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Queries for maximum and minimum difference between Fibonacci numbers in given ranges

  • Last Updated : 02 Jun, 2021

Given an array arr[][] containing N queries of the form [L, R], the task is to find the maximum difference between two Fibonacci numbers in the range for each query. If there are no Fibonacci numbers in the range or only one Fibonacci number, then print 0. 
Note: All the ranges are below 100005.

Examples:  

Input: N = 2, arr[][] = {{2, 2}, {2, 5}} 
Output: 0 3 
Explanation: 
In the first query, there is only one Fibonacci number. So, the answer is 0. 
In the second query, 2 is the minimum and 5 is the maximum Fibonacci number. 
Therefore, the maximum difference = 3. 

Input: N = 2, arr[][] = {{2, 21}, {30, 150}} 
Output: 19 110 
Explanation: 
In the first query, 2 is the minimum and 5 is the maximum Fibonacci number. 
Therefore, the maximum difference = 19. 
In the second query, 34 is the minimum and 144 is the maximum Fibonacci number. 
Therefore, the maximum difference = 110. 

Approach: The idea is to use the concept of hashing and prefix sum array to precompute and store the Fibonacci numbers in two arrays prefix[] and suffix[]



After performing the above precomputation, we can check if a number is a Fibonacci or not in constant time. Therefore, in order to perform the above operations, the following approach is used: 

  1. Find the maximum difference: In order to find the maximum difference, the prefix array which stores the largest Fibonacci number less than ‘i’ for every index and a suffix array that stores the smallest Fibonacci number greater than ‘i’ for every index is used. For every query {L, R}, prefix[R] – suffix[L] is returned.
  2. Find the minimum difference: The difference between the first two numbers in the range {L, R} is the minimum possible difference.

Below is the implementation of the above approach:  

C++




// C++ program to find the maximum differences
// between two Fibonacci numbers in given ranges
 
#include <bits/stdc++.h>
using namespace std;
#define MAX 100005
 
bool isFib[MAX];
int prefix[MAX], suffix[MAX];
 
// Function to precompute the Fibonacci,
// Prefix array and Suffix array
void precompute()
{
    // Initializing it with False
    memset(isFib, false, sizeof(isFib));
    // Variable to store the Fibonacci
    // numbers
 
    // Marking the first two Fibonacci numbers
    // as True in the array
    int prev = 0, curr = 1;
    isFib[prev] = isFib[curr] = true;
 
    // Loop to iterate until the maximum number
    while (curr < MAX) {
        int temp = curr + prev;
        isFib[temp] = true;
        prev = curr;
        curr = temp;
    }
 
    prefix[1] = 1;
    suffix[MAX - 1] = 1e9 + 7;
 
    // Precomputing Prefix array
    for (int i = 2; i < MAX; i++) {
 
        // If the number is a Fibonacci number,
        // then adding it to the prefix array
        if (isFib[i])
            prefix[i] = i;
        else
            prefix[i] = prefix[i - 1];
    }
 
    // Precompute Suffix array
    for (int i = MAX - 1; i > 1; i--) {
        if (isFib[i])
            suffix[i] = i;
        else
            suffix[i] = suffix[i + 1];
    }
}
 
// Function to solve each query
int query(int L, int R)
{
    if (prefix[R] < L || suffix[L] > R)
        return 0;
    else
        return prefix[R] - suffix[L];
}
 
// Function to return the minimum difference
// between any two fibonacci numbers
// from the given range [L, R]
int minDifference(int L, int R)
{
 
    // Find the first Fibonacci numbers
    // from the range
    int fst = 0;
 
    for (int i = L; i <= R; i++) {
 
        if (isFib[i]) {
            fst = i;
            break;
        }
    }
 
    // Find the second Fibonacci numbers
    // from the range
    int snd = 0;
    for (int i = fst + 1; i <= R; i++) {
 
        if (isFib[i]) {
            snd = i;
            break;
        }
    }
 
    // If the number of fibonacci numbers in
    // the given range is < 2
    if (snd == 0)
        return -1;
 
    // To store the minimum difference between
    // two consecutive fibonacci numbers from the range
    int diff = snd - fst;
 
    // Range left to check for fibonacci numbers
    int left = snd + 1;
    int right = R;
 
    // For every integer in the range
    for (int i = left; i <= right; i++) {
 
        // If the current integer is fibonacci
        if (isFib[i]) {
 
            // If the difference between i
            // and snd is minimum so far
            if (i - snd <= diff) {
 
                fst = snd;
                snd = i;
                diff = snd - fst;
            }
        }
    }
 
    return diff;
}
 
// Function to print the answer
// for every query
void findAns(int arr[][2], int q)
{
 
    precompute();
 
