Selection in expected linear time O(n)

Selection in expected linear time

           The general selection problem appears more difficult than the simple problem of finding a minimum. Yet, surprisingly, the asymptotic running time for both problems is the same: theta(n).

          In this section, we present a divide-and-conquer algorithm for the selection problem. The algorithm R ANDOMIZED-SELECT is modeled after the quicksort algorithm of Chapter 7. As in quicksort, we partition the input array recursively. But unlike quicksort, which recursively processes both sides of the partition, R ANDOMIZED -S ELECT works on only one side of the partition. This difference shows up in the analysis: whereas quicksort has an expected running time of theta{n*lg(n)}, the expected running time of RANDOMIZED-S ELECT is O(n), assuming that the elements are distinct.

下面是random_select()的实现：

返回在输入数据中，第i大的数据，时间复杂度是O(n)

"""Code writer : EOFCode date   : 2015.01.15Code file   : random_select.pye-mail      : [e-mail] description :      Here is a implementation for random-select in Python.Function  @random_select(A, p, r, i) will return the i-thelement from the @A which's range is @p to @r."""import randomdef partition(A, p, r) :    x = A[r]    i = p-1    for j in range(p, r) :        if A[j] <= x :           i += 1           # swap A[i] and A[j]           A[i], A[j] = A[j], A[i]     A[i+1], A[r] = A[r], A[i+1]    return i+1def random_partition(A, p, r) :    i = random.randint(p,r)    A[r], A[i] = A[i], A[r]    return partition(A, p, r)def random_select(A, p, r, i) :    if p == r :       return A[p]    q = random_partition(A, p ,r)    k = q - p + 1    if i == k :       return A[q]    elif i < k :       return random_select(A, p, q-1, i)    else :       return random_select(A, q+1, r, i-k)#-------------------testing code---------------------------A = [13,19,9,5,12,8,7,4,21,2,6,11]print "Before sorting A= " , Ax = random_select(A,0,len(A)-1, 5)print "After  sorting A= " , sorted(A), x