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Algorithm Correctness & Analysis

SummaryConfidence in algorithms from

testing and correctness proofCorrectness of recursive algorithms

proved directly by inductionExamples: Fibonacci numbers,

maximum, multiplication

CorrectnessHow do we know that an algorithm works?Logical method for checking correctness

TestingCorrectness proof

Testing vs. Correctness ProofsTesting: try the algorithm on sample inputsCorrectness Proof: Prove mathematically;

testing may not found obscure bugsUsing testing alone can be dangerous

Correctness of Recursive algorithmsTo prove correctness of recursive algorithm:

Prove it by induction on the size of the problem being solved

Base of recursion is base of inductionNeed to prove the recursive calls are given

sub-problems, i.e., no infinite recursionInductive step: assume that recursive calls

work correctly, and use this assumption to prove that the current call works correctly

Recursive Fibonacci NumbersFibonacci numbers: F0 = 0, F1 =1, and for all n ≥ 2,

Fn = Fn-2 + Fn-1

function fib(n) Comment return Fn

1. If n ≤ 1 then return (n)2. else return (fib(n-1) + fib (n-2))

Recursive Fibonacci NumbersClaim: For all n ≥ 0, fib(n) return Fn

Base: For n =0, fib(n ) returns 0 as claimed. For n = 1, fib(n) returns 1 as claimed.Induction: Suppose that n ≥ 2 and for all 0 ≤ m < n, fib (m) returns Fm .

Required to prove fib(n) returns Fn

What does fib(n) returns?

fib(n-1) + fib(n-2) = Fn-1 + Fn-2 (by Ind. Hyp.)

= Fn

Recursive Maximum

Recursive maximum

Recursive Multiplication

Analysis

Summary

Constant Multiples

Analyze the resource usage of an algorithm to within a constant multiple.

Why? Because other constant multiples creep in when translating

from an algorithm to executable code: Programmer abilityProgrammer effectivenessCompilerComputer hardwareRecall: Measure resource usage as a function of input size

Big oh

Example

Most big-Os can be proved by induction.

First Example: log n = O(n).

Claim : For all n >1, log n <n . The proof is by induction on n.

The claim is trivially for n=1,since 0<1. Now suppose n > 1 and

log n < n. Then,

log (n+1)

< log 2n

= log n+1

< n+1 (by ind. hyp.)

Second example

Note that we need

Big Omega

Big Theta

True or false

Adding Big Ohs

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Multiplying Big Ohs

Types of Analysis

Example…

Time Complexity

Multiplication

Bubble sort

Analysis Trick

Example

Lies, Damn Lies, and Big-Os

Algorithms and Problems

Algorithms Analysis 2

The Heap

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To Delete the Minimum

But we have lost the tree structure

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But we have violated the structure condition

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What Does it work?

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To Insert a New Element

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Why Does it Work

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Implementing a Heap

Analysis of Priority Queue Operation

Analysis of l(n)

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Example

Heapsort

Building a Heap Top Down

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Building a Heap Bottom Up

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Algorithms Course Notes Algorithm Analysis 3

summaryAnalysis of recursive algorithms:Recurrence relationsHow to derive themHow to solve them

Deriving Recurrence Relations

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Example

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Analysis of Multiplication

Solving Recurrence Relations

The Multiplication Example

Repeated Substitution

Warning

Reality Check

Merge Sorting

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A General Theorem

Proof Sketch

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Geometric Progressions

Back to the Proof

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Messy Details

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Example

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