Q (Quaternary) Search Algorithm

Transcription

1 Q (Quaternary) Search Algorithm Taranjit Khokhar Abstract In computer science, there are many ways to search the position of the required input value in an array. There are algorithms such as binary search and linear search. In this research paper, I will describe another new search algorithm which has been written to make the search even faster. This algorithm has been named as q (quaternary) search as it divides the array into four parts. After dividing the array, three elements are obtained which are compared to the specified input value. If the specified value matches any of the elements, the result is displayed. If not, then this process is repeated between any of the four parts of the array depending upon the specified value. This process is repeated until the result is found. There is also a disadvantage of this search algorithm as it requires the array to be sorted. Index Terms Search Algorithm I. INTRODUCTION The task of searching in an array is a very common one. For example, we search the name of a person in a telephone directory. We search the meaning of a word in a dictionary. In both cases, the list is a sorted one. It is sorted in an ascending order i.e. from a to z. Working of other Search Algorithms A search algorithm finds the position of a specified input value within an array. Linear Search: - Linear Search checks every element in the array sequentially until the desired element is found. Binary Search: - A binary search locates an item in a sorted array by repeatedly dividing the search interval in half. Working of Q Search A four search locates an item in a sorted array by repeatedly dividing the search interval into four equal parts. Comparison Table: LINEAR SEARCH Works only on sorted and unsorted items. Works with linked lists. Works with arrays. Less efficient. Number of comparisons is more. Detailed Working of Q Search Q SEARCH Works only on sorted items. Does not work with linked lists. Works with arrays. Very efficient. Number of comparisons is least. This algorithm works by dividing the array into four equal parts. Hence, three elements are obtained. The specified element is then compared to those three elements. If any of them match with the required element, the position of that element in the array is obtained. We need a general formula for dividing the array into four equal parts. The formula is obtained as follows: Aim: To find a general formula for dividing an array into four equal parts. Diagram: Proof: Let the largest and lowest element of an array be h and l respectively. Let the middle element be m. Let the element at 1/4 distance from l be a. Let the element at 1/4 distance from h be b. Now, m = (l+h)/2 a = (m+l)/2 = m/2 +l/2 a = [(l+h)/2]/2 + l/2 a = (l+h)/4 + l/2 IJIRT INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 17

2 a = (3l+h)/4 Similarly, b = (m+h)/2 = m/2 + h/2 b = [(h+l)/2]/2 + h/2 b = (h+l)/4 + h/2 b = (3h+l)/4 Inference: The formula for finding the element at 1/4 distance from l is (3l+h)/4 and the formula for finding the element at 1/4 distance from h is (3h+l)/4. We are going to use our derived result for our algorithm. For example In the following array, a[15]=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 Number to be searched (S) = 6. Largest value (h) = 16 Lowest value (l) = 1 Using our derived result, a = 4; m = 8; b = 12; S is not equal to a, m or m. Now, a < S < m Hence, the search is repeated between a and m, i.e. between the numbers 4 and 8. Therefore, h= a; l=m; Therefore, h = 8; l = 4; m = 6; a = 5; b = 7; S is not equal to a or b. But, m is equal to m. Hence, the position of that m is obtained. The number of times the loop executes is given by log 4 n where n is the number of elements of the array. This is because the array is divided into four equal parts every time the loop is executed. Working explained with steps 1. If the array bounds are improper, return error and terminate the process. 2. Else, compute the subscript of the element at 1/4 distance (a), the middle element (m) and the element at 3/4 distance (b). 3. If the target is equal to a or m or b, return the subscript of a or m or b accordingly. 4. Else if the target is less than a, go back to step 1 and search from first to a. 5. Else if the target is less than m, go back to step 1 and search from a to m 6. Else if the target is less than b, go back to step 1 and search from m to b. 7. Else go back to step 1 and search from b to last. Expressing the above steps in code: int search (int a[], int find, int l, int h) int a,m,b; /* Step 1*/ if((find<l) (find>h)) return( error ); while(l<=h) /* Step 2 */ a=((3*l) + h)/4; m=(h + l)/2; b=((3*h) + l)/4; /* Step 3 */ if (a[a]==find) IJIRT INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 18

3 return a; if(a[m]==find) return m; if(a[b]==find) return b; /* Step 4 */ if(find<a[a]) h=a-1; /* Step 5 */ if(find<a[m]) l=a+1; h=m-1; /* Step 6 */ if(find<a[b]) l=m+1; h=b-1; /* Step 7 */ l=b+1 ; Proof that q search is faster than binary search (when not considering the time taken by if- statement) It is nearly impossible to consider the time taken by if statements. R.H.S = O(log 2n ) L.H.S = O(log 4n ) Aim: To prove that L.H.S > R.H.S for any positive value of n. Proof: R.H.S Let O(log 2 n ) = x O(n) = 2 x L.H.S Let O(log 4 n ) = y O(n) = 4 x 4 x > 2 x for any positive value of x. Inference: Hence, it can be seen that the number of times the loop runs on q search is less than binary search. Though q search has more if- statements, the number of times the loop will run is nearly half that of binary search. Search loop comparison: As it can be seen, the number of times a loop runs for a particular value in binary search is double the number of times the loop runs on Q search. Where can Q search be used? Q search can be used when you have a lot of data. It can also be used when you need to save up space. For example, if you want to store 500, bit integers, you can store the integers in a sorted array of 2,000,000 bytes. However, if you want to store the values in a B-tree, it will take up at least 60,000,000 bytes. Hence, it saves up a lot of space compared to other alternatives. IJIRT INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 19

4 C++ code for Q Search and comparison with Binary Search: Q search #include<iostream.h> #include<conio.h> void main() clrscr(); int f, n, a[10000]; cout<<"enter THE NUMBER OF ELEMENTS"; cin>>n; cout<<"enter THE ELEMENTS OF THE ARRAY"; for(int m=0;m<n;m++) int h = n-1; int l = 0; cin>>a[m]; cout<<"enter NUMBER TO BE SEARCHED"; cin>>f; while(h>=l) int j = ((3*l)+h)/4; int m = (h+l)/2; int k = ((3*l)+h)/4; if(a[j]==f) cout<<a[j]; if(a[m]==f) Binary Search #include<iostream.h> #include<conio.h> void main() clrscr(); int n, a[10000]; cout<<"enter THE NUMBER OF ELEMENTS"; cin>>n; cout<<"enter THE ELEMENTS OF THE ARRAY"; for(int m=0;m<n;m++) int f; int h = n-1; int l = 0; cin>>a[m]; cout<<"enter NUMBER TO BE SEARCHED"; cin>>f; while(h>=l) int m = (h+l)/2; if(a[m]==f) cout<<a[m]; if(f<a[m]) IJIRT INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 20

5 getch(); cout<<a[m]; if(a[k]==f) cout<<a[k]; if(f<a[j]) h=j-1; if(f<a[m]) l=j+1; h=m-1; if(f<a[k]) l=m+1; h=k-1; l=k+1; getch(); h=m-1; l=m+1; IJIRT INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 21

6 II. CONCLUSION: Q search is a very efficient search algorithm. It can be used in many real life applications. It requires the data to be sorted in either ascending or descending order. It can be used to for very large amounts of data. Q search is more effective than binary search in most cases. It also saves up a lot of space compared to a binary tree. The basic idea behind this search algorithm is that it repeatedly divides the array into four equal parts and searches for the required number. REFERENCES [1] _PAPER.pdf [2] [3] [4] orithm IJIRT INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 22