Wednesday, 25 August 2010

MongoDB With C++ In Ubuntu

Install standard repository package.

$ sudo aptitude install mongodb

Create data directory. By default mongo uses /data/db.

$ sudo mkdir -p /data/db/
$ sudo chown `id -u` /data/db

Try to run the server. (If you decided to use another path for data run mongod with --dbpath option)

$ mongod

If it runs, the you are ok. But if it throws an error

mongo: error while loading shared libraries: cannot open shared object file: No such file or directory

Then we will have to find that missing library.

$ find /usr/lib/* -name

Now create a symbolic link

$ cd /usr/lib
$ sudo ln -s xulrunner-devel-

Now try to start the server again. It should be working. Start the client and run some queries:

$ mongo
> { a : 1 } )

Installing Libraries For C++

So the database is up and running, now we need some C++ libs to be able to connect to it.

$ sudo aptitude install libboost-all-dev libpcre++-dev

Now we should be able to run this code, to test the connection.

#include <iostream>
#include <mongo/client/dbclient.h>

using namespace std;

void run() {
  mongo::DBClientConnection c;

int main() {
  try {
    cout << "connected ok" << endl;
  } catch( mongo::DBException &e ) {
    cout << "caught " << e.what() << endl;
  return 0;

Build and run this code using:

$ g++ tutorial.cpp -lmongoclient -lboost_thread-mt -lboost_filesystem-mt -lboost_program_options-mt tutorial
$ ./tutorial
connected ok

Wednesday, 18 August 2010

Determine If Number Is Prime

The SICP book is still being read...

Now I am going to introduce some algorithms that allow us to say whether the number is prime or not. Again we will start from the simplest one.

Take numbers 1 by 1 starting from 2 to √n and check, if it divides n. The complexity of the algorithm is Θ(√n).

(define (prime? n)
  (define (divides? k)
    (= (remainder n k) 0))
  (define (find-divisor n test)
    (cond ((> (sqr test) n) n)
          ((divides? test) test)
          (else (find-divisor n (+ test 1)))))

  (= n (find-divisor n 2)))

Fermat Primality Test

There is a test for primality with Θ(log(n)) complexity. It is based on Fermat little theorem.

an ≡ a (mod n), where 1 ≤ a < n

Which means, that if n is prime, then for any integer a, an − a will be evenly divisible by n.

;this procedure calculates baseexp mod m without using big numbers
(define (expmod base exp m)
  (cond ((= exp 0) 1)
        ((even? exp) (remainder (sqr (expmod base (/ exp 2) m)) m))
        (else (remainder (* base (expmod base (- exp 1) m)) m))))

(define (fermat-test n)
  (define (try-it a)
    (= (expmod a n n) a))
  (try-it (+ 1 (random (- n 1)))))

Unfortunately there are some issues with this test - it is probabilistic. If the test fails, n is certainly composite. But if it passes, it does not guarantee that n is prime, although it shows high probability for n to be. So we can run as many tests as we need to make the mistake probability as low as possible.

(define (fermat-prime? n times)
  (cond ((= times 0) true)
        ((fermat-test n) (fermat-prime? n (- times 1)))
        (else false)))

Unfortunately that is not all. There are some numbers that pass the test (for every a!), without being prime. These are called Carmichael numbers. So to determine, if n is prime, we need to check if it passes the Fermat test and is not a Carmichael number, or use a similar Miller–Rabin primality test, that cannot be tricked.

Miller–Rabin Primality Test

It starts from an alternate form of Fermat's Little Theorem:

an-1 ≡ 1 (mod n), where 1 ≤ a < n

The difference is that whenever we perform the squaring step in expmod, we check to see if we have discovered a "nontrivial square root of 1 modulo p", that is, a number not equal to 1 or n - 1 whose square is equal to 1 mod n. It is possible to prove that if such a nontrivial square root of 1 exists, then n is not prime. It is also possible to prove that if n is an odd number that is not prime, then, for at least half the numbers a < n, computing an-1 in this way will reveal a nontrivial square root of 1 mod n.

(define (miller-rabin-prime? n times)
  (cond ((= times 0) true)
        ((miller-rabin-test n) (miller-rabin-prime? n (- times 1)))
        (else false)))

(define (miller-rabin-test n)
  (define (try-it a)
    (= (expmod a (- n 1) n) 1))
  (try-it (+ 1 (random (- n 1)))))

(define (expmod base exp m)
  (define (!= x y) (not (= x y)))

  (cond ((= exp 0) 1)
        ((and (!= base 1) 
              (!= base (- m 1)) 
              (= (remainder( sqr base) m) 1)) 0)
        ((even? exp) (remainder (sqr (expmod base (/ exp 2) m)) m))
        (else (remainder (* base (expmod base (- exp 1) m)) m))))