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DATA STRUCTURES USING C
Chapter 1 – Basic Concepts
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Overview Data: data is a simple value or set of values. Entity: Entity is some thing that has certain
attributes or properties which may assigned values.
Field: It is a single elementary unit of information representing an attribute of an entity.
Record: A record is a collection of field values of a given entity.
File: A file is a collection of records of the entities in a given entity set.
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Why Data Structures? Problem solving is more related to
understanding the problem, designing a solution and Implementing the solution, then
What exactly is a solution? In a very crisp way, it can be
demonstrate as a solution which is equal to a program and it is also approximately equal to algorithm.
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Algorithm An algorithm is a sequence of steps that take us
from the input to the output. An algorithm must be Correct. It should provide a
correct solution according to the specifications. Finite. It should terminate and general. It should work for every instance of a problem is
efficient. It should use few resources (such as time or
memory).
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What is Information? It is an undefined term that cannot be
explained in terms of more elementary concepts.
In computer science we can measure quantities of information.
The basic unit of information is bit. It is a concentration of the term binary
digit.
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Definition ! Data may be organized in many
different ways ; the logical / mathematical model of a particular, organization of data is called Data Structure.
Data structure can be also defined as, it is the mathematical model which helps to store and retrieve the data efficiently from primary memory.
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System Life Cycle Large-Scale program contains many
complex interacting parts. Programs undergo a development
process called the system life cycle. Solid foundation of the methodologies
in1. data abstraction 2. algorithm specification3. performance analysis and measurement
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5 phases of system life cycle:1. Requirements: Begins with purpose of the project. What
inputs (needs), outputs (results)2. Analysis: Break the problem down into manageable
pieces. bottom-up vs. top-down, divide and conquer3. Design: data objects and operations creation of abstract
data types, second the specification of algorithms and algorithm design strategies.
4. Refinement & coding: representations for our data objects and write algorithms for each operation on them.
we should write those algorithms that are independent of the data objects first.
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5. Verification: consists of developing correctness proofs for the program, testing the program with a variety of input data and removing errors.
Correctness proofs: proofs are very time-consuming and difficult to develop fro large projects. Scheduling constraints prevent the development of a complete set of proofs for a large system.
Testing: before and during the coding phase. testing is used at key checkpoints in the overall process to determine whether objectives are being met.
Error Removal: system tests will indicate erroneous code. Errors can be removed depending on the design & coding decisions made earlier.
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Pointers For any type T in C there is a corresponding
type pointer-to-T. The actual value of a pointer type is an address of memory. & the address operator. * the dereferencing ( or indirection ) operator. Declaration
• int i , *pi;then i is an integer variable and pi is a pointer to an integer.
• pi = &i;then &i returns the address of i and assigns it as the value of pi.
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• i = 10 or• *pi = 10; In both cases the integer 10 is stored as the value of i. In second case, the * in front of the pointer pi causes
it to be dereferenced, by which we mean that instead of storing 10 into the pointer, 10 is stored into the location pointed at by the pointer pi.
The size of a pointer can be different on different computers.
The size of a pointer to a char can be longer than a pointer to a float.
Test for the null pointer in C if (pi == NULL) or if(!pi)
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Dynamic Memory Allocation
When you will write program you may not know how much space you will need. C provides a mechanism called a heap, for allocating storage at run-time.
The function malloc is used to allocate a new area of memory. If memory is available, a pointer to the start of an area of memory of the required size is returned otherwise NULL is returned.
When memory is no longer needed you may free it by calling free function.
The call to malloc determines size of storage required to hold int or the float.
The notations (int *) and (float *) are type cast expressions.
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Example…!int i,*pi;float f,*pf;pi = (int *) malloc (sizeof(int));pf = (float *) malloc(sizeof(float));*pi=1024;*pf=3.124;printf(“an integer = %d, a float = %f\n”,*pi,*pf);free(pi);free(pf);
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When programming in C it is a wise practice to set all pointers to NULL when they are not pointing to an object.
