlogN) time complexity. It divides the 1. INTRODUCTION array into two subarrays each with N/2 items. Conquer each Search engine relies on sorting algorithm very much. When you subarray by sorting it. Unless the array is sufficiently small(one search some key word online, the feedback information is element left), use recursion to do this. Combine the solutions to brought to you sorted by the importance of the web page. the subarrays by merging them into single sorted array. 2 Bubble, Selection and Insertion Sort, they all have an O(N2) In Bubble sort, Selection sort and Insertion sort, the O(N ) time time complexity that limits its usefulness to small number of complexity limits the performance when N gets very big. element no more than a few thousand data points. -
Complexity Theory Lecture 9 Co-NP Co-NP-Complete
Complexity Theory 1 Complexity Theory 2 co-NP Complexity Theory Lecture 9 As co-NP is the collection of complements of languages in NP, and P is closed under complementation, co-NP can also be characterised as the collection of languages of the form: ′ L = x y y <p( x ) R (x, y) { |∀ | | | | → } Anuj Dawar University of Cambridge Computer Laboratory NP – the collection of languages with succinct certificates of Easter Term 2010 membership. co-NP – the collection of languages with succinct certificates of http://www.cl.cam.ac.uk/teaching/0910/Complexity/ disqualification. Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 3 Complexity Theory 4 NP co-NP co-NP-complete P VAL – the collection of Boolean expressions that are valid is co-NP-complete. Any language L that is the complement of an NP-complete language is co-NP-complete. Any of the situations is consistent with our present state of ¯ knowledge: Any reduction of a language L1 to L2 is also a reduction of L1–the complement of L1–to L¯2–the complement of L2. P = NP = co-NP • There is an easy reduction from the complement of SAT to VAL, P = NP co-NP = NP = co-NP • ∩ namely the map that takes an expression to its negation. P = NP co-NP = NP = co-NP • ∩ VAL P P = NP = co-NP ∈ ⇒ P = NP co-NP = NP = co-NP • ∩ VAL NP NP = co-NP ∈ ⇒ Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 5 Complexity Theory 6 Prime Numbers Primality Consider the decision problem PRIME: Another way of putting this is that Composite is in NP. -
Computational Complexity: a Modern Approach
i Computational Complexity: A Modern Approach Draft of a book: Dated January 2007 Comments welcome! Sanjeev Arora and Boaz Barak Princeton University [email protected] Not to be reproduced or distributed without the authors’ permission This is an Internet draft. Some chapters are more finished than others. References and attributions are very preliminary and we apologize in advance for any omissions (but hope you will nevertheless point them out to us). Please send us bugs, typos, missing references or general comments to [email protected] — Thank You!! DRAFT ii DRAFT Chapter 9 Complexity of counting “It is an empirical fact that for many combinatorial problems the detection of the existence of a solution is easy, yet no computationally efficient method is known for counting their number.... for a variety of problems this phenomenon can be explained.” L. Valiant 1979 The class NP captures the difficulty of finding certificates. However, in many contexts, one is interested not just in a single certificate, but actually counting the number of certificates. This chapter studies #P, (pronounced “sharp p”), a complexity class that captures this notion. Counting problems arise in diverse fields, often in situations having to do with estimations of probability. Examples include statistical estimation, statistical physics, network design, and more. Counting problems are also studied in a field of mathematics called enumerative combinatorics, which tries to obtain closed-form mathematical expressions for counting problems. To give an example, in the 19th century Kirchoff showed how to count the number of spanning trees in a graph using a simple determinant computation. Results in this chapter will show that for many natural counting problems, such efficiently computable expressions are unlikely to exist. -
Chapter 24 Conp, Self-Reductions
