Optimization by Memoization ( not memorization )

To iterate is Human, to Recurse, to Recurse, Divine - unknown

Solving COINS at spoj was a great lesson in recursion. Beautiful may the recursive code seem, but in some cases it is slow as a snail in worst case scenario. Consider finding nth Fibonacci by recursion as an example! And in some cases as in finding p^q, recursion is truly divine god's gift ;P.
Same is the case with COINS, its pretty easy to figure out the function that gets the job done... but give it something like n = 1000000000, poof! it vanishes into a deep slumber!
So, how can we make it work faster... ?
Have you ever noticed that, kids are asked to remember multiplication tables from 2 to 10 when in primary mathematics classes! Ever wondered why to do this mugging up at such a tender age. The answer is in... how we multiply two big numbers (abcd * pqrs). Here every digit i & j is in range of 0 through 9. So, if you've memorized multiplication table for 2 to 9, it will be very easy to get through (abcd * pqrs).
Thats exactly what we will do here. Basically as the problem gets solved for bigger numbers, loads of steps trickle down to maxm() function for small numbers, just like end of roots of a tree. So, here again we will pre-calculate and store maxm(n) in an array for 1<n<9400. Why 9400? Have a look at http://www.spoj.pl/problems/COINS/, you can't write more that 50KB of code! So this weekend I decide to see the real consequence of memoization. Here's the allegro code I wrote to plot time against my COIN's solution. Memoization 9400 values works for the spoj judge. And it seems even memorizing 450 values works amazingly for my Athlon 64 :P

#include <iostream>
#include <allegro.h>
#include <time.h>

using namespace std;
unsigned int 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double lim = 0;

unsigned int maxm(unsigned int n){
// 9400 - size of array
if (n<lim)
 return a[n];

unsigned int p;
p = n/2 + n/3 + n/4;

if (p>n)
 //sort of binary recursion which you should figure out to solve COINS
else
 return n;
}

int main (){
unsigned int n=10,col2,col;
double x=0,px=1,py=760,y;
clock_t start ,stop;

allegro_init();
set_color_depth(32);
set_gfx_mode (GFX_AUTODETECT_WINDOWED,1024,768,0,0);

clear_to_color (screen,makecol(0,0,0));

while (lim<450){
 col = makecol ((lim/450)*250+5,  (lim/450)*255, 100);
 col2=makecol (((lim/450)*250+5)/3,  (lim/450)*255/3, 100/3);
 while (n<310){
  n+=1;
  start = clock();
  maxm(n*20000);
  stop = clock();
  //  cout << stop <<" "<<start << endl;
  y=760-((stop-start)*5);
  rectfill (screen,px,760,px+3,y,col2);
  line (screen,px,py,px+3,y,col);
 
  px+=3;
  py=y;
 }
 lim=lim+5;
 x=0; px=1; py=760;
 n=10;
 cout << lim<<endl;
}
getchar();
}
END_OF_MAIN();

And here's the plot! amazing visible differences...

checkout the drastic improvement from lim=0 to lim=450!

That's all for the Friday research folks!