给定一个数组arr[]大小为N ，一个整数P和一个二维数组Q[][]由以下类型的查询组成：
- 1 LRB[R – L + 1]：这个查询的任务是用数组B[] b 替换子数组{arr[L], … arr[R]给定任何数组元素最多可以被替换P次.
- 2 X：此查询的任务是打印arr[X] 。
例子：
Input: arr[] = {3, 10, 4, 2, 8, 7}, P = 1, Q[][] = {{1, 0, 3, 3, 2, 1, 11}, {2, 3}, {1, 2, 3, 5, 7}, {2, 2} }
Output: 11 1
Explanation:
Query 1: Replacing the subarray {arr[0], …, arr[3]} with the array {3, 2, 1, 11} modifies arr[] to {3, 2, 1, 11, 8, 7}
Query 2: Print arr[3].
Query 3: Since P = 1, therefore the subarray {arr[2], …, arr[3]} can’t be replaced more than once.
Query 4: Print arr[2].
Input: arr[] = {1, 2, 3, 4, 5}, P = 2, Q[][] = {{2, 0}, {1, 1, 3, 6, 7, 8}, {1, 3, 4, 10, 12}, {2, 4}}
Output: 1 12
方法：该问题可以使用联合查找算法来解决。这个想法是遍历数组Q[][]并检查Q[0]是否等于1 。如果发现为真，则用新数组替换子数组，并且每当任何数组元素被替换 P 次时，然后使用联合查找创建一个新子集。请按照以下步骤解决问题：
- 初始化一个数组，比如visited[] ，其中visited[i]检查索引i是否存在于任何不相交的子集中。
- 初始化一个数组，比如count[] ，其中count[i]存储arr[i]被替换的次数。
- 初始化一个数组，比如last[] ，以存储每个不相交子集的最大元素。
- 初始化一个数组，比如parent[] ，以存储每个不相交子集的最小元素。
- 遍历Q[][]数组并检查每个查询是否Q[i][0] == 1 。如果发现为真则执行以下操作：
- 使用变量low在范围[ Q[i][1], Q[i][2] ] 上迭代并检查visited[low]是否为真。如果发现为真，则找到该子集的父级存在的低点，找到该子集中存在的最大元素。
- 否则，检查count[low]是否小于P。如果发现为真，则将arr[low]的值替换为相应的值。
- 否则，检查count[low]是否等于P。如果发现为真，则创建当前索引的新不相交子集。
- 否则，打印arr[Q[i][1]]] 。
下面是上述方法的实现：
Java
// Java program to implement
// the above approach
import java.io.*;
import java.util.*;
class GFG {
// visited[i]: Check index i is present
// in any disjoint subset or not.
static boolean[] visited;
// Store the smallest element
// of each disjoint subset
static int[] parent;
// count[i]: Stores the count
// of replacements of arr[i]
static int[] last;
// Store the largest element
// of each disjoint subset
static int[] count;
// Function to process all the given Queries
static void processQueries(int[] arr, int P,
List > Q)
{
// Traverse the queries[][] array
for (int i = 0; i < Q.size(); i++) {
// Stores the current query
List query = Q.get(i);
// If query of type is 1
if (query.get(0) == 1) {
// Perform the query of type 1
processTypeOneQuery(query, arr, P);
}
// If query of type is 2
else {
// Stores 2nd element of
// current query
int index = query.get(1);
// Print arr[index]
System.out.println(arr[index]);
}
}
}
// Function to perform the query of type 1
static void processTypeOneQuery(
List query, int[] arr, int P)
{
// Stores the value of L
int low = query.get(1);
// Stores the value of R
int high = query.get(2);
// Stores leftmost index of the
// subarray for which a new
// subset can be generated
int left = -1;
// Stores index of
// the query[] array
int j = 3;
// Iterate over the
// range [low, high]
while (low <= high) {
// If low is present in
// any of the subset
if (visited[low]) {
// If no subset created for
// the subarray arr[left...low - 1]
if (left != -1) {
// Create a new subset
newUnion(left, low - 1,
arr.length);
// Update left
left = -1;
}
// Stores next index to be
// processed
int jump = findJumpLength(low);
// Update low
low += jump;
// Update j
j += jump;
}
// If arr[low] has been
// already replaced P times
else if (count[low] == P) {
