What is the Algorithms

Introduction to Algorithms

lecture 1

What is the Algorithms



A sequence of procedure input -> output



example – finding max value

60 80 70 90 40 20 60 100

Expression

 Natural language

 ^{flow chart}

 pseudo code

 Programming language

Data structure



data structure is important in Algorithms



^array

 ^Pros

 ^Cons



linked list

 ^Pros

 ^{Cons … 10 50 30 40 …}

10 50 30 40 20 15

Data structure



^queue

FIFO (First In First Out)



^stack

LIFO (Last In First Out)

In 40 30 20 10 Out

In Out 40

30 20 10

Insert 10, 20, 30, 40

Data structure



^graph

 ^{G=( V, E )}

V: vertex, E: edge

 undirected graph

 directed graph

 ^loop

 degree, in-degree, out-degree

 ^path

u v

u v

(u,v) = (v,u)

<u,v>

u v

Data structure



^graph

 ^tree

rooted tree binary tree

root

leaf

ancestor descendant parent

child sibling depth height

Evaluation of an Algorithm



^Accuracy



^Efficiency



^{big O}

 O(1) : constant

 O(logn) : logarithmic

 O(n) : linear

 O(nlogn)

 O(n²) :quadratic

 O(n³) :cubic

 O(n^k) : polynomial

 O(2ⁿ) :, exponential

 O(n!)

Exchanging coins

Greedy Algorithm



Choose next step based on current state



_greedy



_{Input: M= 37, c}=(25, 20, 10, 5, 1), d=5

Output: k =

Exchanging coins

Greedy Algorithm

O(Md)

O(d)

GreedyChange ( M, c, d ) GreedyChange2 ( M, c, d )

Evaluation

(correctness)



Coin change problem

M=40, c = (25,20,10,5,1) greedy: k=

Correct answer: k=

Coin change problem

Exhaustive search/Brute force Algorithm) BruteForceChange ( M, c, d )

O(M^d)

Examines every possible alternative to find one particular solution