First Questions for Algorithm Analysis

1. First Questions for Algorithm Analysis

2. Space • Input magnitude (the number of bits needed to represent the input • Number of inputs • Space required to store the inputs, perform intermediate computations, and store outputs • Time Measurement Units

3. Definition of g(n) ε O(f(n)). The function g(n) is in O(f(n)) if there exist non negative constants c and such that g(n) ≤ c f(n) for all n ≥ . Big O(.)

4. Find Algorithms That Are O(f(n))

5. Is the O() = Φ? • Are there decision questions that cannot be computed by any algorithm? Do things ever get worse?

6. Definition: g(n) εΩ(f(n)). What does it mean to be bounded below? • The function g(n) is in Ω(f(n)) if there exist constants c>0 and nonnegative such that g(n) ≥ c f(n) for all n ≥ . • Definition: g(n) εΘ(f(n)). What does it mean to be bounded above and below? • The function g(n) is in Θ(f(n)) if there exist non negative constants c1, c2 and such that c1 f(n) ≤g(n) ≤ c2 f(n)for all n ≥ . Other bounds

7. Example with F(n)=n*n

8. How should input size be measured? • Number of items in a list • Value of a parameter or the number of bits needed to store the parameter • What is the program’s most frequently occurring operation? • Look inside loops • Are there other factors than input size that affect the frequency of the most frequent operation? • Separate best, worst and average case analyses • How to tally the frequency? • Summations • Profiling Questions to answer when Analyzing non-recursive algorithms

9. Find max • Given a list containing n>0 integers (It is a separate question to ask the time for set-up. What would it be for this example?) • Pseudo-code max = list[0] for i = 0 to n-1 if list[i]>max max = list[i] • Analysis? Example 1

10. max = list[0] • O(1) = load + store • if… • Evaluate condition (a difference) • Conditional transfer (a jump, branch or skip depending upon flag values) • Assignment = load + store • O(??) • for… • Iterates n times • Total accumulated time = ? Example 1 Analysis

11. To understand recursion one must first understand recursion • Questions • How will one measure input size? • What is the core operation? • Do other factors than input size influence the number of repetitions? • What is a recurrence expressing the repetition of the core operation? • Solve the recurrence? Analysis of recursive algorithms

12. import random • import math • c = [0, 0] • alist=[] • n = int(input('Number of items for list = ' )) • for i in range(n): • x = random.randint(1, 1000) • alist.append(x) • ##print(len(alist)) • ##print('The initial list = ', alist) • defrecursfindmax(alis): • print(alis) • if len(alis) == 1: • c[0]+=1 • return alis[0] • else: • c[0]+=1 • amax = max(alis[0], recursfindmax(alis[1:len(alis)])) • return(amax) • deffindmax(alis): • amax = alis[0] • for i in range(len(alis)): • c[1]+=1 • if amax < alis[i]: • amax = alis[i] • return (amax) • print('Recursive max = ', recursfindmax(alist)) • print('Iterative max = ', findmax(alist)) • print('Count recursive, iterative ', c) Example 2:A recursive algorithm

13. t(n) = 1 + t(n-1) • t(n-1) = 1+ t(n-2) • t(n-2) = 1 + t(n-3) • t(n-3) = 1 + t(n-4) • … • t(n-(n-2)) = 1 + t(n-1) • t(n-n-1)) = t(1) = 1; this is the anchor • t(n) = 1 + t(n-1) = 2 + t(n-2) = 3 + t(n-3) = … = (n-1) + t(n-(n-1)) = n Analysis

14. Both methods are O(n), where n is the number of items in the list • How do the algorithms compare with respect to space utilization? Comparison

15. Given a homogeneous, second order, linear recurrence with constant coefficients (a≠0) of the form a y[n] + b y[n-1] + c y[n-2] = 0The recurrence has a closed form solution depending upon the roots and of the characteristic equationand the initial conditions for y[0] and y[1] Fun fact

16. If the roots are real and distinct • If there is a repeated root • If the roots are imaginary , and Fun fact (cont’d)

17. Fib(0) = 0 • Fib(1) = 1 • Fib(n) = Fib(n-1) + Fib(n-2) • Produces the homogeneous recurrence:Fib(n) – Fib(n-1) – Fib(n-2) = 0 • With characteristic equation: • Where a = 1, b = -1 and c = -1 Consider the fibonacci number recurrence

18. Thus, • The roots are real and distinct so that the recurrence is expressed by Solving the equation

19. Recall that Fib(0) = 0, Fib(1) = 1, and • Applying the first condition yields: • Applying the second condition yields: • Since , the second becomes: • Thus, and so that Applying initial conditions

20. Given • Recall and note that, • Thus, and notes

21. A(n) = 1 (addition)+1 A(n-1)+ 1A(n-2)=1 + 1 +A(n-2)+A(n-3)+ A(n-2)= 2 + 2 A(n-2) + 1 A(n-3)= 2 + 2(1+A(n-3)+A(n-4))+A(n-3)= 4 + 3 A(n-3) + 2 A(n-4)= 4 + 3(1+A(n-4)+A(n-5))+2A(n-4)= 7 + 5 A(n-4)+3 A(n-5)= 7 + 5(1 + A((n-5)+A(n-6))+ 3 A(n-5)= 12 + 8 A(n-5) + 5 A(n-6)= …= (Fib(i+2) -1)+Fib(i+1)A(n-i)+Fib(i)A(n-(i+1)) • Letting i = n-1, yields A(n) = Fib(n+1)-1,since A(0)=A(1)=0 Counting additions in recursive fibonacci

22. c=[0,0,0,0,0] • def fib(n): • c[0]+=1 • if n>1: • c[1]+=1 • return fib(n-1)+fib(n-2) • elif n==0: • c[2]+=1 • return 0 • else: • c[3]+=1 • return 1 • c[4]+=1 • print(fib(int(input("Supply a number: ")))) • print(c) Empirical Verficiation