Algorithm to find the minimum length of substring having all characters of other string

Question

I have a two strings:
string1 - hello how are you,
String2 - olo (including space character)

Output: lo ho ( hello how are you )

lo ho is the only substring that contain all characters of string2. Can anyone please suggest a good algorithm for this ( i can think og only Brute Force algo - O(n^2).

Also output should be the minimum length string(in case of multiple options).

Answer 1

Keep two pointer l and r, and a hash table M = character -> count for characters in string2 that do not occur in s[l..r].

Initially set l = 0 and r so that string1[l..r] contains all the characters of string2 (if possible). You do that by removing characters from M until it is empty.

Then proceed by incrementing r by one in each step and then incrementing l as much as possible while still keeping M empty. The minimum over all r - l + 1 (the length of the substring s[l..r]) is the solution.

Pythonish pseudocode:

n = len(string1)
M = {}   # let's say M is empty if it contains no positive values
for c in string2:
    M[c]++
l = 0
r = -1
while r + 1 < n and M not empty:
    r++
    M[string1[r]]--
if M not empty: 
    return "no solution"
answer_l, answer_r = l, r
while True:
    while M[string1[l]] < 0:
        M[string1[l]]++
        l++
    if r - l + 1 < answer_r - anwer_l + 1:
        answer_l, answer_r = l, r
    r++
    if r == n:
        break
    M[string1[r]]--
return s[answer_l..answer_r]

The "is empty" checks can be implemented in O(1) if you maintain the number of positive entries when performing the increment and decrement operations.

Let n be the length of string1 and m be the length of string2.

Note that l and r are only ever incremented, so there are at most O(n) increments and thus at most O(n) instructions are executed in the last outer loop.

If M is implemented as an array (I assume the alphabet is constant size), you get runtime O(n + m), which is optimal. If the alphabet is too large, you can use a hash table to get expected O(n + m).

Example execution:

string1 = "abbabcdbcb"
string2 = "cbb"

# after first loop
M = { 'a': 0, 'b': 2, 'c': 1, 'd': 0 }

# after second loop
l = 0
r = 5
M = { 'a': -2, 'b': -1, 'c': 0, 'd': 0 }

# increment l as much as possible:
l = 2
r = 5
M = { 'a': -1, 'b': 0, 'c': 0, 'd': 0 }

# increment r by one and then l as much as possible
l = 2
r = 6
M = { 'a': -1, 'b': 0, 'c': 0, 'd': -1 }

# increment r by one and then l as much as possible
l = 4
r = 7
M = { 'a': 0, 'b': 0, 'c': 0, 'd': -1 }

# increment r by one and then l as much as possible
l = 4
r = 8
M = { 'a': 0, 'b': 0, 'c': -1, 'd': -1 }

# increment r by one and then l as much as possible
l = 7
r = 9
M = { 'a': 0, 'b': 0, 'c': 0, 'd': 0 }

The best solution is s[7..9].