Intro to Algorithms: BIBLIOGRAPHY Intro to Algorithms: BIBLIOGRAPHY Table of Contents BIBLIOGRAPHY...............................................................................................................................................1 CHAPTER 1: INTRODUCTION....................................................................................................................12 1.1 Algorithms...................................................................................................................................................13 Insertion sort..........................................................................................................................................13 Pseudocode conventions........................................................................................................................15 Exercises................................................................................................................................................16 1.2 Analyzing algorithms..................................................................................................................................17 Analysis of insertion sort.......................................................................................................................17 Worst−case and average−case analysis.................................................................................................19 Order of growth......................................................................................................................................20 Exercises................................................................................................................................................20 1.3 Designing algorithms..................................................................................................................................22 1.3.1 The divide−and−conquer approach..............................................................................................22 1.3.2 Analyzing divide−and−conquer algorithms..................................................................................23 Exercises................................................................................................................................................24 Analysis of merge sort........................................................................................................................................25 1.4 Summary......................................................................................................................................................26 Exercises................................................................................................................................................26 Problems............................................................................................................................................................27 Chapter notes....................................................................................................................................................28 CHAPTER 2: GROWTH OF FUNCTIONS..................................................................................................29 2.1 Asymptotic notation....................................................................................................................................30 −notation................................................................................................................................................30 O−notation.............................................................................................................................................32 −notation................................................................................................................................................32 Asymptotic notation in equations..........................................................................................................33 o−notation..............................................................................................................................................34 −notation................................................................................................................................................35 Comparison of functions........................................................................................................................35 Exercises................................................................................................................................................36 2.2 Standard notations and common functions..............................................................................................38 Monotonicity..........................................................................................................................................38 Floors and ceilings.................................................................................................................................38 Polynomials...........................................................................................................................................38 Exponentials...........................................................................................................................................39 Logarithms.............................................................................................................................................40 Factorials................................................................................................................................................41 The iterated logarithm function.............................................................................................................42 Fibonacci numbers.................................................................................................................................42 Exercises................................................................................................................................................43 i Intro to Algorithms: BIBLIOGRAPHY Table of Contents Problems............................................................................................................................................................45 Chapter notes....................................................................................................................................................48 CHAPTER 3: SUMMATIONS........................................................................................................................49 3.1 Summation formulas and properties........................................................................................................50 Linearity.................................................................................................................................................50 Arithmetic series....................................................................................................................................50 Geometric series.....................................................................................................................................51 Harmonic series.....................................................................................................................................51 Integrating and differentiating series.....................................................................................................52 Telescoping series..................................................................................................................................52 Products.................................................................................................................................................52 Exercises................................................................................................................................................53 3.2 Bounding summations................................................................................................................................54 Mathematical induction.........................................................................................................................54 Bounding the terms................................................................................................................................55 Splitting summations.............................................................................................................................56 Approximation by integrals...................................................................................................................57 Exercises................................................................................................................................................59 Technicalities.........................................................................................................................................61 Making a good guess.............................................................................................................................62 Subtleties................................................................................................................................................63 Avoiding pitfalls....................................................................................................................................63 Changing variables.................................................................................................................................65