http://qs321.pair.com?node_id=963138


in reply to Re: [OT] The statistics of hashing.
in thread [OT] The statistics of hashing.

OK, I think I'm done with this diversion. I've built a program that will compute the expected number of collisions with no looping. Example output:

$ ./ana_2.pl 16 4 65536
N=65536, V=4, X=65536 integral(65536)=139415.765849051, integral(0)=13
+6533.333333333
Expected collisions: 2882.43251571807

$ ./ana_2.pl 16 4 16384
N=65536, V=4, X=16384 integral(16384)=136541.854969116, integral(0)=13
+6533.333333333
Expected collisions: 8.52163578287582

$ ./ana_2.pl 14 3 16384
N=16384, V=3, X=16384 integral(16384)=31411.9141476821, integral(0)=30
+037.3333333333
Expected collisions: 1374.58081434877

$ ./ana_2.pl 16 10 32768
N=65536, V=10, X=32768 integral(32768)=191953.190301726, integral(0)=1
+91952.863492063
Expected collisions: 0.326809662627056

[download]

The code is straightforward:

$ cat ana_2.pl
#!/usr/bin/perl
#
#   ana_2.pl N V X
#
#   N=vector size (bits), V=number of vectors, X=sample number
#
use strict;
use warnings;
use feature ':5.10';

my $n=shift; $n = 1<<$n;
my $v=shift;
my $x=shift;

my ($exp1, $exp2, $exp3);

given ($v) {
    when ( 1) { $exp1=ex_1($n, $x), $exp2=ex_1($n, 0) }
    when ( 2) { $exp1=ex_2($n, $x), $exp2=ex_2($n, 0) }
    when ( 3) { $exp1=ex_3($n, $x), $exp2=ex_3($n, 0) }
    when ( 4) { $exp1=ex_4($n, $x), $exp2=ex_4($n, 0) }
    when (10) { $exp1=ex_10($n, $x), $exp2=ex_10($n, 0) }
    default  { die "Need symbolic integral form for $v vectors!\n"; }
}

$exp3 = $exp1-$exp2;

print "N=$n, V=$v, X=$x integral($x)=$exp1, integral(0)=$exp2\n";
print "Expected collisions: $exp3\n";

sub ex_1 {
    my ($N, $X) = @_;
    return      $N  *exp(  -$X/$N)
            + $X;
}

sub ex_2 {
    my ($N, $X) = @_;
    return     -$N/2*exp(-2*$X/$N)
            + 2*$N  *exp(  -$X/$N)
            + $X;
}

sub ex_3 {
    my ($N, $X) = @_;
    return      $N/3*exp(-3*$X/$N)
            - 3*$N/2*exp(-2*$X/$N)
            + 3*$N  *exp(  -$X/$N)
            + $X;
}

sub ex_4 {
   my ($N, $X) = @_;
    return     -$N/4*exp(-4*$X/$N)
            + 4*$N/3*exp(-3*$X/$N)
            - 3*$N  *exp(-2*$X/$N)
            + 4*$N  *exp(  -$X/$N)
            + $X;
}

sub ex_10 {
   my ($N, $X) = @_;
    return       -$N/10*exp(-10*$X/$N)
            +  10*$N/9 *exp( -9*$X/$N)
            -  45*$N/8 *exp( -8*$X/$N)
            + 120*$N/7 *exp( -7*$X/$N)
            -  35*$N   *exp( -6*$X/$N)
            + 252*$N/5 *exp( -5*$X/$N)
            - 105*$N/2 *exp( -4*$X/$N)
            +  40*$N   *exp( -3*$X/$N)
            -  45*$N/2 *exp( -2*$X/$N)
            +  10*$N   *exp( -1*$X/$N)
            + $X;
}

[download]

Notes:

  1. The only testing I did is limited to the cases above. Be sure to run a few more before relying on it.
  2. I entered the ex_1 and ex_2 routines, but didn't test 'em.
  3. For the other numbers of vectors, just expand (1 - exp(-X/N))^V, and integrate it. The ex_1..n functions are coded straight from the integration.

...roboticus

When your only tool is a hammer, all problems look like your thumb.

Replies are listed 'Best First'.
Re^3: [OT] The statistics of hashing.(SOLVED)
by BrowserUk (Patriarch) on Apr 02, 2012 at 23:59 UTC

    Thank you roboticus.

    I ran your algorithm against the results of my 60-hour run, and the correlation is simply stunning! 

