1170 - Counting Perfect BST

Posted 2020-07-24 SJY

1170 - Counting Perfect BST

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Time Limit: 2 second(s)

Memory Limit: 32 MB

BST is the acronym for Binary Search Tree. A BST is a tree data structure with the following properties.

i)        Each BST contains a root node and the root may have zero, one or two children. Each of the children themselves forms the root of another BST. The two children are classically referred to as left child and right child.

ii)      The left subtree, whose root is the left children of a root, contains all elements with key values less than or equal to that of the root.

iii)    The right subtree, whose root is the right children of a root, contains all elements with key values greater than that of the root.

An integer m is said to be a perfect power if there exists integer x > 1 and y > 1 such that m = x^y. First few perfect powers are {4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, ...}. Now given two integer a and b we want to construct BST using all perfect powers between a and b, where each perfect power will form the key value of a node.

Now, we can construct several BSTs out of the perfect powers. For example, given a = 1 and b = 10, perfect powers between a and b are 4, 8, 9. Using these we can form the following five BSTs.

4           4         8          9         9

  \          \      / \      /         /

    8          9   4     9   4         8

      \      /                 \      /

       9   8                     8   4

In this problem, given a and b, you will have to determine the total number of BSTs that can be formed using perfect powers between a and b.

Input

Input starts with an integer T (≤ 20000), denoting the number of test cases.

Each case of input contains two integers: a and b (1 ≤ a ≤ b ≤ 10¹⁰, b - a ≤ 10⁶) as defined in the problem statement.

Output

For each case, print the case number and the total number of distinct BSTs that can be formed by the perfect powers between a and b. Output the result modulo 100000007.

Sample Input	Output for Sample Input
4 1 4 5 10 1 10 1 3	Case 1: 1 Case 2: 2 Case 3: 5 Case 4: 0

 

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Problem Setter: Shamim Hafiz

Special Thanks: Jane Alam Jan

题意：给定一个区间范围a,b，a,b内所以可以表示为x^y的数字可以组成的二叉排序树有多少种；

思路：n个节点能够成的二叉排序树种类是卡特兰数；

定义一个数是基，当且仅当这个数不是另一个数的幂次方。我们可以在近似O(nlogn)的时间内找出[1,100000]内的所有的基。

然后对于每一个幂次k，通过二分找出x^k 在所给范围内的基的个数，累加即可求得。

卡特兰数打表就行，要用费马小定理求逆元。

  1 #include<stdio.h>
  2 #include<algorithm>
  3 #include<iostream>
  4 #include<string.h>
  5 #include<queue>
  6 #include<stdlib.h>
  7 #include<math.h>
  8 #include<stack>
  9 using namespace std;
 10 typedef unsigned long long  LL;
 11 bool pr[100005];
 12 int ans[100005];
 13 LL KTL[1000006];
 14 const int N=1e8+7;
 15 LL quick(LL n,LL m)
 16 {
 17         LL ak=1;
 18         while(m)
 19         {
 20                 if(m&1)
 21                 {
 22                         ak=(ak*n)%N;
 23                 }
 24                 n=(n*n)%N;
 25                 m/=2;
 26         }
 27         return ak;
 28 }
 29 LL qu(LL n,LL m,LL ask)
 30 {
 31         LL ak=1;
 32         while(m)
 33         {
 34                 if(m&1)
 35                 {
 36                         ak*=n;
 37                         if(ak>ask)
 38                                 return 0;
 39                 }
 40                 n*=n;
 41                 if(n>ask&&m!=1)return 0;
 42                 m/=2;
 43         }
 44         if(ak<=ask)
 45         {
 46                 return 1;
 47         }
 48 }
 49 LL qu1(LL n,LL m, LL ac)
 50 {
 51         LL ak=1;
 52         while(m)
 53         {
 54                 if(m&1)
 55                 {
 56                         ak*=n;
 57                         if(ak>ac)
 58                         {
 59                                 return 1;
 60                         }
 61                 }
 62                 n*=n;
 63                 if(n>ac&&m!=1)return 1;
 64                 m/=2;
 65         }
 66         if(ak<ac)
 67         {
 68                 return 0;
 69         }
 70         else return 1;
 71 }
 72 int main(void)
 73 {
 74         int i,j,k;
 75         scanf("%d",&k);
 76         int s;
 77         LL n,m;
 78         memset(pr,0,sizeof(pr));
 79         for(i=2; i<1000; i++)
 80         {
 81                 if(!pr[i])
 82                 {
 83                         for(j=i; i*j<=100000; j*=i)
 84                         {
 85                                 pr[i*j]=true;
 86                         }
 87                 }
 88         }
 89         int cnt=0;
 90         for(i=2; i<=100000; i++)
 91         {
 92                 if(!pr[i])
 93                 {
 94                         ans[cnt++]=i;
 95                 }
 96         }
 97         KTL[1]=1;
 98         KTL[2]=2;
 99         KTL[3]=5;
100         for(i=4; i<=1000000; i++)
101         {
102                 KTL[i]=KTL[i-1]*(4*i-2)%N;
103                 KTL[i]=KTL[i]*(quick((LL)(i+1),(LL)(N-2)))%N;
104         }
105         for(s=1; s<=k; s++)
106         {
107                 int sum=0;
108                 scanf("%lld %lld",&n,&m);
109                 for(i=2; i<=34; i++)
110                 {
111                         int l=0;
112                         int r=cnt-1;
113                         int id=-1;
114                         while(l<=r)
115                         {
116                                 int mid=(l+r)/2;
117                                 int flag=qu((LL)ans[mid],(LL)i,m);
118                                 if(flag)
119                                 {
120                                         id=mid;
121                                         l=mid+1;
122                                 }
123                                 else r=mid-1;
124                         }
125                         l=0;
126                         r=cnt-1;
127                         int id1=-1;
128                         while(l<=r)
129                         {
130                                 int mid=(l+r)/2;
131                                 int flag=qu1((LL)ans[mid],(LL)i,n);
132                                 if(flag)
133                                 {
134                                         id1=mid;
135                                         r=mid-1;
136                                 }
137                                 else l=mid+1;
138                         }
139 
140                         if(id1<=id&&id!=-1)sum+=id-id1+1;
141                 }
142                 printf("Case %d: ",s);
143                 printf("%lld\n",KTL[sum]);
144 
145         }
146         return 0;
147 }