**Google Code Jam, Round 2**

I got pwned. Big time.

That is all.

fejj: the standard way to do
that is to
replace the division with a multiplication by the
reciprocal, multiplied by some power of 2. Then divide the
result by that power of 2, which can be done with a right
shift. In the case of division by 7, you'd end up with the
increasingly more accurate expressions:

`
`

`q = ((a<<3) + a) >> 6
`

`q = ((a<<6) + (a<<3) + a) >> 9
`

`q = ((a<<9) + (a<<6) + (a<<3) + a)
>> 12
`

`q = ((a<<12) + (a<<9) + (a<<6) +
(a<<3) + a) >> 15
`

`...
`

and so on (in binary, 1/7 is .001001001...). Optionally, the result can be fixed with something like

`
r = a - q*7; while (r >= 7) { q++; r -= 7; }
`

The multiplication by 7 can be replaced with shifts and adds, of course.

My performance was dismal. I did the easy problem, then moved straight to the hard one. Then I threw away 20 minutes because I jumped straight to the code before understanding completely the problem statement and then had to start all over again, wasted another 20 minutes tracking down a stupid off-by-one bug, panicked for about 5 minutes, and finally ended up submitting a /wrong/ solution. Guess I need to work on the think-crush-under-pressure thing.

What makes it so frustrating is that the problem was not really that hard, and I was able to concoct a correct solution (here) without major difficulties afterwards, when the time pressure was gone.

Gah.

The qualification round was easier than I expected. It's probably just meant to filter out the curious. I was able to do both problems in about 30 minutes, and qualified to the next phase.

**Tupper's Self-Referential
Formula**

... seems to be making the rounds on the internet. Two different people sent me the link. It's cute, but there's nothing magical about it, really - perhaps if some people look at it as a simple Perl script they'll realize that it's just a way to rasterize the big number, and the bitmap could be anything.

So, tomorrow is Google Code Jam Latin America. I'll give it a try, but I expect to get my arse kicked. Although I've been practicing a bit at SPOJ and acm.uva.es, I really suck at thinking under pressure. Ah well, let's see how it goes.

OpenSpecies:

1.

#define is_odd(x) (((x)&1) != 0) #define is_even(x) (((x)&1) == 0)

2. if you're out to disprove the Collatz conjecture, use GMP or something.

mchirico, fejj: it's the
Look and Say Sequence. See also: Conway's Cosmological Theorem.

It's all demonstrated in this wonderful IOCCC winner (hint file).

[elided]

**Free software**

Haven't touched my personal projects in a while, partly because of lack of time, partly because of the SPOJ addiction, partly because of general pissedoffness, but anyway I packed together the minor changes I did on vulcan in the past couple of months (lots of bugfixes) and called it 0.3.1.

Speaking of SPOJ, I recently ordered Sedgewick's algorithms book with the specific purpose of improving my SPOJ score. Whee.

StevenRainwater: you
rule. Thank you!

Where can we find the updated mod_virgule code?

**SPOJ**

Phew. This took a lot of work. I exchanged the first place with the guy that's currently second several times until finally changing algorithms and getting to the first place with a decent margin.

Changelog, *ex posto facto*:

- 0.1: trial division followed by Fermat primality test.
- 0.2: Rabin-Miller.
- 0.3: accelerated a bit multiplication followed by
division operations in the modular exponentiation code (from
now on referred to as
`expmod`) with x86-specific inline assembly (it has 32-bit -> 64-bit multiplication and a 64-bit -> 32-bit division and remainder instructions). - 0.4: found a "cheat" that allowed me to go back to Fermat test, at a big performance win.
- 0.5: lots of micro-optimizations. Used a sliding bit
window in
`expmod`. Got to the first place for the first time. - 0.6: realized that the only way to get faster would be
to eliminate the division operation in
`expmod`. Replaced the division with multiplication by the inverse (some call this "Barrett reduction"). Was a teeny weeny bit faster on my machine, but slower at spoj. The reduction involved a 64-bit multiplication (three multiplication instructions on 32-bit x86). :/ - 0.7: Montgomery exponentiation!

For future reference, here's a neat trick to find the multiplicative inverse of a number modulo 2^k, lightning fast (this was needed in the pre-calculation phase of Montgomery). First, two facts:

- a^-1 = y*(2 - a*y) (mod 2^k), where y = a^-1 (mod 2^{k-1}). We can use this fact alone to derive a moderately fast algorithm to get a^-1 mod 2^k.
- if q = 2^{k/2}*a + b (with b odd), then q*(2*b - q) = b^2 (mod 2^k), so q^-1 = (2*b - q)*((b^2)^-1) (mod 2^k).

To get q^-1 (mod 2^k), with k even, first get (b^2)^-1 from a table (for e.g. k=32 it will be a small table) and then use fact #2. With k odd, do that for k-1 and then use fact #1.

It would be really sad to see it go away. wingo put it nicely.

While it's still here, I'll keep on rambling...

**SPOJ**

I fell a bit behind on work and personal projects due to a recently-acquired addiction to the oddly-named website Sphere Online Judge. It's full of programming problems, mostly algorithmic. You post a solution and it gets compiled and tested automatically. Each problem has a ranking with the best solutions. It accepts everything from C to Haskell to Common Lisp to Intercal (really). This is my current status.

Hopefully I have the addiction firmly under control now. On the plus side, working on the problems is giving me the motivation I needed to finally do something about the gaping lacunae in my knowledge of algorithms.

**New HTML Parser**: The long-awaited libxml2 based HTML parser
code is live. It needs further work but already handles most
markup better than the original parser.

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If you're a C programmer with some spare time, take a look at the mod_virgule project page and help us with one of the tasks on the ToDo list!