12. Merge Sort

The discovery of generic programming

I have to tell you a story about fish. I was young once. The year was 1976. Some of you were not even born. I was a good programmer. I was writing operating systems and doing a good job. But, I aspired to greater things. When you are young you aspire. There was an opening in one very prestigious institute, Soviet Academy of Science to work on the design of a parallel computer. I wanted this job really badly.

A friend of mine who worked at this Institute recommended me to this group. I knew I was going to go there for an interview and I’d have to say some wonderful things or they would not give me a job. Right before my interview, terrible thing happened. I ate some raw fish and it was very tasty but within eight hours my temperature was 103. I was in a bad situation.

I’m in the hospital flying above my bed. That happens to you when you have high fever. But, what I’m thinking about is this upcoming interview about parallel computers. I know nothing about parallel computers not now, nor ever, but I really want the job. So I need to come up with some brilliant idea. So I’m floating above the bed in space and thinking. Then I suddenly realize how to add four numbers in parallel I said, “oh I could add them in parallel if I add the first two and the second two in parallel”:

     +
    /  \
   /    \
  +      +
 / \    / \
x   y  z   w

I had this vision of this huge addition tree. When you’re sick amazing things happen in your mind. Then I realized the second great thing, while still floating. It could be multiplication and the same thing will work. I started realizing more and more functions will work min, max, but division will not work. Then I realized it’s good to be sick. It has to do something which I almost forgot called abstract algebra. How is this related? I have been teaching the whole course¹ so I remember that there is this thing called associativity. I realized that this idea will work as long as the operation is associative.

This became my central theme. How could I talk about associative operations? How could I write algorithms like that? When I came to United States somewhere along the way, maybe in Austria, I realized that merge was associative It was a very big deal because I never even thought about merge in terms of + and *, but merge is associative. Then I realized I could do merge sort with parallel reduction. That’s the summary. Alex, what have you been doing all your life? This. You might say, “that’s not good enough”. Probably not.

I never talked about that discovery, during the job interview. I did get the job and it wasn’t particularly good. It wasn’t good at all. But, things happen².

Iterators as a concept

Generic programming means to me finding algebraic structures on which algorithms are defined and writing them like that. The second idea is that it will be terribly nice if using beautiful programming at that level wouldn’t cost us anything. Anybody can do beautiful things if it’s very slow. This is why we design languages like Prolog. They don’t care as long as they could write it as Horn clauses, or things like that. So it’s the combination of two things:

1. Doing programming in mathematical theories
2. Not losing efficiency

We are slowly hinting more at concepts. How is a concept different from a class? How is a concept different than an abstract class or interface in Java? One way to see this is try to think of how to specify the concept of an iterator using them. Consider that iterators have a value_type. In an interface, every argument must have a fixed type. Interfaces specify identical interfaces. A concept defines similar interfaces. The concept of a pointer doesn’t define the same interface for every type. The * operator can point to int, or double, or who knows what. Concepts have variance on the types.

Think also about ==. It is a binary predicate. If you use inheritance, you can only modify the type of the first argument. The second argument stays the same. In Java you can kind of fix it by doing lots of casting. But, the other problem is more serious as you can’t fix the fundamental variance on value_type.

People say, “but couldn’t there be a correct inheritance?” Of course, yes. But it’s not the inheritance we have in C++ or Java.

Because we don’t have concepts, when we write functions which take an iterator as an argument you cannot say what requirements it has.

Kinds of iterators

There are several iterator concepts in STL³, all of which we have mentioned before:

• InputIterator: can only advance forward and once advanced, iterators pointing to previous elements become invalid.
• ForwardIterator: can advance forward and have iterators pointing to previous elements.
• BidirectionalIterator: can move iterators forward and backward.
• RandomAccessIterator: can advance iterators by arbitrary steps in constant time (like pointers).

Linked iterator

There are some people who say, “concepts aren’t important because Alex already exhausted them. there are just four kinds of iterators and that’s all.” There is a “great” person who goes around and gives talks saying this. I’m not going to name him⁴.

What I’m going to do now is introduce a new kind of iterator, just to spite this distinguished person. We want to extend the concept of an iterator with a new concept of a LinkedIterator. It is a ForwardIterator which in addition to the normal operations:

• ++: advance
• ==: equality
• *: dereference

it has one additional operation which allows you to set the successor. If you have an iterator and you have another one, you can just make the second iterator the successor of the first. You can literally connect them. Of course, the standard model is a linked list.

