Array Range Queries to count Powerful numbers with updates
Last Updated :
12 Jul, 2025
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Given an array of N integers, the task is to perform the following two operations on the given array:
query(L, R): Print the number of Powerful numbers in the subarray from L to R.
update(i, x) : update the value at index i to x, i.e arr[i] = x
A number N is said to be Powerful Number if, for every prime factor p of it, p2 also divides it.
Prerequisites: Powerful Number, Segment tree
Examples:
Input:
arr = {1, 12, 3, 8, 17, 9}
Query 1: query(L = 1, R = 4)
Query 2: update(i = 1, x = 9)
Query 3: query(L = 1, R = 4)
Output:
1
2
Explanation:
Query 1: Powerful numbers in range arr[1:4] is 8.
Query 2: Powerful numbers in range arr[1:4] after update operation is 9 and 8.
Approach:
Since we need to handle point updates and range queries so we will use a segment tree to solve the problem.
- We will precompute all the Powerful numbers till the maximum value that arr[i] can take, say MAX.
The time complexity of this operation will be O(MAX * sqrt(MAX))
- Building the segment tree:
- The problem can be reduced to subarray sum using segment tree.
- Now we can build the segment tree where the leaf nodes will represent 1(when a number is a Powerful number) or 0(when a number is not a powerful number). All the internal nodes will have the sum of both of its children.
- The problem can be reduced to subarray sum using segment tree.
- Point Updates:
- To update an element we need to look at the interval in which the element is and recurse accordingly on the left or the right child. If the element to be updated is a Powerful number then we update the leaf as 1, else 0.
- To update an element we need to look at the interval in which the element is and recurse accordingly on the left or the right child. If the element to be updated is a Powerful number then we update the leaf as 1, else 0.
- Range Query:
- Whenever we get a query from L to R, then we can query the segment tree for the sum of nodes in range L to R, which in turn represents the number of Powerful numbers in the range L to R.
- Whenever we get a query from L to R, then we can query the segment tree for the sum of nodes in range L to R, which in turn represents the number of Powerful numbers in the range L to R.
Below is the implementation of the above approach:
// C++ Program to find the number
// of Powerful numbers in subarray
// using segment tree
#include <bits/stdc++.h>
using namespace std;
#define MAX 100000
// Size of segment tree = 2^{log(MAX)+1}
int tree[3 * MAX];
int arr[MAX];
bool powerful[MAX + 1];
// Function to check if the
// number is powerful
bool isPowerful(int n)
{
// First divide the number
// repeatedly by 2
while (n % 2 == 0)
{
int power = 0;
while (n % 2 == 0)
{
n /= 2;
power++;
}
// If only 2^1 divides
// n (not higher powers),
// then return false
if (power == 1)
return false;
}
// If n is not a power of 2 then
// this loop will execute
// repeat above process
for (int factor = 3; factor
<= sqrt(n); factor += 2)
{
// Find highest power of
// "factor" that divides n
int power = 0;
while (n % factor == 0)
{
n = n / factor;
power++;
}
// If only factor^1 divides n
// (not higher powers),
// then return false
if (power == 1)
return false;
}
// n must be 1 now if it is not
// a prime numenr. Since prime
// numbers are not powerful,
// we return false if n is not 1.
return (n == 1);
}
// Function to build the array
void BuildArray(int input[], int n)
{
for (int i = 0; i < n; i++)
{
// Check if input[i] is
// a Powerful number or not
if (powerful[input[i]])
arr[i] = 1;
else
arr[i] = 0;
}
return;
}
// A utility function to get the middle
// index from corner indexes.
int getMid(int s, int e)
{
return s + (e - s) / 2;
}
/* A recursive function that constructs
Segment Tree for array[ss..se].
si --> 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]
*/
void constructSTUtil(int si, int ss,
int se)
{
if (ss == se) {
// If there is one element
// in array
tree[si] = arr[ss];
return;
}
// 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
else {
int mid = getMid(ss, se);
constructSTUtil(2 * si + 1,
ss, mid);
constructSTUtil(2 * si + 2,
mid + 1, se);
tree[si] = tree[2 * si + 1]
+ tree[2 * si + 2];
}
}
/* A recursive function to update the
nodes which have the given index
in their range.
