CSC 8301 Design and Analysis of Algorithms: Recursive Analysis

Transcription

1 CSC 8301 Design and Analysis of Algorithms: Recursive Analysis Professor Henry Carter Fall 2016

2 Housekeeping Quiz #1 New TA office hours: Tuesday 1-3 2

3 General Analysis Procedure Select a parameter for n (input size) Identify the algorithm s basic operation Check what other dependencies affect the number of executions of the basic operation (Best? Worst? Average?) Set up a sum counting the basic operation Find a closed formula or establish the order of growth 3

4 Recap: Recursion Solving a problem by solving smaller versions of the same problem In programming: a function that calls itself Composed of a base condition, a basic solution, and a recursive call 4

5 Example: Factorial 5

6 Recap: Recurrence Relations A sequence is an ordered list of numbers A recurrence relation is a formula that recursively defines a sequence (i.e., based on a previous sequence element) An initial condition defines the starting element of a sequence We need a recurrence relation and an initial condition to describe a sequence uniquely 6

7 Example: Factorial 7

8 Example: Factorial Recurrence 8

9 Example: Factorial Operation Count 9

10 General Analysis Procedure: Recursion Select a parameter for n (input size) Identify the algorithm s basic operation Check what other dependencies affect the number of executions of the basic operation (Best? Worst? Average?) Set up a recurrence relation and initial condition counting the basic operation Solve the recurrence or establish the order of growth 10

11 Example: Tower of Hanoi Conceived by Edouard Lucas in 1883 Monks in a temple move 64 discs from one column to another (with a third auxiliary column) They move one at a time They never place a larger disc on top of a smaller disc The world will end when they finish! Iterative and recursive solutions exist requiring the same number of moves 11

12 Example: Tower of Hanoi 12

13 Example: Tower of Hanoi 13

14 Example: Tower of Hanoi 14

15 Example: Counting Digits v. 2 15

16 Example: Counting Digits v. 2 16

17 Example: Counting Digits v. 2 17

18 Fibonacci Numbers 0,1,1,2,3,5,8,13,21,34, Defined by the recurrence: F(n) = F(n 1) + F(n 2) for n > 1 F(0) = 0, F(1) = 1 18

19 Fibonacci Numbers Found in Indian mathematics texts as old as 200 BC Named for Leonardo Fibonacci, who used the sequence to describe the growth of a rabbit population Why is this sequence special? Golden ratio Many ways to solve 19

20 Solving the recurrence F(n) = F(n 1) + F(n 2) Homogeneous second-order linear recurrence with constant coefficients ax(n) + bx(n 1) + cx(n 2) = 0 a, b, c are real fixed numbers where a 0 Characteristic equation: ar 2 + br + c = 0 x(n) = αr n + βr n when the characteristic equation has real, distinct roots r, r 20

21 Characteristic Equation 21

22 Solve 22

23 Fibonacci take 1: Recursion 23

24 Fibonacci take 1: Recursion 24

25 Fibonacci take 2: Storage 25

26 Fibonacci take 3: Direct F(n) (1/ 5) * φ n Efficiency determined by exponentiation technique Naïve approach: Θ(n) = n 2 log n 26

27 Fibonacci take 4: Matrices n F(n 1) F(n) F(n) F(n 2) = For n 1 27

28 Recap Recursive algorithms call themselves for smaller inputs Recurrence relations recursively define a sequence (with some starting condition) Modified 5-steps to analyzing recursive algorithms 28

29 Next Time... Levitin Chapter Remember, you need to read it BEFORE you come to class! Homework: 2.4: 1, 2, 3, 8, : 2, 3, 7, 8 29