data structure
TRANSCRIPT
Data Structures and AlgorithmsLecture # 1
Book: Fundamentals of Data Structures in c++
Horwitz, Sahani, and Mehta
Software Design Quality
• What is good design? - Is it efficient code?
- compact implementation? - Most maintainable?
. For most large, long life time software systems,maintenance cost normally exceeds development cost byfactors ranging from 2 to 3.
Maintainable DesignMaintainable Design
• Cost of system changes is minimal
• Readily adaptable to modify existing functionality and enhance functionality
• Design is understandable
• Changes should be local in effect
• Design should be modular
AbstractionAbstraction. Abstraction is a technique in which we construct a
model of an entity based upon its essentialCharacteristics while ignoring the inessential details.
. The principle of abstraction also helps in handling theinherent complexity of a system by allowing looking at its important external characteristic, and hiding itsinner complexity at the same time.
. Hiding the internal details is called encapsulation.
. Engineers of all fields, including computer science,have been practicing abstraction for masteringcomplexity.
Types of data Abstraction
. Code and Data abstraction
– What is Data ?
- What is code ?
Advantages of data Abstraction
• Simplification of software development
• Testing and Debugging
• Reusability
• Modification to representation of a data type
• etc
• void selectionSort(int a[],int size){
int I,j,min,temp;for(i=0; i<size-1; i++){
min=i; for(j=i; i<size; j++)
{if(a[j]< a[min])
min=j;}temp=a[i];a[i]=a[min];
a[min]=temp;}
}
int minimum(int a[],int from,int to) void swap(int &x, int &y){ {
int min=from; int temp=x;for(int i=from;i<=to;i++) x=y; if(a[i] < a[min]) min=i; y=temp;return min;
} }
void selectionSort(int a[],int size){
int i,j,min;for(i=0;i<size-1;i++){
min=minimum(a,I,size-1)swap(a[i],a[min]);
}}
void selectionSort(int a[], int size)
{
int I,j,min,temp;
for(i=0;i<size-1;i++)
{
min=i; void selectionSort(int a[],int size)
for(j=i;j<size; j++) {
{ int i,j,min;
if(a[j]< a[min]) for(i=0;i<size-1;i++) {
min=j; min=minimum(a,i,size-1);
} swap(a[i],a[min]);
temp=a[i]; }
a[i]=a[min]; }
a[min]=temp;
}
}
Data Abstraction and Abstract Data Types(ADT)
• A data type is a template for objects and a set of
operations that define the behavior of the objects (or
instances) of that type.• An Abstract data type (ADT) is a data type in which
the implementation details are hidden and the user is
concerned with the properties ( or behavior ) of that
type.• An ADT has two commponents:
- Interface – the behavior
- Implementation• Example:
-int,float
Abstract Data Type
• The data structures used to
implement the data type can only
be accessed through the interface.• Any change in the implementation
does not change the user
programs as long as the interface
remains the same.• This is also known as data
encapsulation or data abstraction.
implementation
interface1 interfece2
interface3 interface4
Abstraction Vs.Implementation
• X -> 01000001 01000010 01000011 00000000• X = ?
• If x is CString
-then x -> “ABC”• If x is integer
- then x -> 1094861568
int main(){
int i, *pi;float f, *pf;
i = 1024;pi = &i;
pf = (float *) pi ;
f = *pf;cout << i << “ “ <<f<<“\n”;
f = i ;cout << i << “ “ <<f<<“\n”;
return 0;
}
1024 1.43493e-042 1024 1024press any key to continue
Abstraction Vs. ImplementationTwo dimensional array
1 5 3 6
3 2 38 64
22 76 82 99
0 106 345 54
User’s view (abstraction)
1 5 3 6 3 2 38 64 22 76 82 99 0 106 345 54
System’s view (implementation)
ADTs and C++ Classes
• A class is used to define (and implement) an ADT
in C++.• A class consists of data and functions that operate
on that data.• A class in C++ has two parts – public and private
(let’s ignore the protected members for now).• The data and functions defined in the class are
called the members of the class.
