## CS13002 Programming and Data Structures | ## Spring semester |

## Abstract data types

You are now a master C programmer. You know most of the essential features of C. So, given a problem, you plan to jump to write the code. Yes, I see that you have mentally written the

#includeline.Please wait. Solving a problem is a completely different game. Writing the final code is only a tiny part of it. I admit that even when all the algorithms are fully specified, writing a good program is not always a joke, in particular, when the program is pretty huge and involves cooperation of many programmers. Think of the Linux operating system which was developed by thousands of free-lance programmers all over the globe. The system works pretty harmoniously and reportedly with much less bugs than software from commercial giants like Microsoft.

First notice that your code may be understood and augmented by third parties in your absence. Even if you flood your code with documentation, its readability is not ensured. An important thing you require is a design document. That is not at the programming level, but at a more abstract level.

Data abstraction is the first step. A problem is a problem of its own nature. It deals with input and output in specified formats not related to any computer program. For example, a weather forecast system reads gigantic databases and outputs some prediction. Where is C coming in the picture in this behavioral description? One can use any other computer language, perhaps assembly languages, or even hand calculations, to arrive at the solution.

Assume that you are taught the natural language English pretty well. You are also given a plot. Your problem is to write an attractive detective story in English. Is it a trivial matter? I think it is not quite so, at least for most of us. You have to carefully plan about your characters, the location, the sequence, the suspense, and what not. Each such planning step involves many things that have nothing to do with English. The murderer is to be modeled as a human being, an abstract data, together with a set of behaviors, a set of abstract procedures. There is no English till this point. A language is necessary only when you want to give a specific concrete form to these abstract things.

Still, you cannot perhaps be a Conan Doyle or Christie, neither in plot design nor in expressions. Well, they are geniuses. However, if you plan carefully and master English reasonably well to arrive at a decent and sleek production, who knows, you may be the script-writer for the next Bollywood blockbuster?

## What is an abstract data type?

An abstract data type (

ADT) is an object with a generic description independent of implementation details. This description includes a specification of the components from which the object is made and also the behavioral details of the object. Instances of abstract objects include mathematical objects (like numbers, polynomials, integrals, vectors), physical objects (like pulleys, floating bodies, missiles), animate objects (dogs, Pterodactyls, Indians) and objects (like poverty, honesty, inflation) that are abstract even in the natural language sense. You do not see C in Pterodactyls. Only when you want to simulate a flying Pterodactyl, you would think of using a graphics package in tandem with a computer language. Similarly, inflation is an abstract concept. When you want to model it and want to predict it for the next 10 years, you would think of writing an extrapolation program in C.Specifying only the components of an object does not suffice. Depending on the problem you are going to solve, you should also identify the properties and behaviors of the object and perhaps additionally the pattern of interaction of the object with other objects of same and/or different types. Thus in order to define an ADT we need to specify:

- The components of an object of the ADT.
- A set of procedures that provide the behavioral description of objects belonging to the ADT.
There may be thousands of ways in which a given ADT can be implemented, even when the coding language remains constant. Any such implementation must comply with the content-wise and behavioral description of the ADT.

Examples

Integers:An integer is an abstract data type having the standard mathematical meaning. In order that integers may be useful, we also need to specify operations (arithmetic operations, gcd, square root etc.) and relations (ordering, congruence etc.) on integers.Real numbers:There are mathematically rigorous ways of defining real numbers (Dedekind cuts, completion of rational numbers, etc). To avoid these mathematical details, let us plan to represent real numbers by decimal expansions (not necessarily terminating). Real numbers satisfy standard arithmetic and other operations and the usual ordering.Complex numbers:A complex number may be mathematically treated as an ordered pair of real numbers. An understanding of real numbers is then sufficient to represent complex numbers. However, the complex arithmetic is markedly different from the real arithmetic.Polynomialswith real (or complex or integer or rational) coefficients with the standard arithmetic.Matriceswith real (or complex or integer or rational) entries with the standard matrix arithmetic (which may include dimension, rank, nullity, etc).Setsare unordered collections of elements. We may restrict our study to sets of real (or complex) numbers and talk about union, intersection, complement and other standard operations on sets.- A
multisetis an unordered collection of elements (say, numbers), where each element is allowed to have multiple occurrences. For example, an aquarium is a multiset of fish types. One can add or delete fishes to or from an aquarium.- A
bookis an ADT with attributes like name, author(s), ISBN, number of pages, subject, etc. You may think of relations like comparison of difficulty levels of two books.## How to implement an abstract data type?

