Array range queries to count the number of Fibonacci numbers with updates

Given an array arr[] of N integers, the task is to perform the following two queries: 

• query(start, end): Print the number of fibonacci numbers in the subarray from start to end
• update(i, x): Add x to the array element referenced by array index i, that is: arr[i] = x

Examples:

Input: arr = { 1, 2, 3, 4, 8, 9 } 
Query 1: query(start = 0, end = 4) 
Query 2: update(i = 3, x = 5) 
Query 3: query(start = 0, end = 4) 
Output:4 
5 
Explanation
In Query 1, the subarray [0…4] has 4 fibonacci numbers viz. {1, 2, 3, 8} 
In Query 2, the value at index 3 is updated to 5, the array arr now is, {1, 2, 3, 5, 8, 9} 
In Query 3, the subarray [0…4] has 5 fibonacci numbers viz. {1, 2, 3, 5, 8} 

Approach: To handle both point updates and range queries, a segment tree is optimal for this purpose.
In order to check for Fibonacci numbers, we can build a hash table using dynamic programming containing all the Fibonacci numbers less than or equal to the maximum value arr_i. We can take MAX which will be used to test a number in O(1) time complexity.
Building the segment tree:

• The problem is now reduced to the subarray sum using segment tree problem.
• Now, we can build the segment tree where a leaf node is represented as either 0 (if it is not a prime number) or 1 (if it is a Fibonacci number).
• The internal nodes of the segment tree equal to the sum of its child nodes, thus a node represent the total Fibonacci numbers in the range from L to R with range [L, R] falling under this node and the sub-tree underneath it.

Handling Queries and Point Updates:

• Whenever we receive a query from beginning to end, we can query the segment tree for the sum of nodes in range from start to end, which in turn represent the number of Fibonacci numbers in the range start to end. 
  
• To perform a point update and to update the value at index i to x, we check for the following cases: 
  Let the old value of arr_i be y and the new value be x. 
  1. Case 1: Fibonacci: If x and y both are Fibonacci numbers
     Count of Fibonacci numbers in the subarray does not change so we just update array and do not modify the segment tree
  2. Case 2: If x and y both are not Fibonacci numbers
     Count of Fibonacci numbers in the subarray does not change so we just update array and do not modify the segment tree
  3. Case 3: If y is a Fibonacci number but x is not
     Count of Fibonacci numbers in the subarray decreases so we update array and add -1 to every range. The index i which is to be updated is a part of in the segment tree
  4. Case 4: If y is not a Fibonacci number but x is a Fibonacci number
     Count of Fibonacci numbers in the subarray increases so we update array and add 1 to every range. The index i which is to be updated is a part of in the segment tree

Below is the implementation of the above approach:

C++

// C++ program to find number of fibonacci numbers
// in a subarray and performing updates

#include <bits/stdc++.h>
using namespace std;

#define MAX 1000

// Function to create hash table
// to check Fibonacci numbers
void createHash(set<int>& hash,
                int maxElement)
{
    int prev = 0, curr = 1;
    hash.insert(prev);
    hash.insert(curr);

    while (curr <= maxElement) {
        int temp = curr + prev;
        hash.insert(temp);
        prev = curr;
        curr = temp;
    }
}

// A utility function to get the middle
// index from corner indexes.
int getMid(int s, int e)
{
    return s + (e - s) / 2;
}

// Recursive function to get the number
// of Fibonacci numbers in a given range
/* where
    st    --> Pointer to segment tree
    index --> Index of current node in the
              segment tree. Initially 0 is passed
              as root is always at index 0
    ss & se  --> Starting and ending indexes of
              the segment represented by current
              node, i.e., st[index]
    qs & qe  --> Starting and ending indexes
              of query range
    */
int queryFibonacciUtil(int* st, int ss,
                       int se, int qs,
                       int qe, int index)
{
    // If segment of this node is a part
    // of given range, then return
    // the number of Fibonacci numbers
    // in the segment
    if (qs <= ss && qe >= se)
        return st[index];

    // If segment of this node
    // is outside the given range
    if (se < qs || ss > qe)
        return 0;

    // If a part of this segment
    // overlaps with the given range
    int mid = getMid(ss, se);
    return queryFibonacciUtil(st, ss, mid, qs,
                              qe, 2 * index + 1)
           + queryFibonacciUtil(st, mid + 1, se,
                                qs, qe, 2 * index + 2);
}