si --> 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]
ind --> Index of array to be updated
val --> The new value to be updated
*/
void updateValueUtil(int si, int ss, int se,
int idx, int val)
{
// Leaf node
if (ss == se) {
tree[si] = tree[si] - arr[idx] + val;
arr[idx] = val;
}
else {
int mid = getMid(ss, se);
// If idx is in the left child,
// recurse on the left child
if (ss <= idx and idx <= mid)
updateValueUtil(2 * si + 1, ss,
mid, idx, val);
// If idx is in the right child,
// recurse on the right child
else
updateValueUtil(2 * si + 2, mid + 1,
se, idx, val);
// Internal node will have the sum
// of both of its children
tree[si] = tree[2 * si + 1]
+ tree[2 * si + 2];
}
}
/* A recursive function to get the number
of Powerful numbers in a given
range of array indexes
si --> 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]
l & r --> Starting and ending indexes of
query range
*/
int queryPowerfulUtil(int si, int ss, int se,
int l, int r)
{
// If segment of this node is
// outside the given range
if (r < ss or se < l) {
return 0;
}
// If segment of this node is a part
// of given range, then return the
// number of composites
// in the segment
if (l <= ss and se <= r) {
return tree[si];
}
// If a part of this segment
// overlaps with the given range
int mid = getMid(ss, se);
int p1 = queryPowerfulUtil(2 * si + 1,
ss, mid, l,
r);
int p2 = queryPowerfulUtil(2 * si + 2,
mid + 1,
se, l, r);
return (p1 + p2);
}
void queryPowerful(int n, int l, int r)
{
printf("Number of Powerful numbers between %d to %d = %d\n",
l, r, queryPowerfulUtil(0, 0,
n - 1,
l, r));
}
void updateValue(int n, int ind, int val)
{
// If val is a Powerful number
// we will update 1 in tree
if (powerful[val])
updateValueUtil(0, 0, n - 1,
ind, 1);
else
updateValueUtil(0, 0, n - 1,
ind, 0);
}
void precomputePowerful()
{
memset(powerful, false,
sizeof(powerful));
// Computing all Powerful
// numbers till MAX
for (int i = 1; i <= MAX; i++)
{
// If the number is
// Powerful make
// powerful[i] = true
if (isPowerful(i))
powerful[i] = true;
}
}
// Driver Code
int main()
{
// Precompute all the powerful
// numbers till MAX
precomputePowerful();
// Input array
int input[] = { 4, 5, 18, 27, 40, 144 };
// Size of Input array
int n = sizeof(input) / sizeof(input[0]);
// Build the array.
BuildArray(input, n);
// Build segment tree from
// given array
constructSTUtil(0, 0, n - 1);
// Query 1: Query(L = 0, R = 3)
int l = 0, r = 3;
queryPowerful(n, l, r);
// Query 2: Update(i = 1, x = 9),
// i.e Update input[i] to x
int i = 1;
int val = 9;
updateValue(n, i, val);
// Query 3: Query(L = 0, R = 3)
queryPowerful(n, l, r);
return 0;
}
// C++ Program to find the number// of Powerful numbers in subarray// using segment treeusing namespace std;// Size of segment tree = 2^{log(MAX)+1}int tree[3 * MAX];int arr[MAX];bool powerful[MAX + 1];// Function to check if the // number is powerfulbool isPowerful(int n){ // First divide the number // repeatedly by 2 while (n % 2 == 0) { int power = 0; while (n % 2 == 0) { n /= 2; power++; } // If only 2^1 divides // n (not higher powers), // then return false if (power == 1) return false; } // If n is not a power of 2 then // this loop will execute // repeat above process for (int factor = 3; factor <= sqrt(n); factor += 2) { // Find highest power of // "factor" that divides n int power = 0; while (n % factor == 0) { n = n / factor; power++; } // If only factor^1 divides n // (not higher powers), // then return false if (power == 1) return false; } // n must be 1 now if it is not // a prime numenr. Since prime // numbers are not powerful, // we return false if n is not 1. return (n == 1);}// Function to build the arrayvoid BuildArray(int input[], int n){ for (int i = 0; i < n; i++) { // Check if input[i] is // a Powerful number or not if (powerful[input[i]]) arr[i] = 1; else arr[i] = 0; } return;}// A utility function to get the middle// index from corner indexes.int getMid(int s, int e){ return s + (e - s) / 2;}/* A recursive function that constructs Segment Tree for array[ss..se].si --> 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]*/void constructSTUtil(int