Class 19: Analyzing Algorithms
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This document summarizes a class about analyzing algorithms. It recaps asymptotic operators like Big-O, Omega, and Theta notation. It provides examples of using these notations to analyze algorithms and prove statements about runtime complexity. Specifically, it analyzes a "bigger" algorithm that compares two inputs and returns the larger value. It discusses measuring runtime based on input size rather than input values. The document emphasizes that proofs about algorithms are constructed by finding constants and input sizes that satisfy the definitions of the asymptotic notations.
Class 19: Analyzing Algorithms
- 1. Class 19: Analyzing Algorithms cs1120 Fall 2011 David Evans 5 October 2011
- 2. Plan Asymptotic Operators Recap Analyzing bigger How Bletchley Park Really Broke Lorenz (maybe not until Friday) Assistant Coaches’ Review Sessions for Exam 1: Tuesday (tomorrow), 6:30pm, Rice 442 Wednesday, 7:30pm, Rice 442 2
- 3. A XOR Machine B TRUE FALSE TRUE Design by Alex Shin 3
- 4. XOR Machine Wine Wine Float Wine Float NOT from book Design by Anthony Teate 4
- 5. XOR Machine Wine Wine Float Wine Float Wine Float NOT from book Design by Anthony Teate 5
- 6. Measuring Cost Machines vary, so we want to understand how the cost to execute a procedure scales with the size of the input We use the asymptotic operators to abstract away all the details 6
- 7. Recap Big-O: functions that grow no faster than f Omega ( ): functions that grow no slower than f Theta ( ): functions that grow as fast as f 7
- 8. Example Prove: 73n2 + n (n2) 8
- 9. Real CSists 9
- 10. Proofs are Not Scary! Theorem: There exists a polygon with four sides. Proof: What kind of proof is this?
- 11. Proof by Construction We can prove a “there exists an X with property Y” theorem, by showing an X that has property Y O(f) means “there are positive constants c and n0 such that g(n) cf(n) for all n n0 So, to prove g is in O(f) we need to find c and n0 and show that g(n) cf(n) for all n n0
- 12. Dis-Proof by Construction To prove g is not in O(f): O(f) means: there are positive constants c and n0 such that g(n) cf(n) for all n n0 So, to prove g is not in O(f) we need to find a way given any c and n0, to find an n n0 such that g(n) > cf(n).
- 13. Example Prove: 73n2 + n (n2) 13
- 14. Algorithm Analysis What is the asymptotic running time of the Scheme procedure below: (define (bigger a b) (if (> a b) a b)) 14
- 15. Algorithm Analysis What is the asymptotic running time of (> a b): (define (bigger a b) (if (> a b) a b)) 15
- 16. What are we really measuring? Input size: number of tape squares it takes to write down the input Running time: number of steps it takes before TM enters a final state Input size for (bigger a b) = n Number of tape squares to represent input (not its magnitude) Why doesn’t the alphabet size matter? 16
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- 18. Next Class What if the input is binary instead of unary? a>b# 0# if a > b, 1# if a b where a, b are the numbers a, b represented in binary 18