// Recursive function to update
// the nodes which have the given
// index in their range.
/* where
    st, si, ss and se are same as getSumUtil()
    i --> index of the element to be updated.
          This index is in input array.
   diff --> Value to be added to all nodes
          which have i in range
*/
void updateValueUtil(int* st, int ss,
                     int se, int i,
                     int diff, int si)
{
    // Base Case:
    // If the input index lies outside
    // the range of this segment
    if (i < ss || i > se)
        return;

    // If the input index is in range
    // of this node, then update the value
    // of the node and its children
    st[si] = st[si] + diff;
    if (se != ss) {

        int mid = getMid(ss, se);
        updateValueUtil(st, ss, mid, i,
                        diff, 2 * si + 1);
        updateValueUtil(st, mid + 1, se,
                        i, diff, 2 * si + 2);
    }
}

// Function to update a value in the
// input array and segment tree.
// It uses updateValueUtil() to update
// the value in segment tree
void updateValue(int arr[], int* st,
                 int n, int i,
                 int new_val,
                 set<int> hash)
{
    // Check for erroneous input index
    if (i < 0 || i > n - 1) {
        printf("Invalid Input");
        return;
    }

    int diff, oldValue;

    oldValue = arr[i];

    // Update the value in array
    arr[i] = new_val;

    // Case 1: Old and new values
    // both are Fibonacci numbers
    if (hash.find(oldValue) != hash.end()
        && hash.find(new_val) != hash.end())
        return;

    // Case 2: Old and new values
    // both not Fibonacci numbers
    if (hash.find(oldValue) == hash.end()
        && hash.find(new_val) == hash.end())
        return;

    // Case 3: Old value was Fibonacci,
    // new value is non Fibonacci
    if (hash.find(oldValue) != hash.end()
        && hash.find(new_val) == hash.end()) {
        diff = -1;
    }

    // Case 4: Old value was non Fibonacci,
    // new_val is Fibonacci
    if (hash.find(oldValue) == hash.end()
        && hash.find(new_val) != hash.end()) {
        diff = 1;
    }

    // Update the values of nodes in segment tree
    updateValueUtil(st, 0, n - 1, i, diff, 0);
}

// Return number of Fibonacci numbers
// in range from index qs (query start)
// to qe (query end).
// It mainly uses queryFibonacciUtil()
void queryFibonacci(int* st, int n,
                    int qs, int qe)
{
    int FibonacciInRange
        = queryFibonacciUtil(st, 0, n - 1,
                             qs, qe, 0);

    cout << "Number of Fibonacci numbers "
         << "in subarray from "
         << qs << " to "
         << qe << " = "
         << FibonacciInRange << "\n";
}

// Recursive function that constructs
// Segment Tree for array[ss..se].
// si is index of current node
// in segment tree st
int constructSTUtil(int arr[], int ss,
                    int se, int* st,
                    int si, set<int> hash)
{
    // If there is one element in array,
    // check if it is Fibonacci number
    // then store 1 in the segment tree
    // else store 0 and return
    if (ss == se) {

        // if arr[ss] is fibonacci number
        if (hash.find(arr[ss]) != hash.end())
            st[si] = 1;
        else
            st[si] = 0;

        return st[si];
    }

    // If there are more than one elements,
    // then recur for left and right subtrees
    // and store the sum of the
    // two values in this node
    int mid = getMid(ss, se);
    st[si] = constructSTUtil(arr, ss, mid, st,
                             si * 2 + 1, hash)
             + constructSTUtil(arr, mid + 1, se, st,
                               si * 2 + 2, hash);
    return st[si];
}

// Function to construct a segment tree from given array.
// This function allocates memory for segment tree and
// calls constructSTUtil() to fill the allocated memory
int* constructST(int arr[], int n, set<int> hash)
{
    // Allocate memory for segment tree

    // Height of segment tree
    int x = (int)(ceil(log2(n)));

    // Maximum size of segment tree
    int max_size = 2 * (int)pow(2, x) - 1;

    int* st = new int[max_size];

    // Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, st, 0, hash);

    // Return the constructed segment tree
    return st;
}

// Driver Code
int main()
{

    int arr[] = { 1, 2, 3, 4, 8, 9 };
    int n = sizeof(arr) / sizeof(arr[0]);

    // find the largest node value in the array
    int maxEle = *max_element(arr, arr + n);

    // Creating a set containing all Fibonacci numbers
    // upto the maximum data value in the array
    set<int> hash;
    createHash(hash, maxEle);

    // Build segment tree from given array
    int* st = constructST(arr, n, hash);

    // Query 1: Query(start = 0, end = 4)
    int start = 0;
    int end = 4;
    queryFibonacci(st, n, start, end);

    // Query 2: Update(i = 3, x = 5),
    // i.e Update a[i] to x
    int i = 3;
    int x = 5;
    updateValue(arr, st, n, i, x, hash);