si, int ss, int se){ if (ss == se) { // If there is one element // in array tree[si] = arr[ss]; return; } // 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 else { int mid = getMid(ss, se); constructSTUtil(2 * si + 1, ss, mid); constructSTUtil(2 * si + 2, mid + 1, se); tree[si] = tree[2 * si + 1] + tree[2 * si + 2]; }}/* A recursive function to update the nodes which have the given indexin their range.si --> 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]ind --> Index of array to be updatedval --> The new value to be updated*/void updateValueUtil(int si, int ss, int se, int idx, int val){ // Leaf node if (ss == se) { tree[si] = tree[si] - arr[idx] + val; arr[idx] = val; } else { int mid = getMid(ss, se); // If idx is in the left child, // recurse on the left child if (ss <= idx and idx <= mid) updateValueUtil(2 * si + 1, ss, mid, idx, val); // If idx is in the right child, // recurse on the right child else updateValueUtil(2 * si + 2, mid + 1, se, idx, val); // Internal node will have the sum // of both of its children tree[si] = tree[2 * si + 1] + tree[2 * si + 2]; }}/* A recursive function to get the numberof Powerful numbers in a given range of array indexessi --> 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]l & r --> Starting and ending indexes of query range */int queryPowerfulUtil(int si, int ss, int se, int l, int r){ // If segment of this node is // outside the given range if (r < ss or se < l) { return 0; } // If segment of this node is a part // of given range, then return the // number of composites // in the segment if (l <= ss and se <= r) { return tree[si]; } // If a part of this segment // overlaps with the given range int mid = getMid(ss, se); int p1 = queryPowerfulUtil(2 * si + 1, ss, mid, l, r); int p2 = queryPowerfulUtil(2 * si + 2, mid + 1, se, l, r); return (p1 + p2);}void queryPowerful(int n, int l, int r){ printf("Number of Powerful numbers between %d to %d = %d\n", l, r, queryPowerfulUtil(0, 0, n - 1, l, r));}void updateValue(int n, int ind, int val){ // If val is a Powerful number // we will update 1 in tree if (powerful[val]) updateValueUtil(0, 0, n - 1, ind, 1); else updateValueUtil(0, 0, n - 1, ind, 0);}void precomputePowerful(){ memset(powerful, false, sizeof(powerful)); // Computing all Powerful // numbers till MAX for (int i = 1; i <= MAX; i++) { // If the number is // Powerful make // powerful[i] = true if (isPowerful(i)) powerful[i] = true; }}// Driver Codeint main(){ // Precompute all the powerful // numbers till MAX precomputePowerful(); // Input array int input[] = { 4, 5, 18, 27, 40, 144 }; // Size of Input array int n = sizeof(input) / sizeof(input[0]); // Build the array. BuildArray(input, n); // Build segment tree from // given array constructSTUtil(0, 0, n - 1); // Query 1: Query(L = 0, R = 3) int l = 0, r = 3; queryPowerful(n, l, r); // Query 2: Update(i = 1, x = 9), // i.e Update input[i] to x int i = 1; int val = 9; updateValue(n, i, val); // Query 3: Query(L = 0, R = 3) queryPowerful(n, l, r); return 0;}// Java program to find the number
// of Powerful numbers in subarray
// using segment tree
import java.util.*;
class GFG{
static final int MAX = 100000;
// Size of segment tree = 2^{log(MAX)+1}
static int []tree = new int[3 * MAX];
static int []arr = new int[MAX];
static boolean []powerful = new boolean[MAX + 1];
// Function to check if the
// number is powerful
static boolean isPowerful(int n)
{
// First divide the number
// repeatedly by 2
while (n % 2 == 0)
{
int power = 0;
while (n % 2 == 0)
{
n /= 2;
power++;
}
// If only 2^1 divides
// n (not higher powers),
// then return false
if (power == 1)
return false;
}
// If n is not a power of 2 then
// this loop will execute
// repeat above process
for(int factor = 3;
factor <= Math.sqrt(n);
factor += 2)
{
// Find highest power of
// "factor" that divides n
int power = 0;
while (n % factor == 0)
{
n = n / factor;
power++;
}
// If only factor^1 divides n
// (not higher powers),
// then return false
if (power == 1)
return false;
}
// n must be 1 now if it is not
// a prime numenr. Since prime
// numbers are not powerful,
// we return false if n is not 1.
