V2EX theta-notation

Theta-Notation

定义 Definition

Theta-notation(Θ 记号)用于描述算法时间或空间复杂度的渐近紧确界:如果函数 \(f(n)\) 在足够大的 \(n\) 上同时被某个 \(g(n)\) 的常数倍上界下界夹住,则记为
\[ f(n)=\Theta(g(n)). \]
它表示 \(f(n)\) 与 \(g(n)\) 的增长速度在数量级上“同阶”,比只给上界的 Big-O 更精确。(在复杂度分析里也常写作“Big-Theta”。)

发音 Pronunciation (IPA)

/θit noten/

例句 Examples

The running time of binary search is Θ(log n).
二分查找的运行时间是 Θ(log n)。

Because the inner loop runs n times for each of n iterations, the algorithm’s total work is Θ(n) in the worst and average cases.
由于内层循环在每次外层迭代中运行 n 次,而外层也迭代 n 次,所以该算法在最坏与平均情况下的总工作量是 Θ(n)。

词源 Etymology

Theta 来自希腊字母 Θ/θ(theta),在数学与计算机科学中常用希腊字母做符号记法;notation 意为“记号、表示法”。“Θ 记号”之所以用 theta,源于渐近分中用不同符号区分上界(O)、下界(Ω)与紧确界(Θ)的传统。

相关词 Related Words

文学与名著用例 Literary Works

  • Introduction to Algorithms(Cormen, Leiserson, Rivest, Stein)中系统使用 Θ 记号描述算法复杂度。
  • The Art of Computer Programming(Donald E. Knuth)在渐近分析与算法讨论中广泛采用 Θ 等记号。
  • Algorithms(Robert Sedgewick & Kevin Wayne)在性能分析章节中频繁出现 Θ 表达式。
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