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Problem solution · Python

Count The Number of Winning Sequences

Count The Number of Winning Sequences: a Python solution using dynamic programming. Learn the idea, check the complexity, and read the full code, with credit to walkccc LeetCode Solutions.

Technique
Dynamic programming
Source
walkccc LeetCode Solutions
Length
45 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Dynamic programming

For Count The Number of Winning Sequences, the implementation records answers for smaller states and reuses them to build the requested result without repeating work.

  1. Define precisely what one DP state represents.
  2. Establish the base cases before transitions are evaluated.
  3. Process states in dependency order and combine only already-known values.

Code notes

  • 45 lines of Python from the credited upstream file 3320.py.
  • The implementation visibly relies on cached states.
  • No explicit loop blocks detected.

Complexity

Multiply the number of reachable states by the work performed for each transition, then include the stored state table in memory usage.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from walkccc LeetCode Solutions by P.-Y. Chen (walkccc) and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeCount The Number of Winning Sequences · PythonPython
Use this to learn the idea, then write your own version.
class Solution:  def countWinningSequences(self, s: str) -> int:    MOD = 1_000_000_007     @functools.lru_cache(None)    def dp(i: int, prev: int, bob: int) -> int:      """      Returns the number of distinct sequences Bob can use to beat Alice for      s[i..n), where the previous character is `prev` (0: F, 1: W, 2: E) and the      number of points that Bob is having is `bob`.      """      if i == len(s):        return int(bob > 0)       f = 0  # If Bob summons a Fire Dragon at i.      w = 0  # If Bob summons a Water Serpent at i.      e = 0  # If Bob summons a Earth Golem at i.       match s[i]:        case 'F':          if prev != 0:            f = dp(i + 1, 0, bob) % MOD          if prev != 1:            w = dp(i + 1, 1, bob + 1) % MOD          if prev != 2:            e = dp(i + 1, 2, bob - 1) % MOD        case 'W':          if prev != 0:            f = dp(i + 1, 0, bob - 1) % MOD          if prev != 1:            w = dp(i + 1, 1, bob) % MOD          if prev != 2:            e = dp(i + 1, 2, bob + 1) % MOD        case 'E':          if prev != 0:            f = dp(i + 1, 0, bob + 1) % MOD          if prev != 1:            w = dp(i + 1, 1, bob - 1) % MOD          if prev != 2:            e = dp(i + 1, 2, bob) % MOD       return f + w + e     return (dp(0, 0, 0) + dp(0, 1, 0) + dp(0, 2, 0)) // 2 % MOD

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