return (n == 1);
}
// Function to build the array
static void BuildArray(int input[], int n)
{
for(int i = 0; i < n; i++)
{
// Check if input[i] is
// a Powerful number or not
if (powerful[input[i]])
arr[i] = 1;
else
arr[i] = 0;
}
return;
}
// A utility function to get the middle
// index from corner indexes.
static int getMid(int s, int e)
{
return s + (e - s) / 2;
}
/* A recursive function that constructs
Segment Tree for array[ss..se].
si -. 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]
*/
static void constructSTUtil(int si, int ss,
int se)
{
if (ss == se)
{
// If there is one element
// in array
tree[si] = arr[ss];
return;
}
// 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
else
{
int mid = getMid(ss, se);
constructSTUtil(2 * si + 1,
ss, mid);
constructSTUtil(2 * si + 2,
mid + 1, se);
tree[si] = tree[2 * si + 1] +
tree[2 * si + 2];
}
}
/* A recursive function to update the
nodes which have the given index
in their range.
si -. 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]
ind -. Index of array to be updated
val -. The new value to be updated
*/
static void updateValueUtil(int si, int ss, int se,
int idx, int val)
{
// Leaf node
if (ss == se)
{
tree[si] = tree[si] - arr[idx] + val;
arr[idx] = val;
}
else
{
int mid = getMid(ss, se);
// If idx is in the left child,
// recurse on the left child
if (ss <= idx && idx <= mid)
updateValueUtil(2 * si + 1, ss,
mid, idx, val);
// If idx is in the right child,
// recurse on the right child
else
updateValueUtil(2 * si + 2, mid + 1,
se, idx, val);
// Internal node will have the sum
// of both of its children
tree[si] = tree[2 * si + 1] +
tree[2 * si + 2];
}
}
/* A recursive function to get the number
of Powerful numbers in a given
range of array indexes
si -. 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]
l & r -. Starting and ending indexes of
query range
*/
static int queryPowerfulUtil(int si, int ss,
int se, int l, int r)
{
// If segment of this node is
// outside the given range
if (r < ss || se < l)
{
return 0;
}
// If segment of this node is a part
// of given range, then return the
// number of composites
// in the segment
if (l <= ss && se <= r)
{
return tree[si];
}
// If a part of this segment
// overlaps with the given range
int mid = getMid(ss, se);
int p1 = queryPowerfulUtil(2 * si + 1,
ss, mid, l,
r);
int p2 = queryPowerfulUtil(2 * si + 2,
mid + 1,
se, l, r);
return (p1 + p2);
}
static void queryPowerful(int n, int l, int r)
{
System.out.printf("Number of Powerful numbers " +
"between %d to %d = %d\n", l, r,
queryPowerfulUtil(0, 0, n - 1,
l, r));
}
static void updateValue(int n, int ind, int val)
{
// If val is a Powerful number
// we will update 1 in tree
if (powerful[val])
updateValueUtil(0, 0, n - 1,
ind, 1);
else
updateValueUtil(0, 0, n - 1,
ind, 0);
}
static void precomputePowerful()
{
Arrays.fill(powerful, false);
// Computing all Powerful
// numbers till MAX
for(int i = 1; i <= MAX; i++)
{
// If the number is
// Powerful make
// powerful[i] = true
if (isPowerful(i))
powerful[i] = true;
}
}
// Driver Code
public static void main(String[] args)
{
// Precompute all the powerful
// numbers till MAX
precomputePowerful();
// Input array
int input[] = { 4, 5, 18, 27, 40, 144 };
// Size of Input array
int n = input.length;
// Build the array.
