Add two Number

You are given two linked lists representing two non-negative numbers. The digits are stored in reverse order and each of their nodes contain a single digit. Add the two numbers and return it as a linked list.

Input: (2 -> 4 -> 3) + (5 -> 6 -> 4)
Output: 7 -> 0 -> 8

http://www.leetcode.com/onlinejudge

Below is the working c code

 

#include<stdio.h>
#include<stdlib.h>
struct node
{
        unsigned int val;
        struct node * next;
};
typedef struct node * nodeptr;
void insert(nodeptr * head,unsigned int item)
{
        nodeptr temp = (nodeptr) malloc(sizeof(nodeptr));
        temp->val = item;
        temp->next = *head;
        *head = temp;
}
void print(nodeptr head)
{
        while (head)
        {
                printf("%d ",head->val);
                head=head->next;
        }
        printf("\n");
}
nodeptr add(nodeptr head1,nodeptr head2)
{
        nodeptr temp=NULL;
        unsigned int sum=0,carry = 0;
        while(head1!=NULL && head2!=NULL)
        {
                sum = head1->val + head2->val;
                insert(&temp,(sum%10)+carry);
                carry = sum/10;
                head1=head1->next;
                head2=head2->next;
        }
        while(head1)
        {
                insert(&temp,head1->val+carry);
                carry = 0;
                head1=head1->next;
        }
        while(head2)
        {
                 insert(&temp,head2->val+carry);
                 carry = 0;
                 head2=head2->next;
        }
        if(carry!=0)
                insert(&temp,carry);
        return temp;
}
void reverse(nodeptr *head)
{
       nodeptr h1,h2;
       h1=*head;
       h2=h1->next;
       h1->next=NULL;
       *head=h1;
       while(h2)
       {
                h1=h2->next;
                h2->next=*head;
                *head=h2;
                h2=h1;
       }        
}
int main()
{
        nodeptr root = NULL,root1=NULL,result=NULL;
        insert(&root,2);
        insert(&root,3);
        insert(&root1,9);
        insert(&root1,8);
        result=add(root,root1);
        reverse(&result);
        print(root);
        print(root1);
        print(result);
        return 0;
}

Merge Sort

The merge sort algorithm closely follows the divide-and-conquer paradigm. Intuitively, it operates as follows.
Divide: Divide the n-element sequence to be sorted into two subsequences of n/2 elements each.
Conquer: Sort the two subsequences recursively using merge sort.
Combine: Merge the two sorted subsequences to produce the sorted answer.

We use this technique to sort a file in the following way. Divide the file into n subfiles of size 1 and merge adjacent (disjoint) pairs of files. we then have approximately n/2 files of size 2. Repeat this process until there is only one file remaining of size n.

Below is the nonrecursive procedure expaining merge sort.

#define NUMELTS  .... // .... is to be put as the number of elements in array.

void MergeSort(int x[], int n)
{
    int aux[NUMLETS], i, j, k, l1, l2, size, u1, u2;
   
    size = 1; / * Merge files of size 1;
    while(size < n) {
        l1 = 0; /* initialize lower bounds of first file */
        k = 0; /* k is index for auxiliary array */
        while(l1+size < n) { /* Check to see if there are two files to merge */
            /* Compute remaining indices */
            l2 = l1+size;
            u1 = l2-1;
            u2 = (l2+size-1 < n) > l2+size-1 : n-1;
            /* proceed through the two subfiles */
            for(i=l1,j=l2;i<=u1 && j<=u2; k++)
                /* Enter smaller into the aux array */
                if(x[i] <= x[j])
                    aux[k]=x[i++];
                else
                    aux[k] = x[j++];
            /* At this point, one of the subfiles has been exhausted.
             * Insert any remaining portions of the other file
             */
            for(;i<=u1;k++)
                aux[k] = x[i++];
            for(;j<=u2;k++)
                aux[k] = x[j++];
            /* advance l1 to start of the next pair of files. */
            l1 = u2 +1;
        }
        /* Copy any remaining single file */
        for(i=l1;k<n;i++)
            aux[k++]=x[i];
        /* Copy aux into x and adjust size */
        for(i=0;i<n;i++)
            x[i]=aux[i];
        size *= 2;
    }
}

Average case time complexity of merge sort is O(nlogn) but it requires O(n) additional space for the auxiliary array.