// If already subset
// created for left
if (left == -1) {
// Update left
left = low;
}
// Mark low as an element
// of any subset
visited[low] = true;
// Update low
low++;
// Update j
j++;
}
// If arr[low] has been replaced
// less than P times
else {
// If no subset created for
// the subarray arr[left...low - 1]
if (left != -1) {
// Create a new subset
newUnion(left, low - 1, arr.length);
// Update left
left = -1;
}
// Replace arr[low] with
// the corresponding value
arr[low] = query.get(j);
// Update count[low]
count[low]++;
// Update low
low++;
// Update j
j++;
}
}
// If no subset has been created for
// the subarray arr[left...low - 1]
if (left != -1) {
// Create a new subset
newUnion(left, high, arr.length);
}
}
// Function to find the next index
// to be processed after visiting low
static int findJumpLength(int low)
{
// Stores smallest index of
// the subset where low present
int p = findParent(low);
// Stores next index
// to be processed
int nextIndex = last[p] + 1;
// Stores difference between
// low and nextIndex
int jump = (nextIndex - low);
// Return jump
return jump;
}
// Function to create a new subset
static void newUnion(int low, int high,
int N)
{
// Iterate over
// the range [low + 1, high]
for (int i = low + 1; i <= high;
i++) {
// Perform union operation
// on low
union(low, i);
}
// If just smaller element of low
// is present in any of the subset
if (low > 0 && visited[low - 1]) {
// Perform union on (low - 1)
union(low - 1, low);
}
// If just greater element of high
// is present in any of the subset
if (high < N - 1 && visited[high + 1]) {
// Perform union on high
union(high, high + 1);
}
}
// Function to find the smallest
// element of the subset
static int findParent(int u)
{
// Base Case
if (parent[u] == u)
return u;
// Stores smallest element
// of parent[u
return parent[u]
= findParent(parent[u]);
}
// Function to perform union operation
static void union(int u, int v)
{
// Stores smallest element
// of subset containing u
int p1 = findParent(u);
// Stores smallest element
// of subset containing u
int p2 = findParent(v);
// Update parent[p2]
parent[p2] = p1;
// Update last[p1]
last[p1] = last[p2];
}
// Function to find all the queries
static List > getQueries()
{
// Stores all the queries
List > Q
= new ArrayList >();
// Initialize all queries
Integer[] query1 = { 1, 0, 3, 3, 2,
1, 11 };
Integer[] query2 = { 2, 3 };
Integer[] query3 = { 1, 2, 3, 5, 7 };
Integer[] query4 = { 2, 2 };
// Insert all queries
Q.add(Arrays.asList(query1));
Q.add(Arrays.asList(query2));
Q.add(Arrays.asList(query3));
Q.add(Arrays.asList(query4));
// Return all queries
return Q;
}
// Driver Code
public static void main(String[] args)
{
int[] arr = { 3, 10, 4, 2, 8, 7 };
int N = arr.length;
int P = 1;
parent = new int[N];
last = new int[N];
count = new int[N];
visited = new boolean[N];
// Initialize parent[] and
// last[] array
for (int i = 0; i < parent.length;
i++) {
// Update parent[i]
parent[i] = i;
// Update last[i]
last[i] = i;
}
List > Q = getQueries();
processQueries(arr, P, Q);