    #! perl -slw
use strict;

sub ex_10 {
   my ($N, $X) = @_;
    return       -$N/10*exp(-10*$X/$N)
            +  10*$N/9 *exp( -9*$X/$N)
            -  45*$N/8 *exp( -8*$X/$N)
            + 120*$N/7 *exp( -7*$X/$N)
            -  35*$N   *exp( -6*$X/$N)
            + 252*$N/5 *exp( -5*$X/$N)
            - 105*$N/2 *exp( -4*$X/$N)
            +  40*$N   *exp( -3*$X/$N)
            -  45*$N/2 *exp( -2*$X/$N)
            +  10*$N   *exp( -1*$X/$N)
            + $X;
}

my $exp10_0 = ex_10( 2**32, 0 );

open I, '<', 'L25B32V10I32S1.posns' or die $!;


while( <I> ) {
    my( $inserts, $collisions ) = split;
    my $exp3 = ex_10( 2**32, $inserts ) - $exp10_0;
    printf "Insert  %10d: Predicted  %6d Actual: %6d Delta: %+.3f%%\n"
+,
        $inserts, $exp3, $collisions, ( $collisions - $exp3 ) / $exp3 
+* 100;
}

__END__
C:\test>rbtcs-form-verify
Insert   779967210: Predicted       1 Actual:      1 Delta: -18.051%
Insert   782382025: Predicted       1 Actual:      2 Delta: +58.813%
Insert   830840395: Predicted       2 Actual:      3 Delta: +29.292%
Insert   882115420: Predicted       4 Actual:      4 Delta: -5.961%
Insert   883031614: Predicted       4 Actual:      5 Delta: +16.325%
Insert   897571477: Predicted       5 Actual:      6 Delta: +18.390%
Insert   923155269: Predicted       6 Actual:      7 Delta: +4.086%
Insert   948108745: Predicted       8 Actual:      8 Delta: -8.996%
Insert   954455113: Predicted       9 Actual:      9 Delta: -4.244%
Insert   967783959: Predicted      10 Actual:     10 Delta: -7.404%
Insert   988381482: Predicted      13 Actual:     11 Delta: -17.487%
Insert   992691311: Predicted      13 Actual:     12 Delta: -13.814%
Insert   995935158: Predicted      14 Actual:     13 Delta: -9.624%
Insert  1011301141: Predicted      16 Actual:     14 Delta: -16.457%
Insert  1013742872: Predicted      17 Actual:     15 Delta: -12.616%
Insert  1022258193: Predicted      18 Actual:     16 Delta: -14.242%
Insert  1031874023: Predicted      20 Actual:     17 Delta: -16.989%
Insert  1034026887: Predicted      20 Actual:     18 Delta: -13.909%
Insert  1036254774: Predicted      21 Actual:     19 Delta: -11.051%
Insert  1037064360: Predicted      21 Actual:     20 Delta: -7.093%
Insert  1037193945: Predicted      21 Actual:     21 Delta: -2.569%
Insert  1037309710: Predicted      21 Actual:     22 Delta: +1.957%
...( ~ 52000 ellided }
Insert  2375752842: Predicted   52209 Actual:  52277 Delta: +0.130%
Insert  2375755671: Predicted   52209 Actual:  52278 Delta: +0.131%
Insert  2375756509: Predicted   52209 Actual:  52279 Delta: +0.133%
Insert  2375760656: Predicted   52210 Actual:  52280 Delta: +0.133%
Insert  2375763928: Predicted   52211 Actual:  52281 Delta: +0.134%
Insert  2375785238: Predicted   52215 Actual:  52282 Delta: +0.128%
Insert  2375788721: Predicted   52215 Actual:  52283 Delta: +0.128%
Insert  2375789878: Predicted   52216 Actual:  52284 Delta: +0.130%
Insert  2375790896: Predicted   52216 Actual:  52285 Delta: +0.131%
Insert  2375798283: Predicted   52217 Actual:  52286 Delta: +0.131%

    [download]

    And that still includes the possibility -- though I believe it to be remote -- that there are 1 or more actual dups in there.

    My only regret now is that I wish I'd allowed the run to go to the 3/4 point instead of stopping it half way. The number of false positive would still have been easily manageable for the second pass:

    C:\test>rbtcs-bloom-probb 32 10 30e8
N=4294967296, V=10, X=30e8 integral(30e8)=12580199467.653, integral(0)
+=12579822861.8159
Expected collisions: 376605.837186813

    [download]

    That is an order of magnitude less that my own crude attempt was predicting, hence why I stopped the run.