To make our list_pool iterator a LinkedIterator we simply add the following function:

void set_successor(iterator x, iterator y) {
    x.pool->next(x.node) = y.node;
}

We want it to be a function not a member because primitive/built-in types don’t have member functions⁵. Furthermore, in the case of an iterator, it might be a pointer, not a class. In general, I don’t like member functions. If you have the option of:

x.foo()

Or

foo(x)

There is at least one provable advantage. It has one fewer character⁶. But, since we think generically, a better question is whether the function could operate on a built-in type instead of a class.

Linked iterator is “unsafe”

As long as you don’t do set_successor, the topology remains. But, if you do, things change. This is a very unsafe operation. You can even make circular lists. Some people say this is very bad. It should not be allowed. We have to come up with language mechanisms for not doing this evil. These are the people who believe that there are some syntactic rules that can make programming good.

Are they right? Why not in the case of circular lists? They are very useful. Of course, if you try to get to the end, you will not be successful. But, it is a legitimate data structure.

A long time ago when we were programming in Lisp, there were people saying, “never use set successor (rplcd)”. They were wrong. Use whatever is given to you. But, be wise. You can write bad code just with +.

while (true) a = a + 1;

That doesn’t terminate. Well maybe just while (true) is bad.

a = 1;
while (a > 0) { do stuff }

There are many ways of writing bad code and syntax can’t help you. No smart pointer or syntactic constraint will make a bad programmer into a good programmer. It will just replace one sort of bugs with another. Bad programmers are very creative, they find amazing things.

Reverse linked ranges

To understand how this concept works we will look at a basic algorithm. If I hadn’t already shown you set_successor we can learn its use from this algorithm. It takes two lists, reverses the first, and attaches it to the second. It’s a very important list algorithm. In scheme (append! (reverse! x) tail) or (nconc (nreverse x) tail) in Common Lisp.

template <typename I>
// requires I is LinkedIterator
I reverse_linked(I first, I last, I tail) {
  while (first != last) {
    I next = first;
    ++next;
    set_successor(first, tail);
    tail = first;
    first = next;
  }
  return tail;
}

Merging sorted lists

Simple merge

The following function merges two sorted lists, using the algorithm we just wrote.

template <typename I, typename Compare>
// requires I is Linked Iterator
I merge_linked_simple(I first1, I last1, I first2, I last2, Compare cmp) {
  I result = last1;
  while (first1 != last1 && first2 != last2) {
    I tmp = cmp(*first2, *first1) ? first2++ : first1++;
    set_successor(tmp, result);
    result = tmp;

  return reverse_linked(result, last1, first1 == last1 ? first2 : first1);
}

Notice that a good thing about iterators is it helps us avoid writing nil/null.

It’s called simple because it’s simplistic. Why? In terms of complexity it takes two passes. We have to go through the lists, and merge them, roughly n successors on the way there. Then we do the same number doing the reverse_linked. Also note that algorithmically there is no good reason why we evaluate

while (first1 != last1 && first2 != last2)

On every single iteration, because only the one iterator we moved in the loop could become empty.

Exercise: Write an alternative merge which does not do this extra comparison every loop (solved just below).

Merge with fewer comparisons

I clearly teach you all the wrong things. I tell you not to use inheritance, smart pointers, or read certain books. Now I’m going to teach you to use goto. The greatest authority in computer science wrote a famous letter to communications of ACM called, “Go To Statement Considered Harmful”. There is nothing harmful in the computer. The computer is good. It has an instruction called JMP (or branch). If it’s there, why not use it?

If you are going to use goto, it’s probably because you have something like a finite state machine. The state machine goes between fundamental states. You probably learned about them in school, but they never told you it relates to goto. What are the states which we need for this algorithm? Which list won the comparison and advanced. Either the 1st, or the 2nd.

