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Solution 2: Minimum Inclusive Span (Minimum Window Substring)

Approach Explanation

We use a sliding window where r expands the window and l shrinks it once the window is valid (contains all characters of t).

Step-by-Step Logic

  1. Count characters in t.
  2. l = 0, r = 0. have = 0, need = len(t_count).
  3. Expand r:
    • Update window counts. If char count matches t_count, have += 1.
    • While have == need:
      • Update min window result.
      • Shrink l. update counts. If count drops below t_count, have -= 1.

Complexity

  • Time Complexity: O(N).
  • Space Complexity: O(M).

Code

from collections import Counter

def min_window(s, t):
    if not t: return ""
    
    count_t, window = Counter(t), {}
    have, need = 0, len(count_t)
    res, res_len = [-1, -1], float("infinity")
    l = 0
    
    for r in range(len(s)):
        c = s[r]
        window[c] = 1 + window.get(c, 0)
        
        if c in count_t and window[c] == count_t[c]:
            have += 1
            
        while have == need:
            if (r - l + 1) < res_len:
                res = [l, r]
                res_len = r - l + 1
            
            window[s[l]] -= 1
            if s[l] in count_t and window[s[l]] < count_t[s[l]]:
                have -= 1
            l += 1
            
    l, r = res
    return s[l : r + 1] if res_len != float("infinity") else ""