    The only way I can repay you is to assure you that I will do my very best to try and understand the formula -- which I do not currently.

    And of course, give you credit, when they world comes knocking at my door for my eminently patentable -- at least if you take Apple as your guide -- algorithm :)

    With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'
    Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
    "Science is about questioning the status quo. Questioning authority".
    In the absence of evidence, opinion is indistinguishable from prejudice.

    The start of some sanity?
[reply]
[d/l]
[select]
      I will do my very best to try and understand the formula

      Perhaps this will help some.

      Below is some fairly simple code that does the precise calculations of the odds for a collision in a single hash (and compares those calculations with the formula roboticus proposed). I came up with a simpler implementation than I expected to. I didn't even try to implement this at first (only a little because I expected it to be more cumbersome, but) mostly because I knew it would be impractical for computing odds for such a large number of insertions. It consumes O($inserts) for memory and O($inserts**2) for CPU. 

      $|= 1;
my( $b )= ( @ARGV, 2**32 ); # Total number of bits in the hash.
my $i= 1;                   # Number of insertions done so far.
my @odds = 1;               # $odds[$s] == Odds of their being $s+1 bi
+ts set
while( 1 ) {                # Just hit Ctrl-C when you've seen enough
    my $exp= $b*( 1 - exp(-$i/$b) );
    my $avg = 0;
    $avg += $_ for map $odds[$_]*($_+1), 0..$#odds;
    my $err = sprintf "%.6f%%", 100*($exp-$avg)/$avg;
    print "$i inserts, $err: avg=$avg exp=$exp bits set\n"
        if  $i =~ /^\d0*$/;
    # Update @odds to in preparation for the next value of $i:
    for my $s (  reverse 0..$#odds  ) {
        $odds[$s+1] += $odds[$s]*($b-$s-1)/$b;
        $odds[$s] *= ($s+1)/$b;
    }
    $i++;
}

      [download]

      $i tracks the number of insertions done so far. $odds[$s] represents the odds of there being $s+1 bits set in the hash (after $i insertions). $avg is an average of these values of $s (1..@odds) weighted by the odds. But, more importantly, it is also the odds of getting a single-hash collision when inserting (after $i insertions) except multiplied by $bits ($b). I multiple it and 1-exp(-$i/$b) by $b to normalize to the expected number of set bits instead of the odds of a collision because humans have a much easier time identifying a number that is "close to 14" than a number that is "close to 14/2**32".

      $odds[-1] turns out to exactly match (successively) the values from birthday problem. For low numbers of $inserts, this swamps the calculation of $avg (the other terms just don't add up to a significant addition), which is part of why I was computing it for some values in my first reply. (Since you asked about that privately.)

      I have yet to refresh my memory of the exact power series expansion of exp($x), so what follows is actually half guesses, but I'm pretty confident of them based on vague memory and observed behavior.

      For large $bits, 1-exp(-$inserts/$bits) ends up being close to 1/$bits because 1-exp(-$inserts/$bits) expands (well, "can be expanded") to a power series where 1/$bits is the first term and the next term depends on 1/$bits**2 which is so much smaller that it doesn't matter much (and nor do any of the subsequent terms, even when added together).

      On the other hand, for large values of $inserts, 1-exp(-$inserts/$bits) is close to $avg because the formula for $avg matches the first $inserts terms of the power series expansion.

      I hope the simple code makes it easy for you to see how these calculations match the odds I described above. But don't hesitate to ask questions if the correspondence doesn't seem clear to you. Running the code shows how my calculations match roboticus' formula. Looking up (one of) the power series expansions for computing exp($x) should match the values being computed for @odds, though there might be some manipulation required to make the match apparent (based on previous times I've done such work decades ago).

      - tye        

[reply]
[d/l]
[select]
        Perhaps this will help some.

        I seriously hope this will not offend you, but suspect it will.

        Simply put, your post does not help me at all.

        I am a programmer, not a mathematician, but given a formula, in a form I can understand(*), I am perfectly capable of implementing that formula in code. And perfectly capable of coding a few loops and print statements in order to investigate its properties.

        What I have a problem with -- as evidently you do too -- is deriving those formula.

        Like you (according to your own words above; there is nothing accusatory here), my knowledge of calculus is confined to the coursework I did at college some {mumble mumble} decades ago. Whilst I retain an understanding of the principles of integeration; and recall some of its uses, the details are shrouded in a cloud of disuse.

        Use it or lose it, is a very current, and very applicable aphorism.

