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LISTS
. Single linked lists
. Double linked lists
. Stacks & Queues
. Polynomials
Array◦ sequential representation◦ some operation can be very time-consuming
(data movement)◦ size of data must be predefined◦ static storage allocation and deallocation
Solution to the data movement in sequen-tial representation◦ pointers: often called links
Pointers
For any type T in C, there is corresponding type ‘pointer-to-T’
Actual value of pointer type : an address of memory
Pointer operators in C◦ the address operator : &◦ the dereferencing(or indirection) operator : *
Pointers
Dynamically allocated storage◦ C provides a mechanism, called a heap, for al-
locating storage at run-time
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);
Pointers
Composed of data part and link part◦ link part contains address of the next element
in a list◦ non-sequential representations◦ size of the list is not predefined◦ dynamic storage allocation and deallocation
Singly Linked Lists
bat satcat vat NULL
ptr
To insert “mat” between “cat” and “sat”
1) get a node currently unused(paddr)2) set the data of this node to “mat”3) set paddr’s link to point to the address found in the
link of the node “cat”4) set the link of the node “cat” to point to paddrCaution) step 3 must be done before step 4
Singly Linked Lists
bat satcat vat NULL
ptr
mat
To delete “mat” from the list
1) find the element that immediately precedes “mat”, which is “cat”
2) set its link to point to mat’s link
No data movement in insert and delete opera-tions
Singly Linked Lists
bat satcat vat NULL
ptr
mat
Required capabilities to make linked representations possible
◦ a mechanism for defining a node’s structure, that is, the field it contains
◦ a way to create new nodes when we need them
◦ a way to remove nodes that we no longer need
Singly Linked Lists
[list of words ending in “at”]◦ Define a node structure for the list
data field: character array link field: pointer to the next node self-referential structuretypedef struct list_node *list_ptr;typedef struct list_node {
char data[4];list_ptr link;
};list_ptr ptr = NULL;
Singly Linked Lists
◦ Create new nodes for our list◦ Then place the word “bat” into our list
ptr=(list_ptr)malloc(sizeof(list_node));strcpy(ptr->data,”bat”);ptr->link=NULL;
Singly Linked Lists
b a t \0 NULL
ptr
address of
first nodeptr->data ptr->link
[two-node linked list]◦ Create a linked list of integers
typedef struct list_node *list_ptr;typedef struct list_node {
int data;list_ptr link;
};list_ptr ptr = NULL;
Singly Linked Lists
10 20 NULL
ptr
list_ptr create2() {list_ptr first, second;first = (list_ptr)malloc(sizeof(list_node));second = (list_ptr)malloc(sizeof(list_node));second->link=NULL;second->data=20;first->data=10;first->link=second;return first;
}
Singly Linked Lists
[list insertion]
◦ Determine if we have used all available memory: IS_FULL
Singly Linked Lists
10 20 NULL
ptr
node 50
temp
void insert(list_ptr *pptr,list_ptr node) {list_ptr temp;temp=(list_ptr)malloc(sizeof(list_node));if(IS_FULL(temp)) {
fprintf(stderr,”The momory is full\n”);exit(1);
}temp->data=50;if(*pptr) {
temp->link = node->link;node->link = temp;
}else {
temp->link = NULL; *pptr = temp;}
}
Singly Linked Lists
[list deletion]◦ Deletion depends on the location of the node:
more complicated
◦ Assumption ptr: point to the start of list node: point to the node to be deleted trail: point to the node that precedes node to be
deleted
Singly Linked Lists
1) the node to be deleted is the first node in the list
2) otherwise
Singly Linked Lists
10 50
ptr
20 NULL
trail
10
ptr
20 NULL
node
(b) after deletion(a) before deletion
10 50
ptr
20 NULL
node
50
ptr
20 NULL
trail = NULL
(b) after deletion(a) before deletion
void delete(list_ptr *pptr, list_ptr trail, list_ptr node)
{if(trail)
trail->link = node->link;else
*pptr = (*pptr)->link;free(node);
}
Singly Linked Lists
[printing out a list]while not end of the list do
print out data field;move to the next node;
end;void print_list(list_ptr ptr) {
printf(“The list contains: “);for(; ptr; ptr = ptr->link)
printf(“%4d”, ptr->data);printf(“\n”);
}
Singly Linked Lists
Representing stack / queue by linked list
Dynamically LinkedStacks and Queues
NULL
top
element link
······
NULL
front
element link
······
rear
(a) linked stack
(b) linked queue
Representing N stacks by linked lists
#define MAX_STACKS 10 /* N=MAX_STACKS=10 */typedef struct {
int key;/* other fields */
} element;typedef struct stack *stack_ptr;typedef struct stack {
element item;stack_ptr link;
};stack_ptr top[MAX_STACKS];
Dynamically Linked Stacks and Queues
···
Dynamically LinkedStacks and Queues
NULL
top[0]
element link
······key
NULL
top[1]
······
NULL
top[MAX_STACKS-1]
······
◦ Initial condition for stacks top[i] = NULL, 0 £ i < MAX_STAKCS
◦ Boundary condition for n stacks Empty condition:
top[i] = NULL iff the i-th stack is empty Full condition:
IS_FULL(temp) iff the memory is full
Dynamically LinkedStacks and Queues
◦ Add to a linked stack
void push(stack_ptr *ptop, element item) {stack_ptr temp =
(stack_ptr)malloc(sizeof (stack));if(IS_FULL(temp)) {
fprintf(stderr,”The memory is full\n”);exit(1);
}temp->item=item;temp->link=*ptop;*ptop = temp;
}
Dynamically LinkedStacks and Queues
◦ Delete from a linked stackelement pop(stack_ptr *ptop) {
stack_ptr temp = *ptop;element item;if(IS_EMPTY(temp)) {
fprintf(stderr,”The stack is empty\n”);exit(1);
}item=temp->item;*ptop=temp->link;free(temp);return item;
}
Dynamically LinkedStacks and Queues
Representing M queues by linked lists
#define MAX_QUEUES 10 /* M=MAX_QUEUES=10 */typedef struct queue *queue_ptr;typedef struct queue {
element item;queue_ptr link;
};queue_ptr front[MAX_QUEUES],rear[MAX_QUEUES];
Dynamically LinkedStacks and Queues
Dynamically LinkedStacks and Queues
NULL
front[0]
element link
······
rear[0]
key
NULL
front[1]
······
rear[1]
NULL
front[MAX_QUEUES-1]
······
rear[MAX_QUEUES-1]
·
·
·
◦ Initial conditon for queues front[i]=NULL,0 £ i < MAX_QUEUES
◦ Boundary condition for queues Empty condition:
front[i]= NULL iff the i-th queue is empty Full condition:
IS_FULL(temp) iff the memory is full
Dynamically LinkedStacks and Queues
Add to the rear of a linked queue
void insert(queue_ptr *pfront, queue_ptr *prear, element item) {
queue_ptr temp =(queue_ptr)malloc(sizeof(queue));
if(IS_FULL(temp)) {fprintf(stderr,”The memory is full\n”);exit(1);
}temp->item=item;temp->link=NULL;if (*pfront) (*prear)->link=temp;else *pfront = temp;*prear = temp;
}
Dynamically LinkedStacks and Queues
Delete from the front of a linked queue
element delete(queue_ptr *pfront) {queue_ptr temp=*pfront;element item;if (IS_EMPTY(*front)) {
