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 arri. 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 arri be y and the new value be x.- 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 - 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 - 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 - 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
- Case 1: Fibonacci: If x and y both are Fibonacci numbers
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 numbersvoid 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 treevoid 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 stint 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 memoryint* 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 Codeint 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 updatesimport 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 updatesimport mathMAX = 1000# Function to create hash table# to check Fibonacci numbersdef 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 rangedef 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 rangedef 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 treedef 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 stdef 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 memorydef 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 Codearr = [1, 2, 3, 4, 8, 9]n = len(arr)# find the largest node value in the arraymaxEle = -1for 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 arrayhash = set()createHash(hash, maxEle)# Build segment tree from given arrayst = constructST(arr, n, hash)# Query 1: Query(start = 0, end = 4)start = 0end = 4queryFibonacci(st, n, start, end)# Query 2: Update(i = 3, x = 5),# i.e Update a[i] to xi = 3x = 5updateValue(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 = 0end = 4queryFibonacci(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 updatesclass 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)
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