add longest palindromic substring

Author: tg123

longest-palindromic-substring/README.md

@@ -1,4 +1,67 @@
+## Leetcode Blog Version
 
-## TODO 
-  * write down thinking
+Here is the [Leetcode Blog Version](http://leetcode.com/2011/11/longest-palindromic-substring-part-i.html). 
+This is a classic problem. I write my first leetcode accepted version based on that.
 
+
+## My own words version
+
+Brute force + isPalindromic will run out of time.
+
+### Recursive
+
+ * `length 1 substring`: all are palindromic
+ * `length 2 substring`: only if two char are same
+
+What if `length 3`
+
+```
+
+length 3 = length 1 + two char
+
+0 1 2 3 4
+a b c b a
++   ^    
+|   |
++---+    
+
+
+0 1 2 4 4
+a b c b a
+  +   ^     
+  |   |
+  +---+    
+
+```
+
+ * `length n` = inner `length n - 2` is palindromic AND (first char == last char)
+
+
+### Store `length n` into `P[lenght n][start index]`
+
+ * `P[1][3] = true` means that the substring starts from index 3 and 1 char long is palindromic.
+ * `P[5][2] = true` means that the substring starts from index 2 and 5 char long is NOT palindromic.
+
+
+a matrix for `abcba`
+
+```
+
+  0 1 2 3 4
+  a b c b a
+1 1 1 1 1 1
+2 0 0 0 0 -
+3 0 1 0 - -
+4 0 0 - - -
+5 1 - - - -
+^
+l
+e
+n
+
+```
+
+With the matrix, we can fill up the martix by `P[len][i] = P[len -2][i + 1] && S[i] == S[i + len -1]`.
+
+
+Note: the `length 0` is useless, so just using `P[len - 1]` for `len`.