BuildArray(input, n);
// Build segment tree from
// given array
constructSTUtil(0, 0, n - 1);
// Query 1: Query(L = 0, R = 3)
int l = 0, r = 3;
queryPowerful(n, l, r);
// Query 2: Update(i = 1, x = 9),
// i.e Update input[i] to x
int i = 1;
int val = 9;
updateValue(n, i, val);
// Query 3: Query(L = 0, R = 3)
queryPowerful(n, l, r);
}
}
// This code is contributed by amal kumar choubey
# Python3 program to find the number
# of Powerful numbers in subarray
# using segment tree
import math
MAX = 100000
# Size of segment tree = 2^{log(MAX)+1}
tree = [0]*(3 * MAX)
arr = [0]*(MAX)
powerful = [False]*(MAX + 1)
# Function to check if the
# number is powerful
def isPowerful(n):
# First divide the number
# repeatedly by 2
while (n % 2 == 0):
power = 0
while (n % 2 == 0):
n = int(n / 2)
power += 1
# If only 2^1 divides
# n (not higher powers),
# then return false
if (power == 1):
return False
# If n is not a power of 2 then
# this loop will execute
# repeat above process
for factor in range(3, int(math.sqrt(n)) + 1, 2):
# Find highest power of
# "factor" that divides n
power = 0
while (n % factor == 0):
n = int(n / factor)
power += 1
# If only factor^1 divides n
# (not higher powers),
# then return false
if (power == 1):
return False
# n must be 1 now if it is not
# a prime number. Since prime
# numbers are not powerful,
# we return false if n is not 1.
return (n == 1)
# Function to build the array
def BuildArray(Input, n):
for i in range(n):
# Check if input[i] is
# a Powerful number or not
if (powerful[Input[i]]):
arr[i] = 1
else:
arr[i] = 0
return
# A utility function to get the middle
# index from corner indexes.
def getMid(s, e):
return s + int((e - s) / 2)
""" A recursive function that constructs
Segment Tree for array[ss..se].
si -. 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]
"""
def constructSTUtil(si, ss, se):
if (ss == se):
# If there is one element
# in array
tree[si] = arr[ss]
return
# 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
else:
mid = getMid(ss, se)
constructSTUtil(2 * si + 1, ss, mid)
constructSTUtil(2 * si + 2, mid + 1, se)
tree[si] = tree[2 * si + 1] + tree[2 * si + 2]
""" A recursive function to update the
nodes which have the given index
in their range.
si -. 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]
ind -. Index of array to be updated
val -. The new value to be updated
"""
def updateValueUtil(si, ss, se, idx, val):
# Leaf node
if (ss == se):
tree[si] = tree[si] - arr[idx] + val
arr[idx] = val
else:
mid = getMid(ss, se)
# If idx is in the left child,
# recurse on the left child
if (ss <= idx and idx <= mid):
updateValueUtil(2 * si + 1, ss, mid, idx, val)
# If idx is in the right child,
# recurse on the right child
else:
updateValueUtil(2 * si + 2, mid + 1, se, idx, val)
# Internal node will have the sum
# of both of its children
tree[si] = tree[2 * si + 1] + tree[2 * si + 2]
""" A recursive function to get the number
of Powerful numbers in a given
range of array indexes
si -. 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]
l & r -. Starting and ending indexes of
query range
"""
def queryPowerfulUtil(si, ss, se, l, r):
# If segment of this node is
# outside the given range
if (r < ss or se < l):
return 0
# If segment of this node is a part
# of given range, then return the
# number of composites
# in the segment
if (l <= ss and se <= r):
return tree[si]
# If a part of this segment
# overlaps with the given range
mid = getMid(ss, se)
p1 = queryPowerfulUtil(2 * si + 1, ss, mid, l, r)
p2 = queryPowerfulUtil(2 * si + 2, mid + 1, se, l, r)
return (p1 + p2)
def queryPowerful(n, l, r):
print("Number of Powerful numbers between", l, "to", r, "=", queryPowerfulUtil(0, 0, n - 1, l, r))
def updateValue(n, ind, val):
# If val is a Powerful number
# we will update 1 in tree
if (powerful[val]):
updateValueUtil(0, 0, n - 1, ind, 1)
else:
updateValueUtil(0, 0, n - 1, ind, 0)
def precomputePowerful():
for i in range(MAX + 1):
powerful[i] = False
# Computing all Powerful
# numbers till MAX
for i in range(1, MAX + 1):
# If the number is
# Powerful make
# powerful[i] = true
if (isPowerful(i)):
powerful[i] = True
# Precompute all the powerful
# numbers till MAX
precomputePowerful()
# Input array
Input = [ 4, 5, 18, 27, 40, 144 ]
# Size of Input array
n = len(Input)