    // uncomment to see array after update
    // for(int i = 0; i < n; i++)
    // cout << arr[i] << " ";

    // Query 3: Query(start = 0, end = 4)
    start = 0;
    end = 4;
    queryFibonacci(st, n, start, end);

    return 0;
}

Java

// Java program to find number of fibonacci numbers
// in a subarray and performing updates
import java.util.Arrays;
import java.util.HashSet;
import java.util.Set;
class GFG
{
  static final int MAX = 1000;

  // Function to create hash table
  // to check Fibonacci numbers
  static void createHash(Set<Integer> hash, int maxElement)
  {
    int prev = 0, curr = 1;
    hash.add(prev);
    hash.add(curr);
    while (curr <= maxElement)
    {
      int temp = curr + prev;
      hash.add(temp);
      prev = curr;
      curr = temp;
    }
  }

  // A utility function to get the middle
  // index from corner indexes.
  static int getMid(int s, int e)
  {
    return s + (e - s) / 2;
  }

  // Recursive function to get the number
  // of Fibonacci numbers in a given range
  /*
     * where st --> Pointer to segment tree index --> Index of current node in the
     * segment tree. Initially 0 is passed as root is always at index 0 ss & se -->
     * Starting and ending indexes of the segment represented by current node, i.e.,
     * st[index] qs & qe --> Starting and ending indexes of query range
     */
  static int queryFibonacciUtil(int[] st, int ss, int se,
                                int qs, int qe, int index)
  {

    // If segment of this node is a part
    // of given range, then return
    // the number of Fibonacci numbers
    // in the segment
    if (qs <= ss && qe >= se)
      return st[index];

    // If segment of this node
    // is outside the given range
    if (se < qs || ss > qe)
      return 0;

    // If a part of this segment
    // overlaps with the given range
    int mid = getMid(ss, se);
    return queryFibonacciUtil(st, ss, mid,
                              qs, qe, 2 * index + 1)
      + queryFibonacciUtil(st, mid + 1, se,
                           qs, qe, 2 * index + 2);
  }

  // Recursive function to update
  // the nodes which have the given
  // index in their range.
  /*
     * where st, si, ss and se are same as getSumUtil() i --> index of the element
     * to be updated. This index is in input array. diff --> Value to be added to
     * all nodes which have i in range
     */
  static void updateValueUtil(int[] st, int ss,
                              int se, int i,
                              int diff, int si)
  {

    // Base Case:
    // If the input index lies outside
    // the range of this segment
    if (i < ss || i > se)
      return;

    // If the input index is in range
    // of this node, then update the value
    // of the node and its children
    st[si] = st[si] + diff;
    if (se != ss)
    {

      int mid = getMid(ss, se);
      updateValueUtil(st, ss, mid, i,
                      diff, 2 * si + 1);
      updateValueUtil(st, mid + 1, se, i,
                      diff, 2 * si + 2);
    }
  }

  // Function to update a value in the
  // input array and segment tree.
  // It uses updateValueUtil() to update
  // the value in segment tree
  static void updateValue(int arr[], int[] st, int n,
                          int i, int new_val, Set<Integer> hash)
  {

    // Check for erroneous input index
    if (i < 0 || i > n - 1)
    {
      System.out.printf("Invalid Input");
      return;
    }

    int diff = 0, oldValue;
    oldValue = arr[i];

    // Update the value in array
    arr[i] = new_val;

    // Case 1: Old and new values
    // both are Fibonacci numbers
    if (hash.contains(oldValue) &&
        hash.contains(new_val))
      return;

    // Case 2: Old and new values
    // both not Fibonacci numbers
    if (!hash.contains(oldValue) &&
        !hash.contains(new_val))
      return;

    // Case 3: Old value was Fibonacci,
    // new value is non Fibonacci
    if (hash.contains(oldValue) &&
        !hash.contains(new_val))
    {
      diff = -1;
    }

    // Case 4: Old value was non Fibonacci,
    // new_val is Fibonacci
    if (!hash.contains(oldValue) &&
        hash.contains(new_val))
    {
      diff = 1;
    }

    // Update the values of nodes in segment tree
    updateValueUtil(st, 0, n - 1, i, diff, 0);
  }

  // Return number of Fibonacci numbers
  // in range from index qs (query start)
  // to qe (query end).
  // It mainly uses queryFibonacciUtil()
  static void queryFibonacci(int[] st, int n, int qs, int qe)
  {
    int FibonacciInRange = queryFibonacciUtil(st, 0,
                                              n - 1, qs, qe, 0);
    System.out.printf("Number of Fibonacci numbers in subarray from %d to %d = %d\n", qs, qe, FibonacciInRange);
  }

