Dada una array de N enteros, la tarea es realizar las siguientes dos operaciones en la array dada:
consulta (inicio, fin) : imprime el número de números compuestos en el subarreglo de principio a fin
actualización (i, x) : actualiza el valor en el índice i a x, es decir, arr [i] = x
Ejemplos :
Input : arr = {1, 12, 3, 5, 17, 9}
Query 1: query(start = 0, end = 4)
Query 2: update(i = 3, x = 6)
Query 3: query(start = 0, end = 4)
Output :1
2
Explanation
In Query 1, the subarray [0...4]
has 1 Composite number viz. {12}
In Query 2, the value at index 3
is updated to 6, the array arr now is, {1, 12, 3,
6, 7, 9}
In Query 3, the subarray [0...4]
has 2 Composite Numbers viz. {12, 6}
Dado que necesitamos manejar consultas de rango y actualizaciones de puntos, un método eficiente es usar un árbol de segmentos para resolver el problema. Un árbol de segmentos es el más adecuado para este propósito.
Podemos usar Sieve of Eratosthenes para preprocesar todos los números primos hasta el valor máximo que arr i puede tomar, digamos MAX. La complejidad de tiempo para esta operación será O(MAX log(log(MAX))) .
Construyendo el árbol de segmentos:
El problema se puede reducir a la suma de subarreglos utilizando el árbol de segmentos .
Ahora, podemos construir el árbol de segmentos donde un Node de hoja se representa como 0 (si es un número primo) o 1 (si es un número compuesto).
Los Nodes internos del árbol de segmentos son iguales a la suma de sus Nodes secundarios, por lo tanto, un Node representa los números compuestos totales en el rango de L a R donde el rango 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 la suma de Nodes en el rango de principio a fin , que a su vez representa el número de compuestos 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:
Sea y el antiguo valor de arr i y x el nuevo valor.
Caso 1: Si x e y ambos son compuestos.
El recuento de compuestos en el subarreglo no cambia, por lo que solo actualizamos el arreglo y no
modificamos el árbol de segmentos
Caso 2: si x e y son primos.
El conteo de compuestos en el subarreglo no cambia, así que solo actualizamos el arreglo y no
modificamos el árbol de segmentos
Caso 3: Si y es compuesto pero x es primo.
El recuento de números compuestos en el subarreglo disminuye, por lo que actualizamos el arreglo y agregamos -1 a cada
rango, el índice i que se actualizará forma parte del árbol de segmentos
Caso 4: si y es primo pero x es compuesto.
El recuento de números compuestos en el subarreglo aumenta, por lo que actualizamos el arreglo y agregamos 1 a cada
rango, el índice i que se actualizará es parte del árbol de segmentos
A continuación se muestra la implementación del enfoque anterior:
C++
// C++ program to find number of composite numbers in a
// subarray and performing updates
#include <bits/stdc++.h>
using namespace std;
#define MAX 1000
// Function to calculate primes upto MAX
// using sieve of Eratosthenes
void sieveOfEratosthenes(bool isPrime[])
{
isPrime[1] = true;
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 composites
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 queryCompositesUtil(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 composites
// 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 queryCompositesUtil(st, ss, mid, qs, qe, 2 * index + 1)
+ queryCompositesUtil(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 composite
if ((!isPrime[oldValue]) && (!isPrime[new_val]))
return;
// Case 3: Old value was composite, new value is prime
if (!isPrime[oldValue] && isPrime[new_val]) {
diff = -1;
}
// Case 4: Old value was prime, new_val is composite
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 composite numbers in range
// from index qs (query start) to qe (query end).
// It mainly uses queryCompositesUtil()
void queryComposites(int* st, int n, int qs, int qe)
{
int compositesInRange = queryCompositesUtil(st, 0, n - 1, qs, qe, 0);
cout << "Number of Composites in subarray from " << qs
<< " to " << qe << " = " << compositesInRange << "\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 composite
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 Code
int main()
{
int arr[] = { 1, 12, 3, 5, 17, 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 composite, 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;
queryComposites(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);
// Query 3: Query(start = 0, end = 4)
start = 0;
end = 4;
queryComposites(st, n, start, end);
return 0;
}
Java
// Java program to find number of composite numbers in a
// subarray and performing updates
public class Main
{
static int MAX = 1000;
// Function to calculate primes upto MAX
// using sieve of Eratosthenes
static void sieveOfEratosthenes(boolean[] isPrime)
{
isPrime[1] = true;
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 composites
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 queryCompositesUtil(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 composites
// 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 queryCompositesUtil(st, ss, mid, qs, qe, 2 * index + 1)
+ queryCompositesUtil(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.print("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 primes
if (isPrime[oldValue] && isPrime[new_val])
return;
// Case 2: Old and new values both composite
if ((!isPrime[oldValue]) && (!isPrime[new_val]))
return;
// Case 3: Old value was composite, new value is prime
if (!isPrime[oldValue] && isPrime[new_val]) {
diff = -1;
}
// Case 4: Old value was prime, new_val is composite
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 composite numbers in range
// from index qs (query start) to qe (query end).
// It mainly uses queryCompositesUtil()
static void queryComposites(int[] st, int n, int qs, int qe)
{
int compositesInRange = queryCompositesUtil(st, 0, n - 1, qs, qe, 0);
System.out.println("Number of Composites in subarray from " + qs
+ " to " + qe + " = " + compositesInRange);
}
// 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 composite
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;
}
public static void main(String[] args) {
int[] arr = { 1, 12, 3, 5, 17, 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 composite, else true.
*/
boolean[] isPrime = new boolean[MAX + 1];
for(int a = 0; a < MAX + 1; a++)
{
isPrime[a] = 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;
queryComposites(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);
// Query 3: Query(start = 0, end = 4)
start = 0;
end = 4;
queryComposites(st, n, start, end);
}
}
// This code is contributed by divyeshrabadiya07.
Python3
# Python3 program to find
# number of composite numbers
# in a subarray and performing
# updates
import math
MAX = 1000
# Function to calculate primes
# upto MAX using sieve of Eratosthenes
def sieveOfEratosthenes(isPrime):
isPrime[1] = True;
p = 2
while p * p <= MAX:
# If prime[p] is not
# changed, then
# it is a prime
if (isPrime[p] == True):
# Update all multiples of p
for i in range(p * 2,
MAX + 1, p):
isPrime[i] = False;
p += 1
# 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 composites 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 queryCompositesUtil(st, ss, se, qs,
qe, index):
# If segment of this node is a
# part of given range, then
# return the number of composites
# 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 (queryCompositesUtil(st, ss,
mid, qs,
qe, 2 * index + 1) +
queryCompositesUtil(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):
print("Invalid Input");
return
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 composite
if ((not isPrime[oldValue]) and
(not isPrime[new_val])):
return;
# Case 3: Old value was composite,
# new value is prime
if (not isPrime[oldValue] and
isPrime[new_val]):
diff = -1;
# Case 4: Old value was prime,
# new_val is composite
if (isPrime[oldValue] and
not isPrime[new_val]):
diff = 1;
# Update the values of
# nodes in segment tree
updateValueUtil(st, 0,
n - 1, i,
diff, 0);
# Return number of composite
# numbers in range from index
# qs (query start) to qe (query end).
# It mainly uses queryCompositesUtil()
def queryComposites(st, n, qs, qe):
compositesInRange = queryCompositesUtil(st, 0,
n - 1,
qs, qe, 0);
print("Number of Composites in subarray from ",
qs, " to ", qe, " = ", compositesInRange)
# 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 composite
if (not 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 = (int)(math.ceil(math.log2(n)));
# 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, 12, 3, 5, 17, 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 composite, 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;
queryComposites(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);
# Query 3: Query(start = 0,
# end = 4)
start = 0;
end = 4;
queryComposites(st, n,
start, end)
# This code is contributed by Chitranayal
C#
// C# program to find number of composite numbers in a
// subarray and performing updates
using System;
class GFG {
static int MAX = 1000;
// Function to calculate primes upto MAX
// using sieve of Eratosthenes
static void sieveOfEratosthenes(bool[] isPrime)
{
isPrime[1] = true;
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 composites
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 queryCompositesUtil(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 composites
// 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 queryCompositesUtil(st, ss, mid, qs, qe, 2 * index + 1)
+ queryCompositesUtil(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.Write("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 primes
if (isPrime[oldValue] && isPrime[new_val])
return;
// Case 2: Old and new values both composite
if ((!isPrime[oldValue]) && (!isPrime[new_val]))
return;
// Case 3: Old value was composite, new value is prime
if (!isPrime[oldValue] && isPrime[new_val]) {
diff = -1;
}
// Case 4: Old value was prime, new_val is composite
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 composite numbers in range
// from index qs (query start) to qe (query end).
// It mainly uses queryCompositesUtil()
static void queryComposites(int[] st, int n, int qs, int qe)
{
int compositesInRange = queryCompositesUtil(st, 0, n - 1, qs, qe, 0);
Console.WriteLine("Number of Composites in subarray from " + qs
+ " to " + qe + " = " + compositesInRange);
}
// 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 composite
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;
}
static void Main() {
int[] arr = { 1, 12, 3, 5, 17, 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 composite, else true.
*/
bool[] isPrime = new bool[MAX + 1];
for(int a = 0; a < MAX + 1; a++)
{
isPrime[a] = 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;
queryComposites(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);
// Query 3: Query(start = 0, end = 4)
start = 0;
end = 4;
queryComposites(st, n, start, end);
}
}
// This code is contributed by decode2207.
Javascript
<script>
// Javascript program to find number of composite numbers in a
// subarray and performing updates
let MAX = 1000;
// Function to calculate primes upto MAX
// using sieve of Eratosthenes
function sieveOfEratosthenes(isPrime)
{
isPrime[1] = true;
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 s + parseInt((e - s) / 2, 10);
}
/* A recursive function to get the number of composites
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 queryCompositesUtil(st, ss, se, qs, qe, index)
{
// If segment of this node is a part of given range,
// then return the number of composites
// 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 queryCompositesUtil(st, ss, mid, qs, qe, 2 * index + 1)
+ queryCompositesUtil(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, 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 composite
if ((!isPrime[oldValue]) && (!isPrime[new_val]))
return;
// Case 3: Old value was composite, new value is prime
if (!isPrime[oldValue] && isPrime[new_val]) {
diff = -1;
}
// Case 4: Old value was prime, new_val is composite
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 composite numbers in range
// from index qs (query start) to qe (query end).
// It mainly uses queryCompositesUtil()
function queryComposites(st, n, qs, qe)
{
let compositesInRange = queryCompositesUtil(st, 0, n - 1, qs, qe, 0);
document.write("Number of Composites in subarray from " + qs
+ " to " + qe + " = " + compositesInRange + "</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 composite
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;
}
let arr = [ 1, 12, 3, 5, 17, 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 composite, else true.
*/
let isPrime = new Array(MAX + 1);
for(let a = 0; a < MAX + 1; a++)
{
isPrime[a] = 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;
queryComposites(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);
// Query 3: Query(start = 0, end = 4)
start = 0;
end = 4;
queryComposites(st, n, start, end);
// This code is contributed by suresh07.
</script>
Producción:
Number of Composites in subarray from 0 to 4 = 1 Number of Composites in subarray from 0 to 4 = 2
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.