        The direction my career has taken me means that I've had no more than a couple of occasions when calculus would have been useful. And on both those occasions, I succeeded in finding "a man that can", who could provide me with an understandable formula, and thus, I achieved my goal without having to relive the history of mathematics.

        (*) A big part of the problem is that mathematicians not only have a nomenclature -- which is necessary -- the also have 'historical conventions' -- which are not; and the latter are the absolute bane of the lay-person's life in trying to understand the mathematician's output.

        There you are, happily following along when reach a text that goes something like this:

        We may think intuatively of the Riemann sum: Ʃba f(x) dx

        as the infinite sum: f(x0)dx + f(x1)dx + ... + f(xH - 1)dx + f(xH)(b - xH)

        Where did H come from? Where did a disappear to? Is H (by convention) == to b - a?

        For the answer to this and other questions, tune in next week ..... to the last 400 (or sometimes 4000) years of the history of math

        With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'
        Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
        "Science is about questioning the status quo. Questioning authority".
        In the absence of evidence, opinion is indistinguishable from prejudice.

        The start of some sanity?
[reply]
          A reply falls below the community's threshold of quality. You may see it by logging in.
Re^3: [OT] The statistics of hashing.
by BrowserUk (Patriarch) on Apr 03, 2012 at 12:39 UTC

    With the 'cancelling out' you've performed on the constants in your ex_*() subs, I couldn't see the pattern by which they were derived.

    Undoing that, I now see that they come (directly or not) from Pascal's Triangle.

    I guess at some point I should review some teaching materials to understand why those constants are used here, but for now it is enough to know that I can now write a generic ex_*() function (generator). Thanks.

    With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'
    Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
    "Science is about questioning the status quo. Questioning authority".
    In the absence of evidence, opinion is indistinguishable from prejudice.

    The start of some sanity?
[reply]
[d/l]

      BrowserUk:

      In the iterative solution, we're accumulating f(x)=(1-e^(-x/N))^h over the range of x=0 .. NumSamples. That's a rough form of computing the definite integral of the expression. Integrating over three variables (x, N, h) would be a pain, so I treated N and h as constants.

      So first, we multiply out our f(x) expression to remove the exponent. So using h as 2, we get:

      
f(x) = 1 - 2e^(-x/N) + e^(-2x/N)

integ(f(x)) = integ( 1 - 2e^(-x/N) + e^(-2x/N) )
            = integ( 1 ) - 2*(integ(e^(-x/N))) + integ(e^(-2x/N))

since:
integ(1) = x + C
integ(e^(bx)) = (1/b)e^(bx) + C

integ(f(x)) = [ x+C ] + [ (-2N)e^(-x/N) + C ] + [ (-N/2)e^(-2x/N) + C 
+]

      [download]

      Computing a definite integral over a range is simply integ(f(x)) evaluated at the upper limit less the value evaluated at the lower limit. This causes the C terms to cancel.

      Pascal's triangle comes out because we've got (a+b)^n, and when we multiply it out, we get the binomial expansion which is where the coefficients come into play.

      One point I should mention: You don't have to use 0 as the lower bound. If you wanted the number of collisions you'd experience from sample A to sample B, just evaluate integ(f(B))-integ(f(A)). By using A=0 we compute the number of collisions for the entire run.

      ...roboticus

      When your only tool is a hammer, all problems look like your thumb.

[reply]
[d/l]

        Thanks. As may be (becoming) obvious, much of that is over my head for now :)

        However, Now I know how to derive the constants, I have this which I can substitute for your ex_4() & ex_10() by supplying the power as the first argument. Its output matches those two exactly for all the argument combinations I've tried: 

        sub PTn {
    my @row;
    for( 1 .. shift ) {
        push @row, 1;
        $row [$_] += $row [$_ - 1] for reverse 1 .. @row - 2;
    }
    return @row;
}

sub expN {
    my( $P, $N, $X ) = @_;
    my $xDIVn = $X / $N;
    my @coefs = PTn( $P+1 );
    my $rv = 0;
    my $toggle = -1;
    for my $p ( reverse 1 .. $P ) {
        $rv += $toggle * $coefs[ $p ] * $N / $p * exp( -$p * $xDIVn );
        $toggle *= -1;
    }
    return $rv + $X;
}

        [download]

        That allowed me to investigate the affect of using fewer or more hashes without having to hand code a new function for each.