ADTs and C++ Classes
• Users of the class can only access and
manipulate the class state through the public
members of the class.• Private members can only be used by other
members of the class (let’s also ignore the friends for now).
• Data encapsulation is achieved by declaring
all data members of a class to be private.• The interface of the class is provided through
the use of public member functions.
Data Structures
• The primary objective of programming is to efficiently
process the input to generate the desired output.• We can achieve this objective in an efficient and neat
style if the input data is organized in a way to help us
meet our goal.• Data Structures is nothing but ways and means of
organizing data so that it can be processed easily and
efficiently.• Data structures dictate the manner in which the data
can be processed. In other words, the choice of an
algorithm depends upon the underlying data
organization. ( What is an Algorithm ? )
What is an Algorithm ?
• An algorithm is a well defined list of steps for solving a particular problem
• An algorithm manipulates the data in data structures in various ways, such as inserting a new element, searching for a particular item etc.
• An algorithm must satisfy the following criteria1) Input 2) output 3) Definiteness ( each
instruction should be clear and unambiguous) 4) Fitness (terminates after finite number of steps) 5) Effectiveness (each instruction must be feasible enough)
Data Structure Operations
• Traversing• Searching• Inserting• Deleting• Sorting• Merging• Recursion• To perform operations on various data structures
we use algorithms.
Types of Data Structures
• Premitive/Scalar : data types that can be manipulated as a single quantity or can be represented alone
• Structured/ Non-Premitive (Data type which is collection of other premitive or non-premitive data structures.– Can be further divided into
a) linear b) non-linear- Linear can be further split into a) physically linear b) logically linear
problem: Determine if a key is present in a collection of 10 integers
Organization 1: Data are stored in 10 disjoint ( as opposed to
composite ) variables: A0, A2, A3,……,A9
Algorithmfound=false;
if (key = = A0 ) found = true;
else if (key = = A1 ) found = true;
else if (key = = A2 ) found = true;
else if (key = = A3 ) found = true;
else if (key = = A4 ) found = true;
else if (key = = A5 ) found = true;
else if (key = = A6 ) found = true;
else if (key = = A7 ) found = true;
else if (key = = A8 ) found = true;
else if (key = = A9) found = true;
problem: Determine if a key is present in a collection of 10 integers
Organization 2: Data are stored in an array of 10 elements
Algorithmfound=false;
for (int i =0; i < 10; i ++) if ( A[i] == key) {
found = true;break;
}
• Average number of comparisons in both cases is the same soboth algorithms are equally efficient (or in efficient)
• Organization 2 is better because it yields an algorithms which ismore maintainable. For example, if the collection contains 100 elements. In general, number of elements could be N.
problem: Determine if a key is present in a collection of 10 integers
Organization 3: Data are stored in an array A in ascending order
Algorithmfound=false;
while (( ! Found) && (low<= high))
{
mid = (low + high)/2;
if( A[mid]==key) found=true;
else if ( A[mid] > key) high = mid – 1;
else low = mid +1;
}
• Average number of comparisons ?• Order of “log(N)” as compared to N.
found=false;
while (( ! Found) && (low<= high))
{
mid = (low + high)/2;
if( A[mid]==key) found=true;
else if ( A[mid] > key) high = mid – 1;
else low = mid +1;
}
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 3 5 7 10 12 15 22 28 29 32 47 48 50 55 73
Key=29
Performance Comparison
• NADRA database: ~80,000 records
• Computer which can perform 10,000
comparisons per second- Linear Search: ~2.22 hours
- Binary Search: ~0.005 seconds
- Roughly 1.6 million times less
• Why?