template <typename I, typename Compare>
// I is a linked forward iterator
std::pair<I, std::pair<I, I> >
merge_linked_non_empty(I first1, I last1, I first2, I last2, Compare cmp) {
  I head, tail;
  if (cmp(*first2, *first1)) {
    head = first2;
    tail = first2;
    ++first2;
    goto winner2;
  } else {
    head = first1;
    tail = first1;
    ++first1;
    // goto winner1;
  }
winner1:
  if (first1 == last1) goto empty1;
  if (!cmp(*first2, *first1)) {
    tail = first1;
    ++first1;
    goto winner1;
  } else {
    set_successor(tail, first2); 
    tail = first2;
    ++first2;
    // goto winner2;
  }
winner2:
  if (first2 == last2) goto empty2;
  if (cmp(*first2, *first1)) {
    tail = first2;
    ++first2;
    goto winner2;
  } else {
    set_successor(tail, first1);
    tail = first1;
    ++first1;
    goto winner1;
  }
empty1:
  set_successor(tail, first2);
  return std::make_pair(head, std::make_pair(first2, last2));
empty2:
  set_successor(tail, first1);
  return std::make_pair(head, std::make_pair(first1, last1));
}

Let’s go through the states. Right away, we need to determine who the winner is. For performance this part doesn’t matter because it’s not in the inner loop. winner1 means the smaller element came from the 1st range, and similarly for winner2. We compare just the one that advanced. The two states are symmetric.

I return 3 iterators. Why? It’s specific to linked lists/iterators. If I want to take the resulting list and attach it, I need to be able to modify the successor of the final node. But, if we return the end, it’s probably nil so I don’t have it. You just return all the information. The caller can do whatever they please. If they want to ignore it, ignore it.

Note that it assumes the lists are nonempty, which is perfectly fine for our sort, which is not going to merge empty lists⁷.

Can you write it without goto? Not as efficiently⁸.

Is it worth it?

We now have two programs. One of them is oh so very simple. The other one is elegant, but long and does a minimal number of operations. Which one should we use in practice? We need to do a lot of experiments to establish certainty of what we are doing⁹.

Exercise: Measure and determine whether it’s actually faster than our simple merge.

Can we think of any more performance improvements? In terms of operations we have cut it to the bare bone. But, one thing is branch mispredictions. Modern processors try to predict which way a condition is going to go. If you look at our branches, the probability of going one way or the other is 50 percent. That’s literally the worst thing which could happen. I’m entering the territory where I haven’t done work yet. I haven’t tried see whether some kind of predication avoidance of goto could be done.

Exercise: Try to write a version of this algorithm which is more friendly for branch prediction.

Merge sort

Now that we can merge lists and we have our binary counter, we can write merge sort. For our machinery, we need to wrap this operation in an object. There is no code, it’s just a way of invoking our merge function. It has 5 arguments, and we need a binary function.

template <typename I, typename Compare>
// I is Linked Iterator
struct mergesort_linked_operation
{
  typedef I argument_type;
  I nil;
  Compare cmp;
  mergesort_linked_operation(I nil, const Compare& cmp) : nil(nil), cmp(cmp) {}
  I operator()(I x, I y) { return merge_linked_simple(x, nil, y, nil, cmp); }
};

The purpose of this is not to learn to sort lists. It is of course a useful skill. The purpose is to show you this code. Observe it is the same machine as we had before.

template <typename I, typename Compare>
// I is Linked Iterator
I mergesort_linked(I first, I last, Compare cmp) {
  mergesort_linked_operation<I, Compare> op(last, cmp);
  binary_counter<mergesort_linked_operation<I, Compare> > counter(op, last);
  counter.reserve(16);
  while (first != last) {
    I tmp = first++;
    set_successor(tmp, last);
    counter.add(tmp);
  }
  return counter.reduce();
}

It is very efficient, and doesn’t do any kind of list splitting¹⁰. If you don’t find it’s beautiful I have nothing to teach you. We aren’t on the same wavelength. In the modern world, you meet somebody and say, “Bach is a great composer”. He says, “nah, Lady Gaga is much more gifted”. It’s a free world. People are allowed to say whatever they like. Some people will say, “oh this is not object-oriented”, or “this is not functional”, which it isn’t and I’m very proud of it. But, in some sense this is literally the essence of my life’s work, this piece of code. That’s where it started. That’s where it ends. The majority of people do not get it. The majority of computer scientists do not get it. There is absolutely no indication that getting it will make you rich.

Exercise: Implement a visualizer (such as console output) which shows the contents of the counter at each step of the merge algorithm.

Exercise: Implement merge sort with std::accumulate (left fold) instead of binary counter. This is a very inefficient merge sort, but can be helpful to understand how the binary counter works.

Code

• list_algorithm.h
• test_merge_sort.cpp