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Re: Using Java Classes to Sort a Small Array Quickly

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On 9/12/2011 9:49 PM, markspace wrote:
> That is to say, Big O is the upper bound. Big Theta is the average case
> or the expected case.

The first part of that statement I might accept as a simplification, 
while the second sentence I would definitely not.

f(x) = Theta(g(x)) if and only if lim x->inf f(x)/g(x) is neither 0 nor 
infinity; in other words, f(x) and g(x) have the same asymptotic 
behavior up to a constant.

Big-O really just means that the function will grow no faster than 
another; in a way, it's a pessimistic bound: you can prove that your 
function is O(n^3), but it could still be O(n^2) and you can't prove 
that it is. If a function is Theta(n^3), however, there is no way in 
hell that it could be O(n^2); you have proven that it is exactly cubic 
in all cases.

You can use any of these for best-case, average-case, or worst-case 
performance, although using big-O for the best-case performance is 
somewhat counterintuitive.

> Both of these are also incorrect, I think. The sort time for quick sort
> is O(n log n) for RANDOM data. This can be proven. The worst case is
> O(n^2) and that's for already sorted data, a not unreasonable input. So
> the n log n number for quicksort is for a particular arrangement of
> data, not the actual upper bound.

There are ways to touch up quicksort to achieve Theta(n lg n) time for 
sorted data, and the implementation in the JVM does so.

-- 
Beware of bugs in the above code; I have only proved it correct, not 
tried it. -- Donald E. Knuth