    // Finding the answer for every query
    for (int i = 0; i < q; i++) {
 
        cout << "Maximum Difference: "
            << query(arr[i][0], arr[i][1])
            << endl;
 
        cout << "Minimum Difference: "
            << minDifference(arr[i][0], arr[i][1])
            << endl;
    }
}
 
// Driver code
int main()
{
    int q = 1;
 
    int arr[][2] = { { 21, 100 } };
 
    findAns(arr, q);
 
    return 0;
}

Java




// Java program to find the maximum
// differences between two Fibonacci
// numbers in given ranges
import java.util.*;
import java.lang.*;
 
class GFG{
     
static final int MAX = 100005;
   
static boolean isFib[] = new boolean[MAX];
static int[] prefix = new int[MAX],
             suffix = new int[MAX];
   
// Function to precompute the Fibonacci,
// Prefix array and Suffix array
static void precompute()
{
     
    // Variable to store the Fibonacci
    // numbers
   
    // Marking the first two Fibonacci
    // numbers as True in the array
    int prev = 0, curr = 1;
    isFib[prev] = isFib[curr] = true;
   
    // Loop to iterate until the
    // maximum number
    while (curr + prev < MAX)
    {
        int temp = curr + prev;
        isFib[temp] = true;
        prev = curr;
        curr = temp;
    }
   
    prefix[1] = 1;
    suffix[MAX - 1] = (int)1e9 + 7;
   
    // Precomputing Prefix array
    for(int i = 2; i < MAX; i++)
    {
         
        // If the number is a Fibonacci
        // number, then adding it to the
        // prefix array
        if (isFib[i])
            prefix[i] = i;
        else
            prefix[i] = prefix[i - 1];
    }
   
    // Precompute Suffix array
    for(int i = MAX - 2; i > 1; i--)
    {
        if (isFib[i])
            suffix[i] = i;
        else
            suffix[i] = suffix[i + 1];
    }
}
   
// Function to solve each query
static int query(int L, int R)
{
    if (prefix[R] < L || suffix[L] > R)
        return 0;
    else
        return prefix[R] - suffix[L];
}
   
// Function to return the minimum
// difference between any two
// fibonacci numbers from the
// given range [L, R]
static int minDifference(int L, int R)
{
     
    // Find the first Fibonacci numbers
    // from the range
    int fst = 0;
   
    for(int i = L; i <= R; i++)
    {
        if (isFib[i])
        {
            fst = i;
            break;
        }
    }
   
    // Find the second Fibonacci numbers
    // from the range
    int snd = 0;
    for(int i = fst + 1; i <= R; i++)
    {
        if (isFib[i])
        {
            snd = i;
            break;
        }
    }
   
    // If the number of fibonacci
    // numbers in the given range is < 2
    if (snd == 0)
        return -1;
   
    // To store the minimum difference
    // between two consecutive fibonacci
    // numbers from the range
    int diff = snd - fst;
   
    // Range left to check for
    // fibonacci numbers
    int left = snd + 1;
    int right = R;
   
    // For every integer in the range
    for(int i = left; i <= right; i++)
    {
         
        // If the current integer is fibonacci
        if (isFib[i])
        {
             
            // If the difference between i
            // and snd is minimum so far
            if (i - snd <= diff)
            {
                fst = snd;
                snd = i;
                diff = snd - fst;
            }
        }
    }
    return diff;
}
   
// Function to print the answer
// for every query
static void findAns(int arr[][], int q)
{
    precompute();
     
    // Finding the answer for every query
    for(int i = 0; i < q; i++)
    {
       System.out.println("Maximum Difference: " +
                    query(arr[i][0], arr[i][1]));
             
   
       System.out.println("Minimum Difference: " +
            minDifference(arr[i][0], arr[i][1]));
    }
}
 
// Driver code
public static void main(String[] args)
{
    int q = 1;
     
    int arr[][] = { { 21, 100 } };
     
    findAns(arr, q);
}
}
 
// This code is contributed by offbeat

Python3




# Python3 program to find the maximum differences
# between two Fibonacci numbers in given ranges
 
MAX = 100005
 
isFib = [False]*MAX
prefix = [0]*MAX
suffix = [0]*MAX
 
# Function to precompute the Fibonacci,
# Prefix array and Suffix array
def precompute():
 
    # Marking the first two Fibonacci numbers
    # as True in the array
    prev , curr = 0 , 1
    isFib[prev] = True
    isFib[curr] = True
 
