以最少的比较次数对数组进行排序 [英] Sorting an array with minimal number of comparisons

问题描述

我的 CS 作业需要一些帮助.我需要编写一个排序例程，在最坏的情况下使用 7 次比较对长度为 5 的数组进行排序(我已经证明需要 7 次，因为决策树的高度).

I need some help with my CS homework. I need to write a sorting routine that sorts an array of length 5 using 7 comparisons in the worst case (I've proven that 7 will be needed, because of the height of the decision tree).

我考虑过使用硬编码"决策树，但这意味着算法真的很复杂，而且我的导师暗示这不是应该的方式.

I considered using the decision tree 'hard-coded', but that means the algorithm is really complicated and was hinted by my tutor that's not the way it's supposed to be done.

我检查过快速排序、合并排序、堆排序、d-ary 堆排序、插入排序、选择排序，都没有满足要求，这让我相信需要一个特定的算法来处理长度的数组5.

I've checked quicksort, merge sort, heap sort, d-ary heap sort, insertion sort, selection sort, all do not answer the requirement, which leads me to believe there's a need for a specific algorithm for arrays of length 5.

真的很想得到一些正确方向的提示.

Would really like to get some hints towards the right direction.

推荐答案

是的，它在 Knuth vol 3 第 185 页(第 5.3.1 节)中.这首先记录在博士论文中，所以你的教授对你很苛刻！没有真正简单优雅的方法；您必须将其作为部分有序树进行跟踪.

Yes it is in Knuth vol 3 page 185 (section 5.3.1). This was first documented in a PhD thesis so your Prof is being quite hard on you! There is no really simple elegant method; you have to track it as a partially ordered tree.

这里是 lisp.测试正常(Linux 上的 SBCL)

Here it is in lisp. Tested OK (SBCL on Linux)

(defun small-sort (a)  
  "Sort a vector A of length 5"  
  (if (> (aref a 0) (aref a 1))  
      (rotatef (aref a 0) (aref a 1)))  
  (if (> (aref a 2) (aref a 3))  
      (rotatef (aref a 2) (aref a 3)))  
  (if (> (aref a 0) (aref a 2))  
      (progn  
        (rotatef (aref a 0) (aref a 2))  
        (rotatef (aref a 1) (aref a 3))))  
  (if (> (aref a 4) (aref a 2))  
      (if (> (aref a 4) (aref a 3))  
          (progn)  
          (rotatef (aref a 3) (aref a 4)))  
      (if (> (aref a 4) (aref a 0))  
          (rotatef (aref a 2) (aref a 4) (aref a 3))  
          (rotatef (aref a 0) (aref a 4) (aref a 3) (aref a 2))))  
  (if (> (aref a 1) (aref a 3))  
      (if (> (aref a 1) (aref a 4))  
          (rotatef (aref a 1) (aref a 2) (aref a 3) (aref a 4))  
          (rotatef (aref a 1) (aref a 2) (aref a 3)))  
      (if (> (aref a 1) (aref a 2))  
          (rotatef (aref a 1) (aref a 2))  
          (progn)))  
  a)  

(defun check-sorted (a)  
  (do ((i 0 (1+ i)))  
      ((>= i (1- (array-dimension a 0))))  
    ;;(format t "~S ~S~%" (aref a i) (aref a (+ 1 i)))  
    (assert (<= (aref a i) (aref a (+ 1 i))))))  

(defun rr ()  
  (dotimes (i 100000)  
    (let ((a (make-array 5 :initial-contents (list (random 1.0) (random 1.0) (random 1.0) (random 1.0) (random 1.0) ))))  
      ;;(format t "A=~S~%" a)  
      (let ((res (small-sort a)))  
        (check-sorted res)  
        ;;(format t "Res=~S~%" res)  
        ))))  

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