Searching: Fibonaccian Search-3: N+1 is NOT a perfect Fibonacci number

Algorithm F

Algorithm F (Fibonaccian search). Given a table of records R1R2 … RN whose
keys are in increasing order K1 < K2 < … < KN, this algorithm searches for a
given argument K.
In this case, N + 1 is not a perfect Fibonacci number, Fk+1. It is not difficult to make
the method work for arbitrary N, if a suitable initialization is provided (see exercise 14).
One idea is to find the least M>=0 such that N+M has the form Fk+1-1.
F1. [Initialize.] Set i <– Fk, p <– Fk-1, q <– Fk-2. (Throughout the algorithm,
p and q will be consecutive Fibonacci numbers.)
F1.5.[Very first comparison K:KFk.] If K>KFk, i=i-M and go to F4.(proceeding normally from then on)
F2. [Compare.] If i<=0, go to F4; If K < Ki, go to step F3; if K > Ki, go to F4; and if K = Ki,
the algorithm terminates successfully.
F3. [Decrease i.] If q = 0, the algorithm terminates unsuccessfully. Otherwise
set i <– i-q, and set (p, q) <– (q, p-q); then return to F2.
F4. [Increase i.] If p = 1, the algorithm terminates unsuccessfully. Otherwise
set i <– i + q, p <– p - q, then q <– q - p, and return to F2. |

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Java program

In this program, R1,…,RN were simplified to K1,…,KN.

/** * Created with IntelliJ IDEA. * User: 1O1O * Date: 12/11/13 * Time: 6:52 PM * :)~ * Fibonaccian Search-3:N+1 is NOT a perfect Fibonacci number:Searching */public class Main {    public static void main(String[] args) {        int N = 16;        int[] K = new int[17];        /*Prepare the ordered data table*/        K[1] = 61;        K[2] = 87;        K[3] = 154;        K[4] = 170;        K[5] = 275;        K[6] = 426;        K[7] = 503;        K[8] = 509;        K[9] = 512;        K[10] = 612;        K[11] = 653;        K[12] = 677;        K[13] = 703;        K[14] = 765;        K[15] = 897;        K[16] = 908;        /*Output sorted Ks*/        System.out.println("Sorted Ks:");        for(int i=1; i<=N; i++){            System.out.println(i+":"+K[i]);        }        System.out.println();        /*Kernel of the Algorithm!*/        int Key = 653;                  /*Key to be found*/        /*Fibonacci number:0,1,1,2,3,5,8,13,21,34,55...*/        /*Fn = Fn-1 + Fn-2, F0 = 0, F1 = 1...*/        /*In this case: N+1=17 is not a perfect Fibonacci number        then, find the least M>=0 such that N+M has the form Fk+1-1        and obviously the M is 4, N+M=20=F8-1=Fk+1-1, then k=7        then i=Fk=F7=13, p=Fk-1=F6=8, q=Fk-2=F5=5*/        int i = 13;        int p = 8;        int q = 5;        int M = 4;        if(Key > K[i]){            i-=M;            i = i+q;            p = p-q;            q = q-p;        }        do{            if(Key > K[i]){                if(p == 1){                    System.out.println("Outputs: "+Key+" not found.");                    break;                }else {                    i = i+q;                    p = p-q;                    q = q-p;                }            }else if(Key < K[i]){                if(q == 0){                    System.out.println("Outputs: "+Key+" not found.");                    break;                }else {                    i = i-q;                    int temp = q;                    q = p-q;                    p = temp;                }            }else {                System.out.println("Outputs: "+Key+" in K["+i+"].");                break;            }        }while (true);    }}

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Outputs

Sorted Ks:1:612:873:1544:1705:2756:4267:5038:5099:51210:61211:65312:67713:70314:76515:89716:908Outputs: 653 in K[11].

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Reference

<< The art of computer programming: Sorting and Searching >> VOLUME 3, DONALD E. KNUTH