Answers for Chapter 3 - GNU libavl 2.0.3
Author: Ben Pfaff

Chapter 3 |

#### Section 3.1

1. The following program can be improved in many ways. However, we will implement a much better testing framework later, so this is fine for now.

    597. <seq-test.c 597> =
    <Program License 2>
    #include <stdio.h>

    #define MAX_INPUT 1024

    <Sequentially search an array of ints 17>

    int
    main (void)
    {
      int array[MAX_INPUT];
      int n, i;

      for (n = 0; n < MAX_INPUT; n++)
        if (scanf ("%d", &array[n]) != 1)
          break;

      for (i = 0; i < n; i++)
        {
          int result = seq_search (array, n, array[i]);
          if (result != i)
            printf ("seq_search() returned %d looking for %d  expected %d\n",
                    result, array[i], i);
        }

      return 0;
    }

#### Section 3.4

1. Some types don't have a largest possible value; e.g., arbitrary-length strings.

#### Section 3.5

1. Knuth's name for this procedure is “uniform binary search.” The code below is an almost-literal implementation of his Algorithm U. The fact that Knuth's arrays are 1-based, but C arrays are 0-based, accounts for most of the differences.

The code below uses for (;;) to assemble an “infinite” loop, a common C idiom.

    598. <Uniform binary search of ordered array 598> =
    /* Returns the offset within array[] of an element equal to key,
       or -1 if key is not in array[].
       array[] must be an array of n ints sorted in ascending order,
       with array[-1] modifiable. */
    int
    uniform_binary_search (int array[], int n, int key)
    {
      int i = (n + 1) / 2 - 1;
      int m = n / 2;

      array[-1] = INT_MIN;
      for (;;)
        {
          if (key < array[i])
            {
              if (m == 0)
                return -1;
              i -= (m + 1) / 2;
              m /= 2;
            }
          else if (key > array[i])
            {
              if (m == 0)
                return -1;
              i += (m + 1) / 2;
              m /= 2;
            }
          else
            return i >= 0 ? i : -1;
        }
    }

This code is included in 602.

See also: [Knuth 1998b], section 6.2.1, Algorithm U.

2a. This actually uses blp_bsearch(), implemented in part (b) below, in order to allow that function to be tested. You can replace the reference to blp_bsearch() by bsearch() without problem.

    599. <Binary search using bsearch() 599> =
    <blp's implementation of bsearch() 600>

    /* Compares the ints pointed to by pa and pb and returns positive
       if *pa > *pb, negative if *pa < *pb, or zero if *pa == *pb. */
    static int
    compare_ints (const void *pa, const void *pb)
    {
      const int *a = pa;
      const int *b = pb;

      if (*a > *b)
        return 1;
      else if (*a < *b)
        return -1;
      else
        return 0;
    }

    /* Returns the offset within array[] of an element equal to key,
       or -1 if key is not in array[].
       array[] must be an array of n ints sorted in ascending order. */
    static int
    binary_search_bsearch (int array[], int n, int key)
    {
      int *p = blp_bsearch (&key, array, n, sizeof *array, compare_ints);
      return p != NULL ? p - array : -1;
    }

This code is included in 602.

2b. This function is named using the author of this book's initials. Note that the implementation below assumes that count, a size_t, won't exceed the range of an int. Some systems provide a type called ssize_t for this purpose, but we won't assume that here. (long is perhaps a better choice than int.)

    600. <blp's implementation of bsearch() 600> =
    /* Plug-compatible with standard C library bsearch(). */
    static void *
    blp_bsearch (const void *key, const void *array, size_t count,
                 size_t size, int (*compare) (const void *, const void *))
    {
      int min = 0;
      int max = count;

      while (max >= min)
        {
          int i = (min + max) / 2;
          void *item = ((char *) array) + size * i;
          int cmp = compare (key, item);

          if (cmp < 0)
            max = i - 1;
          else if (cmp > 0)
            min = i + 1;
          else
            return item;
        }

      return NULL;
    }

This code is included in 599.

3. Here's an outline of the entire program:

    601. <srch-test.c 601> =
    <Program License 2>
    #include <limits.h>
    #include <stdio.h>
    #include <stdlib.h>
    #include <time.h>

    <Search functions 602>
    <Array of search functions 603>

    <Timer functions 606>
    <Search test functions 608>
    <Search test main program 611>

We need to include all the search functions we're going to use:

    602. <Search functions 602> =
    <Sequentially search an array of ints 17>
    <Sequentially search an array of ints using a sentinel 18>
    <Sequentially search a sorted array of ints 19>
    <Sequentially search a sorted array of ints using a sentinel 20>
    <Sequentially search a sorted array of ints using a sentinel (2) 21>
    <Binary search of ordered array 22>
    <Uniform binary search of ordered array 598>
    <Binary search using bsearch() 599>
    <Cheating search 605>

This code is included in 601.