# Build the array.
BuildArray(Input, n)
# Build segment tree from
# given array
constructSTUtil(0, 0, n - 1)
# Query 1: Query(L = 0, R = 3)
l, r = 0, 3
queryPowerful(n, l, r)
# Query 2: Update(i = 1, x = 9),
# i.e Update input[i] to x
i = 1
val = 9
updateValue(n, i, val)
# Query 3: Query(L = 0, R = 3)
queryPowerful(n, l, r)
# This code is contributed by divyesh072019.
// C# program to find the number
// of Powerful numbers in subarray
// using segment tree
using System;
class GFG{
static readonly int MAX = 100000;
// Size of segment tree = 2^{log(MAX)+1}
static int []tree = new int[3 * MAX];
static int []arr = new int[MAX];
static bool []powerful = new bool[MAX + 1];
// Function to check if the
// number is powerful
static bool isPowerful(int n)
{
// First divide the number
// repeatedly by 2
while (n % 2 == 0)
{
int power = 0;
while (n % 2 == 0)
{
n /= 2;
power++;
}
// If only 2^1 divides
// n (not higher powers),
// then return false
if (power == 1)
return false;
}
// If n is not a power of 2 then
// this loop will execute
// repeat above process
for(int factor = 3;
factor <= Math.Sqrt(n);
factor += 2)
{
// Find highest power of
// "factor" that divides n
int power = 0;
while (n % factor == 0)
{
n = n / factor;
power++;
}
// If only factor^1 divides n
// (not higher powers),
// then return false
if (power == 1)
return false;
}
// n must be 1 now if it is not
// a prime numenr. Since prime
// numbers are not powerful,
// we return false if n is not 1.
return (n == 1);
}
// Function to build the array
static void BuildArray(int []input, int n)
{
for(int i = 0; i < n; i++)
{
// Check if input[i] is
// a Powerful number or not
if (powerful[input[i]])
arr[i] = 1;
else
arr[i] = 0;
}
return;
}
// A utility function to get the middle
// index from corner indexes.
static int getMid(int s, int e)
{
return s + (e - s) / 2;
}
/* A recursive function that constructs
Segment Tree for array[ss..se].
si -. 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]
*/
static void constructSTUtil(int si, int ss,
int se)
{
if (ss == se)
{
// If there is one element
// in array
tree[si] = arr[ss];
return;
}
// 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
else
{
int mid = getMid(ss, se);
constructSTUtil(2 * si + 1,
ss, mid);
constructSTUtil(2 * si + 2,
mid + 1, se);
tree[si] = tree[2 * si + 1] +
tree[2 * si + 2];
}
}
/* A recursive function to update the
nodes which have the given index
in their range.
