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Internet Engineering Czesław Smutnicki Discrete Mathematics – Computational Complexity

This comprehensive text by Czesław Smutnicki delves into Discrete Mathematics, particularly focusing on Computational Complexity. It covers essential topics such as asymptotic notation, decision and optimization problems, calculation models (including Turing machines), complexity classes, polynomial-time algorithms, and the theory of NP-completeness. Readers will gain insights into approximate methods, quality measures of approximation, and competitive analysis of algorithms. This work is crucial for anyone interested in the foundational concepts of computer science and mathematics.

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Internet Engineering Czesław Smutnicki Discrete Mathematics – Computational Complexity

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  1. Internet Engineering Czesław Smutnicki DiscreteMathematics– ComputationalComplexity

  2. CONTENTS • Asymptotic notation • Decision/optimization problems • Calculation models • Turing machines • Problem, instances, data coding • Complexity classes • Polynomial-time algorithms • Theory of NP-completness • Approximate methods • Quality measures of approximation • Analysis of quality measures • Calculation cost • Competitive analysis (on-line algorithms) • Inapproximality theory

  3. ASYMPTOTIC NOTATION – symbol O(n) Definition Examples

  4. ASYMPTOTIC NOTATION – symbol (n) Definition Examples

  5. ASYMPTOTIC NOTATION – symbol (n) Definition Examples

  6. ASYMPTOTIC NOTATION - symbol o(n) Definition Examples

  7. ASYMPTOTIC NOTATION - symbol (n) Definition Examples

  8. DECISION/OPTIMIZATION PROBLEMS • decision problem: answer yes-no 2-partition problem: givennumbers. Does a set exist such that • optimization problem: find min or max of the goal function value knapsack problem:givennumbers , and . Find the set such that , • any optimization problem can be transformed into decision problem knapsack problem:givennumbers , , and . Does a setexist such that ,

  9. CALCULATION MODELS i o • Simple machine • Finite-state machine • Automata: Mealy Moore • Deterministic/non-deterministic finite automata i o S

  10. DETERMINISTIC TURING MACHINE s -2 -1 0 … 1 2 3 4

  11. NON-DETERMINISTIC TURING MACHINE s -2 -1 0 … 1 2 3 4

  12. CODING • Instance I/ Problem P • Decimal coding of I • Binary coding of I • Unary coding of I • Data string x(I) • Size N(I) of the instance I • Coding of numbers and structural elements

  13. COMPUTATIONAL COMPLEXITY FUNCTION DEPENDS ON: • Coding rule • Model of calculations (DTM)

  14. FUNDAMENTAL COMPLEXITY CLASSES Polynomial time algorithm O(p(n)), p – polynomial, solvable by DTM, P class Exponential time algorithm NP class, solvable in O(p(n)) on NDTM = solvable in O(2p(n)) on DTM

  15. NP COMPLETE PROBLEMS POLYNOMIAL TIME TRANSFORMATION PROBLEM P1 IS NP-COMPLETE IF P1 BELONGS TO NP CLASS AND FOR ANY P2 FROM NP CLASS, P2 IS POLYNOMIALLY TRANSFORMABLE TO P1 PROBLEM IS PSEUDO-POLYNOMIAL (NPI CLASS) IF ITS COMPUTATIONAL COMPLEXITY FUNCTION IS A POLYNOMIAL OF N(I) AND MAX(I)

  16. COMPLEXITY CLASSES NP CLASS NPI CLASS NP COMPLETE CLASS P CLASS STRONGLY NP COMPLETE CLASS

  17. Thank you for your attention DISCRETE MATHEMATICS Czesław Smutnicki

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