We need to make a list of the search functions. We start by defining the array's element type:

    603. <Array of search functions 603> =
    /* Description of a search function. */
    struct search_func
      {
        const char *name;
        int (*search) (int array[], int n, int key);
      };

See also 604.

This code is included in 601.

Then we define the list as an array:

    604. <Array of search functions 603> +=
    /* Array of all the search functions we know. */
    struct search_func search_func_tab[] =
      {
        {"seq_search()", seq_search},
        {"seq_sentinel_search()", seq_sentinel_search},
        {"seq_sorted_search()", seq_sorted_search},
        {"seq_sorted_sentinel_search()", seq_sorted_sentinel_search},
        {"seq_sorted_sentinel_search_2()", seq_sorted_sentinel_search_2},
        {"binary_search()", binary_search},
        {"uniform_binary_search()", uniform_binary_search},
        {"binary_search_bsearch()", binary_search_bsearch},
        {"cheat_search()", cheat_search},
      };

    /* Number of search functions. */
    const size_t n_search_func = sizeof search_func_tab / sizeof *search_func_tab;

We've added previously unseen function cheat_search() to the array. This is a function that “cheats” on the search because it knows that we are only going to search in a array such that array[i] == i. The purpose of cheat_search() is to allow us to find out how much of the search time is overhead imposed by the framework and the function calls and how much is actual search time. Here's cheat_search():

    605. <Cheating search 605> =
    /* Cheating search function that knows that array[i] == i.
       n must be the array size and key the item to search for.
       array[] is not used.
       Returns the index in array[] where key is found,
       or -1 if key is not in array[]. */
    int
    cheat_search (int array[], int n, int key)
    {
      return key >= 0 && key < n ? key : -1;
    }

This code is included in 602.

We're going to need some functions for timing operations. First, a function to “start” a timer:

    606. <Timer functions 606> =
    /* “Starts” a timer by recording the current time in *t. */
    static void
    start_timer (clock_t *t)
    {
      clock_t now = clock ();
      while (now == clock ())
        /* Do nothing. */;
      *t = clock ();
    }

See also 607.

This code is included in 601.

Function start_timer() waits for the value returned by clock() to change before it records the value. On systems with a slow timer (such as PCs running MS-DOS, where the clock ticks only 18.2 times per second), this gives more stable timing results because it means that timing always starts near the beginning of a clock tick.

We also need a function to “stop” the timer and report the results:

    607. <Timer functions 606> +=
    /* Prints the elapsed time since start, set by start_timer(). */
    static void
    stop_timer (clock_t start)
    {
      clock_t end = clock ();

      printf ("%.2f seconds\n", ((double) (end - start)) / CLOCKS_PER_SEC);
    }

The value reported by clock() can “wrap around” to zero from a large value. stop_timer() does not allow for this possibility.

We will write three tests for the search functions. The first of these just checks that the search function works properly:

    608. <Search test functions 608> =
    /* Tests that f->search returns expect when called to search for
       key within array[],
       which has n elements such that array[i] == i. */
    static void
    test_search_func_at (struct search_func *f, int array[], int n,
                         int key, int expect)
    {
      int result = f->search (array, n, key);
      if (result != expect)
        printf ("%s returned %d looking for %d  expected %d\n",
                f->name, result, key, expect);
    }

    /* Tests searches for each element in array[] having n elements such that
       array[i] == i,
       and some unsuccessful searches too, all using function f->search. */
    static void
    test_search_func (struct search_func *f, int array[], int n)
    {
      static const int shouldnt_find[] = {INT_MIN, -20, -1, INT_MAX};
      int i;

      printf ("Testing integrity of %s...  ", f->name);
      fflush (stdout);

      /* Verify that the function finds values that it should. */
      for (i = 0; i < n; i++)
        test_search_func_at (f, array, n, i, i);

      /* Verify that the function doesn't find values it shouldn't. */
      for (i = 0; i < (int) (sizeof shouldnt_find / sizeof *shouldnt_find); i++)
        test_search_func_at (f, array, n, shouldnt_find[i], -1);

      printf ("done\n");
    }

See also 609 and 610.

This code is included in 601.