Use explicit type casts when converting between pointer typespi = malloc(sizeof(int));/* assign to pi a pointer to int */pf = (float *) pi;/* casts an int pointer to float pointer */
In many systems, pointers have the same size as type int.
int is the default type specifier.
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Algorithm Specification An algorithm is a finite set of instructions that, if
followed, accomplishes a particular task. Algorithms must satisfy following criteria,
Input: there are zero or more quantities that are externally supplied.
Output: at least one quantity is produced. Definiteness: Each instruction is clear and
unambiguous. Finiteness: for all cases, the algorithm terminates after
a finite number of steps. Effectiveness: Every instruction must be basic enough
to be carried out, it must also be feasible.
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Difference between an algorithm and a program : the program does not have to satisfy the fourth condition (Finiteness).
Describing an algorithm: natural language such as English will do. However, natural language is wordy to make a statement definite. That is where the Code of a program language fit in.
Flowchart: work well only for algorithm, small and simple.
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Example:Translating a Problem into an Algorithm Problem
Devise a program that sorts a set of n integers, where n>= 1, from smallest to largest.
Solution I: looks good, but it is not an algorithm From those integers that are currently
unsorted, find the smallest and place it next in the sorted list.
Solution II: an algorithm, written in partially C and English for (i= 0; i< n; i++){ Examine list[i] to list[n-1] and suppose that
the smallest integer is list[min]; Interchange list[i] and list[min]; }
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Example: Program
#define swap(x,y,t) ((t)= (x), (x)= (y), (y)= (t))
void sort(int list[], int n){ int i, j, min, temp; for (i= 0; i< n-1; i++){ min= i; for (j= i+1; j< n; j++){ if (list[j]< list[min]) min= j; } swap(list[i], list[min], temp);}
void swap(int *x, int *y){ int temp= *x; *x= *y; *y= temp;}
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Recursive Algorithms Direct recursion
Functions call themselves Indirect recursion
Functions call other functions that invoke the calling function again
Any function that we can write using assignment, if-else, and while statements can be written recursively.
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int binsearch(int list[], int searchnum, int left, int right){// search list[0]<= list[1]<=...<=list[n-1] for searchnum int middle; while (left<= right){ middle= (left+ right)/2; switch(compare(list[middle], searchnum))
{ case -1: left= middle+ 1;
break; case 0: return middle; case 1: right= middle- 1;
} } return -1;}
int compare(int x, int y){ if (x< y) return -1; else if (x== y) return 0; else return 1;}
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Recursive Implementation of Binary Searchint binsearch(int list[], int searchnum, int left, int right){
int middle;if(left<=right)
{middle = (left+right)/2;switch(COMPARE(list[middle],searchnum)){
case -1: return binsearch(list,searchnum,middle+1,right);case 0: return middle;case 1: returnbinsearch(list,searchnum,left,middle-1);
}}return -1;
}
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Towers of Hanoi
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void Hanoi(int n, char x, char y, char z){
if (n > 1) {
Hanoi(n-1,x,z,y);printf("Move disk %d from %c to %c.\n",n,x,z);Hanoi(n-1,y,x,z);
} else{
printf("Move disk %d from %c to %c.\n",n,x,z); }
}
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Data Abstraction Data Type
A data type is a collection of objects and a set of operations that act on those objects.
if program is dealing with predefined or user-defined data types two aspects i,e objects & operations must be considered.
Example of "int" Objects: 0, +1, -1, ..., Int_Max, Int_Min Operations: arithmetic(+, -, *, /, and %), testing
(equality/inequality), assigns, functions
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Data Abstraction & Encapsulation
Abstraction: is extraction of essential information for a particular purpose and ingnoring the remainder of the information
Encapsulation: is the process of binding together of Data and Code. It can also be defined as the concept that an object totally separates its interface from its implementation.
It has been observed by many software designers that hiding the representation of objects of a data type from its users is a good design strategy.
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Abstract Data Type Abstract Data Type
An abstract data type (ADT) is a data type that is organized in such a way that the specification of the objects and the operations on the objects is separated from the representation of the objects and the implementation of the operations.