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Using Java Classes to Sort a Small Array Quickly KevinSimonson <kvnsmnsn@hotmail.com> - 2011-08-31 19:48 -0700
  Re: Using Java Classes to Sort a Small Array Quickly markspace <-@.> - 2011-08-31 20:39 -0700
    Re: Using Java Classes to Sort a Small Array Quickly bugbear <bugbear@trim_papermule.co.uk_trim> - 2011-09-01 09:18 +0100
    Re: Using Java Classes to Sort a Small Array Quickly Roedy Green <see_website@mindprod.com.invalid> - 2011-09-01 02:38 -0700
      Re: Using Java Classes to Sort a Small Array Quickly Patricia Shanahan <pats@acm.org> - 2011-09-01 03:50 -0700
    Re: Using Java Classes to Sort a Small Array Quickly Wanja Gayk <brixomatic@yahoo.com> - 2011-09-11 23:14 +0200
      Re: Using Java Classes to Sort a Small Array Quickly Eric Sosman <esosman@ieee-dot-org.invalid> - 2011-09-11 17:53 -0400
        Re: Using Java Classes to Sort a Small Array Quickly Wanja Gayk <brixomatic@yahoo.com> - 2011-09-12 01:40 +0200
          Re: Using Java Classes to Sort a Small Array Quickly Eric Sosman <esosman@ieee-dot-org.invalid> - 2011-09-11 20:32 -0400
            Re: Using Java Classes to Sort a Small Array Quickly Wanja Gayk <brixomatic@yahoo.com> - 2011-09-17 02:54 +0200
              Re: Using Java Classes to Sort a Small Array Quickly Eric Sosman <esosman@ieee-dot-org.invalid> - 2011-09-16 21:03 -0400
                Re: Using Java Classes to Sort a Small Array Quickly Wanja Gayk <brixomatic@yahoo.com> - 2011-09-17 16:52 +0200
                Re: Using Java Classes to Sort a Small Array Quickly Eric Sosman <esosman@ieee-dot-org.invalid> - 2011-09-17 11:05 -0400
                Re: Using Java Classes to Sort a Small Array Quickly Wanja Gayk <brixomatic@yahoo.com> - 2011-09-18 15:41 +0200
                Re: Using Java Classes to Sort a Small Array Quickly Lew <lewbloch@gmail.com> - 2011-09-18 12:53 -0700
                Re: Using Java Classes to Sort a Small Array Quickly Patricia Shanahan <pats@acm.org> - 2011-09-18 13:20 -0700
          Re: Using Java Classes to Sort a Small Array Quickly Arne Vajhøj <arne@vajhoej.dk> - 2011-09-11 21:10 -0400
            Re: Using Java Classes to Sort a Small Array Quickly Lew <lewbloch@gmail.com> - 2011-09-11 20:30 -0700
              Re: Using Java Classes to Sort a Small Array Quickly Arne Vajhøj <arne@vajhoej.dk> - 2011-09-12 19:13 -0400
                Re: Using Java Classes to Sort a Small Array Quickly Lew <lewbloch@gmail.com> - 2011-09-12 17:42 -0700
                Re: Using Java Classes to Sort a Small Array Quickly Wanja Gayk <brixomatic@yahoo.com> - 2011-09-17 02:54 +0200
                Re: Using Java Classes to Sort a Small Array Quickly Eric Sosman <esosman@ieee-dot-org.invalid> - 2011-09-16 21:13 -0400
                Re: Using Java Classes to Sort a Small Array Quickly Patricia Shanahan <pats@acm.org> - 2011-09-16 23:14 -0700
                Re: Using Java Classes to Sort a Small Array Quickly Wanja Gayk <brixomatic@yahoo.com> - 2011-09-17 16:53 +0200
                Re: Using Java Classes to Sort a Small Array Quickly Eric Sosman <esosman@ieee-dot-org.invalid> - 2011-09-12 21:24 -0400
          Re: Using Java Classes to Sort a Small Array Quickly Gene Wirchenko <genew@ocis.net> - 2011-09-12 10:57 -0700
            Re: Using Java Classes to Sort a Small Array Quickly Arne Vajhøj <arne@vajhoej.dk> - 2011-09-12 19:01 -0400
              Re: Using Java Classes to Sort a Small Array Quickly Eric Sosman <esosman@ieee-dot-org.invalid> - 2011-09-12 21:29 -0400
                Re: Using Java Classes to Sort a Small Array Quickly markspace <-@.> - 2011-09-12 19:49 -0700
                Re: Using Java Classes to Sort a Small Array Quickly Lew <lewbloch@gmail.com> - 2011-09-12 21:51 -0700
                Re: Using Java Classes to Sort a Small Array Quickly Joshua Cranmer <Pidgeot18@verizon.invalid> - 2011-09-13 11:32 -0500
                Re: Using Java Classes to Sort a Small Array Quickly Arne Vajhøj <arne@vajhoej.dk> - 2011-09-13 21:51 -0400
                Re: Using Java Classes to Sort a Small Array Quickly Gene Wirchenko <genew@ocis.net> - 2011-09-14 11:00 -0700
                Re: Using Java Classes to Sort a Small Array Quickly Patricia Shanahan <pats@acm.org> - 2011-09-13 06:33 -0700
              Re: Using Java Classes to Sort a Small Array Quickly Gene Wirchenko <genew@ocis.net> - 2011-09-13 09:48 -0700
  Re: Using Java Classes to Sort a Small Array Quickly Eric Sosman <esosman@ieee-dot-org.invalid> - 2011-09-01 00:25 -0400
    Re: Using Java Classes to Sort a Small Array Quickly Eric Sosman <esosman@ieee-dot-org.invalid> - 2011-09-01 00:54 -0400
    Re: Using Java Classes to Sort a Small Array Quickly bugbear <bugbear@trim_papermule.co.uk_trim> - 2011-09-01 09:20 +0100
      Re: Using Java Classes to Sort a Small Array Quickly Eric Sosman <esosman@ieee-dot-org.invalid> - 2011-09-01 08:06 -0400
    Re: Using Java Classes to Sort a Small Array Quickly Roedy Green <see_website@mindprod.com.invalid> - 2011-09-01 02:43 -0700
  Re: Using Java Classes to Sort a Small Array Quickly Robert Klemme <shortcutter@googlemail.com> - 2011-09-01 08:32 +0200
  Re: Using Java Classes to Sort a Small Array Quickly RedGrittyBrick <RedGrittyBrick@spamweary.invalid> - 2011-09-01 10:08 +0100
    Re: Using Java Classes to Sort a Small Array Quickly Patricia Shanahan <pats@acm.org> - 2011-09-01 04:02 -0700
  Re: Using Java Classes to Sort a Small Array Quickly Roedy Green <see_website@mindprod.com.invalid> - 2011-09-01 02:37 -0700
  Re: Using Java Classes to Sort a Small Array Quickly Roedy Green <see_website@mindprod.com.invalid> - 2011-09-01 04:53 -0700
  Re: Using Java Classes to Sort a Small Array Quickly Joshua Cranmer <Pidgeot18@verizon.invalid> - 2011-09-01 09:05 -0500

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