DP_LCS

 

The Longest Common Subsequence problem

Give two strings a and b. return the length of the longest common subsequence of them.

Note: characters in subsequence needn’t to be consecutive, but in substring, it is

For example, string a = “abcdefg”, string b = “bcdg”, Then, the longest commons subsequence is “bcdg”, but the longest common substring is “bcd”.

 

Analysis:

Most DP problem needs to hold an array to store the partial solutions, so does this one.

Assume t[,] is the array to hold the solution, t[i, j] means the length of the LCS between a0a1…ai-1ai and b0b1…bj-1bj, And we can compute t[i,j] in such way

• If(a[i] == b[j]) then increase the length by 1

t[i, j] = t[i – 1, j – 1] + 1

• else

t[i, j] = max(t[i – 1, j], t[i, j – 1]) ;

 

How to initialize array t? If string a or string b is empty, then the length will be 0. So we initialize the first row and first column of t to 0, and we need an extra row and column to hold these 0s

The following table shows you the detail steps to compute the solution

 

First, initialize array t

	Index a→	0	1	2	3	4	5
Index b ↓			a	b	c	d	e
0		0	0	0	0	0	0
1	b	0
2	c	0
3	e	0

 

Compute the first row

	Index a→	0	1	2	3	4	5
Index b ↓			a	b	c	d	e
0		0	0	0	0	0	0
1	b	0	0	1	1	1	1
2	c	0
3	e	0

 

Compute the second row

	Index a→	0	1	2	3	4	5
Index b ↓			a	b	c	d	e
0		0	0	0	0	0	0
1	b	0	0	1	1	1	1
2	c	0	0	1	2	2	2
3	e	0	0	1	2	2	3

So the result is the last element 3

When you compute any value t[i,j]in the table, first see if a[i] == b[j], it that’s true

Then t[i,j] wil be it’s upper left value in table plus one, t[1,2] is the case

Because b[1] = ‘b’,a[2] = ‘b’, so t[1,2] == t[0,1] + 1 = 0 + 1 = 1. if that’s false(a[i] != b[j]), then t[i,j] will be the max value between it’s left value and up value in the table, t[1,3] is in this case, since b[1] = ‘b’, and a[3] = ‘c’, they are different, so t[1,3] = max(t[1, 2] + t[0,3])=max(1,0)=1 , repeat this process until you get the last element of table t. that’s the value you want.

 

See code

 1 // Compute the length of the longest
 2 // common subsequence of two string
 3 // example a = "abcdefg" , b = "bcze"
 4 // then longest common subsequence is bce
 5 // and return it's length 3
 6 public int Length(string a, string b)
 7 {
 8     // allocate an two dimensional array to hold the solution
 9     // the array must be one more row and one more column than 
10     //the length of the two strings
11     int r = a.Length + 1;
12     int c = b.Length + 1;
13     int[,] t = new int[r, c];
14 
15     // Initialize the first line and first row of the array to 0
16     for (int i = 0; i < c; i++)
17         t[0, i] = 0;
18     for (int i = 0; i < r; i++)
19         t[i, 0] = 0;
20 
21     // Dynamic compute the solution
22     for (int i = 1; i < r; i++)
23     {
24         for (int j = 1; j < c; j++)
25         {
26             if (a[i - 1] == b[j - 1])
27                 t[i, j] = t[i - 1, j - 1] + 1;
28             else
29                 t[i, j] = Math.Max(t[i - 1, j], t[i, j - 1]);
30         }
31     }
32 
33     return t[r - 1, c - 1];
34 }
35