ADT Operations — only the operation name and its parameters are visible to the user — through interface
Why abstract data type ? implementation-independent
Abstraction is … Generalization of operations with unspecified
implementation
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Abstract data type model
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Classifying the Functions of a ADT Creator/constructor
Create a new instance of the designated type Transformers
Also create an instance of the designated type by using one or more other instances
Observers/reporters Provide information about an instance of the
type, but they do not change the instance
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Abstract data type NaturalNumber
ADT NaturalNumber is structure NaturalNumber is (denoted by Nat_No) objects: an ordered subrange of the integers starting at zero and ending at the maximum integer (INT_MAX) on the computer functions: for all x, y Nat_Number; TRUE, FALSE Boolean and where +, -, <, and == are the usual integer operations. Nat_No Zero ( ) ::= 0 Boolean IsZero(x) ::= if (x) return FALSE else return TRUE Nat_No Add(x, y) ::= if ((x+y) <= INT_MAX) return x+y else return INT_MAX Boolean Equal(x,y) ::= if (x== y) return TRUE else return FALSE Nat_No Successor(x) ::= if (x == INT_MAX) return x else return x+1 Nat_No Subtract(x,y) ::= if (x<y) return 0 else return x-y end NaturalNumber
Creator
Observer
Transformer
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Cont’d.. ADT definition begins with name of the
ADT. Two main sections in the definition the
objects and the functions. Objects are defined in terms of integers
but no explicit reference to their representation.
Function denote two elements of the data type NaturalNumber, while TRUE & FALSE are elements of the data type Boolean.
The symbols “::=“ should be read as “is defined as.”
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Cont’d… First function Zero has no arguments &
returns the natural number zero (constructor function).
The function Successor(x) returns the next natural nor in sequence (ex of transformer function).
Other transformer functions are Add & Subtract.
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Performance Analysis Criteria upon which we can judge a program
Does the program meet the original specifications of the task ?
Does it work correctly ? Does the program contain documentation that shows
how to use it and how it works ? Does the program effectively use functions to create
logical units ? Is the program’s code readable ?Space & Time Does the program efficiently use primary and secondary
storage ? Is the program’s running time acceptable for the task ?
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Cont’d… Performance evaluation is loosely
divided into two fields: First field focuses on obtaining estimates
of time and space that are machine independent. (Performance Analysis)
Second field, (Performance measurement), obtains machine-dependent running times, these are used to identify inefficient code segments.
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Cont’d… Evaluate a program
Meet specifications, Work correctly, Good user-interface, Well-documentation,Readable, Effectively use functions, Running time acceptable, Efficiently use space/storage
How to achieve them? Good programming style, experience, and practice Discuss and think
The space complexity of a program is the amount of memory that it needs to run to completion.
The time complexity of a program is the amount of computer time that it needs to run to completion
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Space Complexity The space needed is the sum of
Fixed space and Variable space Fixed space: C
Includes the instructions, variables, and constants
Independent of the number and size of I/O Variable space: Sp(I)
Includes dynamic allocation, functions' recursion
Total space of any program S(P)= C+ Sp(Instance)where C is constant & Sp is instance
characteristics
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Example1 to find space complexity
Algorithm P1(P,Q,R)1. Start2. Total = (P+Q+R*P+Q*R+P)/(P*Q);3. EndIn the above algorithm there is no instance characteristics and space needed by P,Q,R and Total is independent of instance characteristics thereforeS(P1) = 4 + 0where one space is required for each of P,Q,R &
Total
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Ex 2 Algorithm SUM()1. [Sum the values in an array A]
Sum=0Repeat for I=1,2,…..NSum = sum + A[i]
2. [Finished]Exit.In the above algorithm there is an instance characteristic N, since A must be large enough to hold the N elements to be summed & space needed by sum, I, and N is the independent of instance characteristics, we can writeS(Sum) = 3 + N.N for A[] and 3 for N,I and sum
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Program 1.11: Recursive function for summing a list of numbers float rsum(float list[ ], int n)
{ if (n) return rsum(list, n-1) + list[n-1]; return 0; }
Figure 1.1: Space needed for one recursive call of Program 1.11
Ssum(I)=Ssum(n)=6n
Type Name Number of bytes parameter: float parameter: integer return address:(used internally)
list [ ] n
4 4 4(unless a far address)
TOTAL per recursive call 12
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Time Complexity The time T(P) taken by a program, P is the sum of
its compile time and its run (execution) time. It depends on several factors
The input of the program. Time required to generate the object code by the
compiler. The speed to CPU used to execute the program.
