CLRS/9.3.md at master · gzc/CLRS
n the algorithm SELECT, the input elements are divided into groups of 5. Will the algorithm work in linear time if they are divided into groups of 7? Argue that SELECT does not run in linear time if groups of 3 are used.
Answer
假设现在每个group有k个元素.
小于（或者大于）中位数的中位数的个数至少有n/4-k个，在最坏情况下，下一次调用SELECT会递归调3n/4+k个元素.
因此 
假设对所有的n，T(n) <= cn
因此1/k+3/4 <= 1得到k >= 4
Exercises 9.3-2
Analyze SELECT to show that if n ≥ 140, then at least ⌈n/4⌉ elements are greater than the median-of-medians x and at least ⌈n/4⌉ elements are less than x.
Answer
条件满足
Exercises 9.3-3
Show how quicksort can be made to run in O(
) time in the worst case.
Answer
比较直观，我们先利用线性时间去找到中位数，再用这个中位数去做partition.
Exercises 9.3-4
Suppose that an algorithm uses only comparisons to find the ith smallest element in a set of n elements. Show that it can also find the i - 1 smaller elements and the n - i larger elements without performing any additional comparisons.
Answer
只用比较来确定的话，就跟这讲的方法是一样的. 我们既然找到了第i小元素，那么比i大和比i小的都已经被我们分类分好了.
Exercises 9.3-5
Suppose that you have a "black-box" worst-case linear-time median subroutine. Give a simple, linear-time algorithm that solves the selection problem for an arbitrary order statistic.
Answer
code tells everything. Thanks original code here.
主要思想就是找到中位数，然后递归其中一半.
Exercises 9.3-6
The kth quantiles of an n-element set are the k - 1 order statistics that divide the sorted set into k equal-sized sets (to within 1). Give an O(n lg k)-time algorithm to list the kth quantiles of a set.
Answer
code tells everything. Thanks original code here.
- 如果k是偶数，那么取中间一个，再对左右两边进行递归.
- 如果k是奇数，取最接近中间的两个数，再对左右两边进行递归.
Exercises 9.3-7
Describe an O(n)-time algorithm that, given a set S of n distinct numbers and a positive integer k ≤ n, determines the k numbers in S that are closest to the median of S.
Answer
- 计算出中位数median
- 将所有数减去median,再取绝对值
- 用SELECT计算出第k小数字y
- 遍历数组，取出所有绝对值小于等于y的
代码的局限性在于只能接收绝对值不含有相同元素的.
Exercises 9.3-8
Let [1 .. n] and Y [1 .. n] be two arrays, each containing n numbers already in sorted order. Give an O(lg n)-time algorithm to find the median of all 2n elements in arrays X and Y.
Answer
code 这个题目在leetcode就出现过.
Exercises 9.3-9
Professor Olay is consulting for an oil company, which is planning a large pipeline running east to west through an oil field of n wells. From each well, a spur pipeline is to be connected directly to the main pipeline along a shortest path (either north or south), as shown in Figure 9.2. Given x- and y-coordinates of the wells, how should the professor pick the optimal location of the main pipeline (the one that minimizes the total length of the spurs)? Show that the optimal location can be determined in linear time.
Figure 9.2: Professor Olay needs to determine the position of the east-west oil pipeline that minimizes the total length of the north-south spurs.
Answer
只需要找出y的中位数，x是无关的
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