  // Recursive function that constructs
  // Segment Tree for array[ss..se].
  // si is index of current node
  // in segment tree st
  static int constructSTUtil(int arr[], int ss, int se,
                             int[] st, int si, Set<Integer> hash)
  {

    // If there is one element in array,
    // check if it is Fibonacci number
    // then store 1 in the segment tree
    // else store 0 and return
    if (ss == se)
    {

      // if arr[ss] is fibonacci number
      if (hash.contains(arr[ss]))
        st[si] = 1;
      else
        st[si] = 0;
      return st[si];
    }

    // If there are more than one elements,
    // then recur for left and right subtrees
    // and store the sum of the
    // two values in this node
    int mid = getMid(ss, se);
    st[si] = constructSTUtil(arr, ss, mid, st,
                             si * 2 + 1, hash)
      + constructSTUtil(arr, mid + 1, se,
                        st, si * 2 + 2, hash);
    return st[si];
  }

  // Function to construct a segment tree from given array.
  // This function allocates memory for segment tree and
  // calls constructSTUtil() to fill the allocated memory
  static int[] constructST(int arr[], int n, Set<Integer> hash)
  {

    // Allocate memory for segment tree

    // Height of segment tree
    int x = (int) (Math.ceil(Math.log(n) / Math.log(2)));

    // Maximum size of segment tree
    int max_size = 2 * (int) Math.pow(2, x) - 1;
    int[] st = new int[max_size];

    // Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, st, 0, hash);

    // Return the constructed segment tree
    return st;
  }

  // Driver Code
  public static void main(String[] args)
  {

    int arr[] = { 1, 2, 3, 4, 8, 9 };
    int n = arr.length;

    // find the largest node value in the array
    int maxEle = Arrays.stream(arr).max().getAsInt();

    // Creating a set containing all Fibonacci numbers
    // upto the maximum data value in the array
    Set<Integer> hash = new HashSet<>();
    createHash(hash, maxEle);

    // Build segment tree from given array
    int[] st = constructST(arr, n, hash);

    // Query 1: Query(start = 0, end = 4)
    int start = 0;
    int end = 4;
    queryFibonacci(st, n, start, end);

    // Query 2: Update(i = 3, x = 5),
    // i.e Update a[i] to x
    int i = 3;
    int x = 5;
    updateValue(arr, st, n, i, x, hash);

    // uncomment to see array after update
    // for(int i = 0; i < n; i++)
    // cout << arr[i] << " ";

    // Query 3: Query(start = 0, end = 4)
    start = 0;
    end = 4;
    queryFibonacci(st, n, start, end);
  }
}

// This code is contributed by sanjeev2552

Python3

# Python program to find number of fibonacci numbers
# in a subarray and performing updates
import math
MAX = 1000

# Function to create hash table
# to check Fibonacci numbers
def createHash(hash, maxElement):
    prev = 0
    curr = 1
    hash.add(prev)
    hash.add(curr)

    while (curr <= maxElement):
        temp = curr + prev
        hash.add(temp)
        prev = curr
        curr = temp

# A utility function to get the middle
# index from corner indexes.


def getMid(s, e):
    return math.floor(s + (e - s) / 2)

# Recursive function to get the number
# of Fibonacci numbers in a given range
   # where
   #  st    --> Pointer to segment tree
   #  index --> Index of current node in the
   #            segment tree. Initially 0 is passed
   #            as root is always at index 0
   #  ss & se  --> Starting and ending indexes of
   #            the segment represented by current
   #            node, i.e., st[index]
   #  qs & qe  --> Starting and ending indexes
   #            of query range
def queryFibonacciUtil(st, ss, se, qs, qe, index):

    # If segment of this node is a part
    # of given range, then return
    # the number of Fibonacci numbers
    # in the segment
    if (qs <= ss and qe >= se):
        return st[index]

    # If segment of this node
    # is outside the given range
    if (se < qs or ss > qe):
        return 0

    # If a part of this segment
    # overlaps with the given range
    mid = getMid(ss, se)
    return queryFibonacciUtil(st, ss, mid, qs, qe, 2 * index + 1) + queryFibonacciUtil(st, mid + 1, se, qs, qe, 2 * index + 2)

# Recursive function to update
# the nodes which have the given
# index in their range.
   # where
   #  st, si, ss and se are same as getSumUtil()
   #  i --> index of the element to be updated.
   #        This index is in input array.
   # diff --> Value to be added to all nodes
   #        which have i in range
def updateValueUtil(st, ss, se, i, diff, si):
    # Base Case:
    # If the input index lies outside
    # the range of this segment
    if (i < ss or i > se):
        return

