Combninatorial Mathematics 2

8/10/2019 Combninatorial Mathematics 2

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2 Chapter 2 Solving Recurrences


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Section 2.1 Types of Recurrences 3

2.1 TYPES OF RECURRENCES


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Variable Coef cients


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Fig 2.1.1

Table 2.1.1


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Some Non-linear Recurrences


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Section 2.2 Finding an OGF 9

2.2 FINDING AN OGF


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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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Alternative Initial Values


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Repeated Roots


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Non-homogeneous Equations


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Complex Roots


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2.5 SIMULTANEOUS RECURSIONS

Fig 2.5.1


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Section 2.5 Simultaneous Recursions 31

Fibonacci Rabbits


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Ubiquitousness of the Fibonacci Seq

Fig 2.5.2


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Fig 2.5.3

Solving the Fibonacci Recurrence


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Some Tiling Problems

Fig 2.5.4


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2.6 FIBO NUMBER IDENTITIES

Forward-Shift and Subscript Multipliers


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Section 2.6 Fibo Number Identities 39


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Cassinis Identity


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2.7 NON-CONSTANT COEFFICIENTS

A Reduction Strategy


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Section 2.7 Non-Constant Coef cients 45


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Sum in a Recurrence: Quicksort


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Analysis of the Time to Quicksort


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Con rming Small Cases


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Section 2.8 Divide-&-Conquer Relations 53

2.8 DIVIDE-&-CONQUER RELATIONS

Binary Search


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Algorithm 2.8.1:

Analysis of the Time for a Binary Search


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Recursive Mergesort

Algorithm 2.8.3:

Analysis of the Time or a Mergesort


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The Josephus Recurrence

Fig 2.8.1


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Fig 2.8.4


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Combninatorial Mathematics 2

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