►A Data-Centric Introduction to Computing
►VI Data Structures with Analysis
▼15 Directed Acyclic Graphs
►15.2 The Size of a DAG
On this page:
15.2.1 Stage 1
15.2.2 Stage 2
15.2.3 Stage 3
15.2.4 Stage 4
15.2.5 Stage 5
15.2.6 What We’ve Learned
15.2.7 More on Value Printing:   An Aside from Racket
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15.2 The Size of a DAG

    15.2.1 Stage 1

    15.2.2 Stage 2

    15.2.3 Stage 3

    15.2.4 Stage 4

    15.2.5 Stage 5

    15.2.6 What We’ve Learned

    15.2.7 More on Value Printing: An Aside from Racket

Let’s start by defining a function to compute the size of a tree: 

data BT:
  | mt
  | nd(v :: Number, l :: BT, r :: BT)
end

fun size-1(b :: BT) -> Number:
  cases (BT) b:
    | mt => 0
    | nd(v, l, r) => 1 + size-1(l) + size-1(r)
  end
end

This is straightforward enough.

But let’s say that our input isn’t actually a tree, but rather a DAG. For instance: 

#|
     4
    / \
   2   3
    \ /
     1
|#

n1 = nd(1, mt, mt)
n2 = nd(2, mt, n1)
n3 = nd(3, n1, mt)
n4 = nd(4, n2, n3)

where n4 is the DAG. There are two notions of size here. One is like a “print size”: how much space will it occupy when printed. The current size function computes that well. But another is the “allocation” size: how many nodes did we allocate. How do we fare?

15.2.1 Stage 1

check:
  size-1(n1) is 1
  size-1(n2) is 2
  size-1(n3) is 2
  size-1(n4) is 4
end

Clearly the answer should be 4: we can just read off how many nd calls there are. And clearly the function is wrong.

The problem, of course, is that a DAG involves repeating nodes, and we aren’t doing anything to track the repetition. So we need a stronger contract: we’ll split the problem into two parts, a standard interface function that takes just the DAG and returns a number, and a richer helper function, which also takes a memory of the nodes already seen.

15.2.2 Stage 2

fun size-2-h(b :: BT, seen :: List<BT>) -> Number:
  if member-identical(seen, b):
    0
  else:
    cases (BT) b:
      | mt => 0
      | nd(v, l, r) =>
        new-seen = link(b, seen)
        1 + size-2-h(l, new-seen) + size-2-h(r, new-seen)
    end
  end
end

Exercise

Why does this code use member-identical rather than member?

Observe that if we replace every member-identical with member in this chapter, the code still behaves the same. Why?

Make changes to demonstrate the need for member-identical.

Is it odd that we return 0? Not if we reinterpret what the function does: it doesn’t count the size, it counts the additional contribution to the size (relative to what has already been seen) of the BT it is given. A node already in seen makes no marginal contribution; it was already counted earlier.

Finally, we should not export such a function to the user, who has to deal with an unwieldy extra parameter and may send something poorly-formed, thereby causing our function to break. Instead, we should write a wrapper for it: 

fun size-2(b :: BT): size-2-h(b, empty) end

This also enables us to use our old tests (renamed): 

check:
  size-2(n1) is 1
  size-2(n2) is 2
  size-2(n3) is 2
  size-2(n4) is 4
end

Unfortunately, this still doesn’t work!

Do Now!

Use Pyret’s spy construct in size-2-h to figure out why.

15.2.3 Stage 3

Did you remember to use spy? Otherwise you may very well miss the problem! Be sure to use spy (feel free to elide the first few tests for now) to get a feel for the issue.

As you may have noted, the problem is that we want seen to be all the nodes ever seen. However, every time we return from one sub-computation, we also lose track of whatever was seen during its work. Instead, we have to also return everything that was seen, so as to properly preserve the idea that we’re computing the marginal contribution of each node.

