Voder-Vocoder

“Real” is not an acceptible synonym for “80-bit floating-point number.” Real numbers aren't even countable, much less in a bijection to a set of size 2^80.

Even though I’ve studied this algorithm a couple of times, I’ve never had to implement it before. So I assigned it to myself.

/** should have a time-complecity of O(N×log(N))
    and a space-compelcity of O(N) **/
void mergesort(int N, int array[]) {
  int k; // k is the block size.
  int x; //x is which block we are at.
  int i,j; //indices in old[]
  int p; // index in new[]
  int ilimit,jlimit; //end of blocks.
  int *hold = malloc(sizeof(hold) * N);
  if (hold == NULL) {
    fprintf(stderr,"malloc failed\n");
    exit(2);
  }
  int *new = hold;
  int *old = array;
  int *tmp;
  for (k = 1; k < N; k *= 2) {
    p = 0;
    for (x = 0; x < N; x += (2*k)) {
      i = x;
      ilimit = i + k;
      j = ilimit;
      if (ilimit >= N) {
	while (i < N)
	  new[p++] = old[i++];
 	break; //out of for-loop
      }
      jlimit = j + k;
      if (jlimit >= N)
	jlimit = N;
      while (1) {
	if (old[i] < old[j]) {
	  new[p++] = old[i++];
	  if (i == ilimit) {
	    while (j < jlimit)
	      new[p++] = old[j++];
	    break; //out of while-loop
	  }
	} else {
	  new[p++] = old[j++];
	  if (j == jlimit) {
	    while (i < ilimit)
	      new[p++] = old[i++];
	    break; //out of while-loop
	  }
	}
      } // End while loop.
    } // End inner for loop.
    tmp = old; old = new; new = tmp;
  }// End outer for loop.
  if (old != array)
    for (i = 0; i < N; i++)
      array[i] = old[i];
  free(hold);
  return;
}

Next step is to translate to Java and use .compareTo() with arrays of references:

  public static void mergeSort(Comparable array[]) {
    int N = array.length;
    int k; // k is the block size.
    int x; // x is which block we are at.
    int i,j; //indices in old[]
    int p; // index in new[]
    int ilimit,jlimit; // end of blocks.
    Comparable hold [] = new Comparable [N];
    Comparable neww [] = hold;
    Comparable old [] = array;
    Comparable tmp [];
    for (k = 1; k < N; k *= 2) {
      p = 0;
      for (x = 0; x < N; x += (2*k)) {
        i = x;
        ilimit = i + k;
        j = ilimit;
        if (ilimit >= N) {
          while (i < N)
            neww[p++] = old[i++];
          break; //out of for-loop
        }
        jlimit = j + k;
        if (jlimit >= N)
          jlimit = N;
        while (true) {
          if (old[i].compareTo(old[j]) < 0) {
            neww[p++] = old[i++];
            if (i == ilimit) {
              while (j < jlimit)
                neww[p++] = old[j++];
              break; //out of while-loop
            }
          } else {
            neww[p++] = old[j++];
            if (j == jlimit) {
              while (i < ilimit)
                neww[p++] = old[i++];
              break; //out of while-loop
            }
          }
        } // End while loop.
      } // End inner for loop.
      tmp = old; old = neww; neww = tmp;
    }// End outer for loop.
    if (old != array)
      for (i = 0; i < N; i++)
        array[i] = old[i];
  }

About ten years ago, I wrote a C++ program to print out all the prime numbers less than a given number using trial division. I recently went back and looked at the program and realized how little I knew at the time. Even though my first CS class covered object-oriented programming in C++, we never really talked the about simply using the new keyword on arrays to make use of dynamic arrays. The topic was covered in my second CS class, which I took three years later.

int *array;
int array_size = 128;
array = new int[array_size];

/* do somthing to fill the array */

int *temparray = new int[(array_size * 2)];
for (int i = 0; i < array_size; i++)
    temparray[i] = array[i];
array_size = array_size * 2;
delete [] array;
array = temparray;

In the last few years, I have realized that for the simplest progrmas, C is often more efficient and straightforward than C++. In C, the code looks exactly the same, except that new is replaced by malloc() and delete is replaced by free().

int *array;
int *temparray;
int array_size = 128;
int i;
array = malloc(array_size * sizeof(*array));

/* do somthing to fill the array */

temparray = malloc(array_size * 2 * sizeof(*temparray));
for (i = 0; i < array_size; i++)
    temparray[i] = array[i];
array_size = (array_size * 2);
free(array);
array = temparray;

This example of why the right algorithm matters comes directly from my textbook. Here’s the C implementation:

Bad:

#include <stdio.h>
#include <stdlib.h>
long int fib(long int n) {
  if (n==0)
    return 1;
  if (n==1)
    return 1;
  return fib(n-1) + fib(n-2);
}
int main(int argc, char *argv[]) {
  if (argc <= 1) {
    fprintf(stderr, "argument?\n\n");
    exit(1);
  }
  long int n = atol(argv[1]);
  printf("f(%li) = %li\n",n,fib(n));
  return 0;
}

Good:

#include <stdio.h>
#include <stdlib.h>
long int fib(long int n) {
  long a=1, b=1, c;
  int i;
  for (i = 1;i < n; i++){
    c = a + b; a = b; b = c;
  }
  return b;
}
int main(int argc, char *argv[]) {
  if (argc <= 1) {
    fprintf(stderr, "argument?\n\n");
    exit(1);
  }
  long int n = atol(argv[1]);
  printf("f(%li) = %li\n",n,fib(n));
  return 0;
}

Output:

$ time ./fib2 38 ; time ./fib1 38
f(38) = 63245986

real	0m0.002s
user	0m0.000s
sys	0m0.004s
f(38) = 63245986

real	0m1.492s
user	0m1.428s
sys	0m0.004s

And you can show how nicely the good algorithm scales up by pulling out a bigint library, like Java’s BigInteger:

public class fib3 {
  public static String fib(int n) {
    java.math.BigInteger a,b,c;
    int i;
    a = b = java.math.BigInteger.ONE;
    for (i = 1;i < n; i++) {
      c = a.add(b); a = b; b = c;
    }
    return b.toString();
  }
  public static void main(String[] args) {
    if (args.length < 1) {
      System.err.println("argument?");
      System.exit(1);
    }
    int n = Integer.parseInt(args[0]);
    System.out.print("f(" +
      Integer.toString(n) + ") = ");
    System.out.println(fib(n));
  }
}