Run-length encoding is a string compression method that works by replacing consecutive identical characters (repeated 2 or more times) with the concatenation of the character and the number marking the count of the characters (length of the run). For example, to compress the string "aabccc" we replace "aa" by "a2" and replace "ccc" by "c3". Thus the compressed string becomes "a2bc3".

Notice that in this problem, we are not adding '1' after single characters.

Given a string s and an integer k. You need to delete at most k characters from s such that the run-length encoded version of s has minimum length.

Find the minimum length of the run-length encoded version of s after deleting at most k characters.

Example 1:

Input: s = "aaabcccd", k = 2
Output: 4
Explanation: Compressing s without deleting anything will give us "a3bc3d" of length 6. Deleting any of the characters 'a' or 'c' would at most decrease the length of the compressed string to 5, for instance delete 2 'a' then we will have s = "abcccd" which compressed is abc3d. Therefore, the optimal way is to delete 'b' and 'd', then the compressed version of s will be "a3c3" of length 4.

Example 2:

Input: s = "aabbaa", k = 2
Output: 2
Explanation: If we delete both 'b' characters, the resulting compressed string would be "a4" of length 2.


Example 3:

Input: s = "aaaaaaaaaaa", k = 0
Output: 3
Explanation: Since k is zero, we cannot delete anything. The compressed string is "a11" of length 3.


Constraints:

• 1 <= s.length <= 100
• 0 <= k <= s.length
• s contains only lowercase English letters.

## Solution 0: Brute Force DFS (TLE)

Time complexity: O(C(n,k))
Space complexity: O(k)

## Solution1: DP

State:
i: the start index of the substring
last: last char
len: run-length
k: # of chars that can be deleted.

base case:
1. k < 0: return inf # invalid
2. i >= s.length(): return 0 # done

Transition:
1. if s[i] == last: return carry + dp(i + 1, last, len + 1, k)

2. if s[i] != last:
return min(1 + dp(i + 1, s[i], 1, k, # start a new group with s[i]
dp(i + 1, last, len, k -1) # delete / skip s[i], keep it as is.

Time complexity: O(n^3*26)
Space complexity: O(n^3*26)

## State compression

dp[i][k] := min len of s[i:] encoded by deleting at most k charchters.

dp[i][k] = min(dp[i+1][k-1] # delete s[i]
encode_len(s[i~j] == s[i]) + dp(j+1, k – sum(s[i~j])) for j in range(i, n)) # keep

Time complexity: O(n^2*k)
Space complexity: O(n*k)

## Python3

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