Chapter 2 - Solving Recurrences

Download PDF
112 downloads 164 Views 696KB Size Report
Comment
Chapter 2. Solving Recurrences. 2.1 Types of Recurrences. 2.2 Finding Generating Functions. 2.3 Partial Fractions. 2.4 Characteristic Roots. 2.5 Simultaneous ...
Chapter

2

Solving Recurrences 2.1 Types of Recurrences 2.2 Finding Generating Functions 2.3 Partial Fractions 2.4 Characteristic Roots 2.5 Simultaneous Recursions 2.6 Fibonacci Number Identities 2.7 Non-Constant Coefficients 2.8 Divide-and-Conquer Relations

1

2

Chapter 2

Solving Recurrences

Section 2.1 Types of Recurrences

2.1 TYPES OF RECURRENCES

3

4

Chapter 2

Solving Recurrences

Some Linear Recursions

Recurrences without Fixed Degree

Section 2.1 Types of Recurrences

Variable Coefficients

5

6

Chapter 2

Fig 2.1.1

Table 2.1.1

Solving Recurrences

Section 2.1 Types of Recurrences

7

8

Chapter 2

Solving Recurrences

Some Non-linear Recurrences

Section 2.2 Finding an OGF

2.2 FINDING AN OGF

9

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

2.3 PARTIAL FRACTIONS

15

16

Chapter 2

Solving Recurrences

Section 2.3 Partial Fractions

17

18

Chapter 2

Solving Recurrences

Section 2.4 Characteristic Roots

2.4 CHARACTERISTIC ROOTS

Characteristic Equation

19

20

Chapter 2

Solving Recurrences

Section 2.4 Characteristic Roots

Alternative Initial Values

21

22

Repeated Roots

Chapter 2

Solving Recurrences

Section 2.4 Characteristic Roots

Non-homogeneous Equations

23

24

Chapter 2

Solving Recurrences

Section 2.4 Characteristic Roots

25

26

Chapter 2

Solving Recurrences

Section 2.4 Characteristic Roots

27

28

Complex Roots

Chapter 2

Solving Recurrences

Section 2.4 Characteristic Roots

29

30

Chapter 2

Solving Recurrences

2.5 SIMULTANEOUS RECURSIONS

Fig 2.5.1

Section 2.5 Simultaneous Recursions

Fibonacci Rabbits

31

32

Chapter 2

Solving Recurrences

Ubiquitousness of the Fibonacci Seq

Fig 2.5.2

Section 2.5 Simultaneous Recursions

Fig 2.5.3

Solving the Fibonacci Recurrence

33

34

Chapter 2

Solving Recurrences

Section 2.5 Simultaneous Recursions

35

36

Chapter 2

Some Tiling Problems

Fig 2.5.4

Solving Recurrences

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

Cassini’s Identity

Chapter 2

Solving Recurrences

Section 2.6 Fibo Number Identities

41

42

Chapter 2

Fibonacci Number System

Solving Recurrences

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

Sum in a Recurrence: Quicksort

47

48

Chapter 2

Algorithm 2.7.1:

Solving Recurrences

Section 2.7 Non-Constant Coefficients

Analysis of the Time to Quicksort

49

50

Chapter 2

Solving Recurrences

Section 2.7 Non-Constant Coefficients

51

52

Chapter 2

Confirming Small Cases

Solving Recurrences

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

Merging

Algorithm 2.8.2:

57

58

Chapter 2

Solving Recurrences

Section 2.8 Divide-&-Conquer Relations

Iterative Mergesort

59

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

The Josephus Recurrence

Fig 2.8.1

Solving Recurrences

Section 2.8 Divide-&-Conquer Relations

Fig 2.8.2

63

64

Fig 2.8.3

Chapter 2

Solving Recurrences

Section 2.8 Divide-&-Conquer Relations

Fig 2.8.4

65

66

Chapter 2

Solving Recurrences

Section 2.8 Divide-&-Conquer Relations

67