Dada una array de N enteros, la tarea es realizar las siguientes dos consultas:
query(start, end) : Imprime el número de números primos en el subarreglo de principio a fin
update(i, x) : actualiza el valor en el índice i a x, es decir, arr[i] = x
Ejemplos:
Input : arr = {1, 2, 3, 5, 7, 9}
Query 1: query(start = 0, end = 4)
Query 2: update(i = 3, x = 6)
Query 3: query(start = 0, end = 4)
Output :4
3
Explanation
In Query 1, the subarray [0...4]
has 4 primes viz. {2, 3, 5, 7}
In Query 2, the value at index 3
is updated to 6, the array arr now is, {1, 2, 3,
6, 7, 9}
In Query 3, the subarray [0...4]
has 3 primes viz. {2, 3, 7}
Método 1 (Fuerza bruta)
Se puede encontrar un problema similar aquí . Aquí no hay actualizaciones. Podemos modificar esto para manejar las actualizaciones, pero para esto necesitamos construir la array de prefijos siempre que realizamos una actualización, lo que hace que la complejidad del tiempo de este enfoque sea O (Q * N)
Método 2 (Eficiente)
Dado que, necesitamos manejar consultas de rango y actualizaciones de puntos, un árbol de segmentos es el más adecuado para este propósito.
Podemos usar el Tamiz de Eratóstenes para preprocesar todos los números primos hasta el valor máximo que puedo tomar, digamos MAX en O(MAX log(log(MAX)))
Construyendo el árbol de segmentos:
Básicamente reducimos el problema a la suma de subarreglos usando el árbol de segmentos .
Ahora, podemos construir el árbol de segmentos donde un Node de hoja se representa como 0 (si no es un número primo) o 1 (si es un número primo).
Los Nodes internos del árbol de segmentos son iguales a la suma de sus Nodes secundarios, por lo que un Node representa los números primos totales en el rango de L a R donde el rango de L a R cae bajo este Node y el subárbol debajo de él.
Manejo de consultas y actualizaciones de puntos:
cada vez que obtenemos una consulta de principio a fin, podemos consultar el árbol de segmentos para obtener la suma de los Nodes en el rango de principio a fin, que a su vez representan la cantidad de números primos en el rango de principio a fin.
Si necesitamos realizar una actualización de puntos y actualizar el valor en el índice i a x, verificamos los siguientes casos:
Let the old value of arri be y and the new value be x Case 1: If x and y both are primes Count of primes in the subarray does not change so we just update array and donot modify the segment tree Case 2: If x and y both are non primes Count of primes in the subarray does not change so we just update array and donot modify the segment tree Case 3: If y is prime but x is non prime Count of primes 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 non prime but x is prime Count of primes 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
CPP
// C++ program to find number of prime numbers in a
// subarray and performing updates
#include <bits/stdc++.h>
using namespace std;
#define MAX 1000
void sieveOfEratosthenes(bool isPrime[])
{
isPrime[1] = false;
for (int p = 2; p * p <= MAX; p++) {
// If prime[p] is not changed, then
// it is a prime
if (isPrime[p] == true) {
// Update all multiples of p
for (int i = p * 2; i <= MAX; i += p)
isPrime[i] = false;
}
}
}
// 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 to get the number of primes in a given range
of array indexes. The following are parameters for this function.
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 queryPrimesUtil(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 primes 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 queryPrimesUtil(st, ss, mid, qs, qe, 2 * index + 1) +
queryPrimesUtil(st, mid + 1, se, qs, qe, 2 * index + 2);
}
/* A recursive function to update the nodes which have the given
index in their range. The following are parameters
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);
}
}
// The function to update a value in 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,
bool isPrime[])
{
// 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 primes
if (isPrime[oldValue] && isPrime[new_val])
return;
// Case 2: Old and new values both non primes
if ((!isPrime[oldValue]) && (!isPrime[new_val]))
return;
// Case 3: Old value was prime, new value is non prime
if (isPrime[oldValue] && !isPrime[new_val]) {
diff = -1;
}
// Case 4: Old value was non prime, new_val is prime
if (!isPrime[oldValue] && isPrime[new_val]) {
diff = 1;
}
// Update the values of nodes in segment tree
updateValueUtil(st, 0, n - 1, i, diff, 0);
}
// Return number of primes in range from index qs (query start) to
// qe (query end). It mainly uses queryPrimesUtil()
void queryPrimes(int* st, int n, int qs, int qe)
{
int primesInRange = queryPrimesUtil(st, 0, n - 1, qs, qe, 0);
cout << "Number of Primes in subarray from " << qs << " to "
<< qe << " = " << primesInRange << "\n";
}
// 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, bool isPrime[])
{
// If there is one element in array, check if it
// is prime then store 1 in the segment tree else
// store 0 and return
if (ss == se) {
// if arr[ss] is prime
if (isPrime[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, isPrime) +
constructSTUtil(arr, mid + 1, se, st,
si * 2 + 2, isPrime);
return st[si];
}
/* Function to construct 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, bool isPrime[])
{
// 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, isPrime);
// Return the constructed segment tree
return st;
}
// Driver program to test above functions
int main()
{
int arr[] = { 1, 2, 3, 5, 7, 9 };
int n = sizeof(arr) / sizeof(arr[0]);
/* Preprocess all primes till MAX.