If we must know the running time, the best approach is to use the system clock to time the program
Total time T(P)= compile time + run (or execution) time May run it many times without recompilation. Run time
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Cont’d… How to evaluate?
+ - * / … Use the system clock Number of steps performed
machine-independent Instance
Definition of a program step A program step is a syntactically or semantically meaningful
program segment whose execution time is independent of the instance characteristics.
We cannot express the running time in standard time units such as hours, minutes & seconds rather we can write as “the running time of such and such an algorithm is proportional to n”.
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Cont’d.. Time complexity for a given algorithm can
be calculated using two steps1. Separate the particular operations such as
ACTIVE operations that is central to the algorithm that is executed essentially often as any other.
2. Other operations such as assignments, the manipulation of index I and accessing values in an array, are called BOOK KEEPING operations and are not generally counted.
After the active operation is isolated, the number of times that it is executed is counted.
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Ex 1 Consider the algorithm to sum the values in a
given array A that contains N values. I is an integer variable used as index of A
ALGORITHM SUM()1. [Sum the values in an array A]
sum = 0repeat for I=1,2…Nsum=sum+A[I]
2. [Finished]exit.
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Explaination In the above ex, operation to isolate is the
addition that occurs when another array value is added to the partial sum
After active operation is isolated, the nor of time that it is executed is counted.
The number of addition of the values in the alg is N.
Execution time will increases in proportional to the number of times the active operation is executed.
Thus the above algorithm has execution time proportional to N.
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Tabular MethodStep count table for Program Iterative function to sum a list of numbers
Statement s/e Frequency Total steps float sum(float list[ ], int n) { float tempsum = 0; int i; for(i=0; i <n; i++)
tempsum += list[i]; return tempsum; }
0 0 0 0 0 0 1 1 1 0 0 0 1 n+1 n+1 1 n n 1 1 1 0 0 0
Total 2n+3
steps/execution
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Recursive Function to sum of a list of numbers
Statement s/e Frequency Total stepsfloat rsum(float list[ ], int n){ if (n) return rsum(list, n-1)+list[n-1]; return list[0];}
0 0 00 0 01 n+1 n+11 n n1 1 10 0 0
Total 2n+2
Step count table for recursive summing function
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Asymptotic Notation (O, , ) Exact step count
Compare the time complexity of two programs that computing the same function
Difficult task for most of the programs Asymptotic notation
Big “oh” upper bound(current trend)
Omega lower bound
Theta upper and lower bound
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Cont’d… Asymptotic notations are the terminology
that enables meaningful statements about time & space complexity
The time required by the given algorithm falls under three types1. Best-case time or the minimum time
required in executing the program.2. Average case time or the average time
required in executing program.3. Worst-case time or the maximum time
required in executing program.
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Break Even Point If we have two programs with a complexity of c1n2+c2n &
c3n respectively. We know one with complexity c3n will be faster that one
with complexity c1n2+c2n for large values on n. For small values of n, either would be faster. If c1=1,c2=2 & c3=100 then c1n2+c2n for n<=98 If c1=1,c2=2 & c3=1000 then c1n2+c2n for n<=998 No matter what the values of c1,c2 &c3 there will be an n
beyond which the program with complexity c3n will be faster.
