Exploring Pyramids
Archaeologists have discovered a new set of hidden caves in one of the Egyptian pyramids. The
decryption of ancient hieroglyphs on the walls nearby showed that the caves
structure is as follows.
There are n caves in a pyramid, connected by narrow passages, one of the caves
is connected by a
passage to the outer world. The system of the passages is organized in such a
way, that there is exactly
one way to get from outside to each cave along passages. All caves are located
in the basement of the
pyramid, so we can consider them being located in the same plane. Passages do
not intersect. Each
cave has its walls colored in one of several various colors.
The scientists have decided to create a more detailed description of the caves,
so they decided to
use an exploring robot. The robot they are planning to use has two types of
memory — the output
tape, which is used for writing down the description of the caves, and the
operating memory organized
as a stack.
The robot first enters the cave connected to the outer world along the passage.
When it travels
along any passage for the first time, it puts its description on the top of its
stack. When the robot
enters any cave, it prints the color of its walls to its output tape. After
that it chooses the leftmost
passage among those that it has not yet travelled and goes along it. If there
is no such passage, the
robot takes the passage description from the top of its stack and travels along
it in the reverse direction.
The robot’s task is over when it returns to the outside of the pyramid. It is
easy to see that during
its trip the robot visits each cave at least once and travels along each
passage exactly once in each
direction.
The scientists have sent the robot to its mission. After it returned they
started to study the output
tape. What a great disappointment they have had after they have understood that
the output tape
does not describe the cave system uniquely. Now they have a new problem — they
want to know how
many different cave systems could have produced the output tape they have. Help
them to find that
out.
Since the requested number can be quite large, you should output it modulo 1
000 000 000. Please
note, that the absolute locations of the caves are not important, but their
relative locations are important,
so the caves (c) and (d) on the picture below are considered different.
Input
The input file contains the
output tape that the archaeologists have. The output tape is the sequence of
colors of caves in order the robot visited them. The colors are denoted by
capital letters of the English
alphabet. The length of the tape does not exceed 300 characters.
Output
Output one integer number —
the number of different cave systems (modulo 1 000 000 000) that could
produce the output tape.
Sample Input
ABABABA
AB
Sample Output
5
0
题意：
给出一颗多叉树，每个结点的任意两个子结点都有左右之分。从根结点开始，每次尽量往左走，走不通了就回溯，把途中遇到的字母顺次记录下来，就可以得到一个序列。给定一个序列，问有多少棵多叉树与之对应。
分析：
设输入的序列是S，d(i,j)表示子序列Si,Si+1,…,Sj对应的树的个数，那么边界条件就是d(i,i) = 0，且Si不等于Sj时d(i,j) = 0（起点与终点应是同一点）。这道题我们只用记忆化搜索的方式予以实现。属于区间dp。
1 #include <cstdio> 2 #include <cstring> 3 using namespace std; 4 const int maxn = 300; 5 const int MOD = 1e9; 6 typedef long long LL; 7 char S[maxn + 1]; 8 int d[maxn + 1][maxn + 1]; 9 int dp(int i,int j){ 10 if(i == j) return 1; 11 if(S[i] != S[j]) return 0; 12 int &ans = d[i][j]; 13 if(ans >= 0) return ans; 14 ans = 0; 15 for(int k = i + 2 ; k <= j ; k++) 16 if(S[i] == S[k]) 17 ans = (ans + (LL)dp(i + 1,k - 1) * (LL)dp(k,j)) % MOD; 18 return ans; 19 } 20 int main(){ 21 while (scanf("%s",S) == 1){ 22 memset(d,-1,sizeof(d)); 23 printf("%d ",dp(0,strlen(S) - 1)); 24 } 25 return 0; 26 }