The second test function finds the time required for searching for elements in the array:

    609. <Search test functions 608> +=
    /* Times a search for each element in array[] having n elements such that
       array[i] == i, repeated n_iter times, using function f->search. */
    static void
    time_successful_search (struct search_func *f, int array[], int n, int n_iter)
    {
      clock_t timer;

      printf ("Timing %d sets of successful searches...  ", n_iter);
      fflush (stdout);

      start_timer (&timer);
      while (n_iter-- > 0)
        {
          int i;

          for (i = 0; i < n; i++)
            f->search (array, n, i);
        }
      stop_timer (timer);
    }

The last test function finds the time required for searching for values that don't appear in the array:

    610. <Search test functions 608> +=
    /* Times n search for elements not in array[] having n elements such that
       array[i] == i, repeated n_iter times, using function f->search. */
    static void
    time_unsuccessful_search (struct search_func *f, int array[],
                              int n, int n_iter)
    {
      clock_t timer;

      printf ("Timing %d sets of unsuccessful searches...  ", n_iter);
      fflush (stdout);

      start_timer (&timer);
      while (n_iter-- > 0)
        {
          int i;

          for (i = 0; i < n; i++)
            f->search (array, n, -i);
        }
      stop_timer (timer);
    }

Here's the main program:

    611. <Search test main program 611> =
    <Usage printer for search test program 617>

    <String to integer function stoi() 613>

    int
    main (int argc, char *argv[])
    {
      struct search_func *f;        /* Search function. */
      int *array, n;                /* Array and its size. */
      int n_iter;                   /* Number of iterations. */

      <Parse search test command line 612>
      <Initialize search test array 614>
      <Run search tests 615>
      <Clean up after search tests 616>

      return 0;
    }

This code is included in 601.

    612. <Parse search test command line 612> =
    if (argc != 4)
      usage ();

    {
      long algorithm = stoi (argv[1]) - 1;
      if (algorithm < 0 || algorithm > (long) n_search_func)
        usage ();
      f = &search_func_tab[algorithm];
    }

    n = stoi (argv[2]);
    n_iter = stoi (argv[3]);
    if (n < 1 || n_iter < 1)
      usage ();

This code is included in 611.

    613. <String to integer function stoi() 613> =
    /* s should point to a decimal representation of an integer.
       Returns the value of s, if successful, or 0 on failure. */
    static int
    stoi (const char *s)
    {
      long x = strtol (s, NULL, 10);
      return x >= INT_MIN && x <= INT_MAX ? x : 0;
    }

This code is included in 611 and 619.

When reading the code below, keep in mind that some of our algorithms use a sentinel at the end and some use a sentinel at the beginning, so we allocate two extra integers and take the middle part.

    614. <Initialize search test array 614> =
    array = malloc ((n + 2) * sizeof *array);
    if (array == NULL)
      {
        fprintf (stderr, "out of memory\n");
        exit (EXIT_FAILURE);
      }
    array++;

    {
      int i;

      for (i = 0; i < n; i++)
        array[i] = i;
    }

This code is included in 611.

    615. <Run search tests 615> =
    test_search_func (f, array, n);
    time_successful_search (f, array, n, n_iter);
    time_unsuccessful_search (f, array, n, n_iter);

This code is included in 611.

    616. <Clean up after search tests 616> =
    free (array - 1);

This code is included in 611.

    617. <Usage printer for search test program 617> =
    /* Prints a message to the console explaining how to use this program. */
    static void
    usage (void)
    {
      size_t i;

      fputs ("usage: srchtest <algorithm> <arraysize> <niterations>\n"
             "where <algorithm> is one of the following:\n", stdout);

      for (i = 0; i < n_search_func; i++)
        printf ("        %u for %s\n", (unsigned) i + 1, search_func_tab[i].name);

      fputs ("      <arraysize> is the size of the array to search, and\n"
             "      <niterations> is the number of times to iterate.\n", stdout);

      exit (EXIT_FAILURE);
    }

This code is included in 611.

4. Here are the results on the author's computer, a Pentium II at 233 MHz, using GNU C 2.95.2, for 1024 iterations using arrays of size 1024 with no optimization. All values are given in seconds rounded to tenths.

Function | Successful searches | Unsuccessful searches
seq_search() | 18.4 | 36.3
seq_sentinel_search() | 16.5 | 32.8
seq_sorted_search() | 18.6 | 0.1
seq_sorted_sentinel_search() | 16.4 | 0.2
seq_sorted_sentinel_search_2() | 16.6 | 0.2
binary_search() | 1.3 | 1.2
uniform_binary_search() | 1.1 | 1.1
binary_search_bsearch() | 2.6 | 2.4
cheat_search() | 0.1 | 0.1

Results of similar tests using full optimization were as follows:

