CP4151 Advanced data structures and algorithms

Introduction to Algorithms
Chapter 1: The Role of Algorithms in
Computing

Computational problems
 A computational problem specifies an
input-output relationship
 What does the input look like?
 What should the output be for each input?
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 Example:
 Input: an integer number n
 Output: Is the number prime?
 Example:
 Input: A list of names of people
 Output: The same list sorted alphabetically

Algorithms
 A tool for solving a well-specified
computational problem
Algorithm
Input Output
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 Algorithms must be:
 Correct: For each input produce an appropriate output
 Efficient: run as quickly as possible, and use as little
memory as possible – more about this later
Algorithm
Input Output

Algorithms Cont.
 A well-defined computational procedure that
takes some value, or set of values, as input and
produces some value, or set of values, as output.
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 Written in a pseudo code which can be
implemented in the language of programmer’s
choice.

Correct and incorrect algorithms
 Algorithm is correct if, for every input instance, it ends
with the correct output. We say that a correct algorithm
solves the given computational problem.
 An incorrect algorithm might not end at all on some input
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 An incorrect algorithm might not end at all on some input
instances, or it might end with an answer other than the
desired one.
 We shall be concerned only with correct algorithms.

Problems and Algorithms
 We need to solve a computational problem
 “Convert a weight in pounds to Kg”
 An algorithm specifies how to solve it, e.g.:
 1. Read weight-in-pounds
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 1. Read weight-in-pounds
 2. Calculate weight-in-Kg = weight-in-pounds *
0.455
 3. Print weight-in-Kg
 A computer program is a computer-
executable description of an algorithm

The Problem-solving Process
Problem
specification
Analysis
Design
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Algorithm
Program
Executable
(solution)
Implementation
Compilation

From Algorithms to Programs
Problem
Algorithm
Algorithm: A sequence
of instructions describing
how to do a task (or
process)
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C++ Program
C++ Program

Practical Examples
 Internet and Networks
The need to access large amount of information with the
shortest time.
Problems of finding the best routs for the data to travel.
Algorithms for searching this large amount of data to quickly
find the pages on which particular information resides.
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find the pages on which particular information resides.
 Electronic Commerce
The ability of keeping the information (credit card numbers,
passwords, bank statements) private, safe, and secure.
Algorithms involves encryption/decryption techniques.

Hard problems
 We can identify the Efficiency of an algorithm
from its speed (how long does the algorithm take
to produce the result).
 Some problems have unknown efficient solution.
 These problems are called NP-complete
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 These problems are called NP-complete
problems.
 If we can show that the problem is NP-complete,
we can spend our time developing an efficient
algorithm that gives a good, but not the best
possible solution.

Components of an Algorithm
 Variables and values
 Instructions
 Sequences
 A series of instructions
Procedures
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 Procedures
 A named sequence of instructions
 we also use the following words to refer to a
“Procedure” :
 Sub-routine
 Module
 Function

Components of an Algorithm Cont.
 Selections
 An instruction that decides which of two possible
sequences is executed
 The decision is based on true/false condition
Repetitions
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 Repetitions
 Also known as iteration or loop
 Documentation
 Records what the algorithm does

A Simple Algorithm
 INPUT: a sequence of n numbers
 T is an array of n elements
 T[1], T[2], …, T[n]
 OUTPUT: the smallest number among them
min = T[1]
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 Performance of this algorithm is a function of n
min = T[1]
for i = 2 to n do
{
if T[i] < min
min = T[i]
}
Output min

Greatest Common Divisor
 The first algorithm “invented” in history was Euclid’s
algorithm for finding the greatest common divisor
(GCD) of two natural numbers
 Definition: The GCD of two natural numbers x, y is
the largest integer j that divides both (without
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Definition: The GCD of two natural numbers x, y is
the largest integer j that divides both (without
remainder). i.e. mod(j, x)=0, mod(j, y)=0, and j is the
largest integer with this property.
 The GCD Problem:
 Input: natural numbers x, y
 Output: GCD(x,y) – their GCD

Euclid’s GCD Algorithm
GCD(x, y)
{
while (y != 0)
{
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t = mod(x, y)
x = y
y = t
}
Output x
}

Euclid’s GCD Algorithm – sample run
while (y!=0) {
int temp = x%y;
x = y;
y = temp;
}
Example: Computing GCD(72,120)
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Example: Computing GCD(72,120)
temp x y
After 0 rounds -- 72 120
After 1 round 72 120 72
After 2 rounds 48 72 48
Output: 24

Algorithm Efficiency
 Consider two sort algorithms
 Insertion sort
 takes c1n2 to sort n items
 where c1 is a constant that does not depends on n
 it takes time roughly proportional to n2
 Merge Sort
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 takes c2 n lg(n) to sort n items
 where c2 is also a constant that does not depends on n
 lg(n) stands for log2 (n)
 it takes time roughly proportional to n lg(n)
 Insertion sort usually has a smaller constant factor than
merge sort
 so that, c1 < c2
 Merge sort is faster than insertion sort for large input sizes

Algorithm Efficiency Cont.
 Consider now:
 A faster computer A running insertion sort against
 A slower computer B running merge sort
 Both must sort an array of one million numbers
 Suppose
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 Suppose
 Computer A execute one billion (109) instructions per
second
 Computer B execute ten million (107) instructions per
second
 So computer A is 100 times faster than computer B
 Assume that
 c1 = 2 and c2 = 50

Algorithm Efficiency Cont.
 To sort one million numbers
 Computer A takes
2 . (106)2 instructions
109 instructions/second
= 2000 seconds
 Computer B takes
50 . 106 . lg(106) instructions
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50 . 10 . lg(10 ) instructions
107 instructions/second
 100 seconds
 By using algorithm whose running time grows more slowly,
Computer B runs 20 times faster than Computer A
 For ten million numbers
 Insertion sort takes  2.3 days
 Merge sort takes  20 minutes

Pseudo-code conventions
Algorithms are typically written in pseudo-code that is similar to C/C++
and JAVA.
 Pseudo-code differs from real code with:
 It is not typically concerned with issues of software
engineering.
 Issues of data abstraction, and error handling are often
ignored.
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ignored.
 Indentation indicates block structure.
 The symbol " " indicates that the remainder of the line is a
comment.
 A multiple assignment of the form i ← j ← e assigns to both
variables i and j the value of expression e; it should be treated as
equivalent to the assignment j ← e followed by the assignment i ← j.

Pseudo-code conventions
 Variables ( such as i, j, and key) are local to the given procedure.
We shall not us global variables without explicit indication.
 Array elements are accessed by specifying the array name
followed by the index in square brackets. For example, A[i]
indicates the ith element of the array A. The notation “…" is used
to indicate a range of values within an array. Thus, A[1…j]
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to indicate a range of values within an array. Thus, A[1…j]
indicates the sub-array of A consisting of the j elements A[1],
A[2], . . . , A[j].
 A particular attributes is accessed using the attributes name
followed by the name of its object in square brackets.
 For example, we treat an array as an object with the attribute
length indicating how many elements it contains( length[A]).

CP4151 Advanced data structures and algorithms

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