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Re: Comparing strings from the back?

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From Oscar Benjamin <oscar.j.benjamin@gmail.com>
Subject Re: Comparing strings from the back?
Date Fri, 7 Sep 2012 19:10:16 +0000 (UTC)
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On 2012-09-07, Steven D'Aprano <steve+comp.lang.python@pearwood.info> wrote:
> <snip>
>
> After further thought, and giving consideration to the arguments given by 
> people here, I'm now satisfied to say that for equal-length strings, 
> string equality is best described as O(N).
>
> 1) If the strings are equal, a == b will always compare all N 
>    characters in each string.
>
> 2) If the strings are unequal, a == b will *at worst* compare
>    all N characters.
>
> 3) Following usual practice in this field, worst case is the
>    one which conventionally is meant when discussing Big Oh
>    behaviour. See, for example, "Introduction To Algorithms" 
>    by Cormen, Leiserson and Rivest.

Would you say, then, that dict insertion is O(N)?

>
> Also of some interest is the best case: O(1) for unequal strings (they 
> differ at the first character) and O(N) for equal strings.
>
> Also of interest is the case that has caused the majority of the 
> discussion, the average case. I am now satisfied that the average number 
> of comparisons for unequal strings is O(1). To be precise, it is bounded 
> below by 1 comparison (you always have to compare at least one pair of 
> characters) and bounded above by 2 comparisons.

I find this idea of separating into the comparison of equal strings versus the
comparison of unequal strings rather odd. If the strings you compare come from
a distribution where they are guaranteed to be equal (or unequal) then you can
just use the O(0) comparison method.

Since string comparison is only useful if the strings can be equal or unequal,
the average case depends on how often they are equal/unequal as well as the
average complexity of both. For random strings the frequency of equal strings
decreases very fast as N increases so that the comparison of random strings is
O(1).

>
> (I'm talking about the average here -- the actual number of comparisons 
> can range all the way up to N, but the average is <= 2.)
>
> If I've done the maths right, the exact value for the average is:
>
> ((M-1)*sum( (N-i)*M**i for i in range(0, N) ) + N)/(M**N)

I'm not sure where the extra N comes from --------^ but otherwise good.

I would have written that as:

(1 - p) * sum(i * p**(i-1) for i in range(1, N+1))

where p is the probability of a match (1/M for M equally likely characters) or
in closed form:

 ⎛     N                 ⎞
 ⎝1 - p ⋅(1 + N ⋅(1 - p))⎠
 ─────────────────────────
           1 - p

>
> for random strings of length N taken from an alphabet of size M.
>
> For M = 2, that average approaches but never exceeds 2 as N increases; 
> for M = 3, the average approaches 1.5, for M = 4 it approaches 1.333... 
> and so forth.

It approaches 1 / (1 - p) or, if you prefer: M / (M - 1)

