Array Rotation, Timing Tests

March 9, 2018

We have been looking at Section 2.3 of Jon Bentley’s book Programming Pearls in the last two exercises, and have implemented his “juggling” and “block swap” algorithms. Bentley also discusses a third algorithm, which he calls the “reversal” algorithm, and which we implemented several years ago. Bentley goes on to give timing comparisons between the three algorithms.

Your task is to generate timing comparisons similar to Bentley’s, to see what happens with your system, your language and your compiler. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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3 Responses to “Array Rotation, Timing Tests”

  1. gambiteer said

    I compiled the code with Gambit, added declarations

    (declare (standard-bindings)
    (extended-bindings)
    (block)
    (fixnum)
    (not safe))

    except for

    (define (timing rotate vec dist)
    (declare (generic))
    (let ((start (cpu-time)))
    (rotate vec dist)
    (- (cpu-time) start)))

    and got times

    (let ((vec (list->vector (range 1000000))))
    (map (lambda (x) (timing juggling vec 100)) (range 50)))
    (.029591999999993845
    .02927400000000091
    .027829000000011206
    .03642400000001089
    .024536999999995146
    .02830999999999051
    .029009999999999536
    .02474899999999991
    .02875699999999881
    .0312900000000127
    .02902900000000841
    .028691000000009126
    .028640999999993255
    .026204000000007
    .024539000000018518
    .02653800000000217
    .028767000000001985
    .029217000000002713
    .027430000000009613
    .025491999999999848
    .02925899999999615
    .028920999999982655
    .024906000000001427
    .03006200000000092
    .027745999999993387
    .03249999999999886
    .025558000000003744
    .030752000000006774
    .025658000000007064
    .025202000000007274
    .02870499999998799
    .02643400000000895
    .02801700000000551
    .025395000000003165
    .027732999999997787
    .02871199999999874
    .024641000000002578
    .02881400000001122
    .0244689999999963
    .024765000000002146
    .02464700000000164
    .02467299999999284
    .02451000000000647
    .02820400000000234
    .02889599999998893
    .02443200000000445
    .02453400000000272
    .029072999999982585
    .024445999999997525
    .02456900000001383)

    (let ((vec (list->vector (range 1000000))))
    (map (lambda (x) (timing block-swap vec 100)) (range 50)))
    (.005112999999994372
    .006988000000006878
    .005129000000010819
    .00770599999999888
    .006287999999997851
    .005140999999994733
    .005306000000004474
    .006874000000010483
    .0067049999999824195
    .005473999999992429
    .005122999999997546
    .006666999999993095
    .006698999999997568
    .005892000000002895
    .005227000000004978
    .00513300000000072
    .0065400000000011005
    .005146000000010531
    .005233999999987304
    .005146999999993795
    .00519599999999798
    .0053230000000041855
    .006732999999996991
    .00784100000001331
    .0065659999999923
    .005431999999998993
    .005251000000001227
    .005134999999995671
    .005264999999994302
    .005147000000008006
    .0053039999999953125
    .005131999999989034
    .005214999999992642
    .005148999999988746
    .005223000000000866
    .0052220000000176015
    .005243000000007214
    .005140999999994733
    .005228999999999928
    .00518300000000238
    .00517000000000678
    .0060610000000167474
    .005145000000013056
    .0052450000000163755
    .009970000000009804
    .005146000000010531
    .005279999999999063
    .005134999999995671
    .005239000000003102
    .005659999999991783)
    (let ((vec (list->vector (range 1000000))))
    (map (lambda (x) (timing reversal vec 100)) (range 50)))
    (.0030000000000001137
    .0031049999999908096
    .003112000000001558
    .003090000000000259
    .003170999999980495
    .003987000000009289
    .0032210000000105765
    .0033960000000092805
    .0032839999999936254
    .0030319999999903757
    .003071000000005597
    .0032810000000011996
    .003694000000010078
    .0031510000000167793
    .003117000000003145
    .0030380000000178597
    .0031689999999997553
    .003107999999997446
    .0033580000000057453
    .003789999999995075
    .00334400000001267
    .0033090000000015607
    .0031930000000102154
    .0033829999999994698
    .003629000000003657
    .003005999999999176
    .0031090000000091322
    .0031000000000034333
    .0039209999999911815
    .0031829999999928305
    .003528000000002862
    .0032950000000084856
    .0031599999999940565
    .0030909999999977344
    .003120999999993046
    .0034139999999922566
    .003697000000002504
    .00402800000000525
    .003558999999995649
    .0033030000000167092
    .0032910000000043738
    .00314399999999182
    .0032059999999916045
    .002983000000000402
    .003740000000007626
    .003374999999991246
    .0037559999999956517
    .0033249999999895863
    .003296999999989225
    .003203999999996654)

    which is closer to Bentley’s results

  2. Daniel said

    Here’s a solution in C.

    The output times follow the code.

    The swapping approach worked fastest of my implementations.

    I was able to improve the speed of the juggle approach by using subtraction to calculate the remainder instead of using the modulus operator.

    I also added code to validate that my rotation implementations are working correctly.