    # If the input index is in range
    # of this node, then update the value
    # of the node and its children
    st[si] = st[si] + diff
    if (se != ss):
        mid = getMid(ss, se)
        updateValueUtil(st, ss, mid, i, diff, 2 * si + 1)
        updateValueUtil(st, mid + 1, se, i, diff, 2 * si + 2)

# Function to update a value in the
# input array and segment tree.
# It uses updateValueUtil() to update
# the value in segment tree
def updateValue(arr, st, n, i, new_val, hash):
    # Check for erroneous input index
    if (i < 0 or i > n - 1):
        print("Invalid Input")
        return

    diff = 0
    oldValue = 0

    oldValue = arr[i]

    # Update the value in array
    arr[i] = new_val

    # Case 1: Old and new values
    # both are Fibonacci numbers
    if oldValue in hash:
        if new_val in hash:
            return

    # Case 2: Old and new values
    # both not Fibonacci numbers
    if not oldValue in hash:
        if not new_val in hash:
            return

    # Case 3: Old value was Fibonacci,
    # new value is non Fibonacci
    if oldValue in hash:
        if not new_val in hash:
            diff = -1

    # Case 4: Old value was non Fibonacci,
    # new_val is Fibonacci
    if not oldValue in hash:
        if new_val in hash:
            diff = 1

    # Update the values of nodes in segment tree
    updateValueUtil(st, 0, n - 1, i, diff, 0)

# Return number of Fibonacci numbers
# in range from index qs (query start)
# to qe (query end).
# It mainly uses queryFibonacciUtil()
def queryFibonacci(st, n, qs, qe):
    FibonacciInRange = queryFibonacciUtil(st, 0, n - 1, qs, qe, 0)

    print(
        f"Number of Fibonacci numbers in subarray from {qs} to {qe} = {FibonacciInRange}")

# Recursive function that constructs
# Segment Tree for array[ss..se].
# si is index of current node
# in segment tree st
def constructSTUtil(arr, ss, se, st, si, hash):
    # If there is one element in array,
    # check if it is Fibonacci number
    # then store 1 in the segment tree
    # else store 0 and return
    if (ss == se):

        # if arr[ss] is fibonacci number
        if arr[ss] in hash:
            st[si] = 1
        else:
            st[si] = 0

        return st[si]

    # If there are more than one elements,
    # then recur for left and right subtrees
    # and store the sum of the
    # two values in this node
    mid = getMid(ss, se)
    st[si] = constructSTUtil(arr, ss, mid, st, si * 2 + 1, hash) + \
        constructSTUtil(arr, mid + 1, se, st, si * 2 + 2, hash)
    return st[si]

# Function to construct a segment tree from given array.
# This function allocates memory for segment tree and
# calls constructSTUtil() to fill the allocated memory
def constructST(arr, n, hash):
    # Allocate memory for segment tree

    # Height of segment tree
    x = math.floor(math.ceil(math.log2(n)))

    # Maximum size of segment tree
    max_size = 2 * math.floor(math.pow(2, x)) - 1

    st = [0]*max_size

    # Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, st, 0, hash)

    # Return the constructed segment tree
    return st

# Driver Code
arr = [1, 2, 3, 4, 8, 9]
n = len(arr)

# find the largest node value in the array
maxEle = -1
for i in range(len(arr)):
    if arr[i] > maxEle:
        maxEle = arr[i]


# Creating a set containing all Fibonacci numbers
# upto the maximum data value in the array
hash = set()
createHash(hash, maxEle)

# Build segment tree from given array
st = constructST(arr, n, hash)

# Query 1: Query(start = 0, end = 4)
start = 0
end = 4
queryFibonacci(st, n, start, end)

# Query 2: Update(i = 3, x = 5),
# i.e Update a[i] to x
i = 3
x = 5
updateValue(arr, st, n, i, x, hash)

# uncomment to see array after update
# for(int i = 0; i < n; i++)
# cout << arr[i] << " ";

# Query 3: Query(start = 0, end = 4)
start = 0
end = 4
queryFibonacci(st, n, start, end)

# The code is contributed by Gautam goel (gautamgoel962)

C#

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;

// C# program to find number of fibonacci numbers
// in a subarray and performing updates
class HelloWorld {

  public static int MAX = 1000;

  // Function to create hash table
  // to check Fibonacci numbers
  public static void createHash(HashSet<int> hash, int maxElement)
  {
    int prev = 0;
    int curr = 1;
    hash.Add(prev);
    hash.Add(curr);

    while (curr <= maxElement) {
      int temp = curr + prev;
      hash.Add(temp);
      prev = curr;
      curr = temp;
    }
  }

