Foundations in Data Structure and Algorithms Truong Tuan Anh

CSE-HCMUT

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Outline

Basic conceptsRevision

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What is Data?

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What is Data?

DataData is information that has been translated into a form that ismore convenient to calculate and analyze.

ExampleNumbers, words, measurements, observations or descriptions

of things.

Qualitative data: descriptive informationQuantitative data: numerical information (numbers).

Discrete data: can only take certain values (like whole numbers)Continuous data: can take any value (within a range)

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Data Type

Class of data objects that havethe same properties.

Define a data type1. A set of values2. A set of operations on values

Example

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Data Structure

What is a data structure?1. A combination of elements in which each is either a

data type or another data structure2. A set of associations or relationships (structure)

that holds the data together

ExampleAn array is a number of elements of the same type in a specific order.

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Abstract Data Type

AimUsers know what a data type can do.How it is done is hidden.

DefinitionAn abstract data type is a data declaration packaged together with the operations that are meaningful for the

data type.

1. Declaration of data2. Declaration of operations3. Encapsulation of data and operations

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Abstract Data Type

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Example: List

InterfaceData: sequence of elements of a particular data typeOperations: accessing, insertion, deletion

ImplementationArrayLinked list

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Algorithm

What is an algorithm?The logical steps to solve a problem

What is a program?Program = Data structures + Algorithms

(Niklaus Wirth)

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Pseudocode

The most common tool to define algorithmsEnglish-like representation of the algorithm logic

Pseudocode = English + code

English: relaxed syntax being easy to readCode: instructions using basic control structures (sequential, conditional, iterative)

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Pseudocode

PseudocodeAlgorithm HeaderAlgorithm Body

Algorithm HeaderNameParameters and their typesPurpose: what the algorithm doesPrecondition: precursor requirements for the parametersPostcondition: taken action and status of the parametersReturn condition: returned value

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Pseudocode

PseudocodeAlgorithm HeaderAlgorithm Body

Algorithm BodyStatementsStatement numbers: decimal notation to express levelsVariables: important dataAlgorithm analysis: comments to explain salient pointsStatement constructs: sequence, selection, iteration

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Pseudocode: Example

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Data Structures

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Data Structures

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Data StructuresA member of an object can be accessed directly by a dot (.) inserted between the object name and the member name

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Data Structures

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Data Structures

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Exercise

Define a data structure student_t containing a student's name, first name, and age.

Write a code in C++ to take input your data and display it.

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Class

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Class: Example

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Class

Class ConstructorsAutomatically called whenever a new object of a class is created.Initializing member variables or allocate storage of the object.Declared with a name that matches the class name and without any return type; not even void.

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Class

Member initialization:

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ArrayAn array is a series of elements of the same type placed in

contiguous memory locations that can be individually referenced by a unique identifier with an index.

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Class

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Algorithm Efficiency

A problem often has many algorithms/solutionsComparing two different algorithms

How fast an algorithm is?How much memory does it cost?

Compare by calculate the computational complexity: Measure of the difficulty degree (time and/or space) of an algorithm.

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Algorithm Efficiency

How to calculate the efficiency

efficiency = f(n)

n is the size of a problem (the key number that determines the size of input data)

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Linear Loop

for (i = 0; i < 1000; i++)//application code

The number of times the body of the loop is replicated is 1000f(n) = n

for (i = 0; i < 1000; i+=2)//application code

The number of times the body of the loop is replicated is 500f(n) = n/2

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Linear Loop

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Logarithmic Loops

The number of times the body of the loop is replicated is f(n) = log2 n

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Logarithmic Loops

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Nested Loop

Iterations = Outer loop iterations * Inner loop iterations

The number of times the body of the loop is replicated isf(n) = n*log2 n

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Nested Loop

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Quadratic Loop

The number of times the body of the loop is replicated isf(n) = n2

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Dependent Quadratic Loop

The number of times the body of the loop is replicated isf(n) = 1 + 2 + … + n = n*(n+1)/2

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Quadratic Loop

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Asymptotic Complexity

Algorithm efficiency is considered with only big problem sizesWe are not concerned with an exact measurement of an algorithm's efficiencyTerms that do not substantially change the function's magnitude are eliminated

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Big-O Notation

O(g(n)) = f(n) : ∃ positive constants c and n0,

such that ∀n ≥ n0, we have0 ≤ f(n) ≤ c*g(n)

For function g(n), we define O(g(n)), big-O of n, as the set:

g(n) is an asymptotic upper bound for f(n).

Intuitively: Set of all functions whose rate of growth is the same as or lower than that of g(n).

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Examplef(n) = 2n + 5g(n) = n

Consider the condition2n + 5 <= n

will this condition ever hold? No!

How about if we tack a constant to n?2n + 5 <= 3n

the condition holds for values of n greater than or equal to 5This means we can select c = 3 and n0 = 5

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2n+5

n

2n+53n

point where 3n“beats” 2n+5

2n+5 is O(n)

Example

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Big-O Notation

Set the coefficient of the term to 1Keep the largest term and discard the othersSome example of Big-O:

log2n, n, nlog2n, n2, nk, 2n, n!

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arrayMax Example

Algorithm arrayMax(A,n):Pre: An array A storing n integers.Post: The maximum element in A.currentMax ← A[0]

for i ← 1 to n - 1 do

if currentMax < A[i] then

currentMax ← A[i]

return currentMax

What is the running time of arrayMax?

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arrayMax Example

Depending on what operations we decide to count, running time function

f(n) = a*n + bRegardless, f(n) is O(n)Or, equivalently, the running time of algorithm arrayMax is O(n)

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Standard Measures of Efficiency

Assume instruction speed of 1 microsecond and 10 instructions inloop

n = 10000

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Standard Measures of Efficiency

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Big-O Analysis Example

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Big-O Analysis Example

Nested loop

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Big Omega Ω-notation

g(n) is an asymptotic lower bound for f(n).

Intuitively: Set of all functions whose rate of growth is the same as or higher than that of g(n).

Ω(g(n)) = f(n) : ∃ positive constants c and n0,such that ∀n ≥ n0, we have

0 ≤ c*g(n) ≤ f(n)

For function g(n), we define Ω(g(n)), big-Omega of n, as the set:

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Example

5n2 + 7 is Ω(?)

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Time Costing Operations

The most time consuming: data movement to/from memory/storageOperations under consideration:

ComparisonsArithmetic operationsAssignments

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Time Costing

Big-O can be used for categorizing or characterizing functionsFor example, the statements:

2n + 3 is O(n) and 5n is O(n)→ 2n + 3 and 5n are in the same category

→ If the functions are running times of two algorithms, the algorithms are thus comparable

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Read More

Big-ThetaLittle-OLittle-Omega

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Takeaways

Basic conceptsBasic data structuresAlgorithm complexity