Array range queries to find the maximum Fibonacci number with updates

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

  • maximum(start, end): Print the maximum Fibonacci number of elements in the sub-array from start to end
  • update(i, x): Add x to the array element referenced by array index i, that is: arr[i] = x

Note: The below example consists of 0 based indexing.

Example: 



Input: arr = [1, 3, 5, 7, 9, 11] 
Query 1: Maximum (Start = 1, End = 3) 
Query 2: Update (3, 8) i.e. arr[3] = 8
Output: 
Maximum Fibonacci number in given range = 5 
Updated Maximum Fibonacci number in given range = 8
Explanation: 
In the Maximum Query, the sub-array [1…3] 
has 2 Fibonacci 3 and 5 viz. {3, 5, 7} 
Hence, 5 is the maximum Fibonacci number in the given range.
In the Update Query, the value at index 3 is updated 
to 8, the array arr now is, [1, 3, 5, 8, 9, 11]
In Updated Maximum Query, the sub-array [1…3] 
has all 3 Fibonacci numbers 3, 5 and 8 viz. [3, 5, 8] 
Hence, 8 is the maximum Fibonacci number in the given range. 
 

Simple Approach:

A simple solution is to run a loop from l to r and calculate the maximum Fibonacci number out of all the elements in given range. To update a value, simply do arr[i] = x. The first operation takes O(N) time and the second operation takes O(1) time.

Efficient Approach : 
Here, we need to perform operations in O(Log N) time so we can use Segment Tree to do both operations in O(Log N) time.
Representation of Segment trees:
1. Leaf Nodes are the elements of the input array. 
2. Each internal node represents the maximum Fibonacci number of all of its child nodes or -1 if there is no Fibonacci number existing in the range.
An array representation of tree is used to represent Segment Trees. For each node at index i, the left child is at index 2*i+1, right child at index 2*i+2 and the parent is at index (i-1)/2.
Construction of Segment Tree from given array : 
We start with a segment arr[0 . . . n-1], and every time we divide the current segment into two halves(if it has not yet become a segment of length 1), and then call the same procedure on both halves, and for each such segment, we store the maximum Fibonacci number value or -1 in a segment tree node. All levels of the constructed segment tree will be completely filled except the last level. Also, the tree will be a full Binary Tree because we always divide segments into two halves at every level. Since the constructed tree is always a full binary tree with n leaves, there will be n-1 internal nodes. So total nodes will be 2*n – 1. Height of the segment tree will be log2N. Since the tree is represented using array and relation between parent and child indexes must be maintained, size of memory allocated for segment tree will be 2*( 2ceil(log2n) ) – 1.
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 can take say MAX which will be used to test a number in O(1) time.
We then do a range query on the segment tree to find out the max_set_bits for the given range and output the corresponding value.

Below is the implementation of above approach : 
 

CPP

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// CPP code for range maximum query and updates
#include <bits/stdc++.h>
using namespace std;
  
set<int> fibonacci;
  
// A utility function to get the
// middle index of given range.
int getMid(int s, int e)
{
    return s + (e - s) / 2;
}
  
// Function to create hash table
// to check Fibonacci numbers
void createHash(int maxElement)
{
    int prev = 0, curr = 1;
    fibonacci.insert(prev);
    fibonacci.insert(curr);
  
    while (curr <= maxElement) {
        int temp = curr + prev;
        fibonacci.insert(temp);
        prev = curr;
        curr = temp;
    }
}
  
/*  A recursive function to get the sum of
    values in given range of the array. 
    The following are parameters for this
    function.
  
    st       -> Pointer to segment tree
    node     -> Index of current node in 
                the segment tree .
    ss & se  -> Starting and ending indexes 
                of the segment represented
                by current node, i.e., st[node]
    l & r    -> Starting and ending indexes 
                of range query */
int MaxUtil(int* st, int ss, int se, int l,
            int r, int node)
{
    // If segment of this node is completely
    // part of given range, then return
    // the max of segment
    if (l <= ss && r >= se)
        return st[node];
  
    // If segment of this node does not
    // belong to given range
    if (se < l || ss > r)
        return -1;
  
    // If segment of this node is partially
    // the part of given range
    int mid = getMid(ss, se);
  
    return max(MaxUtil(st, ss, mid, l, r,
                       2 * node + 1),
               MaxUtil(st, mid + 1, se, l,
                       r, 2 * node + 2));
}
  
/* A recursive function to update the nodes which
   have the given index in their range. The following
   are parameters st, ss and se are same as defined
   above index -> index of the element to be updated.*/
void updateValue(int arr[], int* st, int ss, int se,
                 int index, int value, int node)
{
    if (index < ss || index > se) {
        cout << "Invalid Input" << endl;
        return;
    }
  
    if (ss == se) {
        // update value in array and in segment tree
        arr[index] = value;
  
        if (fibonacci.find(value) != fibonacci.end())
            st[node] = value;
        else
            st[node] = -1;
    }
    else {
        int mid = getMid(ss, se);
  
        if (index >= ss && index <= mid)
            updateValue(arr, st, ss, mid, index,
                        value, 2 * node + 1);
        else
            updateValue(arr, st, mid + 1, se,
                        index, value, 2 * node + 2);
  
        st[node] = max(st[2 * node + 1],
                       st[2 * node + 2]);
    }
    return;
}
  
// Return max of elements in range from
// index l (query start) to r (query end).
int getMax(int* st, int n, int l, int r)
{
    // Check for erroneous input values
    if (l < 0 || r > n - 1 || l > r) {
        printf("Invalid Input");
        return -1;
    }
  
    return MaxUtil(st, 0, n - 1, l, r, 0);
}
  
// A 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)
{
    // If there is one element in array, store
    // it in current node of segment tree and return
    if (ss == se) {
        if (fibonacci.find(arr[ss])
            != fibonacci.end())
            st[si] = arr[ss];
        else
            st[si] = -1;
        return st[si];
    }
  
    // If there are more than one elements, then
    // recur for left and right subtrees and
    // store the max of values in this node
    int mid = getMid(ss, se);
  
    st[si]
        = max(constructSTUtil(
                  arr, ss, mid, st,
                  si * 2 + 1),
              constructSTUtil(
                  arr, mid + 1, se,
                  st, si * 2 + 2));
  
    return st[si];
}
  
/* Function to construct segment tree
   from given array.
   This function allocates memory
   for segment tree.*/
int* constructST(int arr[], int n)
{
    // Height of segment tree
    int x = (int)(ceil(log2(n)));
  
    // Maximum size of segment tree
    int max_size = 2 * (int)pow(2, x) - 1;
  
    // Allocate memory
    int* st = new int[max_size];
  
    // Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, st, 0);
  
    // Return the constructed segment tree
    return st;
}
  
// Driver code
int main()
{
    int arr[] = { 1, 3, 5, 7, 9, 11 };
    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
    createHash(maxEle);
  
    // Build segment tree from given array
    int* st = constructST(arr, n);
  
    // Print max of values in array
    // from index 1 to 3
    cout << "Maximum fibonacci number"
         << " in given range = "
         << getMax(st, n, 1, 3) << endl;
  
    // Update: set arr[1] = 8 and update
    // corresponding segment tree nodes.
    updateValue(arr, st, 0, n - 1, 3, 8, 0);
  
    // Find max after the value is updated
    cout << "Updated Maximum Fibonacci"
         << " number in given range = "
         << getMax(st, n, 1, 3) << endl;
  
    return 0;
}

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Output: 
Maximum fibonacci number in given range = 5 
Updated Maximum fibonacci number in given range = 8
 

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