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13.7. Optimizing Sort Algorithms with Code Tuning

13.7.1. Code Tuning for Simple Sorting Algorithms

Todo

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Rewrite along these lines: A classic form of code tuning is "test to save work". For each of our three sorting algorithms, we have a potential "test to save work" "optimization". The question is: When is the cost of test worth the work saved? Let's look at each of the three.

Since sorting is such an important application, it is natural for programmers to want to optimize their sorting code to run faster. Of course all quadratic sorts (Insertion Sort, Bubble Sort and Selection Sort) are relatively slow. Each has (as the name "quadratic suggests) \(\Theta(n^2)\) worst case running time. The best way to speed them up is to find a better sorting algorithm. Nonetheless, there have been many suggestions given over the years about how to speed up one or another of these particular algorithms. There are useful lessons to be learned about code tuning by seeing which of these ideas actually turn out to give better performance. It is also interesting to see the relative performance of the three algorithms, as well as how various programming languages compare.

We start by trying to speed up Insertion Sort. Recall that Insertion Sort repeatedly moves an element toward the beginning of the sorted part of the list until it encounters a key with lesser value. In the original code, this is done with a series of swap operations. There is a better alternative than continuously swapping the record to the left until a smaller value is found. This is to move the current record to a temporary variable, and then shift all of the records with greater value one step to the right. Since swap requires three assignments per element and shifting requires only one assignment per element, we can hope that this will yield a big improvement. Of course, the amount of improvement that we actually get will depend on how much movement there is among the records. If the list is already nearly sorted, then there will be few swaps anyway. Here is an implementation for Insertion Sort using this optimization.

// Instead of swapping, "shift" the values down the array
static void inssortshift(int[] A) {
  for (int i=1; i<A.length; i++) { // Insert i'th record
    int j;
    int temp = A[i];
    for (j=i; (j>0) && (temp < A[j-1]); j--)
      A[j] = A[j-1];
    A[j] = temp;
  }
}
// Instead of swapping, "shift" the values down the array
static void inssortshift(int[] A) {
  for (int i=1; i<A.length; i++) { // Insert i'th record
    int j;
    int temp = A[i];
    for (j=i; (j>0) && (temp < A[j-1]); j--)
      A[j] = A[j-1];
    A[j] = temp;
  }
}
// Instead of swapping, "shift" the values down the array
void inssortshift(Comparable* A[], int n) { // Insertion Sort
  for (int i=1; i<n; i++) { // Insert i'th record
    int j;
    Comparable* temp = A[i];
    for (j = i; (j > 0) && (*temp < *A[j-1]); j--)
      A[j] = A[j-1];
    A[j] = temp;
  }
}

Now, you can test whether you understand how this works.

Table 13.7.1

Empirical comparison of proposed optimizations to quadratic sort implementations. Each sorting algorithm is run on a random integer array with 10,000 items. Times are in milliseconds. The arrays being sorted use the Comparable interface in languages that support this.

\[\begin{split}\begin{array}{l|rrrr} \hline \textbf{Sort} & \textbf{Java}& \textbf{Processing} & \textbf{JavaScript}& \textbf{Python}\\ \hline \textbf{Insertion Sort}\\ \textrm{Standard} & 60 & 26 & 118 & 11,220\\ \textrm{Shifting} & 41 & 18 & 77 & 5,100\\ \hline \textbf{Bubble Sort}\\ \textrm{Standard} & 202 & 149 & 303 & 12,700\\ \textrm{Check Swaps} & 230 & 152 & 327 & 13,275\\ \hline \textbf{Selection Sort}\\ \textrm{Standard} & 104 & 65 & 158 & 4,000\\ \textrm{Check Swaps} & 104 & 65 & 155 & 4,050\\ \hline \end{array}\end{split}\]

Table 13.7.1 shows the relative costs for a number of optimizations in four programming languages: Java, JavaScipt, Processing, and Python.

The programming language that you use can have a big influence on the runtime for a program. Perhaps the greatest distinction is whether your language is compiled or not. Java, C++, and Processing are normally compiled, while JavaScript and Python are normally interpreted. This can make a huge difference in whether a given code change will actually speed the program up or not. In the case of the "shift" vs "swap" choice, shifting always turns out to be a big improvement. This is more true for the interpreted languages JavaScript and Python than for Java and Processing, but still an improvement either way. But the biggest effect that we see is that Python takes over 100 times as long to execute the same program as Java.

Some languages have peculiarities that it pays to be aware of. It turns out that there is a big difference in JavaScript between using i < n or i != n to test termination of a loop.