si -. 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]
ind -. Index of array to be updated
val -. The new value to be updated
*/
static void updateValueUtil(int si, int ss, int se,
int idx, int val)
{
// Leaf node
if (ss == se)
{
tree[si] = tree[si] - arr[idx] + val;
arr[idx] = val;
}
else
{
int mid = getMid(ss, se);
// If idx is in the left child,
// recurse on the left child
if (ss <= idx && idx <= mid)
updateValueUtil(2 * si + 1, ss,
mid, idx, val);
// If idx is in the right child,
// recurse on the right child
else
updateValueUtil(2 * si + 2, mid + 1,
se, idx, val);
// Internal node will have the sum
// of both of its children
tree[si] = tree[2 * si + 1] +
tree[2 * si + 2];
}
}
/* A recursive function to get the number
of Powerful numbers in a given
range of array indexes
si -. 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]
l & r -. Starting and ending indexes of
query range
*/
static int queryPowerfulUtil(int si, int ss,
int se, int l, int r)
{
// If segment of this node is
// outside the given range
if (r < ss || se < l)
{
return 0;
}
// If segment of this node is a part
// of given range, then return the
// number of composites
// in the segment
if (l <= ss && se <= r)
{
return tree[si];
}
// If a part of this segment
// overlaps with the given range
int mid = getMid(ss, se);
int p1 = queryPowerfulUtil(2 * si + 1,
ss, mid, l,
r);
int p2 = queryPowerfulUtil(2 * si + 2,
mid + 1,
se, l, r);
return (p1 + p2);
}
static void queryPowerful(int n, int l, int r)
{
Console.WriteLine("Number of Powerful numbers " +
"between " + l +" to "+r+" = "+
queryPowerfulUtil(0, 0, n - 1,
l, r));
}
static void updateValue(int n, int ind, int val)
{
// If val is a Powerful number
// we will update 1 in tree
if (powerful[val])
updateValueUtil(0, 0, n - 1,
ind, 1);
else
updateValueUtil(0, 0, n - 1,
ind, 0);
}
static void precomputePowerful()
{
for(int i = 0; i <= MAX; i++)
powerful[i] = false;
// Computing all Powerful
// numbers till MAX
for(int i = 1; i <= MAX; i++)
{
// If the number is
// Powerful make
// powerful[i] = true
if (isPowerful(i))
powerful[i] = true;
}
}
// Driver Code
public static void Main(String[] args)
{
// Precompute all the powerful
// numbers till MAX
precomputePowerful();
// Input array
int []input = { 4, 5, 18, 27, 40, 144 };
// Size of Input array
int n = input.Length;
// Build the array.
BuildArray(input, n);
// Build segment tree from
// given array
constructSTUtil(0, 0, n - 1);
// Query 1: Query(L = 0, R = 3)
int l = 0, r = 3;
queryPowerful(n, l, r);
// Query 2: Update(i = 1, x = 9),
// i.e Update input[i] to x
int i = 1;
int val = 9;
updateValue(n, i, val);
// Query 3: Query(L = 0, R = 3)
queryPowerful(n, l, r);
}
}
// This code is contributed by Rohit_ranjan
<script>
// JavaScript program to find the number
// of Powerful numbers in subarray
// using segment tree
let MAX = 100000;
// Size of segment tree = 2^{log(MAX)+1}
let tree = new Array(3 * MAX);
let arr = new Array(MAX);
let powerful = new Array(MAX + 1);
// Function to check if the
// number is powerful
function isPowerful(n)
{
// First divide the number
// repeatedly by 2
while (n % 2 == 0)
{
let power = 0;
while (n % 2 == 0)
{
n = parseInt(n / 2, 10);
power++;
}
// If only 2^1 divides
// n (not higher powers),
// then return false
if (power == 1)
return false;
}
// If n is not a power of 2 then
// this loop will execute
// repeat above process
for(let factor = 3;
factor <= Math.sqrt(n);
factor += 2)
{
// Find highest power of
// "factor" that divides n
let power = 0;
while (n % factor == 0)
{
n = parseInt(n / factor, 10);
power++;
}
// If only factor^1 divides n
// (not higher powers),
// then return false
if (power == 1)
return false;
}
// n must be 1 now if it is not
// a prime numenr. Since prime
// numbers are not powerful,
// we return false if n is not 1.
return (n == 1);
}
// Function to build the array
function BuildArray(input, n)
{
for(let i = 0; i < n; i++)
{
// Check if input[i] is
// a Powerful number or not
if (powerful[input[i]])
arr[i] = 1;
else
arr[i] = 0;
}
return;
}
// A utility function to get the middle
// index from corner indexes.
function getMid(s, e)
{
return s + parseInt((e - s) / 2, 10);
}
/* A recursive function that constructs
Segment Tree for array[ss..se].