Quick Sort

Quick sort applies the divide and conquer paradigm. Let x be an array and n the number of elements to be sorted. Choose an element a from a specific position within the array. Suppose that the elements of x are partitioned so that a is placed into position j and the following conditions hold:

• Each of the elements in positions 0 through j-1 is less than or eual to a.
• Each of the elements in position j+1 through n-1 is greater than or equal to a.

If this conditions holds for a particular a and j, a is the jth smallest element of x, so that a remains in position j when the array is completely sorted.If foregoing process is repeated with the subarays x[0] through x[j-1] and x[j+1] through x[n-1] and any subarrays created by the process in successive iterations, the final result is a sorted file.

Below is procedure for quicksort.

void QuickSort(int x[],int lb,int ub)
{
    int j;
    if(lb>=ub)
        return; /* array is sorted */
   
    partition(x,lb,ub,&j);
    /* partition the elements of the subarray such that one of the elements is now at position x[j] and
     * x[i] <= x[j] for lb <=i < j
     * x[i] >= x[j] for j < i <=ub
     */
   
    QuickSort(x,lb,j-1);     /* sort the subarray between positions lb and j-1 */
    QuickSort(x,j+1,ub);    /* sort the subarray between positions j+1 and ub */
}

Now we need to implement the procedure partition.The object of partition is to allow a specific element to find its proper position with respect to the others in the subarray.
For this we need to consider a pivot element which will be at its proper position after partition. For time being consider pivot element a=x[lb]. Two pointers, up and down are initialized to the upper and lower bounds of the subarray, respectively. At any point during execution, each element in a position above up is greater than or equal to a, and each element in a position below down is less than or equal to a. The two pointers are moved towards each other in following fashion.

• Repeatedly increase the pointer down by one position until x[down] > a.
• Repeatedly decrease the pointer up by one position until x[up] <= a.
• if up > down, interchange x[down] with x[up]

The process is repeated until the condition in last step fails(up<=down), at which point x[up] is interchanged with x[lb](which equals pivot element a), whose final position was sought, and j is set to up.

Below is procedure partition.

void partition(int x[],int lb, int ub, int *pj)
{
    int a, down, temp, up;

    a=x[lb];
    up=ub;
    down=lb;

    while(down < up) {
        while(x[down] <= a && down < up)
            down++;
        while(x[up] > a)
            up--;
        if(down < up) {
            temp = x[down];
            x[down] = x[up];
            x[up] = temp;
        }
    }
    x[lb] = x[up];
    x[up] = a;
    *pj = up;
}
   
Average Time Complexity of quick sort is O(nlogn). Its worst case is when the array is sorted and time complexity in that case is O(n2). It requires O(logn) additional space for the stack.

It is possible to speed up quick sort for sorted files by choosing a random elements of each subfile as the pivot element.If file is nearly sorted this might be a good strategy(may choose a middle element as pivot element). However if nothing is known above the file, such a strategy doesnot improve the worst case behaviour.

Lets see quick sort example.If an array is given as
    25 57 48 37 12 92 86 33
pivot element a = 25. so after 1st iteratin, 25 will be at its proper postion, and the resulting array will be.
    12 25 57 48 37 92 86 33
it will divide the array into two subparts (12) and ( 57 48 37 92 86 33) which needs to be sorted.
proceeding in this way following will be output after successive iterations.
    12 25 (48 37 33 57 92 86)
    12 25 (48 37 33) 57(92 86)
    12 25 (37 33) 48 57 (92 86)
    12 25 (33) 37 48 57 (92 86)
    12 25 33 37 48 57 (92 86)
    12 25 33 37 48 57 (86) 92
    12 25 33 37 48 57 86 92

Insertion Sort

An insertion sort is one that sorts a set of records by inserting records into an existing sorted file.

Below is the insertion sort procedure code

void InsertionSort(int x[],int n)
{
    int i,k,y;
    for(k=1;k<n;k++)
    {
        y=x[k];
        for(i=k-1;i>=0 && y<x[i];i--)
            x[i+1] = x[i];
        x[i+1]=y;
    }
}

Time complexity iss O(n2) if the file is initially sorted in reverse order. On average case also its time complexity is O(n2).

