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Chapter 4 Lists Fundamentals of Data Structures in C. Instructors: C. Y. Tang and J. S. Roger Jang. All the material are integrated from the textbook "Fundamentals of Data Structures in C" and some supplement from the slides of Prof. Hsin-Hsi Chen (NTU). 4.1 Pointers.
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Chapter 4 Lists Fundamentals of Data Structures in C Instructors: C. Y. Tang and J. S. Roger Jang All the material are integrated from the textbook "Fundamentals of Data Structures in C" and some supplement from the slides of Prof. Hsin-Hsi Chen (NTU).
4.1 Pointers • Consider the following alphabetized list of three letter English words ending in at: (bat, cat, sat, vat) • We would like to add the word mat to this list. • If we store this list in an array, then we must move sat and vat one position to the right before we insert mat. • Similarly, if we want to remove the word cat from the list, we must move sat and vat one position to the left to maintain our sequential representation. • In general case, arbitrary insertion and deletion from arrays can be very time-consuming.
- Linked representation • In a sequential representation the order of elements is the same as in the ordered list, while in a linked representation these two sequences need not be the same. • To access elements of the list in the correct order with each element, we store the address, or location, of the next element in that list. • Thus, associated with each list element is a node which contains both a data component and a pointer to the next item in the list. The pointers are often called links.
10 i pi - Pointers in C • & the address operator • * the dereferencing (or indirection) operator • Example: int i, *pi; pi = &i; • &i returns the address of i and assigns it as the value of pi. • To assign a value to i we can say: i = 10; • or *pi = 10;
4.1.1 Pointers can be dangerous • Set all pointers to NULL when they are not actually pointing to an object. • Use explicit type cast when converting between pointer types. • Example: pointer int i, *pi; pi = &i; i=10 or *pi=10 pi= malloc(size of(int)); /* assign to pi a pointer to int */ pf=(float *) pi; /* casts an int pointer to a float pointer */ • Define explicit return types for functions. • Pointers may have the same size as type int. The function return type defaults to int which can later be interpreted as a pointer .
4.1.2 Using dynamically allocated storage • We may not know how much space we will need, nor do we wish to allocate some vary large area that may never be required. • C provides a mechanism, called a heap, for allocating storage at run-time. • We can call a function, malloc, to request the amount of memory. • Later we can call another function, free, to return the area of memory to the system.
Program 4.1 : Allocation and deallocation of pointers int i, *pi; float f, *pf; pi = (int *) malloc(sizeof(int)); pf = (float *) malloc (sizeof(float)); *pi =1024; *pf =3.14; printf(”an integer = %d, a float = %f\n”, *pi, *pf); free(pi); free(pf); request memory return memory
bat cat sat vat NULL 4.2 Singly linked lists • Linked lists are drawn as an order sequence of nodes with links represented as arrows. • The name of the pointer to the first node in the list is the name of the list. • -The nodes do not resident in sequential locations. • -The locations of the nodes may change on different runs.
mat • It is easier to make arbitrary insertions and deletions using a linked list rather than a sequential list. • To insert the word mat between cat can sat, we must: • Get a node that is currently unused; let its address be paddr. • Set the data field of this node to mat. • Set paddr’s link field to point to the address found in the link field of the node containing cat. • Set the link field of the node containing cat to point to paddr. bat cat sat vat NULL
cat sat mat • To delete mat from the list. • Find the element that immediately precedes mat, which is cat, and set its link field to point to mat’s link field. bat vat NULL
4.3 Dynamically linked stacks and queues • When several stacks and queues coexisted, there was no efficient way to represent them sequentially. • Direction of links for both stack and the queue facilitate easy insertion and deletion of nodes.