}
} Python3
# Python3 program to implement
# the above approach
# visited[i]: Check index i is present
# in any disjoint subset or not.
visited = []
# Store the smallest element
# of each disjoint subset
parent = []
# count[i]: Stores the count
# of replacements of arr[i]
last = []
# Store the largest element
# of each disjoint subset
count = []
# Function to process all the given Queries
def processQueries(arr, P, Q):
# Traverse the [,]queries array
for i in range(len(Q)):
# Stores the current query
query = Q[i];
# If query of type is 1
if (query[0] == 1):
# Perform the query of type 1
processTypeOneQuery(query, arr, P);
# If query of type is 2
else:
# Stores 2nd element of
# current query
index = query[1];
# Print arr[index]
print(arr[index]);
# Function to perform the query of type 1
def processTypeOneQuery(query, arr, P):
# Stores the value of L
low = query[1];
# Stores the value of R
high = query[2];
# Stores leftmost index of the
# subarray for which a new
# subset can be generated
left = -1;
# Stores index of
# the query[] array
j = 3;
# Iterate over the
# range [low, high]
while (low <= high):
# If low is present in
# any of the subset
if (visited[low]):
# If no subset created for
# the subarray arr[left...low - 1]
if (left != -1):
# Create a new subset
newUnion(left, low - 1,len(arr));
# Update left
left = -1;
# Stores next index to be
# processed
jump = findJumpLength(low);
# Update low
low += jump;
# Update j
j += jump;
# If arr[low] has been
# already replaced P times
elif (count[low] == P):
# If already subset
# created for left
if (left == -1):
# Update left
left = low;
# Mark low as an element
# of any subset
visited[low] = True;
# Update low
low += 1
# Update j
j += 1
# If arr[low] has been replaced
# less than P times
else:
# If no subset created for
# the subarray arr[left...low - 1]
if (left != -1):
# Create a new subset
newUnion(left, low - 1, len(arr));
# Update left
left = -1;
# Replace arr[low] with
# the corresponding value
arr[low] = query[j];
# Update count[low]
count[low] += 1
# Update low
low += 1
# Update j
j += 1
# If no subset has been created for
# the subarray arr[left...low - 1]
if (left != -1):
# Create a new subset
newUnion(left, high, len(arr));
# Function to find the next index
# to be processed after visiting low
def findJumpLength(low):
# Stores smallest index of
# the subset where low present
p = findParent(low);
# Stores next index
# to be processed
nextIndex = last[p] + 1;
# Stores difference between
# low and nextIndex
jump = (nextIndex - low);
# Return jump
return jump;
# Function to create a new subset
def newUnion(low, high,N):
# Iterate over
# the range [low + 1, high]
for i in range(low+1,high+1):
# Perform union operation
# on low
union(low, i);
# If just smaller element of low
# is present in any of the subset
if (low > 0 and visited[low - 1]):
# Perform union on (low - 1)
union(low - 1, low);
# If just greater element of high
# is present in any of the subset
if (high < N - 1 and visited[high + 1]):
# Perform union on high
union(high, high + 1);
# Function to find the smallest
# element of the subset
def findParent(u):
# Base Case
if (parent[u] == u):
return u;
# Stores smallest element
# of parent[u
parent[u]= findParent(parent[u]);
return parent[u]
# Function to perform union operation
def union(u, v):
# Stores smallest element
# of subset containing u
p1 = findParent(u);
# Stores smallest element
# of subset containing u
p2 = findParent(v);
# Update parent[p2]
parent[p2] = p1;
# Update last[p1]
last[p1] = last[p2];
# Function to find all the queries
def getQueries():
# Stores all the queries
Q = []
# Initialize all queries
query1 = [ 1, 0, 3, 3, 2,1, 11 ]
query2 = [ 2, 3 ]
query3 = [ 1, 2, 3, 5, 7 ]
query4 = [ 2, 2 ]
# Insert all queries
Q.append(query1)
Q.append(query2)
Q.append(query3)
Q.append(query4)
# Return all queries
return Q;
# Driver Code
if __name__=='__main__':
arr = [ 3, 10, 4, 2, 8, 7 ]
N = len(arr)
P = 1;
parent = [i for i in range(N)]
last = [i for i in range(N)]
count = [0 for i in range(N)]
visited = [False for i in range(N)]
Q = getQueries();
processQueries(arr, P, Q);
# This code is contributed by rutvik_56.C#
// C# program to implement
// the above approach
using System;
using System.Collections.Generic;
public class GFG {
// visited[i]: Check index i is present
// in any disjoint subset or not.