Selection Sort

A selection sort is one in which successive elements are selected in order and placed into their proper sorted positions. The elements of the input may have to be preprocessed to make the ordered selection possible.
Selection sort consists entirely of a selection phase in which the largest of the remaining elements,large, is repeatedly placed at its proper position i, at the end of the array.

Below is the working procedure code.

void SelectionSort(int x[], int n)
{
    int i,indx,j,large;
   
    for(i=n-1;i>0;i--)
    {
        large = x[0];
        indx = 0;
        for(j=1;j<=i;j++)
            if(x[j]>large)
            {
                large = x[j];
                indx = j;
            }
        x[indx] = x[i];
        x[i] = large;
    }

The above procedure will sort the array in ascending order.
Consider the elements of array are
25 57 48 37 12
Here we will see the state of array after each iteration.
iteration 0     25 57 48 37 12
iteration 1    25 12 48 37 57
iteration 2    25 12 37 48 57
iteration 3    25 12 37 48 57
iteration 4    12 25 37 48 57

Average time complexity of selection sort is O(n2).

Bubble Sort

The basic idea of bubble sort is to pass through the file sequentially several times. Each pass consists of comparing each element in the file with its successor(x[i] with x[i+1]) and interchanging the two elements if they are not in proper order.

Below is the working procedure code.

void BubbleSort(int x[], int n)
{
    int tmp,i,j;
    int switched = 1;
   
    for(i=0;i<n-1 && switched==1;i++)
    {   
        /* idea of using switched variable is if there
         * is no interchanging of element in one complete
         * array iteration means array is sorted .
         */
        switched = 0;
        for(j=0;j<n-i-1;j++)
            if(x[j]>x[j+1])
            {
                switched = 1;
                tmp = x[j];
                x[j] = x[j+1];
                x[j+1] = tmp;
            }
    }
}

The above procedure will sort the array in ascending order.
Consider the elements of array are
25 57 48 37 12
Here we will see the state of array after each iteration.
iteration 0     25 57 48 37 12
iteration 1    25 37 48 12 57
iteration 2    25 37 12 48 57
iteration 3    25 12 37 48 57
iteration 4    12 25 37 48 57

Time complexity of bubble sort in average case is O(n2). But in case if the file is already sorted time complexity of this sort will be O(n).

Sorting

It is rearranging a collection of items into increasing or decreasing order. Sorting is used to preprocess the collection to make searching faster, as well as to identify items that are similar.

A sort can be classified as
internal - if the records that it is sorting are within main memory.
external - if some of the records that it is sorting are in auxiliary storage.

There are large number of sorting techniques. Some of them  with there average time complexity are:
Bubble Sort       O(n2)
Selection Sort    O(n2)
Insertion Sort     O(n2)
Quick Sort         O(nlogn)
Merge Sort        O(nlogn)
Heap Sort          O(nlogn)

An ideal sort is an inplace sort whose additional space requirement is O(1).

A sorting technique is said to be stable if for all records i and j such that k[i] equals k[j], if r[i] precedes r[j] in original file, r[i] precedes r[j] in the sorted file. Here, k[i] refer to key associated with each record r[i].

which sort is better we can't tell it totally depends upon the problem characteristics where which will be more suitable.

Binary Search Trees(BSTs)

A Problem with arrays is adding and deleting elements to an arrays is computationally expensive, particularly when the array needs to stay sorted. BSTs are similar to arrays in that the keys are in a sorted order but they are easier to perform insertions and deletions into. But BSTs require more space than arrays.
The key lookup, insert, and delete operations for BSTs take time proprotional to the height of the tree, which can in worst case be O(n), if inserts and deletes are naively implemented. However there are implementations of insert and delete which gurantee the tree has height O(logn). These requires sorting and updating additional data at the tree nodes. Red-black trees are an example of such balanced BSTs and they are used in the C++ STL library to implement sets.

BSTs are different from trees. Specifically, in a BST, there is positionality as well as order associated with the children of nodes. Furthermore, the values stored at nodes have to respect the BST property - key stored at a node is greater than or equal to the keys stored in the nodes of its left subchild and less than or equal to the values stored in the nodes of its right subchild.