This value of n is called break even point. the break-even point (BEP) is the point at which cost or
expenses and revenue are equal: there is no net loss or gain
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Asymptotic Notation BIG OH(O) Definition
f(n)= O(g(n)) iff there exist two positive constants c and n0 such that f(n)<= cg(n) for all n, n>= n0
Examples 3n+ 2= O(n) as 3n+ 2<= 4n for all n>= 2 10n2+ 4n+ 2= O(n2) as 10n2+ 4n+ 2<= 11n2 for n>=
5 3n+2<> O(1), 10n2+ 4n+ 2<> O(n)
Remarks g(n) is upper bound, the least?
n=O(n2)=O(n2.5)= O(n3)= O(2n) O(1): constant, O(n): linear, O(n2): quadratic, O(n3):
cubic, and O(2n): exponential
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Asymptotic Notation BIG OMEGA()
Definition f(n)= (g(n)) iff there exist two positive constants
c and n0 such that f(n)>= cg(n) for all n, n>= n0 Examples
3n+ 2= (n) as 3n+ 2>= 3n for n>= 1 10n2+ 4n+ 2= (n2) as 10n2+4n+ 2>= n2 for
n>= 1 6*2n+ n2= (2n) as 6*2n+ n2 >= 2n for n>= 1
Remarks lower bound, the largest ? (used for problem)
3n+3= (1), 10n2+4n+2= (n); 6*2n+ n2= (n100)
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Asymptotic Notation BIG THETA() Definition
f(n)= (g(n)) iff there exist positive constants c1, c2, and n0 such that c1g(n)<= f(n) <= c2g(n) for all n, n>= n0
Examples 3n+2=(n) as 3n+2>=3n for n>1 and 3n+2<=4n
for all n>= 2 10n2+ 4n+ 2= (n2); 6*2n+n2= (2n)
Remarks Both an upper and lower bound 3n+2!=(1); 10n2+4n+ 2!= (n)
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Example of Time Complexity Analysis
Statement Asymptotic complexity
void add(int a[][Max.......) 0{ 0 int i, j; 0 for(i= 0; i< rows; i++) (rows) for(j=0; j< cols; j++) (rows*cols) c[i][j]= a[i][j]+ b[i][j]; (rows*cols)} 0
Total (rows*cols)
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Function valueslog n•0•1•2•3•4•5
n•1•2•4•5•16•32
n log n•0•2•8•24•64•160
n2
•1•2•16•64•256•1024
n3
•1•8•64•512•4096•32768
2n
•2•4•16•256•65536•4,….
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Plot of function values
n
logn
nlogn
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Performance Measurement Clocking
Functions we need to time events are part of C’s library & accessed through the statement #include<time.h>
There are actually two different methods for timing events in C.
Figure 1.10 shows the major differences between these two methods.
Method 1 uses clock to time events. This function gives the amount of processor time that has elapsed since the program began running.
Method 2 uses time to time events. This function returns the time, measured in seconds, as the built-in type time t.
The exact syntax of the timing functions varies from computer to computer and also depends on the operating system and compiler in use.
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Method 1 Method 2
Start timing start=clock(); start=time(NULL);
Stop timing stop=clock(); stop=time(NULL);
Type returned clock_t time_t
Result in seconds duration=((double)(stop-start)/CLK_TCK;
duration=(double)difftime(stop, start);
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Generating Test Data It is necessary to use a computer program to
generate the worst-case data. In these cases, another approach to estimating
worst-case performance is taken. For each set of values of the instance
characteristics of interest. We generate a suitably large number of random
test data. The run times for each of these test data are
obtained. The max of these times is used as an estimate of
the worst-case time for a set of values.
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Polynomials Arrays are not only data structures in their
own right. Most commonly found data structures: the
ordered or linear list. Ex: Days of the week: (Sun,Mon,Tue....)
Values in a deck of cards: (Ace,2,3,4,5,6,7) Years Switzerland fought in WorldWarII: ()
Above ex is an empty list we denote as (). The other lists all contain items that are written
in the form (item0,item1,.....item n-1)
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Operations on lists including, Finding the length, n , of a list Reading the items in a list from left to right or either.
A polynomial is a sum of terms where each term has a form axe , where x is the variable, a is the coefficient and e is the exponent.
Ex: A(x) = 3x20+ 2x5+4
The largest exponent of a polynomial is called its degree.