        And the results are quite interesting. This shows that each new (pair?) of hashes added does increase the discrimination substantially, though the gains obviously fall off fairly rapidly. But the really interesting part are the numbers for odd numbers of hashes:

        C:\test>rbtcs-form-verify -H=10 -B=32
Using N x 2**32 vectors:
        12 :         24          0         24          0         24   
+       0         24          0         24          0
        16 :         32          0         31          0         31   
+       0         31          0         31          0
        24 :         48          0         48          0         48   
+       0         48          0         48          0
        32 :         64          0         64          0         64   
+       0         64          0         64          0
        48 :         95          0         95          0         95   
+       0         95          0         96          0
        64 :        127          0        127          0        127   
+       0        128          0        128          0
        96 :        191          0        191          0        192   
+       0        192          0        192          0
       128 :        255          0        256          0        256   
+       0        256          0        256          0
       192 :        383          0        383          0        383   
+       0        384          0        384          0
       256 :        511          0        512          0        512   
+       0        512          0        512          0
       384 :        767          0        768          0        768   
+       0        768          0        768          0
       512 :       1023          0       1024          0       1024   
+       0       1024          0       1024          0
       768 :       1535          0       1536          0       1536   
+       0       1536          0       1536          0
      1024 :       2047          0       2048          0       2048   
+       0       2048          0       2048          0
      1536 :       3071          0       3072          0       3072   
+       0       3072          0       3072          0
      2048 :       4095          0       4096          0       4096   
+       0       4096          0       4096          0
      3072 :       6143          0       6144          0       6144   
+       0       6144          0       6144          0
      4096 :       8191          0       8192          0       8192   
+       0       8192          0       8192          0
      6144 :      12287          0      12288          0      12288   
+       0      12288          0      12288          0
      8192 :      16383          0      16384          0      16383   
+       0      16383          0      16384          0
     12288 :      24575          0      24576          0      24576   
+       0      24576          0      24576          0
     16384 :      32767          0      32767          0      32767   
+       0      32767          0      32768          0
     24576 :      49151          0      49151          0      49152   
+       0      49152          0      49152          0
     32768 :      65535          0      65536          0      65536   
+       0      65536          0      65536          0
     49152 :      98303          0      98304          0      98304   
+       0      98303          0      98303          0
     65536 :     131071          0     131072          0     131071   
+       0     131072          0     131072          0
     98304 :     196606          0     196608          0     196607   
+       0     196608          0     196608          0
    131072 :     262142          0     262144          0     262144   
+       0     262143          0     262144          0
    196608 :     393211          0     393216          0     393216   
+       0     393215          0     393216          0
    262144 :     524280          0     524288          0     524287   
+       0     524287          0     524288          0
    393216 :     786414          0     786431          0     786431   
+       0     786432          0     786432          0
    524288 :    1048544          0    1048575          0    1048576   
+       0    1048576          0    1048576          0
    786432 :    1572792          0    1572863          0    1572864   
+       0    1572864          0    1572864          0
   1048576 :    2097024          0    2097151          0    2097151   
+       0    2097152          0    2097152          0
   1572864 :    3145440          0    3145727          0    3145728   
+       0    3145727          0    3145728          0
   2097152 :    4193792          0    4194303          0    4194304   
+       0    4194304          0    4194304          0
   3145728 :    6290304          0    6291455          0    6291455   
+       0    6291455          0    6291455          0
   4194304 :    8386560          1    8388607          0    8388608   
+       0    8388608          0    8388608          0
   6291456 :   12578306          4   12582911          0   12582912   
+       0   12582912          0   12582912          0
   8388608 :   16769029         10   16777215          0   16777216   
+       0   16777215          0   16777216          0
  12582912 :   25147409         35   25165823          0   25165823   
+       0   25165824          0   25165824          0
  16777216 :   33521706         85   33554431          0   33554431   
+       0   33554431          0   33554432          0
  25165824 :   50258063        286   50331646          0   50331647   
+       0   50331648          0   50331648          0
  33554432 :   66978132        678   67108860          0   67108863   
+       0   67108864          0   67108864          0
  50331648 :  100369532       2283  100663276          0  100663295   
+       0  100663296          0  100663296          0
  67108864 :  133696160       5397  134217665          0  134217727   
+       0  134217728          0  134217728          0
 100663296 :  200156106      18111  201326276          5  201326591   
+       0  201326591          0  201326592          0
 134217728 :  266359979      42681  268434469         24  268435455   
+       0  268435455          0  268435455          0
 201326592 :  398007464     142383  402648282        179  402653177   
+       0  402653183          0  402653184          0
 268435456 :  528654369     333608  536855705        738  536870874   
+       1  536870911          0  536870911          0
 402653184 :  787008255    1100214  805232175       5328  805305969   
+      30  805306365          0  805306367          0
 536870912 : 1041542872    2548692 1073515796      21337 1073739728   
+     211 1073741802          2 1073741823          0
 805306368 : 1539620713    8218836 1609548824     146448 1610591774   
+    3082 1610612273         70 1610612725          1
1073741824 : 2023785226   18623993 2144354575     558473 2147380065   
+   19731 2147479815        755 2147483497         30
1610612736 : 2953695056   57532294 3207472286    3485537 3220308576   
+  247526 3221157361      19018 3221220099       1532
2147483648 : 3837421596  125076314 4257101987   12129165 4290939237   
+ 1371778 4294491373     167491 4294907677      21417
3221225472 : 5487393872  357203949 6295545197   63685472 6413893311   
+13112112 6436324179    2901774 6441061714     670967
4294967296 : 7009904423  721946381 8229596479  188903097 8487725572   
+56541428 8558136669   18112153 8579512265    6047518