Chapter 24 coNP, Self-Reductions CS 473: Fundamental Algorithms, Spring 2013 April 24, 2013 24.1 Complementation and Self-Reduction 24.2 Complementation 24.2.1 Recap 24.2.1.1 The class P (A) A language L (equivalently decision problem) is in the class P if there is a polynomial time algorithm A for deciding L; that is given a string x, A correctly decides if x 2 L and running time of A on x is polynomial in jxj, the length of x. 24.2.1.2 The class NP Two equivalent definitions: (A) Language L is in NP if there is a non-deterministic polynomial time algorithm A (Turing Machine) that decides L. (A) For x 2 L, A has some non-deterministic choice of moves that will make A accept x (B) For x 62 L, no choice of moves will make A accept x (B) L has an efficient certifier C(·; ·). (A) C is a polynomial time deterministic algorithm (B) For x 2 L there is a string y (proof) of length polynomial in jxj such that C(x; y) accepts (C) For x 62 L, no string y will make C(x; y) accept 1 24.2.1.3 Complementation Definition 24.2.1. Given a decision problem X, its complement X is the collection of all instances s such that s 62 L(X) Equivalently, in terms of languages: Definition 24.2.2. Given a language L over alphabet Σ, its complement L is the language Σ∗ n L. 24.2.1.4 Examples (A) PRIME = nfn j n is an integer and n is primeg o PRIME = n n is an integer and n is not a prime n o PRIME = COMPOSITE . -
Succinctness of the Complement and Intersection of Regular Expressions Wouter Gelade, Frank Neven
Succinctness of the Complement and Intersection of Regular Expressions Wouter Gelade, Frank Neven To cite this version: Wouter Gelade, Frank Neven. Succinctness of the Complement and Intersection of Regular Expres- sions. STACS 2008, Feb 2008, Bordeaux, France. pp.325-336. hal-00226864 HAL Id: hal-00226864 https://hal.archives-ouvertes.fr/hal-00226864 Submitted on 30 Jan 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Symposium on Theoretical Aspects of Computer Science 2008 (Bordeaux), pp. 325-336 www.stacs-conf.org SUCCINCTNESS OF THE COMPLEMENT AND INTERSECTION OF REGULAR EXPRESSIONS WOUTER GELADE AND FRANK NEVEN Hasselt University and Transnational University of Limburg, School for Information Technology E-mail address: [email protected] Abstract. We study the succinctness of the complement and intersection of regular ex- pressions. In particular, we show that when constructing a regular expression defining the complement of a given regular expression, a double exponential size increase cannot be avoided. Similarly, when constructing a regular expression defining the intersection of a fixed and an arbitrary number of regular expressions, an exponential and double expo- nential size increase, respectively, can in worst-case not be avoided. -
The Complexity Zoo
The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/. -
When Subgraph Isomorphism Is Really Hard, and Why This Matters for Graph Databases Ciaran Mccreesh, Patrick Prosser, Christine Solnon, James Trimble
When Subgraph Isomorphism is Really Hard, and Why This Matters for Graph Databases Ciaran Mccreesh, Patrick Prosser, Christine Solnon, James Trimble To cite this version: Ciaran Mccreesh, Patrick Prosser, Christine Solnon, James Trimble. When Subgraph Isomorphism is Really Hard, and Why This Matters for Graph Databases. Journal of Artificial Intelligence Research, Association for the Advancement of Artificial Intelligence, 2018, 61, pp.723 - 759. 10.1613/jair.5768. hal-01741928 HAL Id: hal-01741928 https://hal.archives-ouvertes.fr/hal-01741928 Submitted on 26 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Journal of Artificial Intelligence Research 61 (2018) 723-759 Submitted 11/17; published 03/18 When Subgraph Isomorphism is Really Hard, and Why This Matters for Graph Databases Ciaran McCreesh [email protected] Patrick Prosser [email protected] University of Glasgow, Glasgow, Scotland Christine Solnon [email protected] INSA-Lyon, LIRIS, UMR5205, F-69621, France James Trimble [email protected] University of Glasgow, Glasgow, Scotland Abstract The subgraph isomorphism problem involves deciding whether a copy of a pattern graph occurs inside a larger target graph. -
Lecture 9 1 Interactive Proof Systems/Protocols