    #include <assert.h>
    #include <stdio.h>
    #include <stdlib.h>
    #include <string.h>
    #include <time.h>
    
    void rotate_juggling(int* array, size_t n, size_t i) {
      if (n == 0) return;
      size_t n_moved = 0;
      size_t start_idx = 0;
      while (1) {
        int tmp = array[start_idx];
        size_t dest_idx = start_idx;
        size_t source_idx = start_idx + i;
        if (source_idx >= n) source_idx -= n;
        while (1) {
          ++n_moved;
          if (source_idx == start_idx) {
            array[dest_idx] = tmp;
            break;
          }
          array[dest_idx] = array[source_idx];
          dest_idx = source_idx;
          source_idx = source_idx + i;
          if (source_idx >= n) source_idx -= n;
        }
        if (n_moved == n) break;
        ++start_idx;
      }
    }
    
    static void swap(int* a1, int* a2, size_t n) {
      for (size_t i = 0; i < n; ++i) {
        int tmp = a1[i];
        a1[i] = a2[i];
        a2[i] = tmp;
      }
    }
    
    void rotate_swap(int* array, size_t n, size_t i) {
      i %= n;
      while (1) {
        if (n <= 1 || i == 0) break;
        if (i == n - i) {
          swap(array, array + i, i);
          break;
        } else if (i < n - i) {
          swap(array, array + n - i, i);
          n -= i;
        } else {
          swap(array, array + i, n - i);
          array += n - i;
          size_t n_ = n;
          n = i;
          i -= n_ - i;
        }
      }
    }
    
    static void reverse(int* array, size_t n) {
      for (size_t i = 0; i < n / 2; ++i) {
        int tmp = array[i];
        array[i] = array[n - i - 1];
        array[n - i - 1] = tmp;
      }
    }
    
    void rotate_reverse(int* array, size_t n, size_t i) {
      reverse(array, i);
      reverse(array + i, n - i);
      reverse(array, n);
    }
    
    clock_t calc_rotate_time(void (*rotate) (int*, size_t, size_t),
                            int* array,
                            size_t n,
                            size_t i,
                            size_t iterations) {
      int* array_ = malloc(sizeof(int) * n);
      clock_t total = 0;
      for (size_t iter = 0; iter < iterations; ++iter) {
        memcpy(array_, array, n * sizeof(int));
        clock_t start = clock();
        rotate(array, n, i);
        clock_t end = clock();
        total += end - start;
      }
      free(array_);
      return (double)total;
    }
    
    int main(void) {
      size_t iterations = 1000;
      size_t n = 10000;
      int* array = malloc(n * sizeof(int));
      for (size_t idx = 0; idx < n; ++idx) {
        array[idx] = idx;
      }
      size_t increment = 1000;
      void (*rotate_functions[])(int*,size_t,size_t) = {
        rotate_juggling,
        rotate_swap,
        rotate_reverse
      };
      char* rotate_function_names[] = {"juggle", "swap", "reverse"};
      size_t n_rotate_functions = sizeof(rotate_function_names) / sizeof(char*);
      printf("dist");
      for (size_t i = 0; i < n_rotate_functions; ++i) {
        printf("\t%s", rotate_function_names[i]);
      }
      printf("\n");
      for (size_t distance = increment; distance < n; distance += increment) {
        printf("%zu", distance);
        for (size_t i = 0; i < n_rotate_functions; ++i) {
          void (*rotate_function)(int*,size_t,size_t) = rotate_functions[i];
          // ***************************************
          // * Validate Rotation
          // ***************************************
          int* rotated = malloc(n * sizeof(int));
          memcpy(rotated, array, n * sizeof(int));
          rotate_function(rotated, n, distance);
          for (size_t k = 0; k < n; ++k) {
            assert(rotated[k] == array[(k+distance)%n]);
          }
          free(rotated);
          // ***************************************
          // * Time Rotation
          // ***************************************
          clock_t time = calc_rotate_time(
            rotate_function, array, n, distance, iterations);
          printf("\t%lu", time);
        }
        printf("\n");
      }
      free(array);
      return EXIT_SUCCESS;
    }
    

    Output (-O0):

    dist    juggle  swap    reverse
    1000    39265   25692   31132
    2000    32060   19986   29618
    3000    29878   27256   33861
    4000    30236   20669   30122
    5000    32395   12986   29730
    6000    33031   20343   33086
    7000    38557   28355   37712
    8000    35738   21311   33593
    9000    36285   22194   29386
    

    Output (-O2):

    dist    juggle  swap    reverse
    1000    17076   3093    9534
    2000    10190   2641    7591
    3000    9402    2594    7187
    4000    8445    2308    7191
    5000    10205   1757    7581
    6000    9354    2572    7175
    7000    8411    2166    7001
    8000    8927    2396    7330
    9000    8956    2326    7605
    
  3. Daniel said

    My calc_rotate_time function casts the result to a double in the return statement. That should be removed. An earlier implementation was calculating seconds, and the cast to double was there to bypass integer division. Removing the cast to double won’t have a substantive effect on the numbers reported above.

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