Chapter 5Algorithms (2)

Introduction to CS

1st Semester, 2015 Sanghyun Park

Outline Informal Definition of an Algorithm (previous file) FindLargest (previous file) Three Basic Constructs (previous file) Sorting Algorithms (previous file) Searching Algorithms (previous file) Recursion (previous file) Algorithm Performance Time Complexity Asymptotic Notation Growth Rate

Algorithm Performance Algorithm ____________ is the amount of computer

memory and time needed to run an algorithm

How is it determined? Analytically : performance ________ Experimentally : performance ____________

Space complexity is defined as the amount of ________an algorithm needs to run to completion

Time complexity is defined as the amount of computer _____ an algorithm needs to run to completion

Time Complexity How to measure?

Count a particular operation (________ counts) Count the number of steps (_____ counts) ___________ complexity

Running Example: _________ Sort

for (int i = 1; i < n; i++) // n is the number of elements in an array{

// insert a[i] into a[0:i−1]int t = a[i];int j;for (j = i−1; j ≥ 0 && t < a[j]; j--)

a[j+1] = a[j];a[j+1] = t;

}

Operation Count Pick an ________ characteristic … n,

n = the number of elements in case of insertion sort

Determine countas a ________ of this instance characteristic

Comparison Count

How many comparisons are made? The number of comparisons depends on ___ and __

as well as on __

______ case count = maximum count ______ case count = minimum count ______ case count

for (int i = 1; i < n; i++)for (j = i−1; j ≥ 0 && t < a[j]; j--)

a[j+1] = a[j];

Worst Case Comparison Count

for (j = i−1; j ≥ 0 && t < a[j]; j--)a[j+1] = a[j];

a = [1,2,3,4] and t = 0 → __ comparisons a = [1,2,3,4,…,i] and t = 0 → __ comparisons

for (int i = 1; i < n; i++)for (j = i−1; j ≥ 0 && t < a[j]; j--) a[j+1] = a[j];

Total comparisons = 1+2+3+…+(n−1) = ________

Asymptotic Complexity ofInsertion Sort

O(__)

What does this mean? Time or number of operations does not ________ cn2

on any input of size n So, the worst case time is expected to ________

each time n is doubled

Growth Rates It is often necessary to __________ the number of steps

required to execute an algorithm

We can approximate one function with another function using a _______ rate

We will examine the asymptotic behavior of two functions f and g by comparing f(n) and g(n) for_____ positive values of n

Asymptotic Notation Big O is an example of asymptotic notation

Bio O provides an ______ bound for an algorithm’s time performance

Other asymptotic notation Big Omega notation provides a ______-bound for an

algorithm’s time performance Big Theta notation is used when an algorithm can be bounded

_____ from above and below by the ______ function

Big Oh Notation This is written f(n) = O(g(n)) and is stated,

“f(n) is Big Oh g(n)”

A more mathematical definition of Big Oh is:f(n) = O(g(n)), if there are positive numbers c and msuch that ____________ for all n ≥ m

Since 5n3+4n2 <= 7n3 for all n ≥ 2, 5n3+4n2 = O(__)

Big Oh is often called an _______ bound

Big Omega Notation “f has an order greater than or equal to g”

if there are positive numbers c and m such that_____________ for all n ≥ m

This is written f(n) = (g(n)) and is stated as“f(n) is Big Omega g(n)”

Big Omega is often called a ______ bound

Big Theta Notation “f has same growth rate as g”

if we can find a number m and two nonzero numbersc and d such that _________________ for all n ≥ m

Big Theta (θ) notation is used when an algorithm can be bounded both from _______ and _______ by the same function

Common Growth Rate Functions (1/2)

1 (constant)growth is ______________ of the problem size N

log2N (__________)growth increases slowly compared to the problem size(binary search)

N (_______)growth is directly proportional to the size of the problem

N * log2Ntypical of some divide and conquer approaches(_______ sort)

Common Growth Rate Functions (2/2)

N2 (quadratic)typical in _______ loops

N3 (______)more nested loops

2N (___________)growth is extremely rapid and possibly __________

N!

Growth Rate log2N

Growth Rate N2

Practical Complexities

logN N NlogN N2 N3 2N

012345

1 2 4 8 16 32

0 2 8 24 64 160

1 4 16 64 256 1024

1 8 64 512 4096 32768

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