CS 282A/MATH 209A: Foundations of Cryptography Prof. Rafail Ostrovsky Lecture 9 Lecture date: March 7-9, 2005 Scribe: S. Bhattacharyya, R. Deak, P. Mirzadeh 1 Interactive Proof Systems/Protocols 1.1 Introduction The traditional mathematical notion of a proof is a simple passive protocol in which a prover P outputs a complete proof to a verifier V who decides on its validity. The interaction in this traditional sense is minimal and one-way, prover → verifier. The observation has been made that allowing the verifier to interact with the prover can have advantages, for example proving the assertion faster or proving more expressive languages. This extension allows for the idea of interactive proof systems (protocols). The general framework of the interactive proof system (protocol) involves a prover P with an exponential amount of time (computationally unbounded) and a verifier V with a polyno- mial amount of time. Both P and V exchange multiple messages (challenges and responses), usually dependent upon outcomes of fair coin tosses which they may or may not share. It is easy to see that since V is a poly-time machine (PPT), only a polynomial number of messages may be exchanged between the two. P ’s objective is to convince (prove to) the verifier the truth of an assertion, e.g., claimed knowledge of a proof that x ∈ L. V either accepts or rejects the interaction with the P . 1.2 Definition of Interactive Proof Systems An interactive proof system for a language L is a protocol PV for communication between a computationally unbounded (exponential time) machine P and a probabilistic poly-time (PPT) machine V such that the protocol satisfies the properties of completeness and sound- ness. -
NP As Games, Co-NP, Proof Complexity
CS 6743 Lecture 9 1 Fall 2007 1 Importance of the Cook-Levin Theorem There is a trivial NP-complete language: k Lu = {(M, x, 1 ) | NTM M accepts x in ≤ k steps} Exercise: Show that Lu is NP-complete. The language Lu is not particularly interesting, whereas SAT is extremely interesting since it’s a well-known and well-studied natural problem in logic. After Cook and Levin showed NP-completeness of SAT, literally hundreds of other important and natural problems were also shown to be NP-complete. It is this abundance of natural complete problems which makes the notion of NP-completeness so important, and the “P vs. NP” question so fundamental. 2 Viewing NP as a game Nondeterministic computation can be viewed as a two-person game. The players are Prover and Verifier. Both get the same input, e.g., a propositional formula φ. Prover is all-powerful (but not trust-worthy). He is trying to convince Verifier that the input is in the language (e.g., that φ is satisfiable). Prover sends his argument (as a binary string) to Verifier. Verifier is computationally bounded algorithm. In case of NP, Verifier is a deterministic polytime algorithm. It is not hard to argue that the class NP of languages L is exactly the class of languages for which there is a pair (Prover, Verifier) with the property: For inputs in the language, Prover convinces Verifier to accept; for inputs not in the language, any string sent by Prover will be rejected by Verifier. Moreover, the string that Prover needs to send is of length polynomial in the size of the input. -
Simple Doubly-Efficient Interactive Proof Systems for Locally
Electronic Colloquium on Computational Complexity, Revision 3 of Report No. 18 (2017) Simple doubly-efficient interactive proof systems for locally-characterizable sets Oded Goldreich∗ Guy N. Rothblumy September 8, 2017 Abstract A proof system is called doubly-efficient if the prescribed prover strategy can be implemented in polynomial-time and the verifier’s strategy can be implemented in almost-linear-time. We present direct constructions of doubly-efficient interactive proof systems for problems in P that are believed to have relatively high complexity. Specifically, such constructions are presented for t-CLIQUE and t-SUM. In addition, we present a generic construction of such proof systems for a natural class that contains both problems and is in NC (and also in SC). The proof systems presented by us are significantly simpler than the proof systems presented by Goldwasser, Kalai and Rothblum (JACM, 2015), let alone those presented by Reingold, Roth- blum, and Rothblum (STOC, 2016), and can be implemented using a smaller number of rounds. Contents 1 Introduction 1 1.1 The current work . 1 1.2 Relation to prior work . 3 1.3 Organization and conventions . 4 2 Preliminaries: The sum-check protocol 5 3 The case of t-CLIQUE 5 4 The general result 7 4.1 A natural class: locally-characterizable sets . 7 4.2 Proof of Theorem 1 . 8 4.3 Generalization: round versus computation trade-off . 9 4.4 Extension to a wider class . 10 5 The case of t-SUM 13 References 15 Appendix: An MA proof system for locally-chracterizable sets 18 ∗Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel.