给定一个大小为N的数组 arr[]后跟一个Q 查询数组，有以下两种类型：
- 查询类型 1：给定两个整数 L 和 R，找到从索引 L 到 R 的素数元素之和，其中 0 <= L <= R <= N-1。
- 查询类型 2：给定两个整数 i 和 X，更改 arr[i] = X 其中 0 <= i <= n-1。
注意：子查询的每个第一个索引决定了要回答的查询类型。
例子：
Input: arr[] = {1, 3, 5, 7, 9, 11}, Q = { { 1, 1, 3}, {2, 1, 10}, {1, 1, 3 } }
Output:
15
12
Explanation:
First query is of type 1, so answer is (3 + 5 + 7), = 15
Second query is of type 2, so arr[1] = 10
Third query is of type 1, where arr[1] = 10, which is not prime hence answer is (5 + 7) = 12
Input: arr[] = {1, 2, 35, 7, 14, 11}, Q = { {2, 4, 3}, {1, 4, 5 } }
Output: 14
Explanation:
First query is of type 2, So update arr[4] = 3
Second query is of type 1, since arr[4] = 3, which is prime. So answer is (3 + 11) = 14
朴素的方法：这个想法是对 L 到 R 之间的每个查询进行迭代，并对给定的数组执行所需的操作。
时间复杂度： O(Q * N * (O(sqrt(max(arr[i])))
方法：使用 Segment tree 和 Sieve Of Eratosthenes 来优化问题。
- 首先，创建一个布尔数组来标记素数。
- 现在，在制作线段树时，只将那些数组元素添加为素数的叶节点。
C++
// C++ program for the above approach
#include
using namespace std;
int const MAX = 1000001;
bool prime[MAX];
// Function to find the prime numbers
void SieveOfEratosthenes()
{
// Create a boolean array prime[]
// and initialize all entries it as true
// A value in prime[i] will
// finally be false if i is Not a prime
memset(prime, true, sizeof(prime));
for (int p = 2; p * p <= MAX; p++) {
// Check if prime[p] is not
// changed, then it is a prime
if (prime[p] == true) {
// Update all multiples of p
// greater than or equal to
// the square of it numbers
// which are multiple of p
// and are less than p^2 are
// already been marked
for (int i = p * p; i <= MAX; i += p)
prime[i] = false;
}
}
}
// Function to get the middle
// index from corner indexes
int getMid(int s, int e)
{
return s + (e - s) / 2;
}
// Function to get the sum of
// values in the given range
// of the array
int getSumUtil(int* st, int ss,
int se, int qs,
int qe, int si)
{
// If segment of this node is a
// part of given range, then
// return the sum of the segment
if (qs <= ss && qe >= se)
return st[si];
// 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 getSumUtil(st, ss, mid,
qs, qe,
2 * si + 1)
+ getSumUtil(st, mid + 1,
se, qs, qe,
2 * si + 2);
}
// Function to update the nodes which
// have the given index in their range
void updateValueUtil(int* st, int ss,
int se, int i,
int diff, int si)
{
// If the input index lies
// outside the range of
// this segment
if (i < ss || i > se)
return;
// If the input index is in
// range of this node, then update
// the value of the node and its children
st[si] = st[si] + diff;
if (se != ss) {
int mid = getMid(ss, se);
updateValueUtil(st, ss, mid, i,
diff, 2 * si + 1);
updateValueUtil(st, mid + 1,
se, i, diff,
2 * si + 2);
}
}
// Function to update a value in
// input array and segment tree
void updateValue(int arr[], int* st,
int n, int i,
int new_val)
{
// Check for erroneous input index
if (i < 0 || i > n - 1) {
cout << "-1";
return;
}
// Get the difference between
// new value and old value
int diff = new_val - arr[i];
int prev_val = arr[i];
// Update the value in array
arr[i] = new_val;
// Update the values of
// nodes in segment tree
// only if either previous
// value or new value
// or both are prime
if (prime[new_val]
|| prime[prev_val]) {
// If only new value is prime
if (!prime[prev_val])
updateValueUtil(st, 0, n - 1,
i, new_val, 0);
// If only new value is prime
else if (!prime[new_val])
updateValueUtil(st, 0, n - 1,
i, -prev_val, 0);
// If both are prime
else
updateValueUtil(st, 0, n - 1,
i, diff, 0);
}
}
// Return sum of elements in range
// from index qs (quey start) to qe
// (query end). It mainly uses getSumUtil()
int getSum(int* st, int n, int qs, int qe)
{
// Check for erroneous input values
if (qs < 0 || qe > n - 1 || qs > qe) {
cout << "-1";
return -1;
}
return getSumUtil(st, 0, n - 1,
qs, qe, 0);
}
// Function that constructs Segment Tree
int constructSTUtil(int arr[], int ss,
int se, int* st,
int si)
{
// If there is one element in
// array, store it in current node of
// segment tree and return
if (ss == se) {
// Only add those elements in segment
// tree which are prime
if (prime[arr[ss]])
st[si] = arr[ss];
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 values in this node
int mid = getMid(ss, se);
st[si]
= constructSTUtil(arr, ss, mid,
st, si * 2 + 1)
+ constructSTUtil(arr, mid + 1,
se, st,
si * 2 + 2);
return st[si];
}
// Function to construct segment
// tree from given array
int* constructST(int arr[], int n)
{
// Allocate memory for the 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;
// Allocate memory
int* st = new int[max_size];
// Fill the allocated memory st
constructSTUtil(arr, 0, n - 1, st, 0);
// Return the constructed segment tree
return st;
}
// Driver code
int main()
{
int arr[] = { 1, 3, 5, 7, 9, 11 };
int n = sizeof(arr) / sizeof(arr[0]);
int Q[3][3]
= { { 1, 1, 3 },
{ 2, 1, 10 },
{ 1, 1, 3 } };
// Function call
SieveOfEratosthenes();
// Build segment tree from given array
int* st = constructST(arr, n);
// Print sum of values in
// array from index 1 to 3
cout << getSum(st, n, 1, 3) << endl;
// Update: set arr[1] = 10
// and update corresponding
// segment tree nodes
updateValue(arr, st, n, 1, 10);
// Find sum after the value is updated
cout << getSum(st, n, 1, 3) << endl;
return 0;
} Java
// Java program for the above approach
import java.io.*;
import java.util.*;
class GFG
{
static int MAX = 1000001;
static int prime[] = new int[MAX];
// Function to find the prime numbers
static void SieveOfEratosthenes()
{
// Create a boolean array prime[]
// and initialize all entries it as true
// A value in prime[i] will
// finally be false if i is Not a prime
Arrays.fill(prime, 1);
for (int p = 2; p * p <= MAX; p++)
{
// Check if prime[p] is not
// changed, then it is a prime
if (prime[p] == 1)
{
// Update all multiples of p
// greater than or equal to
// the square of it numbers
// which are multiple of p
// and are less than p^2 are
// already been marked
for (int i = p * p; i <= MAX - 1; i += p)
prime[i] = 0;
}
}
}
// Function to get the middle
// index from corner indexes
static int getMid(int s, int e)
{
return s + (e - s) / 2;
}
// Function to get the sum of
// values in the given range
// of the array
static int getSumUtil(int[] st, int ss,
int se, int qs,
int qe, int si)
{
// If segment of this node is a
// part of given range, then
// return the sum of the segment
if (qs <= ss && qe >= se)
return st[si];
// 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 getSumUtil(st, ss, mid,
qs, qe,
2 * si + 1)
+ getSumUtil(st, mid + 1,
se, qs, qe,
2 * si + 2);
}
// Function to update the nodes which
// have the given index in their range
static void updateValueUtil(int[] st, int ss,
int se, int i,
int diff, int si)
{
// If the input index lies
// outside the range of
// this segment
if (i < ss || i > se)
return;
// If the input index is in
// range of this node, then update
// the value of the node and its children
st[si] = st[si] + diff;
if (se != ss)
{
int mid = getMid(ss, se);
updateValueUtil(st, ss, mid, i,
diff, 2 * si + 1);
updateValueUtil(st, mid + 1,
se, i, diff,
2 * si + 2);
}
}
// Function to update a value in
// input array and segment tree
static void updateValue(int arr[], int[] st,
int n, int i, int new_val)
{
// Check for erroneous input index
if (i < 0 || i > n - 1)
{
System.out.print("-1");
return;
}
// Get the difference between
// new value and old value
int diff = new_val - arr[i];
int prev_val = arr[i];
// Update the value in array
arr[i] = new_val;
// Update the values of
// nodes in segment tree
// only if either previous
// value or new value
// or both are prime
if ((prime[new_val] | prime[prev_val]) != 0)
{
// If only new value is prime
if (prime[prev_val] == 0)
updateValueUtil(st, 0, n - 1,
i, new_val, 0);
// If only new value is prime
else if (prime[new_val] == 0)
updateValueUtil(st, 0, n - 1, i, -prev_val, 0);
// If both are prime
else
updateValueUtil(st, 0, n - 1,
i, diff, 0);
}
}
// Return sum of elements in range
// from index qs (quey start) to qe
// (query end). It mainly uses getSumUtil()
static int getSum(int[] st, int n, int qs, int qe)
{
// Check for erroneous input values
if (qs < 0 || qe > n - 1 || qs > qe)
{
System.out.println( "-1");
return -1;
}
return getSumUtil(st, 0, n - 1,
qs, qe, 0);
}
// Function that constructs Segment Tree
static int constructSTUtil(int arr[], int ss,
int se, int[] st, int si)
{
// If there is one element in
// array, store it in current node of
// segment tree and return
if (ss == se) {
// Only add those elements in segment
// tree which are prime
if (prime[arr[ss]] != 0)
st[si] = arr[ss];
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 values in this node
int mid = getMid(ss, se);
st[si] = constructSTUtil(arr, ss, mid,
st, si * 2 + 1)
+ constructSTUtil(arr, mid + 1,
se, st,
si * 2 + 2);
return st[si];
}
// Function to construct segment
// tree from given array
static int[] constructST(int arr[], int n)
{
// Allocate memory for the 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;
// Allocate memory
int[] st = new int[max_size];
// Fill the allocated memory st
constructSTUtil(arr, 0, n - 1, st, 0);
// Return the constructed segment tree
return st;
}
// Driver Code
public static void main(String[] args)
{
int arr[] = { 1, 3, 5, 7, 9, 11 };
int n = arr.length;
int Q[][] = { { 1, 1, 3 },
{ 2, 1, 10 },
{ 1, 1, 3 } };
// Function call
SieveOfEratosthenes();
// Build segment tree from given array
int[] st = constructST(arr, n);
// Print sum of values in
// array from index 1 to 3
System.out.println(getSum(st, n, 1, 3));
// Update: set arr[1] = 10
// and update corresponding
// segment tree nodes
updateValue(arr, st, n, 1, 10);
// Find sum after the value is updated
System.out.println(getSum(st, n, 1, 3));
}
}
// This coe is contributed by susmitakundugoaldangaPython3
# Python3 program for the above approach
import math
MAX = 1000001
# Create a boolean array prime[]
# and initialize all entires it as true
# A value in prime[i] will
# finally be false if i is Not a prime
prime = [True] * MAX
# Function to find prime numbers
def SieveOfEratosthenes():
p = 2
while p * p <= MAX:
# Check if prime[p] is not
# changed, then it is a prime
if prime[p] == True:
# Update all multiples of p
# greater than or equal to
# the square of it numbers
# which are multiple of p
# and are less than p^2 are
# already been marked
for i in range(p * p, MAX, p):
prime[i] = False
p += 1
# Function to get the middle
# index from corner indexes
def getMid(s, e):
return s + (e - s) // 2
# Function to get the sum of
# values in the given range
# of the array
def getSumUtil(st, ss, se, qs, qe, si):
# If segment of this node is a
# part of given range, then
# return the sum of the segment
if qs <= ss and qe >= se:
return st[si]
# 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 (getSumUtil(st, ss, mid, qs,
qe, 2 * si + 1) +
getSumUtil(st, mid + 1, se,
qs, qe, 2 * si + 2))
# Function to update the nodes which
# have the given index in their range
def updateValueUtil(st, ss, se, i, diff, si):
# If the input index lies
# outside the range of
# this segment
if i < ss or i > se:
return
# If the input index is in
# range of this node, then update
# the value of the node and its children
st[si] = st[si] + diff
if se != ss:
mid = getMid(ss, se)
updateValueUtil(st, ss, mid, i,
diff, 2 * si + 1)
updateValueUtil(st, mid + 1, se, i,
diff, 2 * si + 2)
# Function to update a value in
# input array and segment tree
def updateValue(arr, st, n, i, new_val):
# Check for errorneous imput index
if i < 0 or i > n - 1:
print(-1)
return
# Get the difference between
# new value and old value
diff = new_val - arr[i]
prev_val = arr[i]
# Update the value in array
arr[i] = new_val
# Update the values of nodes
# in segment tree only if
# either previous value or
# new value or both are prime
if prime[new_val] or prime[prev_val]:
# If only new value is prime
if not prime[prev_val]:
updateValueUtil(st, 0, n - 1,
i, new_val, 0)
# If only old value is prime
elif not prime[new_val]:
updateValueUtil(st, 0, n - 1, i,
-prev_val, 0)
# If both are prime
else:
updateValueUtil(st, 0, n - 1,
i, diff, 0)
# Return sum of elements in range
# from index qs (quey start) to qe
# (query end). It mainly uses getSumUtil()
def getSum(st, n, qs, qe):
# Check for erroneous input values
if qs < 0 or qe > n-1 or qs > qe:
return -1
return getSumUtil(st, 0, n - 1, qs, qe, 0)
# Function that constructs the Segment Tree
def constructSTUtil(arr, ss, se, st, si):
# If there is one element in
# array, store it in current
# node of segment tree and return
if ss == se:
# Only add those elements in segment
# tree which are prime
if prime[arr[ss]]:
st[si] = arr[ss]
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 values in this node
mid = getMid(ss, se)
st[si] = (constructSTUtil(arr, ss, mid, st,
2 * si + 1) +
constructSTUtil(arr, mid + 1, se,
st, 2 * si + 2))
return st[si]
# Function to construct segment
# tree from given array
def constructST(arr, n):
# Allocate memory for the segment tree
# Height of segment tree
x = int(math.ceil(math.log2(n)))
# Maximum size of segment tree
max_size = 2 * int(pow(2, x)) - 1
# Allocate memory
st = [0] * max_size
# Fill the allocated memory st
constructSTUtil(arr, 0, n - 1, st, 0)
# Return the constructed segment tree
return st
# Driver code
arr = [ 1, 3, 5, 7, 9, 11 ]
n = len(arr)
Q = [ [ 1, 1, 3 ],
[ 2, 1, 10 ],
[ 1, 1, 3 ] ]
# Function call
SieveOfEratosthenes()
# Build segment tree from given array
st = constructST(arr, n)
# Print sum of values in
# array from index 1 to 3
print(getSum(st, n, 1, 3))
# Update: set arr[1] = 10
# and update corresponding
# segment tree nodes
updateValue(arr, st, n, 1, 10)
# Find sum after value is updated
print(getSum(st, n, 1, 3))
# This code is contrinuted by Stuti PathakC#
// C# program for the above approach
using System;
class GFG {
static int MAX = 1000001;
static int[] prime = new int[MAX];
// Function to find the prime numbers
static void SieveOfEratosthenes()
{
// Create a boolean array prime[]
// and initialize all entries it as true
// A value in prime[i] will
// finally be false if i is Not a prime
Array.Fill(prime, 1);
for (int p = 2; p * p <= MAX; p++)
{
// Check if prime[p] is not
// changed, then it is a prime
if (prime[p] == 1)
{
// Update all multiples of p
// greater than or equal to
// the square of it numbers
// which are multiple of p
// and are less than p^2 are
// already been marked
for (int i = p * p; i <= MAX - 1; i += p)
prime[i] = 0;
}
}
}
// Function to get the middle
// index from corner indexes
static int getMid(int s, int e)
{
return s + (e - s) / 2;
}
// Function to get the sum of
// values in the given range
// of the array
static int getSumUtil(int[] st, int ss,
int se, int qs,
int qe, int si)
{
// If segment of this node is a
// part of given range, then
// return the sum of the segment
if (qs <= ss && qe >= se)
return st[si];
// 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 getSumUtil(st, ss, mid,
qs, qe,
2 * si + 1)
+ getSumUtil(st, mid + 1,
se, qs, qe,
2 * si + 2);
}
// Function to update the nodes which
// have the given index in their range
static void updateValueUtil(int[] st, int ss,
int se, int i,
int diff, int si)
{
// If the input index lies
// outside the range of
// this segment
if (i < ss || i > se)
return;
// If the input index is in
// range of this node, then update
// the value of the node and its children
st[si] = st[si] + diff;
if (se != ss)
{
int mid = getMid(ss, se);
updateValueUtil(st, ss, mid, i,
diff, 2 * si + 1);
updateValueUtil(st, mid + 1,
se, i, diff,
2 * si + 2);
}
}
// Function to update a value in
// input array and segment tree
static void updateValue(int[] arr, int[] st,
int n, int i, int new_val)
{
// Check for erroneous input index
if (i < 0 || i > n - 1)
{
Console.Write("-1");
return;
}
// Get the difference between
// new value and old value
int diff = new_val - arr[i];
int prev_val = arr[i];
// Update the value in array
arr[i] = new_val;
// Update the values of
// nodes in segment tree
// only if either previous
// value or new value
// or both are prime
if ((prime[new_val] | prime[prev_val]) != 0)
{
// If only new value is prime
if (prime[prev_val] == 0)
updateValueUtil(st, 0, n - 1,
i, new_val, 0);
// If only new value is prime
else if (prime[new_val] == 0)
updateValueUtil(st, 0, n - 1, i, -prev_val, 0);
// If both are prime
else
updateValueUtil(st, 0, n - 1,
i, diff, 0);
}
}
// Return sum of elements in range
// from index qs (quey start) to qe
// (query end). It mainly uses getSumUtil()
static int getSum(int[] st, int n, int qs, int qe)
{
// Check for erroneous input values
if (qs < 0 || qe > n - 1 || qs > qe)
{
Console.WriteLine( "-1");
return -1;
}
return getSumUtil(st, 0, n - 1,
qs, qe, 0);
}
// Function that constructs Segment Tree
static int constructSTUtil(int[] arr, int ss,
int se, int[] st, int si)
{
// If there is one element in
// array, store it in current node of
// segment tree and return
if (ss == se) {
// Only add those elements in segment
// tree which are prime
if (prime[arr[ss]] != 0)
st[si] = arr[ss];
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 values in this node
int mid = getMid(ss, se);
st[si] = constructSTUtil(arr, ss, mid,
st, si * 2 + 1)
+ constructSTUtil(arr, mid + 1,
se, st,
si * 2 + 2);
return st[si];
}
// Function to construct segment
// tree from given array
static int[] constructST(int[] arr, int n)
{
// Allocate memory for the segment tree
// Height of segment tree
int x = (int)(Math.Ceiling(Math.Log(n,2)));
// Maximum size of segment tree
int max_size = 2 * (int)Math.Pow(2, x) - 1;
// Allocate memory
int[] st = new int[max_size];
// Fill the allocated memory st
constructSTUtil(arr, 0, n - 1, st, 0);
// Return the constructed segment tree
return st;
}
// Driver code
static void Main()
{
int[] arr = { 1, 3, 5, 7, 9, 11 };
int n = arr.Length;
// Function call
SieveOfEratosthenes();
// Build segment tree from given array
int[] st = constructST(arr, n);
// Print sum of values in
// array from index 1 to 3
Console.WriteLine(getSum(st, n, 1, 3));
// Update: set arr[1] = 10
// and update corresponding
// segment tree nodes
updateValue(arr, st, n, 1, 10);
// Find sum after the value is updated
Console.WriteLine(getSum(st, n, 1, 3));
}
}
// This code is contributed by divyesh072019.Javascript
输出：
15
12时间复杂度： O(Q * log N)
辅助空间： O(N)
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