        [download]

        I suspect it is a coding error on my behalf, but I guess it could be a quirk of the numbers?

        Here's a set using 1 .. 16 2**16 bit hashes:

        C:\test>rbtcs-form-verify -H=16 -B=16
Using N x 2**16 vectors:
        12 :     23      0     23      0     24      0     24      0  
+   24      0     23      0     23      0     23      0
        16 :     31      0     31      0     31      0     32      0  
+   32      0     32      0     32      0     32      0
        24 :     47      0     47      0     48      0     47      0  
+   48      0     47      0     47      0     47      0
        32 :     63      0     63      0     64      0     64      0  
+   64      0     64      0     64      0     64      0
        48 :     95      0     95      0     95      0     95      0  
+   95      0     96      0     96      0     96      0
        64 :    127      0    127      0    128      0    128      0  
+  128      0    127      0    128      0    127      0
        96 :    191      0    191      0    192      0    192      0  
+  192      0    192      0    191      0    191      0
       128 :    255      0    255      0    256      0    255      0  
+  256      0    256      0    256      0    256      0
       192 :    383      0    383      0    383      0    384      0  
+  384      0    384      0    384      0    384      0
       256 :    511      0    511      0    511      0    511      0  
+  512      0    511      0    512      0    512      0
       384 :    766      0    767      0    767      0    768      0  
+  768      0    767      0    768      0    767      0
       512 :   1022      0   1023      0   1023      0   1024      0  
+ 1024      0   1024      0   1024      0   1024      0
       768 :   1531      0   1535      0   1535      0   1536      0  
+ 1536      0   1536      0   1536      0   1536      0
      1024 :   2040      0   2047      0   2047      0   2048      0  
+ 2048      0   2048      0   2048      0   2048      0
      1536 :   3054      0   3071      0   3071      0   3071      0  
+ 3072      0   3072      0   3072      0   3072      0
      2048 :   4064      0   4095      0   4095      0   4095      0  
+ 4095      0   4096      0   4096      0   4096      0
      3072 :   6073      2   6143      0   6143      0   6143      0  
+ 6144      0   6144      0   6144      0   6144      0
      4096 :   8066      5   8191      0   8191      0   8191      0  
+ 8191      0   8191      0   8191      0   8192      0
      6144 :  12008     16  12286      0  12287      0  12287      0  
+12287      0  12287      0  12288      0  12287      0
      8192 :  15892     38  16380      0  16383      0  16383      0  
+16383      0  16383      0  16383      0  16383      0
     12288 :  23492    125  24559      2  24575      0  24575      0  
+24575      0  24575      0  24575      0  24576      0
     16384 :  30880    284  32720      8  32766      0  32767      0  
+32767      0  32767      0  32767      0  32767      0
     24576 :  45069    877  48942     53  49138      3  49150      0  
+49151      0  49151      0  49151      0  49151      0
     32768 :  58554   1908  64958    185  65474     20  65528      2  
+65535      0  65535      0  65535      0  65535      0
     49152 :  83730   5450  96062    971  97868    200  98210     44  
+98282     10  98299      2  98302      0  98303      0
     65536 : 106962  11016 125573   2882 129512    862 130586    276 1
+30912     92 131018     31 131053     11 131065      3

        [download]

        Sorry for the wrapping.

        With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'
        Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
        "Science is about questioning the status quo. Questioning authority".
        In the absence of evidence, opinion is indistinguishable from prejudice.

        The start of some sanity?
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