    # Loop to iterate until the maximum number
    while (curr < MAX):
        temp = curr + prev
        if temp<MAX:
            isFib[temp] = True
        prev = curr
        curr = temp
 
    prefix[1] = 1
    suffix[MAX - 1] = 1000000007
 
    # Precomputing Prefix array
    for i in range(2, MAX):
 
        # If the number is a Fibonacci number,
        # then adding it to the prefix array
        if (isFib[i]):
            prefix[i] = i
        else:
            prefix[i] = prefix[i - 1]
 
    # Precompute Suffix array
    for i in range(MAX - 2, 1, -1):
        if (isFib[i]):
            suffix[i] = i
        else:
            suffix[i] = suffix[i + 1]
 
# Function to solve each query
def query(L, R):
 
    if (prefix[R] < L or suffix[L] > R):
        return 0
    else:
        return prefix[R] - suffix[L]
 
# Function to return the minimum difference
# between any two fibonacci numbers
# from the given range [L, R]
def minDifference(L, R):
 
    # Find the first Fibonacci numbers
    # from the range
    fst = 0
    for i in range(L, R + 1):
        if (isFib[i]):
            fst = i
            break
 
    # Find the second Fibonacci numbers
    # from the range
    snd = 0
    for i in range(fst + 1, R + 1 ):
 
        if (isFib[i]):
            snd = i
            break
 
    # If the number of fibonacci numbers in
    # the given range is < 2
    if (snd == 0):
        return -1
 
    # To store the minimum difference between
    # two consecutive fibonacci numbers from the range
    diff = snd - fst
 
    # Range left to check for fibonacci numbers
    left = snd + 1
    right = R
 
    # For every integer in the range
    for i in range(left, right + 1):
 
        # If the current integer is fibonacci
        if (isFib[i]):
            # If the difference between i
            # and snd is minimum so far
            if (i - snd <= diff):
                fst = snd
                snd = i
                diff = snd - fst
    return diff
 
# Function to print the answer
# for every query
def findAns(arr, q):
 
    precompute()
 
    # Finding the answer for every query
    for i in range(q):
 
        print( "Maximum Difference: "
            , query(arr[i][0], arr[i][1]))
 
        print("Minimum Difference: "
            , minDifference(arr[i][0], arr[i][1]))
 
# Driver code
if __name__ == "__main__":
     
    q = 1
 
    arr = [ [ 21, 100 ] ]
 
    findAns(arr, q)
 
# This code is contributed by chitranayal

C#




using System;
 
// C# program to find the maximum
// differences between two Fibonacci
// numbers in given ranges
public class GFG{
 
  static int MAX = 100005;
  static bool[] isFib = new bool[MAX];
 
  static int[] prefix = new int[MAX],
  suffix = new int[MAX];
 
  // Function to precompute the Fibonacci,
  // Prefix array and Suffix array
  static void precompute()
  {
 
    // Variable to store the Fibonacci
    // numbers
 
    // Marking the first two Fibonacci
    // numbers as True in the array
    int prev = 0, curr = 1;
    isFib[prev] = isFib[curr] = true;
 
    // Loop to iterate until the
    // maximum number
    while (curr + prev < MAX)
    {
      int temp = curr + prev;
      isFib[temp] = true;
      prev = curr;
      curr = temp;
    }
 
    prefix[1] = 1;
    suffix[MAX - 1] = (int)1e9 + 7;
 
    // Precomputing Prefix array
    for(int i = 2; i < MAX; i++)
    {
 
      // If the number is a Fibonacci
      // number, then adding it to the
      // prefix array
      if (isFib[i])
        prefix[i] = i;
      else
        prefix[i] = prefix[i - 1];
    }
 
    // Precompute Suffix array
    for(int i = MAX - 2; i > 1; i--)
    {
      if (isFib[i])
        suffix[i] = i;
      else
        suffix[i] = suffix[i + 1];
    }
  }
 
  // Function to solve each query
  static int query(int L, int R)
  {
    if (prefix[R] < L || suffix[L] > R)
      return 0;
    else
      return prefix[R] - suffix[L];
  }
 
  // Function to return the minimum
  // difference between any two
  // fibonacci numbers from the
  // given range [L, R]
  static int minDifference(int L, int R)
  {
 
    // Find the first Fibonacci numbers
    // from the range
    int fst = 0;
 
    for(int i = L; i <= R; i++)
    {
      if (isFib[i])
      {
        fst = i;
        break;
      }
    }
 
    // Find the second Fibonacci numbers
    // from the range
    int snd = 0;
    for(int i = fst + 1; i <= R; i++)
    {
      if (isFib[i])
      {
        snd = i;
        break;
      }
    }
 
    // If the number of fibonacci
    // numbers in the given range is < 2
    if (snd == 0)
      return -1;
 