Turning to Bubble Sort, the first thing we should notice from this table is that it is far slower on random input than Insertion Sort. Let's consider a possible improvement that is sometimes suggested for Bubble Sort. That is to check during each iteration of the outer loop to see if any swaps took place during that iteration, and quit if not (since we know the list is ordered at this point). We can improve on this idea even more by recognizing that if the last swap done affects the values at positions \(i\) and \(i+1\), no swaps could happen to values at positions greater than \(i\). Thus, we never need to check higher-positioned values again, which could save many iterations even if there are a few swaps lower down. Here is code to implement this approach.

static <T extends Comparable<T>> void bubblecheckswap(T[] A) {
  int n = A.length - 1;
  while (n > 0) {
    int newn = 0;
    for (int i = 0; i < n; i++) {
      /* if this pair is out of order */
      if (A[i].compareTo(A[i+1]) > 0) {
        Swap.swap(A, i, i+1);
        newn = i;
      }
    }
    n = newn;
  }
}
static <T extends Comparable<T>> void bubblecheckswap(T[] A) {
  int n = A.length - 1;
  while (n > 0) {
    int newn = 0;
    for (int i = 0; i < n; i++) {
      /* if this pair is out of order */
      if (A[i].compareTo(A[i+1]) > 0) {
        swap(A, i, i+1);
        newn = i;
      }
    }
    n = newn;
  }
}
void bubblecheckswap(Comparable* A[], int n) {
  n = n-1;
  while (n > 0) {
    int newn = 0;
    for (int i = 0; i < n; i++) {
      /* if this pair is out of order */
      if (*A[i] > *A[i+1]) {
        swap(A, i, i+1);
        newn = i;
      }
    }
    n = newn;
  }
}

The problem with this idea is that a considerable amount of effort (relatively speaking) is required to track the position for the last swap within the inner loop. This tracking process has a cost, and that cost is worthwhile only if the amount of work it saves is greater than the amout of work that it causes. Unfortunately, as the table shows, in the average case it just is not worth the time. Modifying the code simply by removing the tracking steps (and so not getting either the cost of tracking or the benefit of avoiding some of the key comparisons) is faster in the average case. Of course, whether this is always true will depend on how much it costs to extract the record keys and compare them, which depends on the details of the record type and the sort implementation. In our test implementation we are sorting integer values and so the cost to compare records is lower than it would be if we had to get a field out of a more complex object.

It is also true that tracking the last swap position can substantially improve the best case cost. In fact, tracking the last swap position makes the best case cost of Bubble Sort to be only \(\Theta(n)\). But going out of one's way to artificially improve the best case has dubious value if doing so imposes additional cost on nearly all other inputs. Note that we could nominally convert any sorting algorithm to have a best-case cost of \(\Theta(n)\) by simply adding code at the beginning that checks if the list is already sorted. It should be obvious that this is a waste of time, even though it has the (small) possibility of winning big. Unlike Insertion Sort whose best case cost is naturally \(\Theta(n)\) and whose time increases in proportion to how "out of order" the list is, the number of iterations avoided by swap checking in Bubble Sort is sensitive to the detailed placements of the out-of-order records. In fact, if we took a sorted list and moved the smallest value to the end, then there would be no benefit from swap checking whatsoever.

Finally, let's consider Selection Sort. The table shows foremost that Selection Sort can be viewed as a far better optimization to Bubble Sort than tracking the last swap position. That is, tracking the position of the largest element and performing one swap to put it into place is a far better optimization to Bubble Sort than tracking the position of the last swap seen. The table also shows that Selection Sort is faster in the average case than Insertion Sort when implemented in Python. Evidently, the cost to swap is high for Python.

Our original Selection Sort implementation is written to make a call to swap even if the current record is already in its correct location. For example, if the record with the largest value is already in the rightmost array position, then selsort will still call swap with the two position parameters being the same. The net effect is that the work done by swap will not change anything in the array, and this is a waste of time. Thus, the total number of swaps done by Selection Sort is always \(n-1\) in the best, average and worst cases. It might seem like a good idea to test if the positions are the same before calling swap, especially since Selection Sort's claim to fame is its low number of swaps. Actually, we can't expect this to ever make much difference since we are talking about \(\Theta(n)\) actions within \(\Theta(n^2)\) total steps, an inconsequential fraction. The other consideration is whether this is could typically be expected to save time even when just considering the time needed to do the swaps. Doing the check to see if a swap is necessary also takes some time. It is only worthwhile to test if the time required by the test is more than made up for by the work saved when the unnecessary swap was avoided. For randomly ordered input, it is probably more expensive to test this condition before every swap than to just do the swap. If the input records are already sorted, then all of the swaps are unnecessary and it would be (trivially) faster to test. But in the average case, few swaps will be saved this way and the "optimization" might actually slow down the program (but only slightly).

For all of these sorting algorithms, the swap function call might be a key part of the cost since it is called so many times. A simple way to speed things up is to replace this function call with the code that the function would perform. Depending on the language, compiler, and operating system, one might expect to save between 5 and 10 percent of the total time by doing so.

Another important consideration is the type of data object being used. For Processing and Java, we use a simple Integer wrapper object that supports the Comparable interface. This means that some dereferencing of the key value from an object is required, which is a typical expectation in a realistic application of a sorting function. However, if we were to sort a simple array of int values, the cost for all sorting algorithms will be less than half that shown. If we use a the more complicated KVPair objects, the costs will more than double over those shown in the table.

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