static bool[] visited;
// Store the smallest element
// of each disjoint subset
static int[] parent;
// count[i]: Stores the count
// of replacements of arr[i]
static int[] last;
// Store the largest element
// of each disjoint subset
static int[] count;
// Function to process all the given Queries
static void processQueries(int[] arr, int P,
List > Q)
{
// Traverse the [,]queries array
for (int i = 0; i < Q.Count; i++) {
// Stores the current query
List query = Q[i];
// If query of type is 1
if (query[0] == 1) {
// Perform the query of type 1
processTypeOneQuery(query, arr, P);
}
// If query of type is 2
else {
// Stores 2nd element of
// current query
int index = query[1];
// Print arr[index]
Console.WriteLine(arr[index]);
}
}
}
// Function to perform the query of type 1
static void processTypeOneQuery(
List query, int[] arr, int P)
{
// Stores the value of L
int low = query[1];
// Stores the value of R
int high = query[2];
// Stores leftmost index of the
// subarray for which a new
// subset can be generated
int left = -1;
// Stores index of
// the query[] array
int j = 3;
// Iterate over the
// range [low, high]
while (low <= high) {
// If low is present in
// any of the subset
if (visited[low]) {
// If no subset created for
// the subarray arr[left...low - 1]
if (left != -1) {
// Create a new subset
newUnion(left, low - 1,
arr.Length);
// Update left
left = -1;
}
// Stores next index to be
// processed
int jump = findJumpLength(low);
// Update low
low += jump;
// Update j
j += jump;
}
// If arr[low] has been
// already replaced P times
else if (count[low] == P) {
// If already subset
// created for left
if (left == -1) {
// Update left
left = low;
}
// Mark low as an element
// of any subset
visited[low] = true;
// Update low
low++;
// Update j
j++;
}
// If arr[low] has been replaced
// less than P times
else {
// If no subset created for
// the subarray arr[left...low - 1]
if (left != -1) {
// Create a new subset
newUnion(left, low - 1, arr.Length);
// Update left
left = -1;
}
// Replace arr[low] with
// the corresponding value
arr[low] = query[j];
// Update count[low]
count[low]++;
// Update low
low++;
// Update j
j++;
}
}
// If no subset has been created for
// the subarray arr[left...low - 1]
if (left != -1) {
// Create a new subset
newUnion(left, high, arr.Length);
}
}
// Function to find the next index
// to be processed after visiting low
static int findJumpLength(int low)
{
// Stores smallest index of
// the subset where low present
int p = findParent(low);
// Stores next index
// to be processed
int nextIndex = last[p] + 1;
// Stores difference between
// low and nextIndex
int jump = (nextIndex - low);
// Return jump
return jump;
}
// Function to create a new subset
static void newUnion(int low, int high,
int N)
{
// Iterate over
// the range [low + 1, high]
for (int i = low + 1; i <= high;
i++) {
// Perform union operation
// on low
union(low, i);
}
// If just smaller element of low
// is present in any of the subset
if (low > 0 && visited[low - 1]) {
// Perform union on (low - 1)
union(low - 1, low);
}
// If just greater element of high
// is present in any of the subset
if (high < N - 1 && visited[high + 1]) {
// Perform union on high
union(high, high + 1);
}
}
// Function to find the smallest
// element of the subset
static int findParent(int u)
{
// Base Case
if (parent[u] == u)
return u;
// Stores smallest element
// of parent[u
return parent[u]
= findParent(parent[u]);
}
// Function to perform union operation
static void union(int u, int v)
{
// Stores smallest element
// of subset containing u
int p1 = findParent(u);
// Stores smallest element
// of subset containing u
int p2 = findParent(v);
// Update parent[p2]
parent[p2] = p1;
// Update last[p1]
last[p1] = last[p2];
}
// Function to find all the queries
static List > getQueries()
{
// Stores all the queries
List > Q
= new List >();
// Initialize all queries
int[] query1 = { 1, 0, 3, 3, 2,
1, 11 };
int[] query2 = { 2, 3 };
int[] query3 = { 1, 2, 3, 5, 7 };
int[] query4 = { 2, 2 };
// Insert all queries
Q.Add(new List(query1));
Q.Add(new List(query2));
Q.Add(new List(query3));
Q.Add(new List(query4));
// Return all queries
return Q;
}
// Driver Code
public static void Main(String[] args)
{
int[] arr = { 3, 10, 4, 2, 8, 7 };
int N = arr.Length;
int P = 1;
parent = new int[N];
last = new int[N];
count = new int[N];
visited = new bool[N];
// Initialize parent[] and
// last[] array
for (int i = 0; i < parent.Length;
i++) {
// Update parent[i]
parent[i] = i;
// Update last[i]
last[i] = i;
}
List > Q = getQueries();
processQueries(arr, P, Q);
}
}
// This code is contributed by Amit Katiyar 输出：
11
1时间复杂度： O(N + |Q| * P)
辅助空间： O(N)
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