463
个编辑
更改
跳到导航
跳到搜索
第585行：
第585行： − − Prove that <math>\left[ \frac{1}{2} (n^2 + n) \right] T_6 + \left[ \frac{1}{2} (n^2 + 3n) \right] T_5 + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7 \le cn^2,\ n \ge n_0</math> − <br /><br /> − <math>\begin{align} − &\left[ \frac{1}{2} (n^2 + n) \right] T_6 + \left[ \frac{1}{2} (n^2 + 3n) \right] T_5 + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7\\ − \le &( n^2 + n )T_6 + ( n^2 + 3n )T_5 + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7 \ (\text{for } n \ge 0 ) − \end{align}</math> − <br /><br /> − Let ''k'' be a constant greater than or equal to [''T''<sub>1</sub>..''T''<sub>7</sub>] − <br /><br /> − <math>\begin{align} − &T_6( n^2 + n ) + T_5( n^2 + 3n ) + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7 \le k( n^2 + n ) + k( n^2 + 3n ) + kn + 5k\\ − = &2kn^2 + 5kn + 5k \le 2kn^2 + 5kn^2 + 5kn^2 \ (\text{for } n \ge 1) = 12kn^2 − \end{align}</math> − <br /><br /> − Therefore <math>\left[ \frac{1}{2} (n^2 + n) \right] T_6 + \left[ \frac{1}{2} (n^2 + 3n) \right] T_5 + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7 \le cn^2, n \ge n_0 \text{ for } c = 12k, n_0 = 1</math> 第616行：
第600行： − Therefore <math>\left[ \frac{1}{2} (n^2 + n) \right] T_6 + \left[ \frac{1}{2} (n^2 + 3n) \right] T_5 + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7 \le cn^2, n \ge n_0 \text{ for } c = 12k, n_0 = 1</math>+ 第622行：
第606行： − + − 分析此算法的一个更优雅的方法是声明[ t 子1 / sub。 . T 小于7 / sub ]都等于一个时间单位，在一个选择的单位系统中，一个单位大于或等于这些步骤的实际时间。这意味着该算法的运行时间分解如下:+
→运行时复杂性度量 Evaluating run-time complexity
根据'''<font color="#ff8000">经验法则 Rule-of-thumb</font>'''，我们可以假设任意给定函数中的最高阶项主导其增长率，从而定义其运行时顺序。在这个例子中，最高阶项为n<sup>2</sup>，因此可以得出f(n) = O(n<sup>2</sup>)。证明如下：
根据'''<font color="#ff8000">经验法则 Rule-of-thumb</font>'''，我们可以假设任意给定函数中的最高阶项主导其增长率，从而定义其运行时顺序。在这个例子中，最高阶项为n<sup>2</sup>，因此可以得出f(n) = O(n<sup>2</sup>)。证明如下：
证明：<math>\left[ \frac{1}{2} (n^2 + n) \right] T_6 + \left[ \frac{1}{2} (n^2 + 3n) \right] T_5 + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7 \le cn^2,\ n \ge n_0</math>
证明：<math>\left[ \frac{1}{2} (n^2 + n) \right] T_6 + \left[ \frac{1}{2} (n^2 + 3n) \right] T_5 + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7 \le cn^2,\ n \ge n_0</math>
\end{align}</math>
\end{align}</math>
<br /><br />
<br /><br />
因此： <math>\left[ \frac{1}{2} (n^2 + n) \right] T_6 + \left[ \frac{1}{2} (n^2 + 3n) \right] T_5 + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7 \le cn^2, n \ge n_0 \text{ for } c = 12k, n_0 = 1</math>
A more [[elegance|elegant]] approach to analyzing this algorithm would be to declare that [''T''<sub>1</sub>..''T''<sub>7</sub>] are all equal to one unit of time, in a system of units chosen so that one unit is greater than or equal to the actual times for these steps. This would mean that the algorithm's running time breaks down as follows:<ref>This approach, unlike the above approach, neglects the constant time consumed by the loop tests which terminate their respective loops, but it is [[Trivial (mathematics)|trivial]] to prove that such omission does not affect the final result</ref>
A more [[elegance|elegant]] approach to analyzing this algorithm would be to declare that [''T''<sub>1</sub>..''T''<sub>7</sub>] are all equal to one unit of time, in a system of units chosen so that one unit is greater than or equal to the actual times for these steps. This would mean that the algorithm's running time breaks down as follows:<ref>This approach, unlike the above approach, neglects the constant time consumed by the loop tests which terminate their respective loops, but it is [[Trivial (mathematics)|trivial]] to prove that such omission does not affect the final result</ref>
A more elegant approach to analyzing this algorithm would be to declare that [T<sub>1</sub>..T<sub>7</sub>] are all equal to one unit of time, in a system of units chosen so that one unit is greater than or equal to the actual times for these steps. This would mean that the algorithm's running time breaks down as follows:
A more elegant approach to analyzing this algorithm would be to declare that [T1..T7] are all equal to one unit of time, in a system of units chosen so that one unit is greater than or equal to the actual times for these steps. This would mean that the algorithm's running time breaks down as follows:
分析这个算法的一个更好的方法，是声明[''T''<sub>1</sub>..''T''<sub>7</sub>]都等于一个时间单位，在同一个单位制中，选择这样一个单位大于或等于这些步骤的实际时间。这意味着该算法的运行时间分解如下：
{{quote|<math>4+\sum_{i=1}^n i\leq 4+\sum_{i=1}^n n=4+n^2\leq5n^2 \ (\text{for } n \ge 1) =O(n^2).</math>}}
{{quote|<math>4+\sum_{i=1}^n i\leq 4+\sum_{i=1}^n n=4+n^2\leq5n^2 \ (\text{for } n \ge 1) =O(n^2).</math>}}