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Problem solution · Python

Stone Game V

Stone Game V: a Python solution using dynamic programming. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Dynamic programming
Source
Kamyu LeetCode Solutions
Length
84 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 Stone Game V, 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

  • 84 lines of Python from the credited upstream file stone-game-v.py.
  • The implementation visibly relies on sequence storage, 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 Kamyu LeetCode Solutions by kamyu104 and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeStone Game V · PythonPython
Use this to learn the idea, then write your own version.
# Time:  O(n^2)# Space: O(n^2) class Solution(object):    def stoneGameV(self, stoneValue):        """        :type stoneValue: List[int]        :rtype: int        """        n = len(stoneValue)        prefix = [0]        for v in stoneValue:            prefix.append(prefix[-1] + v)         mid = range(n)         dp = [[0]*n for _ in xrange(n)]        for i in xrange(n):            dp[i][i] = stoneValue[i]         max_score = 0        for l in xrange(2, n+1):            for i in xrange(n-l+1):                j = i+l-1                while prefix[mid[i]]-prefix[i] < prefix[j+1]-prefix[mid[i]]:                    mid[i] += 1  # Time: O(n^2) in total                p = mid[i]                max_score = 0                if prefix[p]-prefix[i] == prefix[j+1]-prefix[p]:                    max_score = max(dp[i][p-1], dp[j][p])                else:                    if i <= p-2:                        max_score = max(max_score, dp[i][p-2])                    if p <= j:                        max_score = max(max_score, dp[j][p])                dp[i][j] = max(dp[i][j-1], (prefix[j+1]-prefix[i]) + max_score)                dp[j][i] = max(dp[j][i+1], (prefix[j+1]-prefix[i]) + max_score)        return max_score  # Time:  O(n^2)# Space: O(n^2)class Solution2(object):    def stoneGameV(self, stoneValue):        """        :type stoneValue: List[int]        :rtype: int        """        n = len(stoneValue)        prefix = [0]        for v in stoneValue:            prefix.append(prefix[-1] + v)         mid = [[0]*n for _ in xrange(n)]        for l in xrange(1, n+1):            for i in xrange(n-l+1):                j = i+l-1                p = i if l == 1 else mid[i][j-1]                while prefix[p]-prefix[i] < prefix[j+1]-prefix[p]:                    p += 1  # Time: O(n^2) in total                mid[i][j] = p                rmq = [[0]*n for _ in xrange(n)]        for i in xrange(n):            rmq[i][i] = stoneValue[i]         dp = [[0]*n for _ in xrange(n)]        for l in xrange(2, n+1):            for i in xrange(n-l+1):                j = i+l-1                p = mid[i][j]                max_score = 0                if prefix[p]-prefix[i] == prefix[j+1]-prefix[p]:                    max_score = max(rmq[i][p-1], rmq[j][p])                else:                    if i <= p-2:                        max_score = max(max_score, rmq[i][p-2])                    if p <= j:                        max_score = max(max_score, rmq[j][p])                dp[i][j] = max_score                rmq[i][j] = max(rmq[i][j-1], (prefix[j+1]-prefix[i]) + max_score)                rmq[j][i] = max(rmq[j][i+1], (prefix[j+1]-prefix[i]) + max_score)        return dp[0][n-1]

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