Posted in Python onDecember 18, 2013
# -*- coding: utf-8 -*- from random import randint from math import ceil, floor def _partition(A, l, r, i): """以A[i]为主元划分数组A[l..r],使得: A[l..m-1] <= A[m] < A[m+1..r] """ A[i], A[r] = A[r], A[i] # i交换到末位r,作为主元 pivot = A[r] # 主元 m = l # 索引标记 for n in xrange(l, r): # l..r-1 if A[n] <= pivot: A[m], A[n] = A[n], A[m] # 交换 m += 1 # 后移 A[m], A[r] = A[r], A[m] # 主元到m位 return m def _rand(A, l, r): """随机划分主元""" return randint(l, r) # A[l..r]随机取一个 def _select(A, l, r, k, pivot_selector = _rand): """利用快排,得A[l..r]中第k小的数,k in [l+1,r+1]: 其尾递归方式,伪码如下: SELECT(A, l, r, k) 1 while true: 2 i ← ? // 划分主元位置 3 m ← PARTITION(A, l, r, i) // 数组划分 4 n ← m - l + 1 // A[l..m]元素个数 5 if k = n // 检查A[m]是否是第k小的元素 6 then return A[m] 7 elseif k < n // 左划分区 8 r = m - 1 9 else // 右划分区 10 k = k - n 11 l = m + 1 Args: pivot_selector(Function): 主元选取方法,默认随机方式 """ if not A: return None if l == r: return A[l] while True: i = pivot_selector(A, l, r) m = _partition(A, l, r, i) n = m - l + 1 if k == n: return A[m] elif k < n: r = m - 1 else: k = k - n l = m + 1 def rand_select(A, k): """默认随机划分主元方式,k in [1, len(A)] E[T(n)] = O(n) """ return _select(A, 0, len(A) - 1, k); def _median(A, l, r): """对A[l..r]插入排序(原地)后选取其中位数位置""" for j in xrange(l, r + 1): k = A[j] i = j while i > l and A[i-1] > k: A[i] = A[i-1] i -= 1 A[i] = k return l + int((r - l) * 0.5) # 下中位数 def _medianOfMedians(A, l, r): """中位数的中位数方式: 1. 划分为floor(n/5)个5元组,剩下(n%5)组成最后一组。 2. 找出ceil(n/5)个组各自的中位数。先对每组插入排序,再从中选出中位数。 3. 对第2步中找出的ceil(n/5)个中位数重复上述操作,直到仅有一个中位数。 """ if l == r: return l n = r - l + 1 # 元素个数 m = int(ceil(n / 5.0)) # 划分组数,每组5个元素 for i in xrange(m): # 每组起始位和结束位 sub_l = l + i * 5 sub_r = sub_l + 4 if sub_r > r: sub_r = r # 对每组元素插入排序后,选取中位数 sub_m = _median(A, sub_l, sub_r) # 中位数索引 # 交换中位数到前几位 j = l + i A[j], A[sub_m] = A[sub_m], A[j] return _medianOfMedians(A, l, l + m - 1) # 中位数的中位数 def bfprt_select(A, k): """中位数的中位数方式(BFPRT算法) T(n) = O(n) """ return _select(A, 0, len(A) - 1, k, _medianOfMedians); def _median3(A, l, r): """三数中位数方式,取l,r,(l+r)/2三数中位数""" c = (l + r) / 2 keys = [l, c, r] i = _median(keys, 0, 2) return keys[i] def median_select(A, k): """三数中位数方式,以消除最坏情况""" return _select(A, 0, len(A) - 1, k, _median3); if __name__ == '__main__': import random, time from copy import copy print('preparing data...') n = 1000000 nums = range(n) random.shuffle(nums) print('ready go!') def timeit(fnc, *args, **kargs): print('%s starts processing' % fnc.__name__) begtime = time.clock() retval = fnc(*args, **kargs) endtime = time.clock() print('%s takes time : %f' % (fnc.__name__, endtime - begtime)) return retval test_methods = [rand_select, bfprt_select, median_select] k = random.randrange(n) + 1 dashes = '---' * 10 for test in test_methods: print(dashes) nums_new = copy(nums) result = timeit(test, nums_new, k) print('the %dth smallest element: %d' % (k, result))
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