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

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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

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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

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Section 2.2 Finding an OGF 11

12 Chapter 2 Solving Recurrences

Section 2.2 Finding an OGF 13

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Section 2.3 Partial Fractions 15

2.3 PARTIAL FRACTIONS

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Section 2.3 Partial Fractions 17

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Section 2.4 Characteristic Roots 19

2.4 CHARACTERISTIC ROOTS

Characteristic Equation

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Section 2.4 Characteristic Roots 21

Alternative Initial Values

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Repeated Roots

Section 2.4 Characteristic Roots 23

Non-homogeneous Equations

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Section 2.4 Characteristic Roots 25

26 Chapter 2 Solving Recurrences

Section 2.4 Characteristic Roots 27

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Complex Roots

Section 2.4 Characteristic Roots 29

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2.5 SIMULTANEOUS RECURSIONS

Fig 2.5.1

Section 2.5 Simultaneous Recursions 31

Fibonacci Rabbits

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Ubiquitousness of the Fibonacci Seq

Fig 2.5.2

Section 2.5 Simultaneous Recursions 33

Fig 2.5.3

Solving the Fibonacci Recurrence

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Section 2.5 Simultaneous Recursions 35

36 Chapter 2 Solving Recurrences

Some Tiling Problems

Fig 2.5.4

Section 2.5 Simultaneous Recursions 37

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2.6 FIBO NUMBER IDENTITIES

Forward-Shift and Subscript Multipliers

Section 2.6 Fibo Number Identities 39

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Cassini’s Identity

Section 2.6 Fibo Number Identities 41

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Fibonacci Number System

Section 2.6 Fibo Number Identities 43

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2.7 NON-CONSTANT COEFFICIENTS

A Reduction Strategy

Section 2.7 Non-Constant Coefficients 45

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Section 2.7 Non-Constant Coefficients 47

Sum in a Recurrence: Quicksort

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Algorithm 2.7.1:

Section 2.7 Non-Constant Coefficients 49

Analysis of the Time to Quicksort

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Section 2.7 Non-Constant Coefficients 51

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Confirming Small Cases

Section 2.8 Divide-&-Conquer Relations 53

2.8 DIVIDE-&-CONQUER RELATIONS

Binary Search

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Section 2.8 Divide-&-Conquer Relations 55

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Algorithm 2.8.1:

Analysis of the Time for a Binary Search

Section 2.8 Divide-&-Conquer Relations 57

Merging

Algorithm 2.8.2:

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Section 2.8 Divide-&-Conquer Relations 59

Iterative Mergesort

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Recursive Mergesort

Algorithm 2.8.3:

Analysis of the Time or a Mergesort

Section 2.8 Divide-&-Conquer Relations 61

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The Josephus Recurrence

Fig 2.8.1

Section 2.8 Divide-&-Conquer Relations 63

Fig 2.8.2

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Fig 2.8.3

Section 2.8 Divide-&-Conquer Relations 65

Fig 2.8.4

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Section 2.8 Divide-&-Conquer Relations 67