# Find the longest subsequence of two strings

###
linxea
*Updated on *
・4 min read

### Today coding question is,

find the length of the longest subsequence of two strings.

Given "abc" and "aec", the longest subsequence string is "ac".

### Quick FYI,

if you are confused about substring and subsequence.

Given two strings: "iamnice", "iamtired"

e.g. of a longest substring: "iam"

e.g. of a longest subsequence: "iamie"

In this question, we only care about subsequence.

### So naively I can,

list down (space complexity is kaboom over here) all the possible combinations of both strings (kaboom time complexity), then search for the longest common string (O(mn) where m and n are the lengths of the strings). Altogether, extremely terrible time and space complexity.

```
def findCombinationsOfString(string, list, combi):
if len(string) == 0:
return
newCombi = combi + string[0]
# don't add common strings
if newCombi not in list:
list.append(newCombi)
findCombinationsOfString(string[1:], list, newCombi)
for i in range(1, len(string)):
findCombinationsOfString(string[i:], list, combi)
return list
def nativelyFindLongestSubsequenceOfStrings(str1, str2):
lengthStr1, lengthStr2 = len(str1), len(str2)
if lengthStr1 == 0 or lengthStr2 == 0:
return 0
if str1 == str2:
return lengthStr1
list1 = findCombinationsOfString(str1, [], "")
# ['a', 'ab', 'abc', 'ac', 'b', 'bc', 'c']
list1 = sorted(list1, key=len, reverse=True)
# ['abc', 'ab', 'ac', 'bc', 'a', 'b', 'c']
list2 = findCombinationsOfString(str2, [], "")
# ['a', 'ae', 'aec', 'ac', 'e', 'ec', 'c']
list2 = sorted(list2, key=len, reverse=True)
# ['aec', 'ae', 'ac', 'ec', 'a', 'e', 'c']
for i in range(len(list1)):
for j in range(len(list2)):
if list1[i] == list2[j]:
return len(list1[i])
return 0
print nativelyFindLongestSubsequenceOfStrings("", "")
```

### The better way is to,

put two strings into a 2D array. (I tried to draw my explanation, refer to narrative explanation in the next section)

### More explanation in words.

We put two strings into a 2D array. Iterate through the 2D array to find all common characters. If the characters from both strings are the same, increment 1 (an indicator of 1 character in the subsequence).

To get the longest subsequence, we need to consider all possible combinations of common characters that come before. So we increment 1 + get whatever (possible combinations) value that comes before.

Soon we realise there are 3 possible scenarios of the possible combination values; it could be from the top left box, the top box, and the left box. Since we are finding the longest subsequence, we get the maximum value of the 3 possible values.

If the current index shows that both characters from the two strings are the same, we increment 1 + max(top left value, top value, left value).

Until it reaches the last index at bottom right. It should contain the longest subsequence of two strings.

Tada.

### Code,

of course in Python.

Time complexity: O(mg), just kidding. It's O(mn), where m and n are the lengths of the strings

Space complexity: O(mn), the 2D array takes up space.

```
def findLongestSubsequenceString(str1, str2):
lengthStr1, lengthStr2 = len(str1), len(str2)
# if one of the strings is empty, return length of non empty string
# we can easily use a variable for magic number 0. e.g. EMPTY_STRING = 0
if lengthStr1 == 0 or lengthStr2 == 0:
return 0
# do you think it's a good idea to catch specific case like this?
# if the possibility of both strings being same is high, this is a legit case I think.
if str1 == str2:
return lengthStr1
# create 2D array with length of str1 and str2
# I think this syntax is cool AF.
memo = [[0 for i in range(lengthStr2)] for i in range(lengthStr1)]
for i in range(lengthStr1):
for j in range(lengthStr2):
topLeft = top = left = 0
# need to check if the indices are within range of 2D array,
# else the code will throw you into the sea (I meant throw out of range error)
if i > 0 and j > 0:
topLeft = memo[i - 1][j - 1]
if i > 0:
top = memo[i - 1][j]
if j > 0:
left = memo[i][j - 1]
# get the maximum value of possible subsequence combinations
memo[i][j] = max(topLeft, top, left)
# increment 1 if current characters are the same
# we can do it up there but I think it looks clearer if we separate out this logic
if str1[i] == str2[j]:
memo[i][j] = memo[i][j] + 1
return memo[lengthStr1 - 1][lengthStr2 - 1]
print findLongestSubsequenceString('abc', 'bca')
# 2
# 2D Array looks likes this:
# [1, 0, 0]
# [0, 1, 1]
# [0, 1, 2]
```

### What are we missing out,

this piece of (shit) code only caters for unique strings. What if there are duplicated characters (opps). I just started playing with dynamic programming so I need to study more on it. So far this is the best solution from my side. We could also do it recursively. I have not thought of that approach yet. (Next part?)

Comment if I miss out any edge cases. :<

### I did a quick peek (googled for solution :> ),

there is a better solution using `suffix`

thingy but I believe my brain has enough of this question. We will come back to this `suffix`

thingy soon.

### Ok,

bye.

p/s: are we able to color code the code?

You can add the language directly after the first three back-ticks at the beginning of your code-block. Like so:

ohhh, thanks peter! colourrssss

Your algorithm is not correct in some cases. Just launch it for 'aa' and 'a' strings ;)