It is now and only now when you think about writing C codes. Carefully investigate the specification of the ADT and possible target applications where this ADT is going to be used. Plan for suitable C constructs to provide the appropriate functionality with good performance. Try to exploit your experience with C. But fully understand what you are going to implement, the limitations, the expected performance figures, the ease of code maintenance, and a lot of related issues. After all, you have to market your product.

Examples

Integers:Oh, my! C provides so many integer variables and still I have to write my integers. Yep! You may have to. For most common-place applications C's built-in integer data types are sufficient. But not always. Suppose my target application is designing a cryptosystem, where one deals with very big integers, like those of bit-sizes one to several thousand bits. Our C's maximum integer length is 64 bits. That is grossly inadequate to address the cryptosystem designer's problem. ANSI standards dictate use of integers of length at most 32 bits, which are even poorer for cryptography, but at the minimum portable across platforms. At any rate, you need your customized integer data types.A common strategy is to break big integers into pieces and store each piece in a built-in data type. To an inexperienced user breaking with respect to the decimal representation seems easy and intuitive. But computer's world is binary. So breaking with respect to the binary representation is much more efficient in terms of space and running time. So we plan to use an array of

unsigned longvariables to store the bits of a big integer. Each such variable is a 32-bit word and is capable of storing 32 bits of a big integer. Therefore, if we plan to work with integers of size no larger than 10,000 bits, we require an array of size no more than 313unsigned longvariables. The zeroth location of the array holds the least significant 32 bits of a big integer, the first location the next 32 bits, and so on. Since all integers are not necessarily of size 10,000 bits, it is also necessary to store the actual word-size of a big integer. Finally, if we also plan to allow negative integers, we should also reserve a location for storing the sign information. So here is a possible implementation of the big integer data type.typedef struct { unsigned long words[313]; unsigned int wordSize; unsigned char sign; } bigint;This sounds okay, but has an efficiency problem. When you pass a

bigintdata to a function, the entirewordsarray is copied element-by-element. That leads to unreasonable overheads during parameter passing. We can instead use an array of 315unsigned longvariables and use its 313-th and 314-th locations to store the size and sign information. The first 313 locations (at indexes 0 through 312) represent the magnitude of the integer as before.#define SIZEIDX 313 #define SIGNIDX 314 typedef unsigned long goodbigint[315];Now

goodbigintis a simple array and so passing it to a function means only a pointer is passed. Quite efficient, right?These big integers are big enough for cryptographic applications, but cannot represent integers bigger than big, for example, integers of bit-size millions to billions. Whenever we use static arrays, we have to put an upper limit on the size. If we have to deal with integers of arbitrary sizes (as long as memory permits), we have no option other than using dynamic memory and allocate the exact amount of memory needed to store a very big integer. But then since the maximum index of the dynamic array is not fixed, we have to store the size and sign information at the beginning of the array. Thus the magnitude of the very big integer is stored starting from the second array index. This leads to somewhat clumsy translation between word indices and array indices.

#define SIZEIDX 0 #define SIGNIDX 1 typedef unsigned long *verybigint;A better strategy is to use a structure with a dynamic words pointer.

typedef struct { unsigned long *words; unsigned int size; unsigned char sign; } goodverybigint;So you have to pay a hell lot of attention, when implementation issues come. Good solutions come from experience and innovativeness.

Being able to define integers for a variety of applications is not enough. We need to do arithmetic (add, subtract, multiply etc.) on these integers. It is beyond the scope of this elementary course to go into the details of these arithmetic routines. It suffices here only to highlight the difference between abstract specifications and application-specific implementations. Both are important.