Performance Analysis
1. Does the program efficiently use primary and secondary storage?
2. Is the program’s running time acceptable forthe task?
3. Space Complexity:. The space complexity of a program is the measure of the amount of memory that it needs to run to
completion.4. Time Complexity:
. The time complexity of a program is the measure of the amount of computer time it needs to run to completion.
Performance Estimation
• How to determine which algorithm is
better?
• We need some mechanism to predict
the performance without actually
executing the program.
• Mechanism should be independent of
the compiler and underlying hardware.
Step Count
• Program Step: A program step is a
meaningful program segment.
• We can consider each statement as a
single step.a = 2;
a = 2 * b + c + 3 * c / d – e;
Step Count
To count total number of steps, we must
determine:1. Step count for each statement –
steps/execution or s/e
2. Frequency of each statement
3. Total steps for each statement
4. Finally sum these counts to get the total step
count
Example 1 – Summing of a list ofnumbers (Step count Table)
Statement S/e Frequency Total Steps
Float sum (float list[],int n) 0 0 0
{ 0 0 0
float temp=0; 1 1 1
int i ; 0 0 0
for (i=0;i<n; i++) 1 n + 1 n + 1
temp+=list[i] ; 1 n N
return temp; 1 1 1
} 0 0 0
Total 2n+3
ProblemsDetermining the exact step count of a program can be a
Very difficult task
- inexactness of the definition of a step, exact step count is not very
useful for - comparative purposes.e.g. which one is better 45n+3 or 100n+10
- Frequency of execution
. How many steps are executed?
if (condition)
{
step1;step2step3;step4;step5;
}
else
step6;
We need some asymptotic notation as a measure of growth
Big Oh (O)
Big Oh is defined as:
F(n)=O(g(n)) iff there exists positive constants c and n0
such that f(n)<= cg(n) for all values of n > = n0
• No matter what the value of c1, c2,c3 there will be an n
beyond which the program with complexity c3n will be faster than the one with complexity c1n2+c2n
Example: 1) 3n +3 = O(n) as 3n +3 <=4n for all n >=3
2) 10n2 + 4n + 2 = O(n2) as 10n2 + 4n +2 <=11n2
• An estimate of how will the running time grow as a function
of the problem size.
• There are infinitely many functions g for a given function f:
- N = O(N); N = O(N2) N = O(N3)
- Choose the smallest function g.
• Theorem:If f(n)=amnm+…..a1n+a0, then f(n)=O(nm)
-Choose the largest term in the polynomial
----------------------n
growth
Big Oh (O)
• Summing of list of numbers O(n)
• Matrix addition O(rows,cols)
• Searching a key in an array O(n)
• Binary Search O(n log n)
Big Oh (O)
int search_min(int list[],int from, int to){
int I; int min=from;for ( i=from;I <= to; i++)
If (list[i] < list[min]) min=I;return min;
}// O(n)
void swap(int &a, int &b)
{
int temp=a;
a=b;
b=temp;
}
// O (1)
Big Oh (O)
void Selection_sort(int list[], int size)
{int I,j;for(i=0;i<size-1;i++){ //O(size) or O(n)
j= search_min(list, i+1 , size-1) //O(n).O(n)=O(n2)
swap (list[i], list[j]) //O(1).O(n)=O(n)
}//O(n)+O(n2)+O(n)+O(n)= O(n2)
}
Big Oh (O)
• O(1) - constant
• O(log n) - logarithmic
• O(n) - linear
• O(n log n)- log linear
• O(n2) - quadratic
• O(n3) - cubic
• O(2n) - exponential
Assignment# 1Last Submission Day : 07-09-2009 (Monday)
• Given the following code segment int x=0; for (int i= 1; i<=n; i++)
for (int j = 1; j<=I; j++) for (int k = 1; k<=j; k++)
x++;a) What will be the value of x in terms of n after the following code is
executed ?b) Give the time complexity of the above code in big Oh notation• Prove the following i) 5n2 – 6n = O(n2) ii) 6*2n + n2 = O (2n)