The term with exponent equal to zero does not show the variable since x raised to a power of zero is 1.
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Standard Mathematical definitions for the sum & product of polynomials.
Assume that we have two polynomial... Similarly we can define subtraction &
division on polynomials as well other operations
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Structure Polynomial is objects:; a set of ordered pairs of <ei,ai> where ai in Coefficients and ei in Exponents, ei are integers >= 0functions:for all poly, poly1, poly2 Polynomial, coef Coefficients, expon ExponentsPolynomial Zero( ) ::= return the polynomial, p(x) = 0
Coefficient Coef(poly, expon) ::= if (expon poly) return its coefficient else return Zero Exponent Lead_Exp(poly) ::= return the largest exponent in poly
Polynomial Attach(poly,coef, expon) ::= if (expon poly) return error else return the polynomial poly with the term <coef, expon> inserted
Polynomials A(X)=3X20+2X5+4, B(X)=X4+10X3+3X2+1
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Polynomial Remove(poly, expon) ::= if (expon poly) return the polynomial poly with the term whose exponent is expon deleted else return errorPolynomial SingleMult(poly, coef, expon) ::= return the polynomial poly • coef • xexpon
Polynomial Add(poly1, poly2) ::= return the polynomial poly1 +poly2Polynomial Mult(poly1, poly2) ::= return the polynomial poly1 • poly2
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Polynomial representation Very reasonable first decision requires
unique exponents arranged in decreasing order.
This algorithm works by comparing terms from the two polynomials until one of both of the polynomials become empty.
The switch statement performs the comparison and adds the proper term to the new polynomial d.
One way to represent polynomials in C is to typedef to create the type polynomial..
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#define MAX_DEGREE 101typedef struct {
int degree;float coef[MAX_DEGREE];} polynomial;
/* d =a + b, where a, b, and d are polynomials */d = Zero( )while (! IsZero(a) && ! IsZero(b)) do { switch COMPARE (Lead_Exp(a), Lead_Exp(b)) { case -1: d = Attach(d, Coef (b, Lead_Exp(b)), Lead_Exp(b)); b = Remove(b, Lead_Exp(b)); break; case 0: sum = Coef (a, Lead_Exp (a)) + Coef ( b, Lead_Exp(b)); if (sum) { Attach (d, sum, Lead_Exp(a)); a = Remove(a , Lead_Exp(a)); b = Remove(b , Lead_Exp(b)); } break;
case 1: d = Attach(d, Coef (a, Lead_Exp(a)), Lead_Exp(a)); a = Remove(a, Lead_Exp(a)); } }insert any remaining terms of a or b into d
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Array representation of two polynomials
A(X)=2X1000+1 B(X)=X4+10X3+3X2+1
2 1 1 10 3 1
1000 0 4 3 2 0
starta finisha startb finishb avail
coef
exp
0 1 2 3 4 5 6 specification representationpoly <start, finish>A <0,1>B <2,5>
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Polynomial AdditionAdd two polynomials: D = A + B
void padd (int starta, int finisha, int startb, int finishb, int * startd, int *finishd){/* add A(x) and B(x) to obtain D(x) */ float coefficient; *startd = avail; while (starta <= finisha && startb <= finishb) switch (COMPARE(terms[starta].expon, terms[startb].expon)) { case -1: /* a expon < b expon */ attach(terms[startb].coef, terms[startb].expon); startb++ break;
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case 0: /* equal exponents */ coefficient = terms[starta].coef + terms[startb].coef; if (coefficient) attach (coefficient, terms[starta].expon); starta++; startb++; break;case 1: /* a expon > b expon */ attach(terms[starta].coef, terms[starta].expon); starta++;}
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/* add in remaining terms of A(x) */for( ; starta <= finisha; starta++) attach(terms[starta].coef, terms[starta].expon);/* add in remaining terms of B(x) */for( ; startb <= finishb; startb++) attach(terms[startb].coef, terms[startb].expon);*finishd =avail -1;}
Analysis: O(n+m)where n (m) is the number of nonzeros in A(B).
