UVALive 3516 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

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区间DP~

（并不觉得它像数论……）

对于每一个区间l~r，如果l==r，则答案为1；否则枚举右边界i（l+2<=i<=r），如果s[i]==s[l]，表示这一段可以产生一个分支，就往下DP。

#include<cstdio>#include<cstring>#define modd 1000000000#define ll long longll f[301][301];char s[301];ll findd(int l,int r){if(l==r) return f[l][r]=1;if(s[l]!=s[r]) return f[l][r]=0;if(f[l][r]>=0) return f[l][r];f[l][r]=0;for(ll i=l+2;i<=r;i++)  if(s[i]==s[l]) f[l][r]=(f[l][r]+findd(l+1,i-1)*findd(i,r))%modd;return f[l][r];}int main(){while(scanf("%s",s+1)!=EOF){memset(f,-1,sizeof(f));printf("%lld\n",findd(1,strlen(s+1)));}return 0;}