Create a boolean array "isPrime[0..MAX]".
A value in prime[i] will finally be false
if i is Not a prime, else true. */
bool isPrime[MAX + 1];
memset(isPrime, true, sizeof isPrime);
sieveOfEratosthenes(isPrime);
// Build segment tree from given array
int* st = constructST(arr, n, isPrime);
// Query 1: Query(start = 0, end = 4)
int start = 0;
int end = 4;
queryPrimes(st, n, start, end);
// Query 2: Update(i = 3, x = 6), i.e Update
// a[i] to x
int i = 3;
int x = 6;
updateValue(arr, st, n, i, x, isPrime);
// 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;
queryPrimes(st, n, start, end);
return 0;
}
Java
// Java program to find number of prime numbers in a
// subarray and performing updates
import java.io.*;
import java.util.*;
class GFG
{
static int MAX = 1000 ;
static void sieveOfEratosthenes(boolean isPrime[])
{
isPrime[1] = false;
for (int p = 2; p * p <= MAX; p++)
{
// If prime[p] is not changed, then
// it is a prime
if (isPrime[p] == true)
{
// Update all multiples of p
for (int i = p * 2; i <= MAX; i += p)
isPrime[i] = false;
}
}
}
// 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 to get the number of primes in a given range
of array indexes. The following are parameters for this function.
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 queryPrimesUtil(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 primes 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 queryPrimesUtil(st, ss, mid, qs, qe, 2 * index + 1) +
queryPrimesUtil(st, mid + 1, se, qs, qe, 2 * index + 2);
}
/* A recursive function to update the nodes which have the given
index in their range. The following are parameters
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);
}
}
// The function to update a value in 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, boolean isPrime[])
{
// Check for erroneous input index
if (i < 0 || i > n - 1) {
System.out.println("Invalid Input");
return;
}
int diff = 0;
int oldValue;
oldValue = arr[i];
// Update the value in array
arr[i] = new_val;
// Case 1: Old and new values both are primes
if (isPrime[oldValue] && isPrime[new_val])
return;
// Case 2: Old and new values both non primes
if ((!isPrime[oldValue]) && (!isPrime[new_val]))
return;
// Case 3: Old value was prime, new value is non prime
if (isPrime[oldValue] && !isPrime[new_val])
{
diff = -1;
}
// Case 4: Old value was non prime, new_val is prime
if (!isPrime[oldValue] && isPrime[new_val])
{
diff = 1;
}
// Update the values of nodes in segment tree
updateValueUtil(st, 0, n - 1, i, diff, 0);
}
// Return number of primes in range from index qs (query start) to
// qe (query end). It mainly uses queryPrimesUtil()
static void queryPrimes(int[] st, int n, int qs, int qe)
{
int primesInRange = queryPrimesUtil(st, 0, n - 1, qs, qe, 0);
System.out.println("Number of Primes in subarray from " +
qs + " to " + qe + " = " + primesInRange);
}
// A 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, boolean isPrime[])
{
// If there is one element in array, check if it
// is prime then store 1 in the segment tree else
// store 0 and return
if (ss == se) {
// if arr[ss] is prime
if (isPrime[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, isPrime) +
constructSTUtil(arr, mid + 1, se, st, si * 2 + 2, isPrime);
return st[si];
}
/* Function to construct 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, boolean isPrime[])
{
// 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, isPrime);
// Return the constructed segment tree
return st;
}
// Driver code
public static void main (String[] args)
{
int arr[] = { 1, 2, 3, 5, 7, 9 };
int n = arr.length;
/* Preprocess all primes till MAX.
Create a boolean array "isPrime[0..MAX]".
A value in prime[i] will finally be false
if i is Not a prime, else true. */
boolean[] isPrime = new boolean[MAX + 1];
Arrays.fill(isPrime, Boolean.TRUE);
sieveOfEratosthenes(isPrime);
// Build segment tree from given array
int[] st = constructST(arr, n, isPrime);
// Query 1: Query(start = 0, end = 4)
int start = 0;
int end = 4;
queryPrimes(st, n, start, end);
// Query 2: Update(i = 3, x = 6), i.e Update
// a[i] to x
int i = 3;
int x = 6;
updateValue(arr, st, n, i, x, isPrime);
// 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;
queryPrimes(st, n, start, end);
}
}
// This code is contributed by avanitrachhadiya2155
Python3
# Python3 program to find number of prime numbers in a
# subarray and performing updates
from math import ceil, floor, log
MAX = 1000
def sieveOfEratosthenes(isPrime):
isPrime[1] = False
for p in range(2, MAX + 1):
if p * p > MAX:
break
# If prime[p] is not changed, then
# it is a prime
if (isPrime[p] == True):
# Update all multiples of p
for i in range(2 * p, MAX + 1, p):
isPrime[i] = False
# A utility function to get the middle index from corner indexes.
def getMid(s, e):
return s + (e - s) // 2
#
# /* A recursive function to get the number of primes in a given range
# of array indexes. The following are parameters for this function.
#
# 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 queryPrimesUtil(st, ss, se, qs, qe, index):
# If segment of this node is a part of given range, then return
# the number of primes 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 queryPrimesUtil(st, ss, mid, qs, qe, 2 * index + 1) + \
queryPrimesUtil(st, mid + 1, se, qs, qe, 2 * index + 2)
# /* A recursive function to update the nodes which have the given
# index in their range. The following are parameters
# 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)
# The function to update a value in input array and segment tree.
# It uses updateValueUtil() to update the value in segment tree
def updateValue(arr,st, n, i, new_val,isPrime):
# Check for erroneous input index
if (i < 0 or i > n - 1):
printf("Invalid Input")
return
diff, oldValue = 0, 0
oldValue = arr[i]
# Update the value in array
arr[i] = new_val
# Case 1: Old and new values both are primes
if (isPrime[oldValue] and isPrime[new_val]):
return
# Case 2: Old and new values both non primes
if ((not isPrime[oldValue]) and (not isPrime[new_val])):
return
# Case 3: Old value was prime, new value is non prime
if (isPrime[oldValue] and not isPrime[new_val]):
diff = -1
# Case 4: Old value was non prime, new_val is prime
if (not isPrime[oldValue] and isPrime[new_val]):
diff = 1
# Update the values of nodes in segment tree
updateValueUtil(st, 0, n - 1, i, diff, 0)
# Return number of primes in range from index qs (query start) to
# qe (query end). It mainly uses queryPrimesUtil()
def queryPrimes(st, n, qs, qe):
primesInRange = queryPrimesUtil(st, 0, n - 1, qs, qe, 0)
print("Number of Primes in subarray from ", qs," to ", qe," = ", primesInRange)
# A 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,isPrime):
# If there is one element in array, check if it
# is prime then store 1 in the segment tree else
# store 0 and return
if (ss == se):
# if arr[ss] is prime
if (isPrime[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
mid = getMid(ss, se)
st[si] = constructSTUtil(arr, ss, mid, st,si * 2 + 1, isPrime) + \
constructSTUtil(arr, mid + 1, se, st,si * 2 + 2, isPrime)
return st[si]
# /* Function to construct segment tree from given array.
# This function allocates memory for segment tree and
# calls constructSTUtil() to fill the allocated memory */
def constructST(arr, n, isPrime):
# Allocate memory for segment tree
# Height of segment tree
x = ceil(log(n, 2))
# Maximum size of segment tree
max_size = 2 * pow(2, x) - 1
st = [0]*(max_size)
# Fill the allocated memory st
constructSTUtil(arr, 0, n - 1, st, 0, isPrime)
# Return the constructed segment tree
return st
# Driver code
if __name__ == '__main__':
arr= [ 1, 2, 3, 5, 7, 9]
n = len(arr)
# /* Preprocess all primes till MAX.
# Create a boolean array "isPrime[0..MAX]".
# A value in prime[i] will finally be false
# if i is Not a prime, else true. */
isPrime = [True]*(MAX + 1)
sieveOfEratosthenes(isPrime)
# Build segment tree from given array
st = constructST(arr, n, isPrime)
# Query 1: Query(start = 0, end = 4)
start = 0
end = 4
queryPrimes(st, n, start, end)
# Query 2: Update(i = 3, x = 6), i.e Update
# a[i] to x
i = 3
x = 6
updateValue(arr, st, n, i, x, isPrime)
# uncomment to see array after update
# for(i = 0 i < n i++) cout << arr[i] << " "
# Query 3: Query(start = 0, end = 4)
start = 0
end = 4
queryPrimes(st, n, start, end)
# This code is contributed by mohit kumar 29
C#
// C# program to find number of prime numbers in a
// subarray and performing updates
using System;
public class GFG
{
static int MAX = 1000 ;
static void sieveOfEratosthenes(bool[] isPrime)
{
isPrime[1] = false;
for (int p = 2; p * p <= MAX; p++)
{
// If prime[p] is not changed, then
// it is a prime
if (isPrime[p] == true)
{
// Update all multiples of p
for (int i = p * 2; i <= MAX; i += p)
isPrime[i] = false;
}
}
}
// 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 to get the number of primes in a given range
of array indexes. The following are parameters for this function.
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 queryPrimesUtil(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 primes 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 queryPrimesUtil(st, ss, mid, qs,
qe, 2 * index + 1) +
queryPrimesUtil(st, mid + 1, se,
qs, qe, 2 * index + 2);
}
/* A recursive function to update the nodes which have the given
index in their range. The following are parameters
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);
}
}
// The function to update a value in 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, bool[] isPrime)
{
// Check for erroneous input index
if (i < 0 || i > n - 1)
{
Console.WriteLine("Invalid Input");
return;
}
int diff = 0;
int oldValue;
oldValue = arr[i];
// Update the value in array
arr[i] = new_val;
// Case 1: Old and new values both are primes
if (isPrime[oldValue] && isPrime[new_val])
return;
// Case 2: Old and new values both non primes
if ((!isPrime[oldValue]) && (!isPrime[new_val]))
return;
// Case 3: Old value was prime, new value is non prime
if (isPrime[oldValue] && !isPrime[new_val])
{
diff = -1;
}
// Case 4: Old value was non prime, new_val is prime
if (!isPrime[oldValue] && isPrime[new_val])
{
diff = 1;
}
// Update the values of nodes in segment tree
updateValueUtil(st, 0, n - 1, i, diff, 0);
}
// Return number of primes in range from index qs (query start) to
// qe (query end). It mainly uses queryPrimesUtil()
static void queryPrimes(int[] st, int n, int qs, int qe)
{
int primesInRange = queryPrimesUtil(st, 0, n - 1, qs, qe, 0);
Console.WriteLine("Number of Primes in subarray from " +
qs + " to " + qe + " = " + primesInRange);
}
// A 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, bool[] isPrime)
{
// If there is one element in array, check if it
// is prime then store 1 in the segment tree else
// store 0 and return
if (ss == se)
{
// if arr[ss] is prime
if (isPrime[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, isPrime) +
constructSTUtil(arr, mid + 1, se, st, si * 2 + 2, isPrime);
return st[si];
}
/* Function to construct 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, bool[] isPrime)
{
// Allocate memory for segment tree
// Height of segment tree
int x = (int)(Math.Ceiling(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, isPrime);
// Return the constructed segment tree
return st;
}
// Driver code
static public void Main ()
{
int[] arr = { 1, 2, 3, 5, 7, 9 };
int n = arr.Length;
/* Preprocess all primes till MAX.
Create a boolean array "isPrime[0..MAX]".
A value in prime[i] will finally be false
if i is Not a prime, else true. */
bool[] isPrime = new bool[MAX + 1];
Array.Fill(isPrime, true);
sieveOfEratosthenes(isPrime);
// Build segment tree from given array
int[] st = constructST(arr, n, isPrime);
// Query 1: Query(start = 0, end = 4)
int start = 0;
int end = 4;
queryPrimes(st, n, start, end);
// Query 2: Update(i = 3, x = 6), i.e Update
// a[i] to x
int i = 3;
int x = 6;
updateValue(arr, st, n, i, x, isPrime);
// 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;
queryPrimes(st, n, start, end);
}
}
// This code is contributed by rag2127
Javascript
<script>
// Javascript program to find number of prime numbers in a
// subarray and performing updates
let MAX = 1000;
function sieveOfEratosthenes(isPrime) {
isPrime[1] = false;
for (let p = 2; p * p <= MAX; p++) {
// If prime[p] is not changed, then
// it is a prime
if (isPrime[p] == true) {
// Update all multiples of p
for (let i = p * 2; i <= MAX; i += p)
isPrime[i] = false;
}
}
}
// A utility function to get the middle index from corner indexes.
function getMid(s, e) { return Math.floor(s + (e - s) / 2); }
/* A recursive function to get the number of primes in a given range
of array indexes. The following are parameters for this function.
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 queryPrimesUtil(st, ss, se, qs, qe, index) {
// If segment of this node is a part of given range, then return
// the number of primes 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 queryPrimesUtil(st, ss, mid, qs, qe, 2 * index + 1) +
queryPrimesUtil(st, mid + 1, se, qs, qe, 2 * index + 2);
}
/* A recursive function to update the nodes which have the given
index in their range. The following are parameters
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);
}
}
// The function to update a value in input array and segment tree.
// It uses updateValueUtil() to update the value in segment tree
function updateValue(arr, st, n, i, new_val, isPrime) {
// Check for erroneous input index
if (i < 0 || i > n - 1) {
document.write("Invalid Input");
return;
}
let diff = 0;
let oldValue;
oldValue = arr[i];
// Update the value in array
arr[i] = new_val;
// Case 1: Old and new values both are primes
if (isPrime[oldValue] && isPrime[new_val])
return;
// Case 2: Old and new values both non primes
if ((!isPrime[oldValue]) && (!isPrime[new_val]))
return;
// Case 3: Old value was prime, new value is non prime
if (isPrime[oldValue] && !isPrime[new_val]) {
diff = -1;
}
// Case 4: Old value was non prime, new_val is prime
if (!isPrime[oldValue] && isPrime[new_val]) {
diff = 1;
}
// Update the values of nodes in segment tree
updateValueUtil(st, 0, n - 1, i, diff, 0);
}
// Return number of primes in range from index qs (query start) to
// qe (query end). It mainly uses queryPrimesUtil()
function queryPrimes(st, n, qs, qe) {
let primesInRange = queryPrimesUtil(st, 0, n - 1, qs, qe, 0);
document.write("Number of Primes in subarray from " +
qs + " to " + qe + " = " + primesInRange + "<br>");
}
// A 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, isPrime) {
// If there is one element in array, check if it
// is prime then store 1 in the segment tree else
// store 0 and return
if (ss == se) {
// if arr[ss] is prime
if (isPrime[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, isPrime) +
constructSTUtil(arr, mid + 1, se, st, si * 2 + 2, isPrime);
return st[si];
}
/* Function to construct segment tree from given array.
This function allocates memory for segment tree and
calls constructSTUtil() to fill the allocated memory */
function constructST(arr, n, isPrime) {
// 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, isPrime);
// Return the constructed segment tree
return st;
}
// Driver code
let arr = [1, 2, 3, 5, 7, 9];
let n = arr.length;
/* Preprocess all primes till MAX.
Create a boolean array "isPrime[0..MAX]".
A value in prime[i] will finally be false
if i is Not a prime, else true. */
let isPrime = new Array(MAX + 1);
isPrime.fill(true);
sieveOfEratosthenes(isPrime);
// Build segment tree from given array
let st = constructST(arr, n, isPrime);
// Query 1: Query(start = 0, end = 4)
let start = 0;
let end = 4;
queryPrimes(st, n, start, end);
// Query 2: Update(i = 3, x = 6), i.e Update
// a[i] to x
let i = 3;
let x = 6;
updateValue(arr, st, n, i, x, isPrime);
// uncomment to see array after update
// for(let i = 0; i < n; i++) cout << arr[i] << " ";
// Query 3: Query(start = 0, end = 4)
start = 0;
end = 4;
queryPrimes(st, n, start, end);
// This code is contributed by gfgking
</script>
Producción:
Number of Primes in subarray from 0 to 4 = 4 Number of Primes in subarray from 0 to 4 = 3
La complejidad de tiempo de cada consulta y actualización es O(
logn ) y la de construir el árbol de segmentos es O(n) log(MAX))) donde MAX es el valor máximo que puedo tomar
Tema relacionado: Árbol de segmentos