si -. 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]
*/
function constructSTUtil(si, ss, se)
{
if (ss == se)
{
// If there is one element
// in array
tree[si] = arr[ss];
return;
}
// 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
else
{
let mid = getMid(ss, se);
constructSTUtil(2 * si + 1,
ss, mid);
constructSTUtil(2 * si + 2,
mid + 1, se);
tree[si] = tree[2 * si + 1] +
tree[2 * si + 2];
}
}
/* A recursive function to update the
nodes which have the given index
in their range.
si -. 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]
ind -. Index of array to be updated
val -. The new value to be updated
*/
function updateValueUtil(si, ss, se, idx, val)
{
// Leaf node
if (ss == se)
{
tree[si] = tree[si] - arr[idx] + val;
arr[idx] = val;
}
else
{
let mid = getMid(ss, se);
// If idx is in the left child,
// recurse on the left child
if (ss <= idx && idx <= mid)
updateValueUtil(2 * si + 1, ss,
mid, idx, val);
// If idx is in the right child,
// recurse on the right child
else
updateValueUtil(2 * si + 2, mid + 1,
se, idx, val);
// Internal node will have the sum
// of both of its children
tree[si] = tree[2 * si + 1] +
tree[2 * si + 2];
}
}
/* A recursive function to get the number
of Powerful numbers in a given
range of array indexes
si -. 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]
l & r -. Starting and ending indexes of
query range
*/
function queryPowerfulUtil(si, ss, se, l, r)
{
// If segment of this node is
// outside the given range
if (r < ss || se < l)
{
return 0;
}
// If segment of this node is a part
// of given range, then return the
// number of composites
// in the segment
if (l <= ss && se <= r)
{
return tree[si];
}
// If a part of this segment
// overlaps with the given range
let mid = getMid(ss, se);
let p1 = queryPowerfulUtil(2 * si + 1,
ss, mid, l,
r);
let p2 = queryPowerfulUtil(2 * si + 2,
mid + 1,
se, l, r);
return (p1 + p2);
}
function queryPowerful(n, l, r)
{
document.write("Number of Powerful numbers " +
"between " + l +" to "+r+" = "+
queryPowerfulUtil(0, 0, n - 1,
l, r) + "</br>");
}
function updateValue(n, ind, val)
{
// If val is a Powerful number
// we will update 1 in tree
if (powerful[val])
updateValueUtil(0, 0, n - 1,
ind, 1);
else
updateValueUtil(0, 0, n - 1,
ind, 0);
}
function precomputePowerful()
{
for(let i = 0; i <= MAX; i++)
powerful[i] = false;
// Computing all Powerful
// numbers till MAX
for(let i = 1; i <= MAX; i++)
{
// If the number is
// Powerful make
// powerful[i] = true
if (isPowerful(i))
powerful[i] = true;
}
}
// Precompute all the powerful
// numbers till MAX
precomputePowerful();
// Input array
let input = [ 4, 5, 18, 27, 40, 144 ];
// Size of Input array
let n = input.length;
// Build the array.
BuildArray(input, n);
// Build segment tree from
// given array
constructSTUtil(0, 0, n - 1);
// Query 1: Query(L = 0, R = 3)
let l = 0, r = 3;
queryPowerful(n, l, r);
// Query 2: Update(i = 1, x = 9),
// i.e Update input[i] to x
let i = 1;
let val = 9;
updateValue(n, i, val);
// Query 3: Query(L = 0, R = 3)
queryPowerful(n, l, r);
</script>
Output:
Number of Powerful numbers between 0 to 3 = 2 Number of Powerful numbers between 0 to 3 = 3
Time Complexity: O(logN) per query
Space Complexity: O(MAX*log2MAX). This is because the size of the segment tree used in the program is 3 times the value of MAX, and the maximum height of the segment tree is log2(MAX)+1. Therefore, the total space used by the segment tree is 3 times the value of MAX multiplied by the maximum height of the segment tree, which is O(MAX log2MAX).
In addition to the segment tree, the program also uses three arrays: arr, powerful, and tree. The size of arr and powerful is MAX, and the size of tree is 3 times MAX. Therefore, the total space used by these arrays is O(MAX).
