Click here to load reader

Chapter 2 - Solving Recurrences

  • View
    226

  • Download
    2

Embed Size (px)

Text of Chapter 2 - Solving Recurrences

  • Chapter 2Solving Recurrences

    2.1 Types of Recurrences2.2 Finding Generating Functions2.3 Partial Fractions2.4 Characteristic Roots2.5 Simultaneous Recursions2.6 Fibonacci Number Identities2.7 Non-Constant Coefficients2.8 Divide-and-Conquer Relations

    1

  • 2 Chapter 2 Solving Recurrences

  • Section 2.1 Types of Recurrences 3

    2.1 TYPES OF RECURRENCES

  • 4 Chapter 2 Solving Recurrences

    Some Linear Recursions

    Recurrences without Fixed Degree

  • Section 2.1 Types of Recurrences 5

    Variable Coefficients

  • 6 Chapter 2 Solving Recurrences

    Fig 2.1.1

    Table 2.1.1

  • Section 2.1 Types of Recurrences 7

  • 8 Chapter 2 Solving Recurrences

    Some Non-linear Recurrences

  • Section 2.2 Finding an OGF 9

    2.2 FINDING AN OGF

  • 10 Chapter 2 Solving Recurrences

  • Section 2.2 Finding an OGF 11

  • 12 Chapter 2 Solving Recurrences

  • Section 2.2 Finding an OGF 13

  • 14 Chapter 2 Solving Recurrences

  • Section 2.3 Partial Fractions 15

    2.3 PARTIAL FRACTIONS

  • 16 Chapter 2 Solving Recurrences

  • Section 2.3 Partial Fractions 17

  • 18 Chapter 2 Solving Recurrences

  • Section 2.4 Characteristic Roots 19

    2.4 CHARACTERISTIC ROOTS

    Characteristic Equation

  • 20 Chapter 2 Solving Recurrences

  • Section 2.4 Characteristic Roots 21

    Alternative Initial Values

  • 22 Chapter 2 Solving Recurrences

    Repeated Roots

  • Section 2.4 Characteristic Roots 23

    Non-homogeneous Equations

  • 24 Chapter 2 Solving Recurrences

  • Section 2.4 Characteristic Roots 25

  • 26 Chapter 2 Solving Recurrences

  • Section 2.4 Characteristic Roots 27

  • 28 Chapter 2 Solving Recurrences

    Complex Roots

  • Section 2.4 Characteristic Roots 29

  • 30 Chapter 2 Solving Recurrences

    2.5 SIMULTANEOUS RECURSIONS

    Fig 2.5.1

  • Section 2.5 Simultaneous Recursions 31

    Fibonacci Rabbits

  • 32 Chapter 2 Solving Recurrences

    Ubiquitousness of the Fibonacci Seq

    Fig 2.5.2

  • Section 2.5 Simultaneous Recursions 33

    Fig 2.5.3

    Solving the Fibonacci Recurrence

  • 34 Chapter 2 Solving Recurrences

  • Section 2.5 Simultaneous Recursions 35

  • 36 Chapter 2 Solving Recurrences

    Some Tiling Problems

    Fig 2.5.4

  • Section 2.5 Simultaneous Recursions 37

  • 38 Chapter 2 Solving Recurrences

    2.6 FIBO NUMBER IDENTITIES

    Forward-Shift and Subscript Multipliers

  • Section 2.6 Fibo Number Identities 39

  • 40 Chapter 2 Solving Recurrences

    Cassinis Identity

  • Section 2.6 Fibo Number Identities 41

  • 42 Chapter 2 Solving Recurrences

    Fibonacci Number System

  • Section 2.6 Fibo Number Identities 43

  • 44 Chapter 2 Solving Recurrences

    2.7 NON-CONSTANT COEFFICIENTS

    A Reduction Strategy

  • Section 2.7 Non-Constant Coefficients 45

  • 46 Chapter 2 Solving Recurrences

  • Section 2.7 Non-Constant Coefficients 47

    Sum in a Recurrence: Quicksort

  • 48 Chapter 2 Solving Recurrences

    Algorithm 2.7.1:

  • Section 2.7 Non-Constant Coefficients 49

    Analysis of the Time to Quicksort

  • 50 Chapter 2 Solving Recurrences

  • Section 2.7 Non-Constant Coefficients 51

  • 52 Chapter 2 Solving Recurrences

    Confirming Small Cases

  • Section 2.8 Divide-&-Conquer Relations 53

    2.8 DIVIDE-&-CONQUER RELATIONS

    Binary Search

  • 54 Chapter 2 Solving Recurrences

  • Section 2.8 Divide-&-Conquer Relations 55

  • 56 Chapter 2 Solving Recurrences

    Algorithm 2.8.1:

    Analysis of the Time for a Binary Search

  • Section 2.8 Divide-&-Conquer Relations 57

    Merging

    Algorithm 2.8.2:

  • 58 Chapter 2 Solving Recurrences

  • Section 2.8 Divide-&-Conquer Relations 59

    Iterative Mergesort

  • 60 Chapter 2 Solving Recurrences

    Recursive Mergesort

    Algorithm 2.8.3:

    Analysis of the Time or a Mergesort

  • Section 2.8 Divide-&-Conquer Relations 61

  • 62 Chapter 2 Solving Recurrences

    The Josephus Recurrence

    Fig 2.8.1

  • Section 2.8 Divide-&-Conquer Relations 63

    Fig 2.8.2

  • 64 Chapter 2 Solving Recurrences

    Fig 2.8.3

  • Section 2.8 Divide-&-Conquer Relations 65

    Fig 2.8.4

  • 66 Chapter 2 Solving Recurrences

  • Section 2.8 Divide-&-Conquer Relations 67