  // A utility function to get the middle
  // index from corner indexes.
  public static int getMid(int s, int e)
  {
    return s + ((e - s) / 2);
  }

  // Recursive function to get the number
  // of Fibonacci numbers in a given range
  /* where
        st    --> Pointer to segment tree
        index --> Index of current node in the
                  segment tree. Initially 0 is passed
                  as root is always at index 0
        ss & se  --> Starting and ending indexes of
                  the segment represented by current
                  node, i.e., st[index]
        qs & qe  --> Starting and ending indexes
                  of query range
        */
  public static int queryFibonacciUtil(List<int> st, int ss,
                                       int se, int qs, int qe, int index)
  {
    // If segment of this node is a part
    // of given range, then return
    // the number of Fibonacci numbers
    // in the segment
    if (qs <= ss && qe >= se)
      return st[index];

    // If segment of this node
    // is outside the given range
    if (se < qs || ss > qe)
      return 0;

    // If a part of this segment
    // overlaps with the given range
    int mid = getMid(ss, se);
    return queryFibonacciUtil(st, ss, mid, qs, qe, 2 * index + 1) +
      queryFibonacciUtil(st, mid + 1, se, qs, qe, 2 * index + 2);
  }

  // Recursive function to update
  // the nodes which have the given
  // index in their range.
  /* where
        st, si, ss and se are same as getSumUtil()
        i --> index of the element to be updated.
              This index is in input array.
       diff --> Value to be added to all nodes
              which have i in range
    */
  public static void updateValueUtil(List<int> st, int ss,
                                     int se, int i,
                                     int diff, int si)
  {
    // Base Case:
    // If the input index lies outside
    // the range of this segment
    if (i < ss || i > se)
      return;

    // If the input index is in range
    // of this node, then update the value
    // of the node and its children
    st[si] = st[si] + diff;
    if (se != ss) {

      int mid = getMid(ss, se);
      updateValueUtil(st, ss, mid, i, diff, 2 * si + 1);
      updateValueUtil(st, mid + 1, se, i, diff, 2 * si + 2);
    }
  }

  // Function to update a value in the
  // input array and segment tree.
  // It uses updateValueUtil() to update
  // the value in segment tree
  public static void updateValue(int[] arr, List<int> st,
                                 int n, int i, int new_val,
                                 HashSet<int> hash)
  {
    // Check for erroneous input index
    if (i < 0 || i > n - 1) {
      Console.WriteLine("Invalid Input");
      return;
    }

    int diff = 0;
    int oldValue = 0;

    oldValue = arr[i];

    // Update the value in array
    arr[i] = new_val;

    // Case 1: Old and new values
    // both are Fibonacci numbers
    if (hash.Contains(oldValue) == true && hash.Contains(new_val) == true)
      return;

    // Case 2: Old and new values
    // both not Fibonacci numbers
    if (hash.Contains(oldValue) == false && hash.Contains(new_val) == false)
      return;

    // Case 3: Old value was Fibonacci,
    // new value is non Fibonacci
    if (hash.Contains(oldValue) == true && hash.Contains(new_val) == false){
      diff = -1;
    }


    // Case 4: Old value was non Fibonacci,
    // new_val is Fibonacci
    if (hash.Contains(oldValue) == false && hash.Contains(new_val) == true){
      diff = 1;
    }

    // Update the values of nodes in segment tree
    updateValueUtil(st, 0, n -1, i, diff, 0);
  }



  // Return number of Fibonacci numbers
  // in range from index qs (query start)
  // to qe (query end).
  // It mainly uses queryFibonacciUtil()
  public static void queryFibonacci(List<int> st, int n, int qs, int qe)
  {
    int FibonacciInRange = queryFibonacciUtil(st, 0, n - 1, qs, qe, 0);

    Console.WriteLine("Number of Fibonacci numbers in subarray from " + qs + " to " + qe + " = " + FibonacciInRange);
  }


  // Recursive function that constructs
  // Segment Tree for array[ss..se].
  // si is index of current node
  // in segment tree st
  public static int constructSTUtil(int[] arr, int ss, int se,
                                    List<int> st, int si,
                                    HashSet<int> hash)
  {
    // If there is one element in array,
    // check if it is Fibonacci number
    // then store 1 in the segment tree
    // else store 0 and return
    if (ss == se) {