fprintf(stderr,”The queue is empty\n”);exit(1);
}item=temp->item;*pfront=temp->link;free(temp);return item;
}
Dynamically LinkedStacks and Queues
Representing stacks/queues by linked lists
◦ no data movement is necessary : O(1)◦ no full condition check is necessary◦ size is growing and shrinking dynamically
Dynamically LinkedStacks and Queues
Representing polynomials as singly linked lists
◦ A(x) = am-1xem-1 + ··· + a0xe0
typedef struct poly_node *poly_ptr;typedef struct poly_node {
int coef;int expon;poly_ptr link;
};poly_ptr a,b,d;
Polynomials
◦ poly_node
◦ a = 3x14 + 2x8 + 1
◦ b = 8x14 - 3x10 + 10x6
Polynomials
coef expon link
3 14 2 8 1 0 NULL
a
8 14 -3 10 10 6 NULL
b
◦ Adding polynomials
(a) a->expon == b->expon
Polynomials
3 14 2 8 1 0 NULL
8 14 -3 10 10 6 NULL
b
a
11 14 NULL
d rear
(b) a->expon < b->expon
Polynomials
3 14 2 8 1 0 NULL
8 14 -3 10 10 6 NULL
b
a
-3 10 NULL
d
11 14
rear
(c) a->expon > b->expon
Polynomials
3 14 2 8 1 0 NULL
8 14 -3 10 10 6 NULL
b
a
2 8 NULL
rear
11 14 -3 10
d
(d) a->expon < b->expon
Polynomials
3 14 2 8 1 0 NULL
8 14 -3 10 10 6 NULL
b
a
2 811 14 -3 10
10 6 NULL
rear
d
(e) b == NULL;
Polynomials
3 14 2 8 1 0 NULL
8 14 -3 10 10 6 NULL
b
a
2 811 14 -3 10
rear
10 6 1 0 NULL
d
poly_ptr padd(poly_ptr a,poly_ptr b) {poly_ptr front,rear,temp;int sum;rear=(poly_ptr)malloc(sizeof(poly_node));if(IS_FULL(rear)) {
fprintf(stderr,”The memory is full\n”);exit(1);
}front = rear;
(to be continued)
Polynomials
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;
}
(to be continued)
Polynomials
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;
}
Polynomials
Function attach() to create a new node and append it to the end of d
void attach(float coe,int exp,poly_ptr *pptr){
poly_ptr temp;temp=(poly_ptr)malloc(sizeof(poly_node));if(IS_FULL(temp)) {
fprintf(stderr,”The memory is full\n”);exit(1);
}temp->coef = coe;temp->expon = exp;(*pptr)->link = temp;*pptr=temp;
}
Polynomials
◦ Analysis of padd where m, n : number of terms in each polynomial coefficient additions: O(min{m, n}) exponent comparisons: O(m + n) creation of new nodes for d: O(m + n)
◦ Time complexity: O(m + n)
Polynomials
◦ Erasing polynomials
void erase(poly_ptr *pptr) {poly_ptr temp;while (*pptr) {
temp = *pptr;*pptr = (*pptr)->link;free(temp);
}}
◦ Useful to reclaim the nodes that are being used to represent partial result such as temp(x)
Polynomials
Allocating/deallocating nodes◦ How to preserve free node in a storage pool ?
initially link together all free nodes into a list in a stor-age pool
avail: variable of type poly_ptr that points to the first node in list of freed nodes
Polynomials
NULLavail ······
1 2 n
initial available space list
storage pool
Allocating nodespoly_ptr get_node(void) {
poly_ptr node;if (avail) {
node = avail; avail = avail->link;
}else {
node =(poly_ptr)malloc(sizeof(poly_node));
if (IS_FULL(node)) {fprintf(stderr,”The memory is full\n”);exit(1);
}
}return node;
}
Polynomials
Deallocating nodes
void ret_node(poly_ptr ptr) {ptr->link = avail;avail = ptr;
}
void erase(poly_ptr *pptr) {poly_ptr temp;while (*pptr) {
temp = *pptr;*pptr = (*pptr)->link;ret_node(temp);
}}
Polynomials
Traverse to the last node in the list:◦ O(n), where n: number of terms
How to erase polynomial efficiently? How to return n used nodes to storage pool ?
