Coursera Algorithms week1 算法分析练习测验: Egg drop 扔鸡蛋问题

题目原文：

Suppose that you have an n-story building (with floors 1 through n) and plenty of eggs. An egg breaks if it is dropped from floor T or higher and does not break otherwise. Your goal is to devise a strategy to determine the value of T given the following limitations on the number of eggs and tosses:

Version 0: 1 egg, ≤T tosses.
Version 1: ∼1lgn eggs and ∼1lgn tosses.
Version 2: ∼lgT eggs and ∼2lgT tosses.
Version 3: 2 eggs and ∼2$ \sqrt{n} $ tosses.
Version 4: 2 eggs and ≤c$ \sqrt{T} $ tosses for some fixed constant c

分析：

version0 ：拿着一个鸡蛋从1~n依次扔就可以，到floor T会碎，故复杂度为≤T

version 1: 采用二分查找，首先从n/2层开始扔:

　　　　　　if(鸡蛋碎) 从(n/2)/2层开始扔；

　　　　　　else 从n/2+(n/2)/2层开始扔

　　　　　二分方法需要lgn个鸡蛋尝试lgn次

version 2: 依次从1, 2, 4, 8, 16, 32,...2^k开始扔，如果鸡蛋在2^k碎了，那么2^k-1≤T≤2^k，这时已经使用了 lgT 次步，接下来在[2^k-1+1,2^k）区间进行version1的二分查找方法，需要花费lgT步。这两种操作加起来总共花费2lgT步

version 3: 将0~n层楼分成[1, $ \sqrt{n} $-1], [$ \sqrt{n} $, 2 $ \sqrt{n} $-1], [2$ \sqrt{n} $,3 $ \sqrt{n} $-1]...[k$ \sqrt{n} $, (k+1)$ \sqrt{n} $-1]..个区间，用一个鸡蛋分布从1开始在各个区间的起始楼层扔，如果在k$ \sqrt{n} $层碎了，那就从(k-1)$ \sqrt{n} $+1开始逐层扔。第一步区间选择用了 $ \sqrt{n} $的复杂度，第二步区间内部扔鸡蛋用了 $ \sqrt{n} $的复杂度，总共用了 2$ \sqrt{n} $

version 4: 尝试从1, 4, 9, 16, 25,...(k-1)², k²....楼层扔鸡蛋，加入鸡蛋在楼层k²碎了，意味着(k-1)²≤T≤k²，这一步尝试了$ \sqrt{T} $次(k=$ \sqrt{T} $)。接着从楼层(k-1)²+1开始逐层扔，最多尝试至k²-1结束，这一步需要尝试k²-1-(k-1)²-1=2$ \sqrt{T} $-1=2$ \sqrt{T} $-2次。总共用了3$ \sqrt{T} $-2次

转载于:https://www.cnblogs.com/evasean/p/7208986.html

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