      // if arr[ss] is fibonacci number
      if (hash.Contains(arr[ss]) == true)
        st[si] = 1;
      else
        st[si] = 0;

      return st[si];
    }

    // If there are more than one elements,
    // then recur for left and right subtrees
    // and store the sum of the
    // two values in this node
    int mid = getMid(ss, se);
    st[si] = constructSTUtil(arr, ss, mid, st, si * 2 + 1, hash) +
      constructSTUtil(arr, mid + 1, se, st, si * 2 + 2, hash);
    return st[si];
  }

  // Function to construct a segment tree from given array.
  // This function allocates memory for segment tree and
  // calls constructSTUtil() to fill the allocated memory
  public static List<int> constructST(int[] arr, int n, HashSet<int> hash)
  {
    // Allocate memory for segment tree

    // Height of segment tree
    int x = Convert.ToInt32((Math.Ceiling(Math.Log(n)*2.303)));

    // Maximum size of segment tree
    int max_size = 2 * Convert.ToInt32(Math.Pow(2, x)) - 1;

    List<int> st = new List<int>();
    for(int i = 0; i < max_size; i++){
      st.Add(0);
    }

    // Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, st, 0, hash);

    // Return the constructed segment tree
    return st;
  }

  static void Main() {

    int[] arr = { 1, 2, 3, 4, 8, 9 };
    int n = arr.Length;

    // find the largest node value in the array
    int maxEle = 0;
    for(int j = 0; j < n; j++){
      maxEle = Math.Max(maxEle, arr[j]);
    }

    // Creating a set containing all Fibonacci numbers
    // upto the maximum data value in the array
    HashSet<int> hash = new HashSet<int>();

    createHash(hash, maxEle);

    // Build segment tree from given array
    List<int> st = constructST(arr, n, hash);

    // Query 1: Query(start = 0, end = 4)
    int start = 0;
    int end = 4;
    queryFibonacci(st, n, start, end);

    // Query 2: Update(i = 3, x = 5),
    // i.e Update a[i] to x
    int i = 3;
    int x = 5;
    updateValue(arr, st, n, i, x, hash);

    // uncomment to see array after update
    // for(int i = 0; i < n; i++)
    // cout << arr[i] << " ";

    // Query 3: Query(start = 0, end = 4)
    start = 0;
    end = 4;
    queryFibonacci(st, n, start, end);
  }
}

// The code is contributed by Nidhi goel.

Javascript

<script>
    // Javascript program to find number of fibonacci numbers
    // in a subarray and performing updates

    let MAX = 1000;

    // Function to create hash table
    // to check Fibonacci numbers
    function createHash(hash, maxElement)
    {
      let prev = 0, curr = 1;
      hash.add(prev);
      hash.add(curr);
      while (curr <= maxElement)
      {
        let temp = curr + prev;
        hash.add(temp);
        prev = curr;
        curr = temp;
      }
    }

    // A utility function to get the middle
    // index from corner indexes.
    function getMid(s, e)
    {
      return s + parseInt((e - s) / 2, 10);
    }

    // Recursive function to get the number
    // of Fibonacci numbers in a given range
    /*
       * where st --> Pointer to segment tree index --> Index of current node in the
       * segment tree. Initially 0 is passed as root is always at index 0 ss & se -->
       * Starting and ending indexes of the segment represented by current node, i.e.,
       * st[index] qs & qe --> Starting and ending indexes of query range
       */
    function queryFibonacciUtil(st, ss, se, qs, qe, index)
    {

      // If segment of this node is a part
      // of given range, then return
      // the number of Fibonacci numbers
      // in the segment
      if (qs <= ss && qe >= se)
        return st[index];

      // If segment of this node
      // is outside the given range
      if (se < qs || ss > qe)
        return 0;

      // If a part of this segment
      // overlaps with the given range
      let mid = getMid(ss, se);
      return queryFibonacciUtil(st, ss, mid,
                                qs, qe, 2 * index + 1)
        + queryFibonacciUtil(st, mid + 1, se,
                             qs, qe, 2 * index + 2);
    }

    // Recursive function to update
    // the nodes which have the given
    // index in their range.
    /*
       * where st, si, ss and se are same as getSumUtil() i --> index of the element
       * to be updated. This index is in input array. diff --> Value to be added to
       * all nodes which have i in range
       */
    function updateValueUtil(st, ss, se, i, diff, si)
    {

      // Base Case:
      // If the input index lies outside
      // the range of this segment
      if (i < ss || i > se)
        return;

      // If the input index is in range
      // of this node, then update the value
      // of the node and its children
      st[si] = st[si] + diff;
      if (se != ss)
      {

        let mid = getMid(ss, se);
        updateValueUtil(st, ss, mid, i,
                        diff, 2 * si + 1);
        updateValueUtil(st, mid + 1, se, i,
                        diff, 2 * si + 2);
      }
    }