Oscar

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Thread

Comparing strings from the back? Roy Smith <roy@panix.com> - 2012-09-03 21:54 -0400
  Re: Comparing strings from the back? Chris Angelico <rosuav@gmail.com> - 2012-09-04 12:07 +1000
  Re: Comparing strings from the back? Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2012-09-04 02:17 +0000
    Re: Comparing strings from the back? Dan Sommers <dan@tombstonezero.net> - 2012-09-03 21:56 -0700
    Re: Comparing strings from the back? Mark Lawrence <breamoreboy@yahoo.co.uk> - 2012-09-04 08:50 +0100
    Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-04 18:32 +0200
      Re: Comparing strings from the back? Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2012-09-04 18:07 +0000
        Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-05 11:17 +0200
      Re: Comparing strings from the back? Chris Angelico <rosuav@gmail.com> - 2012-09-05 07:59 +1000
        Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-05 11:24 +0200
          Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-05 11:43 +0200
            Re: Comparing strings from the back? Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2012-09-05 14:30 +0000
              Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-05 16:51 +0200
                Re: Comparing strings from the back? Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2012-09-05 16:24 +0000
                Re: Comparing strings from the back? Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2012-09-05 22:47 +0000
                Re: Comparing strings from the back? Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2012-09-06 08:33 +0000
                Re: Comparing strings from the back? Dave Angel <d@davea.name> - 2012-09-06 06:07 -0400
                Re: Comparing strings from the back? Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2012-09-07 04:06 +0000
                Re: Comparing strings from the back? Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2012-09-07 19:10 +0000
                Re: Comparing strings from the back? Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2012-09-08 00:55 +0000
                Re: Comparing strings from the back? Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2012-09-08 11:53 +0000
                RE: Comparing strings from the back? "Prasad, Ramit" <ramit.prasad@jpmorgan.com> - 2012-09-13 18:39 +0000
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-13 15:37 -0400
                Re: Comparing strings from the back? alex23 <wuwei23@gmail.com> - 2012-09-13 20:48 -0700
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-14 00:46 -0400
                Re: Comparing strings from the back? alex23 <wuwei23@gmail.com> - 2012-09-13 21:54 -0700
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-14 01:38 -0400
                Re: Comparing strings from the back? alex23 <wuwei23@gmail.com> - 2012-09-13 23:06 -0700
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-14 04:03 -0400
                Re: Comparing strings from the back? alex23 <wuwei23@gmail.com> - 2012-09-14 01:20 -0700
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-14 04:53 -0400
                Re: Comparing strings from the back? alex23 <wuwei23@gmail.com> - 2012-09-14 03:26 -0700
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-14 07:36 -0400
                Re: Comparing strings from the back? Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2012-09-14 10:16 +0000
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-14 07:43 -0400
                RE: Comparing strings from the back? "Prasad, Ramit" <ramit.prasad@jpmorgan.com> - 2012-09-14 22:43 +0000
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-14 23:10 -0400
                Re: Comparing strings from the back? alex23 <wuwei23@gmail.com> - 2012-09-16 18:11 -0700
                Re: Comparing strings from the back? Chris Angelico <rosuav@gmail.com> - 2012-09-17 14:05 +1000
                Re: Comparing strings from the back? alex23 <wuwei23@gmail.com> - 2012-09-16 23:06 -0700
                Re: Comparing strings from the back? Ethan Furman <ethan@stoneleaf.us> - 2012-09-17 13:35 -0700
                Re: Comparing strings from the back? Neil Hodgson <nhodgson@iinet.net.au> - 2012-09-18 09:14 +1000
                Re: Comparing strings from the back? Ethan Furman <ethan@stoneleaf.us> - 2012-09-18 08:12 -0700
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-18 11:55 -0400
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-18 11:59 -0400
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-18 12:17 -0400
                Re: Comparing strings from the back? Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2012-09-19 00:47 +0000
                Re: Comparing strings from the back? Chris Angelico <rosuav@gmail.com> - 2012-09-19 02:20 +1000
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-18 16:40 -0400
                Re: Comparing strings from the back? Mark Lawrence <breamoreboy@yahoo.co.uk> - 2012-09-19 00:48 +0100
                Re: Comparing strings from the back? Mark Lawrence <breamoreboy@yahoo.co.uk> - 2012-09-13 20:53 +0100
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-13 17:06 -0400