top front element link element link NULL NULL (a) Linked Stack (b) Linked queue rear
- Represent n stacks • Declarations #define MAX_STACKS 10 /* max number of stacks */ typedef struct { int key; /* other fields */ } element; typedef struct stack *stack_pointer; typedef struct stack { element item; stack_pointer link; }; stack_pointer top[MAX_STACKS]; • Initial condition: top[i] = NULL, 0 ≦ i < MAX_STACKS • Boundary conditions: top[i] = NULL iff the ith stack is empty and IS_FULL(temp) iff the memory is full
Program 4.6 : Push in a linked stack • void add(stack_pointer *top, element item){ /* add an element to the top of the stack */ stack_pointer temp = (stack_pointer) malloc (sizeof (stack)); if (IS_FULL(temp)) { fprintf(stderr, “ The memory is full\n”); exit(1); } temp->item = item; temp->link = *top; *top= temp; }
Program 4.7 : Pop from a linked stack • element delete(stack_pointer *top) {/* delete an element from the stack */ stack_pointer temp = *top; element item; if (IS_EMPTY(temp)) { fprintf(stderr, “The stack is empty\n”); exit(1); } item = temp->item; *top = temp->link; free(temp); return item;}
-Represent n queues • Declarations #define MAX_QUEUES 10 /* maximum number of queues */ typedef struct queue *queue_pointer; typedef struct queue { element item; queue_pointer link; }; queue_pointer front[MAX_QUEUE], rear[MAX_QUEUES]; • Initial conditions: front[i] = NULL, 0 ≦ i < MAX_QUEUES • Boundary conditions: front[i] = NULL iff the ith queue is empty and IS_FULL(temp) iff the memory is full
Program 4.8 : Add to the rear of a linked queue • void addq(queue_pointer *front, queue_pointer *rear, element item){ /* add an element to the rear of the queue */ queue_pointer temp = (queue_pointer) malloc(sizeof (queue)); if (IS_FULL(temp)) { fprintf(stderr, “ The memory is full\n”); exit(1); } temp->item = item; temp->link = NULL; • if (*front) (*rear) -> link = temp; • else *front = temp; • *rear = temp; }
Program 4.9 : Delete from the front of a linked queue • element deleteq(queue_pointer *front) {/* delete an element from the queue */ queue_pointer temp = *front; element item; if (IS_EMPTY(*front)) { fprintf(stderr, “The queue is empty\n”); exit(1); } item = temp->item; *front = temp->link; free(temp); return item;}
4.4 Polynomials • 4.4.1- Representing Polynomials As Singly Linked Lists • We want to represent the polynomial:
Where the ai are nonzero coefficients and the eiare nonnegative integer exponents such that em-1> em-2> … >e1> e0≧ 0. • We represent each term as a node containing coefficient and exponent fields, as well as a pointer to the next term.
coef expon link • Declarations typedef struct poly_node *poly_pointer; typedef struct poly_node { int coef; int expon; poly_pointer link; }; poly_pointer a, b, c;
-Polynomial representation a null 1 0 3 14 2 8 b null 8 14 -3 10 10 6
4.4.2 Adding polynomials • To add two polynomials, we examine their terms starting at the nodes pointed to by a and b. • If the exponents of the two terms are equal, we add the two coefficients and create a new term for the result. • If the exponent of the current term in a is less than the exponent of the current term in b, then we create a duplicate term of b, attach this term to the result, called d, and advance the pointer to the next term in b. • Take a similar action on a if a->expon > b->expon.
(a) a->expon == b->expon 2 8 1 0 3 14 a -3 10 10 6 8 14 b 11 14 d a->expon == b->expon
(b) a->expon < b->expon 2 8 1 0 3 14 a -3 10 10 6 8 14 b -3 10 11 14 d a->expon < b->expon
(c) a->expon > b->expon 2 8 1 0 3 14 a -3 10 10 6 8 14 b -3 10 11 14 2 8 a->expon > b->expon
Program 4.10 : Add two polynomials • poly_pointer padd(poly_pointer a, poly_pointer b) { poly_pointer front, rear, temp; int sum; rear =(poly_pointer)malloc(sizeof(poly_node)); if (IS_FULL(rear)) { fprintf(stderr, “The memory is full\n”); exit(1); } front = rear; while (a && b) { switch (COMPARE(a->expon, b->expon)) {
case -1: /* a->expon < b->expon */ attach(b->coef, b->expon, &rear); b= b->link; break; case 0: /* a->expon == b->expon */ sum = a->coef + b->coef; if (sum) attach(sum,a->expon,&rear); a = a->link; b = b->link; break; case 1: /* a->expon > b->expon */ attach(a->coef, a->expon, &rear); a = a->link; } }
for (; a; a = a->link) attach(a->coef, a->expon, &rear); for (; b; b=b->link)attach(b->coef, b->expon, &rear); rear->link = NULL; temp = front; front = front->link; free(temp); return front; } Delete extra initial node.