    // Function to update a value in the
    // input array and segment tree.
    // It uses updateValueUtil() to update
    // the value in segment tree
    function updateValue(arr, st, n, i, new_val, hash)
    {

      // Check for erroneous input index
      if (i < 0 || i > n - 1)
      {
        document.write("Invalid Input");
        return;
      }

      let diff = 0, oldValue;
      oldValue = arr[i];

      // Update the value in array
      arr[i] = new_val;

      // Case 1: Old and new values
      // both are Fibonacci numbers
      if (hash.has(oldValue) &&
          hash.has(new_val))
        return;

      // Case 2: Old and new values
      // both not Fibonacci numbers
      if (!hash.has(oldValue) &&
          !hash.has(new_val))
        return;

      // Case 3: Old value was Fibonacci,
      // new value is non Fibonacci
      if (hash.has(oldValue) &&
          !hash.has(new_val))
      {
        diff = -1;
      }

      // Case 4: Old value was non Fibonacci,
      // new_val is Fibonacci
      if (!hash.has(oldValue) &&
          hash.has(new_val))
      {
        diff = 1;
      }

      // Update the values of nodes in segment tree
      updateValueUtil(st, 0, n - 1, i, diff, 0);
    }

    // Return number of Fibonacci numbers
    // in range from index qs (query start)
    // to qe (query end).
    // It mainly uses queryFibonacciUtil()
    function queryFibonacci(st, n, qs, qe)
    {
      let FibonacciInRange = queryFibonacciUtil(st, 0,
                                                n - 1, qs, qe, 0);
      document.write("Number of Fibonacci numbers in subarray from " + qs + " to " + qe + " = "  + FibonacciInRange + "</br>");
    }

    // Recursive function that constructs
    // Segment Tree for array[ss..se].
    // si is index of current node
    // in segment tree st
    function constructSTUtil(arr, ss, se, st, si, hash)
    {

      // If there is one element in array,
      // check if it is Fibonacci number
      // then store 1 in the segment tree
      // else store 0 and return
      if (ss == se)
      {

        // if arr[ss] is fibonacci number
        if (hash.has(arr[ss]))
          st[si] = 1;
        else
          st[si] = 0;
        return st[si];
      }

      // If there are more than one elements,
      // then recur for left and right subtrees
      // and store the sum of the
      // two values in this node
      let mid = getMid(ss, se);
      st[si] = constructSTUtil(arr, ss, mid, st,
                               si * 2 + 1, hash)
        + constructSTUtil(arr, mid + 1, se,
                          st, si * 2 + 2, hash);
      return st[si];
    }

    // Function to construct a segment tree from given array.
    // This function allocates memory for segment tree and
    // calls constructSTUtil() to fill the allocated memory
    function constructST(arr, n, hash)
    {

      // Allocate memory for segment tree

      // Height of segment tree
      let x = (Math.ceil(Math.log(n) / Math.log(2)));

      // Maximum size of segment tree
      let max_size = 2 * Math.pow(2, x) - 1;
      let st = new Array(max_size);

      // Fill the allocated memory st
      constructSTUtil(arr, 0, n - 1, st, 0, hash);

      // Return the constructed segment tree
      return st;
    }

    let arr = [ 1, 2, 3, 4, 8, 9 ];
    let n = arr.length;

    // find the largest node value in the array
    let maxEle = Number.MIN_VALUE;
    for(let i = 0; i < n; i++)
    {
        maxEle = Math.max(arr[i], maxEle);
    }

    // Creating a set containing all Fibonacci numbers
    // upto the maximum data value in the array
    let hash = new Set();
    createHash(hash, maxEle);

    // Build segment tree from given array
    let st = constructST(arr, n, hash);

    // Query 1: Query(start = 0, end = 4)
    let start = 0;
    let end = 4;
    queryFibonacci(st, n, start, end);

    // Query 2: Update(i = 3, x = 5),
    // i.e Update a[i] to x
    let i = 3;
    let x = 5;
    updateValue(arr, st, n, i, x, hash);

    // uncomment to see array after update
    // for(int i = 0; i < n; i++)
    // cout << arr[i] << " ";

    // Query 3: Query(start = 0, end = 4)
    start = 0;
    end = 4;
    queryFibonacci(st, n, start, end);

// This code is contributed by divyesh072019.
</script>

Output:

Number of Fibonacci numbers in subarray from 0 to 4 = 4
Number of Fibonacci numbers in subarray from 0 to 4 = 5

Time Complexity: The time complexity of each query and update is O(log n) and that of building the segment tree is O(n)

Space Complexity: O(n)