Q：Hot and Cold Binary Search Game
Hot or cold.
I think you have to do some sort of binary search but I'm not sure how.
I've been racking my brain and I can't seem to come up with a a lg N + O(1).
I found this: http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_cs;action=display;num=1316188034 but could not understand the diagram and it did not describe other possible cases.
我一直绞尽脑汁，我似乎无法拿出一一的LG N + O（1）。
我发现：HTTP：/ / www.ocf。伯克利。edu / ~ WWU / cgi-bin / yabb / yabb.cgi？板= riddles_cs；行动=显示；Num = 1316188034但不能理解图中并没有描述其他可能的情况。
For numbers between (1~N):
1st guess: (1/3)N
2nd guess: (2/3)N -> tells us the answer is either in 1~(1/2)N or (1/2)N +1 ~ N
3rd guess: in 1~(1/2)N: try (1/6)N; in (1/2)N +1 ~ N: try (5/6)N
Starting from 2nd guess, each step cuts the range in 1/2. So it's step 1 + lgN
第一猜测：（1 / 3）N
第二猜：（2 / 3）N - >；告诉我们答案是1 ~（1 / 2）或（1 / 2）n + 1 ~ n
第三猜测：在1 ~（1 / 2）N：尝试（1 / 6）N；在（1 / 2）N + 1；尝试（5 / N）
从第二猜开始，每一步削减范围在1 / 2。所以它的步骤1 + lgn
This is a repost of my answer at Hot or Cold Guess Secret Number, which unfortunately has no upvoted or accepted answer:
This was a task at IOI 2010, for which I sat on the Host Scientific Committee. (We asked for an optimal solution instead of simply lg N + O(1), and what follows is not quite optimal.)
Not swinging outside -N .. 2N and using lg N + 2 guesses is straightforward; all you need to do is show that the obvious translation of binary search works.
Once you have something that doesn't swing outside -N .. 2N and takes lg N + 2 guesses, do this:
Guess N/2, then N/2+1. This tells you which half of the array the answer is in. Then guess the end of that half-array. You're either in one of the two "middle" quarters or you're in one of the two "end" quarters. If you're in a middle quarter, do the thing before and you win in lg N + 4 guesses. The ends are slightly trickier.
Suppose I need to guess a number in 1 .. K without straying outside 1 .. N and my last guess was 1. If I guess K/2 and I'm colder, then I next guess 1; I spent two guesses to get a similar subproblem that's 1/4 the size. If K/2 is hotter, I know the answer is in K/4 .. K. Guess K/2-1 next. The two subcases are K/4 .. K/2-1 and K/2 .. K, both of which are nice. But it took me three guesses to (in the worst-case) halve the size of the problem; if I ever do this, I wind up doing lg N + 6 guesses.
这是在IOI 2010的任务，而我坐在主人的科学委员会。（我们要求最佳的解决方案，而不是简单的LG N + O（1），然后就是不太理想。）
不摆动外- N…2n和使用LG N + 2的猜测是直截了当的；所有你需要做的是表明二进制搜索明显的翻译作品。
一旦你有东西不摆在外面- N…LG N + 2次2N和需要，这样做：
猜N / 2，然后N / 2 + 1。这告诉你答案的数组中的哪一半。然后猜到一半的结束。你要么在其中的两个“中间”宿舍或你在两个“最后”宿舍之一。如果你在一个季度中，做事情前和LG N + 4的猜测你赢了。两端稍麻烦。
假设我需要猜一个数字在1…K不偏离外1。我最后的猜测是1。如果我想K / 2我冷，我一猜1；我花了两个猜测得到一个相似的子问题，1 / 4的大小。如果K / 2是热，我知道答案是在K / 4。K K／2-1猜下。两个子K / 4。K / 2-1和K / 2。这两个都不错。但我花了三次（最坏的）一半大小的问题；如果我这样做，我所做的LG N + 6的猜测。
Suppose you know that your secret integer is in [a,b], and that your last guess is c.
You want to divide your interval by two, and to know whether your secret integer lies in between [a,m] or [m,b], with m=(a+b)/2.
The trick is to guess d, such that (c+d)/2 = (a+b)/2.
Without loss of generality, we can suppose that d is bigger than c. Then, if d is hotter than c, your secret integer will be bigger than (c+d)/2 = (a+b)/2 = m, and so your secret integer will lie in [m,b]. If d is cooler than c, your secret integer will belong to [a,m].
You need to be able to guess between -N and 2N because you can't guarantee that c and d as defined above will always be [a,b]. Your two first guess can be 1 and N.
So, your are dividing your interval be two at each guess, so the complexity is log(N) + O(1).
A short example to illustrate this (results chosen randomly):
Edit, suggested by @tmyklebu:
We still need to prove that our guess will always fall in bewteen [-N,2N]
By recurrence, suppose that c (our previous guess) is in [a-(a+b), b+(a+b)] = [-b,a+2b]
Then d = a+b-c <= a+b-(-b) <= a+2b and d = a+b-c >= a+b-(a+2b) >= -b
Initial case: a=1, b=N, c=1, c is indeed in [-b,a+2*b]
你想把你的区间分为两个，并知道你的秘密整数是否位于[ A，M ]或[ M，B ]，与m =（A + B）/ 2。
诀窍是猜测D，这样（C + D）/ 2 =（A + B）/ 2。
没有一般性的损失，我们可以假设D是大于C。然后，如果D是热比C，你的秘密整数将大于（C + D）/ 2 =（A + B）/ 2 = m，所以你的秘密整数将躺在[ M，B ]。如果D比C更酷，你的秘密整数将属于[ a，m ]。
我们还需要证明我们的猜测总是落在在[ N，2n ]
复发，假设C（我们以前的猜测）是[一个（A + B），B（A + B）] = [ B，一个+ 2B ]
然后D = A + B & lt；= A + B（B）& lt；= a + b和d = A + B & gt；= A + B（A + B）& gt；= B
初始情况：A = 1，B = N，C = 1，C确实在[ B，A + 2 * b ]
The solution is close to binary search. At each step you have an interval that the number can be in. Start with the whole interval [1, N]. First guess both ends - that is the numbers 1 and N. One of them will be closer, thus you will know that now the number you are searching for is in [1, N/2] or in [N/2 + 1, N](considering N even for simplicity). Now you go to the next step having a twice smaller interval. Continue using the same approach. Keep in mind that you've already probed one of the ends, however it may not be your last guess.
I am not sure what you mean by lg N + O(1), but the approach I suggest will perform O(log(N)) operations and in the worst case it will do exactly log4(N) probes.
该解决方案接近二进制搜索。在每个步骤中，你有一个区间，数字可以在。开始与整个间隔[ 1，N ]。第一猜测两端-这是数字1和N.其中之一将更接近，因此，你会知道，现在你正在寻找的数字是[ 1，N / 2 ]或在N / 2 +，N（考虑N甚至简单）。现在你进入下一步有两个较小的间隔。继续使用相同的方法。请记住，你已经探索了其中的一个目的，但它可能不是你最后的猜测。
我不知道你说的什么LG N + O（1），但我建议的方法将执行O（log（n））的操作和在最坏的情况下会做log4（N）探针。