查找处理Q查询后可以形成的字符串数

给定数字N (1 <= N <= 2000)。，任务是找到大小为N的数字字符串，该数字字符串可以在使用从’a’到’z’的字符并处理给定的q (1 < = q <= 200000)查询。

对于每个查询，给定两个整数L，R(0 <= L <= R <= N)，以使生成的大小为N的字符串的子字符串[L，R]必须是回文。的任务是处理所有查询和生成大小的字符串N，使得由所有查询定义的该字符串的子串是回文。

答案可能非常大。因此，输出答案取模1000000007。

注意：字符串考虑从1开始的索引。

例子：

Input : N = 3
query 1: (1, 2)
query 2: (2, 3)
Output : 26
Explanation : Substring 1 to 2 should be 
palindrome and substring 2 to 3 should be palindrome. so, all 
three characters should be same. so, we can obtain 26 such strings.

Input : N = 4
query 1: (1, 3)
query 2: (2, 4)
Output : 676
Explanation : substring 1 to 3 should be 
palindrome and substring 2 to 4 should be a palindrome. 
So, a first and third character should be the same and 
second and the fourth should be the same. So, we can 
obtain 26*26 such strings.

方法：一种有效的解决方案是使用联合查找算法。

• 找到每个范围(查询)的中点，如果有很多查询具有相同的中点，则仅保留长度最大的查询，即(其中r – l为最大)。
• 由于长度为N的字符串中点数为2 * N，因此查询数量最多可以减少到2 * N。
• 现在，对于每个查询，将元素l与r合并，将(l + 1)与(r – 1)合并，将(l + 2)与(r – 2)合并，依此类推。我们这样做是因为要放在索引l上的字符与我们要放在索引r上的字符相同。将此逻辑扩展到所有查询，我们需要维护不交集的数据结构。因此，基本上，不相交集的一个组成部分的所有元素上都应具有相同的字母。
• 处理完所有查询后，让不交集的组件数为x，则答案为26 ^ x

  。

  下面是上述方法的实现：

  C++

  // C++ program to implement above approach
    
  #include 
  using namespace std;
  #define N 2005
  #define mod (int)(1e9 + 7)
    
  // To store the size of string and
  // number of queries
  int n, q;
    
  // To store parent and rank of ith place
  int par[N], Rank[N];
    
  // To store maximum interval
  map m;
    
  // Function for initialization
  void initialize()
  {
      for (int i = 0; i <= n; i++) {
          par[i] = i;
          Rank[i] = 0;
      }
  }
    
  // Function to find parent
  int find(int x)
  {
      if (par[x] != x)
          par[x] = find(par[x]);
    
      return par[x];
  }
    
  // Function to make union
  void Union(int x, int y)
  {
      int xpar = find(x);
      int ypar = find(y);
    
      if (Rank[xpar] < Rank[ypar])
          par[xpar] = ypar;
      else if (Rank[xpar] > Rank[ypar])
          par[ypar] = xpar;
      else {
          par[ypar] = xpar;
          Rank[xpar]++;
      }
  }
    
  // Power function to calculate a raised to m1
  // under modulo 10000007
  int power(long long a, long long m1)
  {
      if (m1 == 0)
          return 1;
      else if (m1 == 1)
          return a;
      else if (m1 == 2)
          return (1LL * a * a) % mod;
      else if (m1 & 1)
          return (1LL * a * power(power(a, m1 / 2), 2)) % mod;
      else
          return power(power(a, m1 / 2), 2) % mod;
  }
    
  // Function to take maxmium interval
  void query(int l, int r)
  {
      m[l + r] = max(m[l + r], r);
  }
    
  // Function to find different possible strings
  int possiblestrings()
  {
      initialize();
    
      for (auto i = m.begin(); i != m.end(); i++) {
          int x = i->first - i->second;
          int y = i->second;
    
          // make union of all chracters which
          // are meant to be same
          while (x < y) {
              Union(x, y);
              x++;
              y--;
          }
      }
    