Below is video lecture related to red-black tree.

Hashing

Hashing is another approach to searching. The idea of hashing is to store keys in an array of length m. keys are stored in array locations based on the "hash code" of the key. The hash code is an integer computed from the key by a hash function. If the hash function is chosen well, the keys are distributed across the array locations uniformly randomly.

There is always the possibility of two keys mapping to the same location, in which case a "collision" is said to occur. The standard mechanism to deal with collisions is to maintain a linked list of keys at each location. Look ups, inserts, and deletes take O(1+n/m) complexity, where n is the number of keys. If the "load" n/m grows large, the table can be rehashed to one with a larger number of locations; the keys are moved to the new tables. Rehashing is expensive.

One disadvantage of hashing is the need for a good hash function.


Here below is lecture which discuss hashing in details.

Suppose that in addition to being sorted, the entries of A are distinct integers. Design an efficient algorithm for finding an index i such that A[i]=i or indicating that no such index exists. 

We can apply binary search technique for this too as we are aware that array is sorted and have distinct elements.

Below is the working code in C

#include<stdio.h>
int bsearch(int a[],int size)
{
    int low=0;
    int up=size-1;
    int mid;
    while(low <= up) {
        mid=(low+up)/2;
        if(a[mid]==mid)
            return mid;
        else if(a[mid]<mid)
            low = mid+1;
        else
            up = mid-1;
    }
    return -1;
}
int main()
{
    int arr[10]={-1,0,2,4,6,8,10,12,14,15};
    int pos;
    pos=bsearch(arr,10);
    if(pos == -1)
        printf("element for a[i] == i is not found\n");
    else
        printf("element for a[i] == i is at location %d\n",pos+1);
    return 0;
}
 

Design an efficient algorithm that finds the index of first occurance of an element larger than a specified key k; return -1 if every element is less than or equal to k.

This can also be solved by doing little modification in binary search program. Below is the fully working code in C.

#include<stdio.h>
int bsearch(int a[],int size,int elem)
{
    int low=0;
    int up=size-1;
    int mid;
    while(low <= up) {
        mid=(low+up)/2;
        if(a[mid]==elem)
        {        if(mid+1 <= up)
                    if(a[mid+1]>elem)
                        return mid+1;
                    else
                        low = mid+1;
                else
                    return -1;
        }
        else if(a[mid]<elem)
        {
            low = mid + 1;
            if(a[low]>elem)
                return low;
        }
        else
        {
            up = mid -1;
            if(a[up]<=elem)
                return up+1;
        }
    }
    return -1;
}
int main()
{
    int arr[10]={1,2,3,3,4,6,9,9,9,10};
    int pos,srch_elem;
    printf("Enter element to search : ");
    scanf("%d",&srch_elem);
    pos=bsearch(arr,10,srch_elem);
    if(pos == -1)
        printf("element greater than %d is not in array\n",srch_elem);
    else
        printf("first element greater than %d is %d at location %d\n",srch_elem,arr[pos],pos+1);
    return 0;
}

Write a program that takes a sorted array A and a key elem and returns the index of first occurance of key in A, and returns -1 if key doesnot appears in A.

Above problem can be solved by doing simple modification of binary search program. Below is the working code in C.

#include<stdio.h>
int bsearch(int a[],int size,int elem)
{
        int low=0;
        int up=size-1;
        int mid;
        while(low <= up) {
                mid=(low+up)/2;
                if(a[mid]==elem)
                {       if(a[mid-1] == elem)
                                up=mid-1;
                        else
                                return mid;
                }
                else if(a[mid]<elem)
                        low=mid+1;
                else
                        up=mid-1;
        }
        return -1;
}
int main()
{
        int arr[10]={1,3,3,3,3,6,9,9,9,10};
        int pos,srch_elem;
        printf("Enter element to search : ");
        scanf("%d",&srch_elem);
        pos=bsearch(arr,10,srch_elem);
        if(pos == -1)
                printf("%d is not in array\n",srch_elem);
        else
                printf("first occurance of %d is at location %d\n",srch_elem,pos+1);
        return 0;
}