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Runtime complexity analysis of generating effective bracket combination algorithm
Runtime complexity analysis of generating effective bracket combination algorithm
This article will explore in-depth the runtime complexity of recursive algorithms that generate effective bracket combinations. By analyzing the structure of the recursive tree and the number of nodes per layer, common complexity misjudgment was corrected, and it was clearly stated that the runtime complexity of the algorithm is O(4^n) rather than O(2^n).
Recursive algorithm implementation
First, we review the recursive algorithm that generates effective bracket combinations. The algorithm finally generates all possible combinations of valid brackets by recursively adding the left and close brackets and making validity judgments.
class Solution:
def generateParenthesis(self, n: int) -> List[str]:
resultList = []
comboList = []
self.generate(resultList, n, comboList, 0, 0)
return resultList
def generate(self, resultList, n, comboList, openCount, closeCount):
# are we done?
if (openCount == n and closeCount == n):
resultList.append(''.join(comboList))
# can we open?
if openCount closeCount:
comboList.append(')')
self.generate(resultList, n, comboList, openCount, closeCount 1)
comboList.pop()
The core of this code lies in the generate function. It recursively tries to add the left and close brackets, and maintains openCount and closeCount to track the number of left and close brackets that have been used. When both openCount and closeCount are equal to n, a valid combination of brackets is generated.
Complexity analysis
The key is to understand the structure of the recursive tree. The depth of the recursive tree is indeed 2n, because recursion will stop when both openCount and closeCount reach n. However, the number of nodes per layer does not simply increase by 2 times.
At each level, the generate function can call itself up to twice: once to add the left bracket and once to add the close bracket. However, the condition for adding the close bracket is openCount > closeCount, which limits the addition of the close bracket. Therefore, not every node will produce two child nodes.
More precisely, the number of nodes in a recursive tree is closely related to the number of Catlands. For a given n, the number of effective bracket combinations is the nth Catlan number, denoted as C_n = (1/(n 1)) * (2n choose n). The growth rate of the Catland number is approximately 4^n / (n * sqrt(πn)).
Therefore, the runtime complexity of the algorithm is O(4^n), not O(2^n). Even if the operation of each node is constant time complexity, the total number of nodes is at the 4^n level.
Notes:
- Constant factors cannot be ignored at will: in complexity analysis, constant factors cannot be ignored at will. O(2^(2n)) is equivalent to O(4^n), and the two are different on the order of magnitude.
- Catland Number: Understanding Catland Numbers is critical to analyzing the complexity of many recursive algorithms, especially those involving balance or pairing problems.
Summarize
The runtime complexity of the recursive algorithm that generates effective bracket combinations is O(4^n). Understanding the concept of the structure of the recursive tree and the Catland number is crucial to accurately analyzing the complexity of the algorithm. When performing complexity analysis, be sure to pay attention to the constant factor and consider the specific constraints of the algorithm.
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