*Program 2.5: Function to add two polynomial (p.64)
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void attach(float coefficient, int exponent){ /* add a new term to the polynomial */ if (avail >= MAX_TERMS) { fprintf(stderr, “Too many terms in the polynomial\n”); exit(1); } terms[avail].coef = coefficient; terms[avail++].expon = exponent;}
Problem: Compaction is requiredwhen polynomials that are no longer needed.(data movement takes time.)
*Program 2.6:Function to add anew term (p.65)
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Sparse Matrices Some of the problems require lot of zeros to
be stored as a part of solution. Hence storing more number of zeros is a
waste of memory. Matrices with relatively high proportion of
zero entries are called sparse matrices. Matrix that contains more number of zero
elements are called sparse matrices. Sparse matrix is used in almost all areas of
the natural sciences
CHAPTER 272
000280000000910000000060000003110
150220015col0 col1 col2 col3 col4 col5
row0
row1
row2
row3
row4
row5
8/36
6*65*315/15
Sparse Matrix
sparse matrixdata structure?
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SPARSE MATRIX ABSTRACT DATA TYPE
Structure Sparse_Matrix is objects: a set of triples, <row, column, value>, where row and column are integers and form a unique combination, and value comes from the set item. functions: for all a, b Sparse_Matrix, x item, i, j, max_col, max_row index
Sparse_Marix Create(max_row, max_col) ::= return a Sparse_matrix that can hold up to max_items = max _row max_col and whose maximum row size is max_row and whose maximum column size is max_col.
74 Sparse_Matrix Transpose(a) ::=
return the matrix produced by interchanging the row and column value of every triple.Sparse_Matrix Add(a, b) ::= if the dimensions of a and b are the same return the matrix produced by adding corresponding items, namely those with identical row and column values. else return errorSparse_Matrix Multiply(a, b) ::= if number of columns in a equals number of rows in b return the matrix d produced by multiplying a by b according to the formula: d [i] [j] = (a[i][k]•b[k][j]) where d (i, j) is the (i,j)th element else return error.
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Sparse Matrix Representation
Consider a normal matrix containing more number of zero entries.
Few steps are…as the first step first row of the sparse matrix stores the following information1. Location of[0,0] stores the row size of the
original matrix2. Location of [0,1] stores the column size of
the original matrix3. Location of [0,2] stores the number non zero
entries of original matrix.
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row col value row col value
a[0] 6 6 8 b[0] 6 6 8 [1] 0 0 15 [1] 0 0 15 [2] 0 3 22 [2] 0 4 91 [3] 0 5 -15 [3] 1 1 11 [4] 1 1 11 [4] 2 1 3 [5] 1 2 3 [5] 2 5 28 [6] 2 3 -6 [6] 3 0 22 [7] 4 0 91 [7] 3 2 -6 [8] 5 2 28 [8] 5 0 -15
(a) (b)
*Figure 2.4:Sparse matrix and its transpose stored as triples (p.69)
(1) Represented by a two-dimensional array. Sparse matrix wastes space.(2) Each element is characterized by <row, col, value>.
row, column in ascending order
# of rows (columns)# of nonzero terms
transpose
77Create Operation
Sparse_matrix Create(max_row, max_col) ::= #define MAX_TERMS 101 /* maximum number of terms +1*/ typedef struct { int col; int row; int value; } term; term a[MAX_TERMS]
# of rows (columns)# of nonzero terms
78
Transposing a matrix(1) for each row i take element <i, j, value> and store it in element <j, i, value> of the transpose. difficulty: where to put <j, i, value> (0, 0, 15) ====> (0, 0, 15) (0, 3, 22) ====> (3, 0, 22) (0, 5, -15) ====> (5, 0, -15)
(1, 1, 11) ====> (1, 1, 11) Move elements down very often.