      // find number of different sets formed
      int ans = 0;
      for (int i = 1; i <= n; i++)
          if (par[i] == i)
              ans++;
    
      // return the required answer
      return power(26, ans) % mod;
  }
    
  // Driver Code
  int main()
  {
      n = 4;
    
      // queries
      query(1, 3);
      query(2, 4);
    
      cout << possiblestrings();
    
      return 0;
  }

  Java

  // Java program to implement above approach
  import java.util.*;
    
  class GFG 
  {
      static final int N = 2005;
      static final int mod = 1000000007;
    
      // To store the size of string and
      // number of queries
      static int n, q;
    
      // To store parent and rank of ith place
      static int[] par = new int[N], Rank = new int[N];
    
      // To store maximum interval
      static HashMap m = new HashMap<>();
    
      // Function for initialization
      static void initialize()
      {
          for (int i = 0; i <= n; i++)
          {
              par[i] = i;
              Rank[i] = 0;
          }
      }
    
      // Function to find parent
      static int find(int x)
      {
          if (par[x] != x)
              par[x] = find(par[x]);
    
          return par[x];
      }
    
      // Function to make union
      static void Union(int x, int y) 
      {
          int xpar = find(x);
          int ypar = find(y);
    
          if (Rank[xpar] < Rank[ypar])
              par[xpar] = ypar;
          else if (Rank[xpar] > Rank[ypar])
              par[ypar] = xpar;
          else 
          {
              par[ypar] = xpar;
              Rank[xpar]++;
          }
      }
    
      // Power function to calculate a raised to m1
      // under modulo 10000007
      static long power(long a, long m1) 
      {
          if (m1 == 0)
              return 1;
          else if (m1 == 1)
              return a;
          else if (m1 == 2)
              return (1L * a * a) % mod;
          else if (m1 % 2 == 1)
              return (1L * a * power(power(a, m1 / 2), 2)) % mod;
          else
              return power(power(a, m1 / 2), 2) % mod;
      }
    
      // Function to take maxmium interval
      static void query(int l, int r)
      {
          if (m.containsKey(l + r))
              m.put(l + r, Math.max(m.get(l + r), r));
          else
              m.put(l + r, r);
      }
    
      // Function to find different possible strings
      static long possiblestrings() 
      {
          initialize();
    
          for (Integer i : m.keySet())
          {
              int x = i - m.get(i);
              int y = m.get(i);
    
              // make union of all chracters which
              // are meant to be same
              while (x < y) 
              {
                  Union(x, y);
                  x++;
                  y--;
              }
          }
    
          // find number of different sets formed
          int ans = 0;
          for (int i = 1; i <= n; i++)
              if (par[i] == i)
                  ans++;
    
          // return the required answer
          return power(26, ans) % mod;
      }
    
      // Driver Code
      public static void main(String[] args) 
      {
          n = 4;
    
          // queries
          query(1, 3);
          query(2, 4);
    
          System.out.println(possiblestrings());
      }
  }
    
  // This code is contributed by
  // sanjeev2552

  Python3

  # Python3 program to implement above approach
  N = 2005
  mod = 10**9 + 7
    
  # To store the size of string and
  # number of queries
  n, q = 0, 0
    
  # To store parent and rank of ith place
  par = [0 for i in range(N)]
  Rank = [0 for i in range(N)]
    
  # To store maximum interval
  m = dict()
    
  # Function for initialization
  def initialize():
    
      for i in range(n + 1):
          Rank[i], par[i] = 0, i
    
  # Function to find parent
  def find(x):
      if (par[x] != x):
          par[x] = find(par[x])
    
      return par[x]
    
  # Function to make union
  def Union(x, y):
    
      xpar = find(x)
      ypar = find(y)
    
      if (Rank[xpar] < Rank[ypar]):
          par[xpar] = ypar
      elif (Rank[xpar] > Rank[ypar]):
          par[ypar] = xpar
      else:
          par[ypar] = xpar
          Rank[xpar] += 1
    
