chapter 2 - solving recurrences
TRANSCRIPT
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
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
Cassini’s 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