Function | Successful searches | Unsuccessful searches
seq_search() | 6.3 | 12.4
seq_sentinel_search() | 4.8 | 9.4
seq_sorted_search() | 9.3 | 0.1
seq_sorted_sentinel_search() | 4.8 | 0.2
seq_sorted_sentinel_search_2() | 4.8 | 0.2
binary_search() | 0.7 | 0.5
uniform_binary_search() | 0.7 | 0.6
binary_search_bsearch() | 1.5 | 1.2
cheat_search() | 0.1 | 0.1

Observations:

- In general, the times above are about what we might expect them to be: they decrease as we go down the table.
- Within sequential searches, the sentinel-based searches have better search times than non-sentinel searches, and other search characteristics (whether the array was sorted, for instance) had little impact on performance.
- Unsuccessful searches were very fast for sorted sequential searches, but the particular test set used always allowed such searches to terminate after a single comparison. For other test sets one might expect these numbers to be similar to those for unordered sequential search.
- Either of the first two forms of binary search had the best overall performance. They also have the best performance for successful searches and might be expected to have the best performance for unsuccessful searches in other test sets, for the reason given before.
- Binary search using the general interface bsearch() was significantly slower than either of the other binary searches, probably because of the cost of the extra function calls. Items that are more expensive to compare (for instance, long text strings) might be expected to show less of a penalty.

Here are the results on the same machine for 1,048,576 iterations on arrays of size 8 with full optimization:

Function | Successful searches | Unsuccessful searches
seq_search() | 1.7 | 2.0
seq_sentinel_search() | 1.7 | 2.0
seq_sorted_search() | 2.0 | 1.1
seq_sorted_sentinel_search() | 1.9 | 1.1
seq_sorted_sentinel_search_2() | 1.8 | 1.2
binary_search() | 2.5 | 1.9
uniform_binary_search() | 2.4 | 2.3
binary_search_bsearch() | 4.5 | 3.9
cheat_search() | 0.7 | 0.7

For arrays this small, simple algorithms are the clear winners. The additional complications of binary search make it slower. Similar patterns can be expected on most architectures, although the “break even” array size where binary search and sequential search are equally fast can be expected to differ.

#### Section 3.6

1. Here is one easy way to do it:

    618. <Initialize smaller and larger within binary search tree 618> =
    /* Initializes larger and smaller within range min...max of
       array[],
       which has n real elements plus a (n + 1)th sentinel element. */
    int
    init_binary_tree_array (struct binary_tree_entry array[], int n,
                            int min, int max)
    {
      if (min <= max)
        {
          /* The `+ 1' is necessary because the tree root must be at n / 2,
             and on the first call we have min == 0 and max == n - 1. */
          int i = (min + max + 1) / 2;
          array[i].larger = init_binary_tree_array (array, n, i + 1, max);
          array[i].smaller = init_binary_tree_array (array, n, min, i - 1);
          return i;
        }
      else
        return n;
    }

This code is included in 619.

2.

    619. <bin-ary-test.c 619> =
    <Program License 2>
    #include <limits.h>
    #include <stdio.h>
    #include <stdlib.h>

    <Binary search tree entry 23>
    <Search of binary search tree stored as array 24>
    <Initialize smaller and larger within binary search tree 618>
    <Show bin-ary-test usage message 621>
    <String to integer function stoi() 613>
    <Main program to test binary_search_tree_array() 620>

    620. <Main program to test binary_search_tree_array() 620> =
    int
    main (int argc, char *argv[])
    {
      struct binary_tree_entry *array;
      int n, i;

      /* Parse command line. */
      if (argc != 2)
        usage ();
      n = stoi (argv[1]);
      if (n < 1)
        usage ();

      /* Allocate memory. */
      array = malloc ((n + 1) * sizeof *array);
      if (array == NULL)
        {
          fprintf (stderr, "out of memory\n");
          return EXIT_FAILURE;
        }

      /* Initialize array. */
      for (i = 0; i < n; i++)
        array[i].value = i;
      init_binary_tree_array (array, n, 0, n - 1);

      /* Test successful and unsuccessful searches. */
      for (i = -1; i < n; i++)
        {
          int result = binary_search_tree_array (array, n, i);
          if (result != i)
            printf ("Searching for %d: expected %d, but received %d\n",
                    i, i, result);
        }

      /* Clean up. */
      free (array);

      return EXIT_SUCCESS;
    }

This code is included in 619.

    621. <Show bin-ary-test usage message 621> =
    /* Print a helpful usage message and abort execution. */
    static void
    usage (void)
    {
      fputs ("Usage: binarytest <arraysize>\n"
             "where <arraysize> is the size of the array to test.\n",
             stdout);
      exit (EXIT_FAILURE);
    }

This code is included in 619.