664. Strange Printer


There is a strange printer with the following two special requirements:

  1. The printer can only print a sequence of the same character each time.
  2. At each turn, the printer can print new characters starting from and ending at any places, and will cover the original existing characters.

Given a string consists of lower English letters only, your job is to count the minimum number of turns the printer needed in order to print it.

Example 1:

Input: "aaabbb"
Output: 2
Explanation: Print "aaa" first and then print "bbb".

Example 2:

Input: "aba"
Output: 2
Explanation: Print "aaa" first and then print "b" from the second place of the string, which will cover the existing character 'a'.

Hint: Length of the given string will not exceed 100.


Approach #1: Dynamic Programming [Accepted]

Intuition

It is natural to consider letting dp(i, j) be the answer for printing S[i], S[i+1], ..., S[j], but proceeding from here is difficult. We need the following sequence of deductions:

  • Whatever turn creates the final print of S[i] might as well be the first turn, and also there might as well only be one print, since any later prints on interval [i, k] could just be on [i+1, k].

  • Say the first print is on [i, k]. We can assume S[i] == S[k], because if it wasn't, we could print up to the last occurrence of S[i] in [i, k] for the same result.

  • When correctly printing everything in [i, k] (with S[i] == S[k]), it will take the same amount of steps as correctly printing everything in [i, k-1]. This is because if S[i] and S[k] get completed in separate steps, we might as well print them first in one step instead.

Algorithm

With the above deductions, the algorithm is straightforward.

To compute a recursion for dp(i, j), for every i <= k <= j with S[i] == S[k], we have some candidate answer dp(i, k-1) + dp(k+1, j), and we take the minimum of these candidates. Of course, when k = i, the candidate is just 1 + dp(i+1, j).

To avoid repeating work, we memoize our intermediate answers dp(i, j).

Complexity Analysis

  • Time Complexity: where is the length of s. For each of possible states representing a subarray of s, we perform work iterating through k.

  • Space Complexity: , the size of our memo.

Analysis written by: @awice.

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