(2) For all elements in column j, place element <i, j, value> in element <j, i,
value>
CHAPTER 2
79
void transpose (term a[], term b[])/* b is set to the transpose of a */{ int n, i, j, currentb; n = a[0].value; /* total number of elements */ b[0].row = a[0].col; /* rows in b = columns in a */ b[0].col = a[0].row; /*columns in b = rows in a */ b[0].value = n; if (n > 0) { /*non zero matrix */ currentb = 1; for (i = 0; i < a[0].col; i++) /* transpose by columns in a */ for( j = 1; j <= n; j++) /* find elements from the current column */ if (a[j].col == i) { /* element is in current column, add it to b */
80
b[currentb].row = a[j].col; b[currentb].col = a[j].row; b[currentb].value = a[j].value; currentb++ } }}
elements
columns
81 void fast_transpose(term a[ ], term b[ ]) { /* the transpose of a is placed in b */ int row_terms[MAX_COL], starting_pos[MAX_COL]; int i, j, num_cols = a[0].col, num_terms = a[0].value; b[0].row = num_cols; b[0].col = a[0].row; b[0].value = num_terms; if (num_terms > 0){ /*nonzero matrix*/ for (i = 0; i < num_cols; i++) row_terms[i] = 0; for (i = 1; i <= num_terms; i++) row_term [a[i].col]++ starting_pos[0] = 1; for (i =1; i < num_cols; i++) starting_pos[i]=starting_pos[i-1] +row_terms [i-1];
columns
elements
columns
82 for (i=1; i <= num_terms, i++) { j = starting_pos[a[i].col]++; b[j].row = a[i].col; b[j].col = a[i].row; b[j].value = a[i].value; } }}
Compared with 2-D array representationO(columns+elements) vs. O(columns*rows)
elements --> columns * rowsO(columns+elements) --> O(columns*rows)
Cost: Additional row_terms and starting_pos arrays are required. Let the two arrays row_terms and starting_pos be shared.
83
Sparse Matrix MultiplicationDefinition: [D]m*p=[A]m*n* [B]n*pProcedure: Fix a row of A and find all elements in column j of B for j=0, 1, …, p-1.
111111111
000000111
001001001
84
void mmult (term a[ ], term b[ ], term d[ ] )/* multiply two sparse matrices */{ int i, j, column, totalb = b[].value, totald = 0; int rows_a = a[0].row, cols_a = a[0].col, totala = a[0].value; int cols_b = b[0].col, int row_begin = 1, row = a[1].row, sum =0; int new_b[MAX_TERMS][3]; if (cols_a != b[0].row){ fprintf (stderr, “Incompatible matrices\n”); exit (1);}
85
fast_transpose(b, new_b);/* set boundary condition */a[totala+1].row = rows_a;new_b[totalb+1].row = cols_b;new_b[totalb+1].col = 0;for (i = 1; i <= totala; ) { column = new_b[1].row; for (j = 1; j <= totalb+1;) { /* mutiply row of a by column of b */ if (a[i].row != row) { storesum(d, &totald, row, column, &sum); i = row_begin; for (; new_b[j].row == column; j++) ; column =new_b[j].row }
86
else switch (COMPARE (a[i].col, new_b[j].col)) { case -1: /* go to next term in a */ i++; break; case 0: /* add terms, go to next term in a and b */ sum += (a[i++].value * new_b[j++].value); break; case 1: /* advance to next term in b*/ j++ } } /* end of for j <= totalb+1 */ for (; a[i].row == row; i++) ; row_begin = i; row = a[i].row; } /* end of for i <=totala */ d[0].row = rows_a; d[0].col = cols_b; d[0].value = totald;}
87
We store the matrices A,B & D in the arrays a,b & d respectively.
To place a triple in d to reset sum to 0, mmult uses storeSum.
In addition mmult uses several local variables that we will describe….
The variable row is the row of A that we are currently multiplying with the columns in B.
The variable rowbegin is the position in a of the first element of the current row, and the variable column in the column of B that we are currently multiplying with a row in A.
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Compared with matrix multiplication using array
for (i =0; i < rows_a; i++) for (j=0; j < cols_b; j++) { sum =0; for (k=0; k < cols_a; k++) sum += (a[i][k] *b[k][j]); d[i][j] =sum; }