    // To store the minimum difference
    // between two consecutive fibonacci
    // numbers from the range
    int diff = snd - fst;
 
    // Range left to check for
    // fibonacci numbers
    int left = snd + 1;
    int right = R;
 
    // For every integer in the range
    for(int i = left; i <= right; i++)
    {
 
      // If the current integer is fibonacci
      if (isFib[i])
      {
 
        // If the difference between i
        // and snd is minimum so far
        if (i - snd <= diff)
        {
          fst = snd;
          snd = i;
          diff = snd - fst;
        }
      }
    }
    return diff;
  }
 
  // Function to print the answer
  // for every query
  static void findAns(int[,] arr, int q)
  {
    precompute();
 
    // Finding the answer for every query
    for(int i = 0; i < q; i++)
    {
      Console.WriteLine("Maximum Difference: " +
                        query(arr[i,0], arr[i,1]));
 
 
      Console.WriteLine("Minimum Difference: " +
                        minDifference(arr[i,0], arr[i,1]));
    }
  }
 
  // Driver code
  static public void Main ()
  {
 
    int q = 1;
 
    int[,] arr = { { 21, 100 } };
 
    findAns(arr, q);
 
  }
}
 
// This code is contributed by avanitrachhadiya2155

Javascript




<script>
 
// JavaScript program to find the maximum differences
// between two Fibonacci numbers in given ranges
 
let MAX = 100005
 
let isFib = new Array(MAX);
let prefix = new Array(MAX)
let suffix = new Array(MAX);
 
// Function to precompute the Fibonacci,
// Prefix array and Suffix array
function precompute()
{
    // Initializing it with False
    isFib.fill(false);
    // Variable to store the Fibonacci
    // numbers
 
    // Marking the first two Fibonacci numbers
    // as True in the array
    let prev = 0, curr = 1;
    isFib[prev] = isFib[curr] = true;
 
    // Loop to iterate until the maximum number
    while (curr < MAX) {
        let temp = curr + prev;
        isFib[temp] = true;
        prev = curr;
        curr = temp;
    }
 
    prefix[1] = 1;
    suffix[MAX - 1] = 1e9 + 7;
 
    // Precomputing Prefix array
    for (let i = 2; i < MAX; i++) {
 
        // If the number is a Fibonacci number,
        // then adding it to the prefix array
        if (isFib[i])
            prefix[i] = i;
        else
            prefix[i] = prefix[i - 1];
    }
 
    // Precompute Suffix array
    for (let i = MAX - 1; i > 1; i--) {
        if (isFib[i])
            suffix[i] = i;
        else
            suffix[i] = suffix[i + 1];
    }
}
 
// Function to solve each query
function query(L, R)
{
    if (prefix[R] < L || suffix[L] > R)
        return 0;
    else
        return prefix[R] - suffix[L];
}
 
// Function to return the minimum difference
// between any two fibonacci numbers
// from the given range [L, R]
function minDifference(L, R)
{
 
    // Find the first Fibonacci numbers
    // from the range
    let fst = 0;
 
    for (let i = L; i <= R; i++) {
 
        if (isFib[i]) {
            fst = i;
            break;
        }
    }
 
    // Find the second Fibonacci numbers
    // from the range
    let snd = 0;
    for (let i = fst + 1; i <= R; i++) {
 
        if (isFib[i]) {
            snd = i;
            break;
        }
    }
 
    // If the number of fibonacci numbers in
    // the given range is < 2
    if (snd == 0)
        return -1;
 
    // To store the minimum difference between
    // two consecutive fibonacci numbers from the range
    let diff = snd - fst;
 
    // Range left to check for fibonacci numbers
    let left = snd + 1;
    let right = R;
 
    // For every integer in the range
    for (let i = left; i <= right; i++) {
 
        // If the current integer is fibonacci
        if (isFib[i]) {
 
            // If the difference between i
            // and snd is minimum so far
            if (i - snd <= diff) {
 
                fst = snd;
                snd = i;
                diff = snd - fst;
            }
        }
    }
 
    return diff;
}
 
// Function to print the answer
// for every query
function findAns(arr, q)
{
 
    precompute();
 
    // Finding the answer for every query
    for (let i = 0; i < q; i++) {
 
        document.write("Maximum Difference: "
            + query(arr[i][0], arr[i][1])
            + "<br>");
 
        document.write("Minimum Difference: "
            + minDifference(arr[i][0], arr[i][1])
            + "<br>");
    }
}
 
// Driver code
 
let q = 1;
 
let arr = [ [ 21, 100 ] ];
 
findAns(arr, q);
 
</script>

Output:

Maximum Difference: 68
Minimum Difference: 13




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