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+\subsection{二分与三分} +\paragraph{二分} 玄学算法之一,什么都可以往上套,会使时间复杂度乘上$\log{n}$。但是二分法仅适用于求解具有单调性的问题,所以做题前要大胆猜想,小(bu)心(yong)求证。 +\paragraph{二分的写法} +\subparagraph{整数二分的写法} + 我被STL荼毒了 + \begin{verbatim}while (L < R) check(m = (L + R) >> 1) ? L = m + 1 : R = m;\end{verbatim} + 注意答案的取值是L还是R还是其他什么的,这是一个大问题,因题而异,不可描述。 +\subparagraph{实数二分的写法} \begin{verbatim}while (R - L > eps) check(m = (L + R) / 2) ? L = m : R = m;\end{verbatim} +\paragraph{二分法常见模型} +\subparagraph{二分答案} + 最小值最大(或是最大值最小)问题,这类双最值问题常常使用二分法求解,也就是确定答案后,配合贪心或DP等其他算法来检验这个答案是否合法,将最优化问题转变为判定性问题。例如:将长度为$n$的序列$a_i$分成最多$m$个连续段,求所有分法中每段和的最大值最小能是多少。 +\subparagraph{二分查找} + 具有单调性的布尔表达式求解分界点,比如在有序数列中求解数字 x 的排名。 +\subparagraph{二分导数代替三分} + 有时对于一些单峰函数,我们可以通过二分导函数的方法求解函数极值,这样使用时通常定义域为整数域比较方便,因为此时$dx$可以直接取整数$1$。 \end{document} \ No newline at end of file