Real numbers:Again C provides built-in implementations of real numbers:float,doubleandlong double. If one has to use floating point numbers of higher precision, one has to go for private floating point data types and write arithmetic routines for these new data types. These are again topics too advanced for this course.Complex numbers:If we are happy with real numbers ofdoubleprecision, the most natural way to define a complex number is the following:typedef struct { double real; double imag; } complex;Let us also illustrate the implementation of some arithmetic routines on complex numbers:

complex cadd ( complex z1 , complex z2 ) { complex z; z.real = z1.real + z2.real; z.imag = z1.imag + z2.imag; return z; } complex cmul ( complex z1 , comple z2 ) { complex z; z.real = z1.real * z2.real - z1.imag * z2.imag; z.imag = z1.real * z2.imag + z1.imag * z2.real; return z; } complex conj ( complex z1 ) { complex z; z.real = z1.real; z.imag = -z1.imag; return z; } void cprn ( complex z ) { printf("(%lf) + i(%lf)", z.real, z.imag); }Matrices:Suppose we want to work with matrices having complex entries and suppose that thecomplexADT has been defined as above. We may define matrices of bounded sizes as:#define MAXROW 10 #define MAXCOL 15 typedef struct { int rowdim; int coldim; complex entry[MAXROW][MAXCOL]; } matrix;Let us now implement some basic arithmetic operations on these matrices.

matrix msetid ( int n ) { matrix C; int i, j; if ((n > MAXROW) || (n > MAXCOL)) { fprintf(stderr, "msetid: Matrix too big\n"); C.rowdim = C.coldim = 0; return C; } C.rowdim = C.coldim = n; for (i = 0; i < C.rowdim; ++i) { for (j = 0; j < C.coldim; ++j) { A.entry[i][j].real = (i == j) ? 1 : 0; A.entry[i][j].imag = 0; } } return C; } matrix madd ( matrix A , matrix B ) { matrix C; int i, j; if ((A.rowdim != B.rowdim) || (A.coldim != B.coldim)) { fprintf(stderr, "madd: Matrices of incompatible dimensions\n"); C.rowdim = C.coldim = 0; return C; } C.rowdim = A.rowdim; C.coldim = A.coldim; for (i = 0; i < C.rowdim; ++i) for (j = 0; j < C.coldim; ++j) C.entry[i][j] = cadd(A.entry[i][j],B.entry[i][j]); return C; } matrix mmul ( matrix A , matrix B ) { matrix C; int i, j, k; complex z; if (A.coldim != B.rowdim) { fprintf(stderr, "mmul: Matrices of incompatible dimensions\n"); C.rowdim = C.coldim = 0; return C; } C.rowdim = A.rowdim; C.coldim = B.coldim; for (i = 0; i < A.rowdim; ++i) { for (j = 0; j < B.coldim; ++j) { C.entry[i][j].real = 0; C.entry[i][j].imag = 0; for (k = 0; k < A.coldim; ++k) { z = cmul(A.entry[i][k], B.entry[k][j]); C.entry[i][j] = cadd(C.entry[i][j],z); } } } return C; }## A complete example : the ordered list ADT

Let us now define a new ADT which has not been encountered earlier in your math courses. We call this ADT the ordered list. It is a list of elements, say characters, in which elements are ordered, i.e., there is a zeroth element, a first element, a second element, and so on, and in which repetitions of elements are allowed. For an ordered list L, let us plan to have the following functionality:

L = init();- Initialize
Lto an empty list.L = insert(L,ch,pos);- Insert the character
chat positionposin the listLand return the modified list. Report error ifposis not a valid position inL.delete(L,pos);- Delete the character at position
posin the listL. Report error ifposis not a valid position inL.isPresent(L,ch);- Check if the character
chis present in the listL. If no match is found, return -1, else return the index of the leftmost match.getElement(L,pos);- Return the character at position
posin the listL. Report error ifposis not a valid position inL.print(L);- Print the list elements from start to end.
We will provide two complete implementations of this ADT. We assume that the element positions are indexed starting from 0.