                Re: Comparing strings from the back? Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2012-09-14 03:39 +0000
                Re: Comparing strings from the back? Chris Angelico <rosuav@gmail.com> - 2012-09-14 14:15 +1000
                Re: Comparing strings from the back? Mark Lawrence <breamoreboy@yahoo.co.uk> - 2012-09-13 22:17 +0100
                Re: Comparing strings from the back? Dwight Hutto <dwightdhutto@gmail.com> - 2012-09-13 17:35 -0400
                RE: Comparing strings from the back? "Prasad, Ramit" <ramit.prasad@jpmorgan.com> - 2012-09-14 21:32 +0000
                Re: Comparing strings from the back? Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2012-09-07 19:40 +0000
                Re: Comparing strings from the back? Gelonida N <gelonida@gmail.com> - 2012-09-08 17:52 +0200
                Re: Comparing strings from the back? Duncan Booth <duncan.booth@invalid.invalid> - 2012-09-10 08:59 +0000
                Re: Comparing strings from the back? Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2012-09-10 13:45 +0000
                Re: Comparing strings from the back? Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2012-09-10 14:06 +0000
                Re: Comparing strings from the back? Duncan Booth <duncan.booth@invalid.invalid> - 2012-09-11 09:51 +0000
                Re: Comparing strings from the back? Terry Reedy <tjreedy@udel.edu> - 2012-09-11 11:55 -0400
                Re: Comparing strings from the back? Chris Angelico <rosuav@gmail.com> - 2012-09-11 00:26 +1000
                Re: Comparing strings from the back? Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2012-09-10 14:32 +0000
                Re: Comparing strings from the back? Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2012-09-10 14:43 +0000
                Re: Comparing strings from the back? Chris Angelico <rosuav@gmail.com> - 2012-09-11 00:56 +1000
                Re: Comparing strings from the back? Duncan Booth <duncan.booth@invalid.invalid> - 2012-09-11 09:41 +0000
                Re: Comparing strings from the back? Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2012-09-06 12:04 +0100
                Re: Comparing strings from the back? Steve Howell <showell30@yahoo.com> - 2012-09-14 17:51 -0700
                Re: Comparing strings from the back? Roy Smith <roy@panix.com> - 2012-09-06 08:13 -0400
                Re: Comparing strings from the back? Chris Angelico <rosuav@gmail.com> - 2012-09-06 22:29 +1000
                Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-06 15:43 +0200
                Re: Comparing strings from the back? Dave Angel <d@davea.name> - 2012-09-06 10:23 -0400
                Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-06 16:33 +0200
                Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-06 16:42 +0200
                Re: Comparing strings from the back? Dave Angel <d@davea.name> - 2012-09-06 11:54 -0400
                Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-06 16:34 +0200
                Re: Comparing strings from the back? Gelonida N <gelonida@gmail.com> - 2012-09-08 17:50 +0200
                Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-06 15:37 +0200
                Re: Comparing strings from the back? Chris Angelico <rosuav@gmail.com> - 2012-09-07 00:39 +1000
                Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-06 17:36 +0200
                Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-06 17:44 +0200
                Re: Comparing strings from the back? Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2012-09-07 04:07 +0000
  Re: Comparing strings from the back? Terry Reedy <tjreedy@udel.edu> - 2012-09-04 01:13 -0400
  Re: Comparing strings from the back? Mark Lawrence <breamoreboy@yahoo.co.uk> - 2012-09-04 08:56 +0100
  Re: Comparing strings from the back? Alain Ketterlin <alain@dpt-info.u-strasbg.fr> - 2012-09-04 11:58 +0200
  Re: Comparing strings from the back? Neil Hodgson <nhodgson@iinet.net.au> - 2012-09-05 12:18 +1000
    Re: Comparing strings from the back? MRAB <python@mrabarnett.plus.com> - 2012-09-05 03:39 +0100
    Re: Comparing strings from the back? Roy Smith <roy@panix.com> - 2012-09-04 22:48 -0400
    Re: Comparing strings from the back? Johannes Bauer <dfnsonfsduifb@gmx.de> - 2012-09-05 16:33 +0200
  Re: Comparing strings from the back? Peter Otten <__peter__@web.de> - 2012-09-05 10:29 +0200
  Re: Comparing strings from the back? Chris Angelico <rosuav@gmail.com> - 2012-09-05 18:33 +1000
  Re: Comparing strings from the back? Peter Otten <__peter__@web.de> - 2012-09-05 11:48 +0200
  Re: Comparing strings from the back? Peter Otten <__peter__@web.de> - 2012-09-05 17:45 +0200
  Re: Comparing strings from the back? Dan Goodman <dg.gmane@thesamovar.net> - 2012-09-10 18:07 +0200
  Re: Comparing strings from the back? Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2012-09-10 16:33 +0000
  Re: Comparing strings from the back? Dan Goodman <dg.gmane@thesamovar.net> - 2012-09-10 19:32 +0200
  Re: Comparing strings from the back? Dan Goodman <dg.gmane@thesamovar.net> - 2012-09-10 19:44 +0200
  Re: Comparing strings from the back? Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2012-09-10 21:52 +0000

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