- Analysis • (1) coefficient additions • 0 additions min(m, n) • where m (n) denotes the number of terms in A (B). • (2) exponent comparisons • extreme case • em-1 > fm-1 > em-2 > fm-2 > … > e0 > f0 • m+n-1 comparisons • (3) creation of new nodes • extreme case • m + n new nodes • summary O(m+n)
4.4.3 Erasing polynomials • Program 4.12 : Erasing a polynomial void erase(poly_pointer *ptr) { /* erase the polynomial pointed to by ptr */ poly_pointer temp; while (*ptr) { temp = *ptr; *ptr = (*ptr)->link; free(temp); } }
ptr 3 14 2 8 1 0 4.4.4 Representing Polynomials As Circularly linked lists • If the link field of the last node points to the first node in the list, all the nodes of a polynomial can be freed more efficiently. ptr =3X 14+2X 8+1
Chain: A singly linked list in which the last node has a null link. • Nodes that are no longer in use are freed so that we can reuse these nodes later. • Maintaining a list (as a chain) of nodes that have been “freed”.
Program 1.13 : get_node function • poly_pointer get_node(void) { poly_pointer node; if (avail) { node = avail; avail = avail->link: } else { node = (poly_pointer)malloc(sizeof(poly_node)); if (IS_FULL(node)) { printf(stderr, “The memory is full\n”); exit(1); } } return node; }
Program 4.14 : ret_node function • void ret_node(poly_pointer ptr) { ptr->link = avail; avail = ptr; }
Program 4.15 : Erasing a circular list void erase(poly_pointer *ptr) { /* erase the circular list ptr */ poly_pointer temp; if (*ptr) { temp = (*ptr)->link; (*ptr)->link = avail; avail = temp; *ptr = NULL; } }
NULL avail ptr temp avail
To avoid the special case of zero polynomial, each polynomial contains one additional head node. • The expon and coef fields of this node are irrelevant.
(a) Zero polynomial -1 a Zero polynomial (b) a -1 2 8 1 0 3 14
Program 4.16 : Adding circularly represented polynomials • poly_pointer cpadd(poly_pointer a, poly_pointer b) { poly_pointer starta, d, lastd; int sum, done = FALSE; starta = a; a = a->link; b = b->link; d = get_node(); d->expon = -1; lastd = d; do { switch (COMPARE(a->expon, b->expon)) { case -1: attach(b->coef, b->expon, &lastd); b = b->link; break; Set expon field of head node to -1.
case 0: if (starta == a) done = TRUE; else { sum = a->coef + b->coef; if (sum) attach(sum,a->expon,&lastd); a = a->link; b = b->link; } break; case 1: attach(a->coef,a->expon,&lastd); a = a->link; } } while (!done); lastd->link = d; return d; } Link last node to first
4.5 Additional list operations typedef struct list_node *list_pointer; typedef struct list_node { char data; list_pointer link; };
Use two extra pointers: middle and trail. • Program 4.17: Inverting a singly linked list list_pointer invert(list_pointer lead) { /* invert the chain pointed to by lead */ list_pointer middle, trail; middle = NULL; while (lead) { trail = middle; middle = lead; lead = lead->link; middle->link = trail } return middle; }
a x1 x1 x2 x2 x3 x3 a 4.5.2 Operations for circularly linked lists • Need to change the link field of the last node when insert a new node at the front of the list. • It is more convenient if the name of the circular list points to the last node.
Program 4.18 : Concatenating singly linked lists • list_pointer concatenate(list_pointer ptr1, list_pointer ptr2) { list_pointer temp; if (IS_EMPTY(ptr1)) return ptr2; else { if (!IS_EMPTY(ptr2)) { for (temp=ptr1;temp->link;temp=temp->link); temp->link = ptr2; } return ptr1; } } O(m) where m is # of elements in the first list
X2 X1 void insert_front (list_pointer *ptr, list_pointer node) { if (IS_EMPTY(*ptr)) { *ptr= node; node->link = node; } else { node->link = (*ptr)->link; (1) (*ptr)->link = node; (2) } } X3 ptr (2) (1)
4.6 Equivalence relations • A relation over a set, S, is said to be an equivalence relation over S iff it is symmertric, reflexive, and transitive over S. • reflexive, x=x • symmetric, if x=y, then y=x • transitive, if x=y and y=z, then x=z
0=4, 3=1, 6=10, 8=9, 7=4, 6=8, 3=5, 2=11, 11=0 • three equivalent classes{0,2,4,7,11}; {1,3,5}; {6,8,9,10}
A Rough Algorithm to find Equivalence Classes • Program 4.21 : First pass at equvalence algorithm • void equivalenec() { initialize; while (there are more pairs) { read the next pair <i,j>; process this pair; } initialize the output; do { output a new equivalence class; } while (not done); }
Program 4.22 : A more detailed version of the equivalence algorithm #include <stdio.h> #include <alloc.h> #define MAX_SIZE 24 #define IS_FULL(ptr) (!(ptr)) #define FALSE 0 #define TRUE 1