## Implementation using static arrays

Let us restrict the number of elements in the ordered list to be

<= 100. One can then use an array of characters of this size. Moreover, one needs to maintain the current size of the list. Thus the list data type can be defined as:#define MAXLEN 100 typedef struct { int len; char element[MAXLEN]; } olist;Let us now implement all the associated functions one by one.olist init () { olist L; L.len = 0; return L; } olist insert ( olist L , char ch , int pos ) { int i; if ((pos < 0) || (pos > L.len)) { fprintf(stderr, "insert: Invalid index %d\n", pos); return L; } if (L.len == MAXLEN) { fprintf(stderr, "insert: List already full\n"); return L; } for (i = L.len; i > pos; --i) L.element[i] = L.element[i-1]; L.element[pos] = ch; ++L.len; return L; } olist delete ( olist L , int pos ) { int i; if ((pos < 0) || (pos >= L.len)) { fprintf(stderr, "delete: Invalid index %d\n", pos); return L; } for (i = pos; i <= L.len - 2; ++i) L.element[i] = L.element[i+1]; --L.len; return L; } int isPresent ( olist L , char ch ) { int i; for (i = 0; i < L.len; ++i) if (L.element[i] == ch) return i; return -1; } char getElement ( olist L , int pos ) { if ((pos < 0) || (pos >= L.len)) { fprintf(stderr, "getElement: Invalid index %d\n", pos); return '\0'; } return L.element[pos]; } void print ( olist L ) { int i; for (i = 0; i < L.len; ++i) printf("%c", L.element[i]); }Here is a possible

main()function with these calls.int main () { olist L; L = init(); L = insert(L,'a',0); printf("Current list is : "); print(L); printf("\n"); L = insert(L,'b',0); printf("Current list is : "); print(L); printf("\n"); L = delete(L,5); printf("Current list is : "); print(L); printf("\n"); L = insert(L,'c',1); printf("Current list is : "); print(L); printf("\n"); L = insert(L,'b',3); printf("Current list is : "); print(L); printf("\n"); L = delete(L,2); printf("Current list is : "); print(L); printf("\n"); L = insert(L,'z',8); printf("Current list is : "); print(L); printf("\n"); L = delete(L,2); printf("Current list is : "); print(L); printf("\n"); printf("Element at position 1 is %c\n", getElement(L,1)); }Here is the complete program.

## Animation example : Implementation of the ordered list ADT with static memory

## Implementation using linked lists

Let us now see an implementation based on dynamic linked lists. We use the same prototypes for function calls. But we define the basic data type

olistin a separate manner. For the sake of ease of writing the functions, we maintain a dummy node at the beginning of the linked list.typedef struct _node { char element; struct _node *next; } node; typedef node *olist; olist init () { olist L; /* Create the dummy node */ L = (node *)malloc(sizeof(node)); L -> element = '\0'; L -> next = NULL; return L; } olist insert ( olist L , char ch , int pos ) { int i; node *p, *n; if (pos < 0) { fprintf(stderr, "insert: Invalid index %d\n", pos); return L; } p = L; i = 0; while (i < pos) { p = p -> next; if (p == NULL) { fprintf(stderr, "insert: Invalid index %d\n", pos); return L; } ++i; } n = (node *)malloc(sizeof(node)); n -> element = ch; n -> next = p -> next; p -> next = n; return L; } olist delete ( olist L , int pos ) { int i; node *p; if (pos < 0) { fprintf(stderr, "delete: Invalid index %d\n", pos); return L; } p = L; i = 0; while ((i < pos) && (p -> next != NULL)) { p = p -> next; ++i; } if (p -> next == NULL) { fprintf(stderr, "delete: Invalid index %d\n", pos); return L; } p -> next = p -> next -> next; return L; } int isPresent ( olist L , char ch ) { int i; node *p; i = 0; p = L -> next; while (p != NULL) { if (p -> element == ch) return i; p = p -> next; ++i; } return -1; } char getElement ( olist L , int pos ) { int i; node *p; i = 0; p = L -> next; while ((i < pos) && (p != NULL)) { p = p -> next; ++i; } if (p == NULL) { fprintf(stderr, "getElement: Invalid index %d\n", pos); return '\0'; } return p -> element; } void print ( olist L ) { node *p; p = L -> next; while (p != NULL) { printf("%c", p -> element); p = p -> next; } }The

main()function of the static array implementation can be used without any change under this implementation. Here is the complete program.

## Animation example : Implementation of the ordered list ADT with dynamic memory

This exemplifies that the abstract properties and functional behaviors are independent of the actual implementation, or stated in another way, our two implementations of the ordered list ADT correctly and consistently tally with the abstract specification.

And why should we stop here? There could be thousand other ways in which the same ADT can be implemented, and in all these cases the function prototypes may be so chosen that the same

main()function will work. This is the precise difference between an abstract specification and particular implementations.