Polynomials
Representing polynomials as circularly linked list
◦ to free all the node of a polynomial more effi-ciently modify list structure
◦ the link of the last node points to the first node in the list called circular list (« chain)
Polynomials
ptr
◦ Maintain our own list (as a chain) of nodes that has been freed obtain effective erase algorithmvoid cerase(poly_ptr *pptr) {
if (*pptr) {temp = (*pptr)->link;(*pptr)->link = avail;avail = temp;*pptr = NULL;
}
}
◦ Independent on the number of nodes in a list: O(1)
Polynomials
Circular list with head nodes◦ handle zero polynomials in the same way as
nonzero polynomials
Polynomials
ptrhead node
- -
ptrhead node
- -(empty list)
Inverting(or reversing) a chain “in place” by using three pointers : lead, middle, trail
typedef struct list_node *list_ptr;typedef struct list_node {
char data;list_ptr link;
};
Operations for Chains
Inverting a chainlist_ptr invert(list_ptr lead) {
list_ptr middle, trail;middle = NULL;while(lead) {
trail = middle;middle = lead;lead = lead->link;middle->link = trail;
}return middle;
}
◦ time: O(length)
Operations for Chains
Operations for Chains
NULL
leadmiddletrail
NULL
leadtrail middle
NULL
lead NULL
middle
Operations for Chains
NULL
leadmiddletrail
NULL
NULL
leadmiddletrail
NULL
NULL
leadmiddletrail
NULL
Concatenate two chains◦ Produce a new list that contains ptr1 fol-
lowed by ptr2
list_ptr concat(list_ptr ptr1,list_ptr ptr2) {list_ptr 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;
}
}
Operations for Chains
Finding the length of a list(chain)
int length(list_ptr ptr) {int count = 0;while (ptr) {
count++;ptr = ptr->link;
}return count;
}
Operations for Chains
Insert a new node at the front or at the rear of the chain
move down the entire length of ptr to insert rear:◦ front-insert : O(1)◦ rear-insert : O(n)
Operations for Chains
ptr x1 x2 x3 NULL
Insert a new node at the front of a list(chain)
void insert_front(list_ptr *pptr, list_ptr node) {if (IS_EMPTY(*pptr)) { *pptr = node; node->link = NULL;}else { node->link = *pptr; *pptr = node;}
}
Operations for Chains
◦ (singly) circularly linked lists
◦ insert a new node at the front or at the rear move down the entire length of ptr to insert both
front and rear: insert-front : O(n) insert-rear : O(n)
Operations forCircularly Linked Lists
ptr x1 x2 x3
◦ Improve this better: make ptr points to the last node
◦ Insert a new node at the front or at the rear front-insert : O(1) rear-insert : O(1)
Operations forCircularly Linked Lists
ptrx1 x2 x3
Insert a new node at the rear of a circular list
void insert_rear(list_ptr *pptr, list_ptr node) {if (IS_EMPTY(*pptr)) { *pptr = node; node->link = node;}else { node->link = (*pptr)->link; (*pptr)->link = node;
*pptr = node; /* for front-insert, remove this line */}
}
Operations forCircularly Linked Lists
Finding the length of a circular list
int length(list_ptr ptr) {list_ptr temp;int count = 0;if (ptr) {
temp = ptr;do {
count++;temp = temp->link;
} while (temp != ptr);
}return count;
}
Operations forCircularly Linked Lists
◦ Problems of singly linked lists move to only one way direction hard to find the previous node hard to delete the arbitrary node
because finding the previous node is hard
◦ Doubly linked circular list doubly lists + circular lists allow two links two way direction
Doubly linked lists
typedef struct node *node_ptr;typedef struct node {
node_ptr llink;element item;node_ptr rlink;
};
◦ Suppose that ptr points to any node in a doubly linked list
ptr = ptr->llink->rlink = ptr->rlink->llink
Doubly linked lists
◦ Introduce dummy node, called, head to represent empty list make easy to implement operations contains no information in item field
empty doubly linked circular list with head node
Doubly linked lists
ptr
doubly linked circular list with head node
Doubly linked lists
head node
llink item rlink
Insertion into a doubly linked circular list
void dinsert(node_ptr node,node_ptr newnode) {/* insert newnode to the right of node */newnode->llink = node;newnode->rlink = node->rlink;node->rlink->llink = newnode;node->rlink = newnode;
}
time: O(1)
Doubly linked lists
insertion into an empty doubly linked circular list
Doubly linked lists
newnode
node node
Deletion from a doubly linked circular listvoid ddelete(node_ptr node,node_ptr deleted){
/* delete from the doubly linked list */if (node == deleted)
printf(“Deletion of head node ” “not permitted.\n”);
else {deleted->llink->rlink = deleted->rlink;deleted->rlink->llink = deleted->llink;free(deleted);
}
} time complexity : O(1)
Doubly linked lists
deletion in a doubly linked list with a single node
Doubly linked lists
deleted
node
node
◦ Doubly linked circular list
don’t have to traverse a list : O(1) insert(delete) front/middle/rear is all the same
Doubly linked lists
