### leetcode Question 27: Distinct Subsequences

Distinct Subsequences

Given a string S and a string T, count the number of distinct subsequences of T in S.
A subsequence of a string is a new string which is formed from the original string by deleting some (can be none) of the characters without disturbing the relative positions of the remaining characters. (ie, "ACE" is a subsequence of "ABCDE"while "AEC" is not).
Here is an example:
S = "rabbbit"T = "rabbit"
Return 3.

Analysis:
In other word, this question ask how many time T has occurred in S, where some char can be added in between T, but the order of T cannot change.
An easy but time-consuming idea is recursion. Details can be found in the source code. However, this algorithm cannot pass the large test.
A better way is

### (Updated 201308)

The idea is using dynamic programming (DP).
First we redefine the problem to another form.
e.g.  S= "aabb" T="aab" , we need to find the occurrence of T in S, so,
(1) We store the positions of each letter of T in S, in a table:
a [0, 1]
a [0, 1]
b [2, 3]
(2) The problem then can be viewed as "find how many paths from top to the bottom with an ascending order".  In other words, from the table above, for "a" we have only one choice 0,  the other "a" we have only one choice 1, and for b we can choose 2 or 3, thus the result is 2.

How to get the path number? you can use dfs, but it might be time consuming.
Here, we use DP, define res[i][j] the number of path can have for table[i][j]
res[i][j] = sum(res[i-1][k]), where table[i-1][k]<table[i][j].
The final result is the sum(res[last row]).

The code (All test cases passed):

class Solution {
public:
void cond(vector<vector<int> > &table, int &res){
int row = table.size();
vector<vector<int> > r(row);
for (int i=0;i<table[0].size();i++){
r[0].push_back(1);
}
for (int i=1;i<table.size();i++){
int tmp=0;
int bf=0;
for (int j=0;j<table[i].size();j++){
for (int k=bf;k<table[i-1].size();k++){
if (table[i][j]>table[i-1][k]){
tmp=tmp+r[i-1][k];
bf=k+1;
}else{break;}
}
r[i].push_back(tmp);
}
}
for (int i=0;i<table[row-1].size();i++){
res = res+r[row-1][i];
}
}

int numDistinct(string S, string T) {
// Start typing your C/C++ solution below
// DO NOT write int main() function
if (S.empty()||T.empty()){return 0;}
int res=0;
vector<vector<int> > table(T.size());
map<char,vector<int> >mp;
vector<int> r;
for(int i=0;i<T.size();i++){mp[T[i]]=r;}
for (int i=0;i<S.size();i++){mp[S[i]].push_back(i);}
for(int i=0;i<T.size();i++){table[i]=mp[T[i]];}

cond(table,res);
return res;
}
};


The old source code(only can pass small test):

class Solution {
public:
int cond(string S, string T){
if (S.size()==0){return 0;}
if (T.size()==0){return 1;}
if ((S.size()==1)&&(T.size()==1)){
if (S[0]==T[0]) {return 1;}
else{return 0;}
}
if (S[0]==T[0]) {
return cond(S.substr(1),T.substr(1))+cond(S.substr(1),T);
}else{
return cond(S.substr(1),T);
}
}

int numDistinct(string S, string T) {
// Start typing your C/C++ solution below
// DO NOT write int main() function
if (S.empty()||T.empty()){return 0;}
return cond(S,T);
}
};


1. Nice solution.

One small bug: line 20, if T is empty then it should return 1 instead of 0.

Also, line 5 seems to be unnecessary.

I think probably another special case to check is that S and T are both empty.

Don't know if this is necessary but I think empty string should be a substring of any string, including empty string itself.

Yes. Does that make sense if line 5 change the "or" to "and"?

And this is just an answer that cannot pass the large test, I'll post the better answer ASAP.

Thanks.

2. I think it should be:

if (T.empty())
return true;

if (S.empty())
return false;

If T is empty, then it should be always true.
If T is not empty and S is empty, then it should be false.

2. class Solution {
public:
int numDistinct(string S, string T) {
int m=S.size();
int n=T.size();
vector> memo(m+1, vector(n+1, 0));
for(int i=0; i<=m; i++)
memo[i][0] = 1;
for(int i=1; i<=m; i++)
{
for(int j=1; j<=n; j++)
{
if(S[i-1]==T[j-1])
memo[i][j] += memo[i-1][j-1];
memo[i][j] += memo[i-1][j];
}
}
return memo[m][n];
}
};

3. int numDistinct(string S, string T)
{
vector f(T.size() + 1);
f[0] = 1;
for (int i = 0; i < S.size(); ++i)
{
for (int j = T.size() - 1; j >= 0; --j)
{
f[j + 1] += S[i] == T[j] ? f[j] : 0;
}
}
return f[T.size()];
}

4. can you help me to understand problem.
S = "rabbbit", T = "rabbit"
T has many distinct subsequence Eg. rabbit, rab, rait , rabit, ait, bit, ans many more
ans all of them present in S.
Then how the answer is 3.

5. I have a solution using less space:

public class Solution {
public int numDistinct(String s, String t) {
if(s == null || t == null || t.length() == 0) return 0;
int[] dp = new int[t.length()];

for(int i = 0; i=0; j–){
if(c == t.charAt(j)){
dp[j] = dp[j] + (j!=0?dp[j-1]: 1);
}
}
}
return dp[t.length()-1];
}
}
URL: http://traceformula.blogspot.com/2015/08/distinct-subsequences.html

6. I have a solution using less space:

public class Solution {
public int numDistinct(String s, String t) {
if(s == null || t == null || t.length() == 0) return 0;
int[] dp = new int[t.length()];

for(int i = 0; i=0; j–){
if(c == t.charAt(j)){
dp[j] = dp[j] + (j!=0?dp[j-1]: 1);
}
}
}
return dp[t.length()-1];
}
}
URL: http://traceformula.blogspot.com/2015/08/distinct-subsequences.html