  # Power function to calculate a raised to m1
  # under modulo 10000007
  def power(a, m1):
    
      if (m1 == 0):
          return 1
      elif (m1 == 1):
          return a
      elif (m1 == 2):
          return (a * a) % mod
      elif (m1 & 1):
          return (a * power(power(a, m1 // 2), 2)) % mod
      else:
          return power(power(a, m1 // 2), 2) % mod
    
  # Function to take maxmium interval
  def query(l, r):
      if l + r in m.keys():
          m[l + r] = max(m[l + r], r)
      else:
          m[l + r] = max(0, r)
    
  # Function to find different possible strings
  def possiblestrings():
      initialize()
    
      for i in m:
          x = i - m[i]
          y = m[i]
    
          # make union of all chracters which
          # are meant to be same
          while (x < y):
              Union(x, y)
              x += 1
              y -= 1
    
      # find number of different sets formed
      ans = 0
      for i in range(1, n + 1):
          if (par[i] == i):
              ans += 1
    
      # return the required answer
      return power(26, ans) % mod
    
  # Driver Code
  n = 4
    
  # queries
  query(1, 3)
  query(2, 4)
    
  print(possiblestrings())
    
  # This code is contributed by Mohit Kumar

  C#

  // C# program to implement above approach
  using System;
  using System.Collections.Generic; 
    
  class GFG{
        
  const int N = 2005;
  const int mod = 1000000007;
    
  // To store the size of string and
  // number of queries
  static int n;
  //static int q;
    
  // To store parent and rank of ith place
  static int[]par = new int[N];
  static int[]Rank = new int[N];
    
  // To store maximum interval
  static Dictionary m = new Dictionary();
    
  // Function for initialization
  static void initialize()
  {
      for(int i = 0; i <= n; i++)
      {
          par[i] = i;
          Rank[i] = 0;
      }
  }
    
  // Function to find parent
  static int find(int x)
  {
      if (par[x] != x)
          par[x] = find(par[x]);
    
      return par[x];
  }
    
  // Function to make union
  static void Union(int x, int y) 
  {
      int xpar = find(x); 
      int ypar = find(y); 
    
      if (Rank[xpar] < Rank[ypar]) 
          par[xpar] = ypar; 
      else if (Rank[xpar] > Rank[ypar]) 
          par[ypar] = xpar; 
      else
      { 
          par[ypar] = xpar; 
          Rank[xpar]++; 
      } 
  }
    
  // Power function to calculate a raised to m1
  // under modulo 10000007
  static long power(long a, long m1) 
  {
      if (m1 == 0)
          return 1;
      else if (m1 == 1)
          return a;
      else if (m1 == 2)
          return (1L * a * a) % mod;
      else if (m1 % 2 == 1)
          return (1L * a * power(
                 power(a, m1 / 2), 2)) % mod;
      else
          return power(power(a, m1 / 2), 2) % mod;
  }
    
  // Function to take maxmium interval
  static void query(int l, int r)
  {
      if (m.ContainsKey(l + r))
          m[l + r] = Math.Max(m[l + r], r);
      else
          m[l + r] = r;
  }
    
  // Function to find different possible strings
  static long possiblestrings() 
  {
      initialize();
    
      foreach(KeyValuePair i in m)
      {
          int x = i.Key - m[i.Key];
          int y = m[i.Key];
    
          // Make union of all chracters 
          // which are meant to be same
          while (x < y) 
          {
              Union(x, y);
              x++;
              y--;
          }
      }
    
      // Find number of different sets formed
      int ans = 0;
      for(int i = 1; i <= n; i++)
          if (par[i] == i)
              ans++;
    
      // Return the required answer
      return power(26, ans) % mod;
  }
    
  // Driver Code
  public static void Main(string[] args) 
  {
      n = 4;
    
      // Queries
      query(1, 3);
      query(2, 4);
    
      Console.Write(possiblestrings());
  }
  }
    
    
  // This code is contributed by rutvik_56

  输出：

  676
  

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