Đề thi cấu trúc dữ liệu và giải thuật dsa ch2 complexity algorithm (2)

Complexity of Algorithms Dr Nguyen Ho Man Rang Algorithm Efficiency Asymptotic Analysis Problems and common complexities P and NP Problems 2 1 Chapter 2 Complexity of Algorithms Data Structures and Al[.]

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Dr Nguyen HoMan Rang

AlgorithmEfficiencyAsymptoticAnalysisProblems andcommoncomplexities

P and NPProblems

Chapter 2

Complexity of Algorithms

Data Structures and Algorithms

Dr Nguyen Ho Man Rang Faculty of Computer Science and Engineering

University of Technology, VNU-HCM

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Outcomes

• L.O.1.1 - Define concept “computational complexity”

and its sepcial cases, best, average, and worst.

• L.O.1.2 - Analyze algorithms and use Big-O notation to

characterize the computational complexity of algorithms

composed by using the following control structures:

sequence, branching, and iteration (not recursion).

• L.O.1.3 - List, give examples, and compare complexity

classes, for examples, constant, linear, etc.

• L.O.1.4 - Be aware of the trade-off between space and

time in solutions.

• L.O.1.5 - Describe strategies in algorithm design and

problem solving.

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Sources of Materials

This materials reused some slides from the following source:

• course CS2210 Data Structures and Algorithms

instructed by Professor Olga Veksler Department of

Computer Science, University of Western Ontario.

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Algorithm Efficiency

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Compare Algorithms

• Given 2 or more algorithms to solve the

same problem, how do we select the best

one?

• Some criteria for selecting an algorithm

• Is it easy to implement, understand, modify?

• How long does it take to run it to completion?

• How much of computer memory does it use?

• Software engineering is primarily concerned

with the first criteria

• In this course we are interested in the

second and third criteria

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Algorithm Efficiency

that an algorithm needs to run to

completion.

an algorithm needs to run.

• We will occasionally look at space

complexity, but we are mostly interested in

time complexity in this course.

• Thus in this course the better algorithm is

the one which runs faster (has smaller time

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How to Calculate Running time

• Most algorithms transform input objects

into output objects.

sorting algorithm

input object output object

• The running time of an algorithm typically

grows with the input size f (n).

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

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How to Calculate Running Time

running time can be very different.

• Example: algorithm that searches an array

containing n integers to find the one with a

particular value K (assume that K appears

exactly once in the array)

• best case

• worst case

• average case

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How to Calculate Running Time

useless

often difficult to determine

time

• Easier to analyze

• Crucial to applications such as games,

finance and robotics

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Evaluations of Running Time

running time between agorithms:

• Experimental evaluation

• Theoretical evaluation

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Experimental Evaluation of Running Time

implementing the

algorithm

inputs of varying size

and composition

actual running time

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

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Limitations of Experiments

• It is necessary to implement the algorithm,

which may be difficult

• Results may not be indicative of the

running time on other inputs not included

in the experiment

• In order to compare two algorithms, the

same hardware and software environments

must be used

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Theoretical Analysis of Running Time

algorithm instead of an

implementation

function of the input size, n

algorithm independent of the

hardware/software environment

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Primitive Operations

primitive or basic operations, which

are simple computations performed by

an algorithm

• Evaluating an expression: x 2 + 3x

• Assigning a value to a variable: x = y

• Indexing into an array: a[10]

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Counting Primitive Operations

• By inspecting the pseudocode, we can determine the

maximum number of primitive operations executed by

an algorithm, as a function of the input size

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Counting Primitive Operations

• By inspecting the pseudocode, we can determine the

maximum number of primitive operations executed by

an algorithm, as a function of the input size

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Estimating Running Time

primitive operations in the worst case.

two linear functions

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Growth Rate of Running Time

environment

• Thus we focus on the big-picture which is

the growth rate of an algorithm

• The linear growth rate of the running time

f (n) is an intrinsic property of algorithm

findMax()

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Dependent Quadratic Loops

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factors to focus on the essential part

of the running time?

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Asymptotic Analysis

measurement of an algorithm’s

efficiency.

change the function’s magnitude are

eliminated

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Asymptotic Analysis

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Big-Oh notation

characterize running times and space

bounds

ignore constant factors and lower

order terms and focus on the main

components of a function which

affect its growth

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Big-Oh notation

say that f (n) is O(g(n)) if there are

• Why?

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Big-Oh Rules

Example

f (n) = c.n ⇒ f (n) = O(n)

f (n) = n(n + 1)/2 = n 2 /2 + n/2 ⇒ f (n) = O(n 2 )

others.

Some example of Big-O:

1 log 2 n n n log 2 n n 2 n k 2 n

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Big-Oh and Growth Rate

bound on the growth rate of a

function

means that the growth rate of f (n) is

no more than the growth rate of g(n)

rank functions according to their

growth rate

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Standard Measures of Efficiency

Assume instruction speed of 1 microsecond and 10

instructions in loop.

n = 10000

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Standard Measures of Efficiency

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Relatives of Big-Oh

• f (n) is Ω(g(n)) if there is a constant c > 0 and

• f (n) is Θ(g(n)) if there are constants c 0 > 0

and c 00 > 0 and an integer constant n 0 ≥ 1

such that c 0 g(n) ≥ f (n) ≥ c 00 g(n) for n ≥ n 0

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Intuition for Asymptotic Notation

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Problems and common

complexities

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Binary search

Recurrence Equation (Phương trình hồi quy)

An equation or inequality that describes a

function in terms of its value on smaller input

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Sequential search

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Quick sort

Recurrence Equation

T (n) = O(n) + 2T (n/2)

• Best case: T (n) = O(n log 2 n)

• Worst case: T (n) = O(n 2 )

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P and NP Problems

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machine).

(can be solved in polynomial time on

a nondeterministic machine).

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P and NP Problems

Travelling Salesman Problem:

A salesman has a list of cities, each of which he must visit

exactly once There are direct roads between each pair of

cities on the list.

Find the route the salesman should follow for the shortest

possible round trip that both starts and finishes at any one

6

8

9 11

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P and NP Problems

Travelling Salesman Problem:

Deterministic machine: f (n) = n(n − 1)(n − 2) 1 = O(n!)

6

8

9 11

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P and NP Problems

NP-complete : NP and every other problem in NP is

polynomially reducible to it.

NP

P

NP-complete

P = NP?

Tiêu đề	Complexity of Algorithms
Tác giả	Dr. Nguyen Ho Man Rang
Người hướng dẫn	Professor Olga Veksler
Trường học	University of Technology, VNU-HCM
Chuyên ngành	Data Structures and Algorithms
Thể loại	lecture notes
Thành phố	Ho Chi Minh City

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