We can do this with the following data structure: 

data Ret: ret(sz :: Number, sn :: List<BT>) end

which is returned by the helper function: 

fun size-3-h(b :: BT, seen :: List<BT>) -> Ret:
  if member-identical(seen, b):
    ret(0, seen)
  else:
    cases (BT) b:
      | mt => ret(0, seen)
      | nd(v, l, r) =>
        new-seen = link(b, seen)
        rl = size-3-h(l, new-seen)
        rr = size-3-h(r, rl.sn)
        ret(1 + rl.sz + rr.sz, rr.sn)
    end
  end
end

Note, crucially, how the seen argument for the right branch is rl.sn: i.e., everything that was already seen in the left branch. This is the crucial step that avoids the bug.

Because of this richer return type, we have to extract the actual answer for the purpose of testing: 

fun size-3(b :: BT): size-3-h(b, empty).sz end

check:
  size-3(n1) is 1
  size-3(n2) is 2
  size-3(n3) is 2
  size-3(n4) is 4
end

Exercise

Must seen be a list? What else can it be?

15.2.4 Stage 4

Observe that the Ret data structure is only of local interest. It’s purely internal to the size-3-h function; even size-3 ignores one half, and it will never be seen by the rest of the program. That is a good use of tuples, as we have seen before: Using Tuples!

fun size-4-h(b :: BT, seen :: List<BT>) -> {Number; List<BT>}:
  if member-identical(seen, b):
    {0; seen}
  else:
    cases (BT) b:
      | mt => {0; seen}
      | nd(v, l, r) =>
        new-seen = link(b, seen)
        {lsz; lsn} = size-4-h(l, new-seen)
        {rsz; rsn} = size-4-h(r, lsn)
        {1 + lsz + rsz; rsn}
    end
  end
end

fun size-4(b :: BT): size-4-h(b, empty).{0} end

check:
  size-4(n1) is 1
  size-4(n2) is 2
  size-4(n3) is 2
  size-4(n4) is 4
end

The notation {0; seen} makes an actual tuple; {Number; List<BT>} declares the contract of a tuple. Also, .{0} extracts the 0th element (the leftmost one) of a tuple.

15.2.5 Stage 5

Notice that we have the two instances of the code {0; seen}. Do they have to be that? What if we were to return {0; empty} instead in both places? Does anything break?

We might expect it to break in the case where member-identical returns true, but perhaps not in the mt case.

Do Now!

Make each of these changes. Does the outcome match your expectations?

Curiously, no! Making the change in the mt case has an effect but making it in the member-identical case doesn’t! This almost seems counter-intuitive. How can we diagnose this?

Do Now!

Use spy to determine what is going on!

Okay, so it seems like returning empty when we revisit a node doesn’t seem to do any harm. Does that mean it’s okay to make that change?

Observe that nothing has actually depended on that seen-list being empty. That’s why it appears to not matter. How can we make it matter? By making it “hurt” the computation by visiting a previously seen, but now forgotten, node yet again. So we need to visit a node at least three times: the first time to remember it; the second time to forget it; and a third time to incorrectly visit it again. Here’s a DAG that will do that: 

#|
    10
    / \
   11 12
  / \ /
 13<--
|#

n13 = nd(13, mt, mt)
n11 = nd(11, n13, n13)
n12 = nd(12, n13, mt)
n10 = nd(10, n11, n12)

check:
  size-4(n10) is 4
end

Sure enough, if either tuple now returns empty, this test fails. Otherwise it succeeds.

15.2.6 What We’ve Learned

We have learned three important principles here:

  • A pattern for dealing with programs that need “memory”. This is called threading (not in the sense of “multi-threading”, which is a kind of parallel computation, but rather the pattern of how the seen list gets passed through the program).

  • A good example of the use of tuples: local, where the documentation benefit of datatypes isn’t necessary (and the extra datatype probably just clutters up the program), as opposed to distant, where it is. In general, it’s always okay to make a new datatype; it’s only sometimes okay to use tuples in their place.

  • An important software-engineering principle, called mutation testing. This is an odd name because it would seem to be the name of a technique to test programs. Actually, it’s a technique to test test suites. You have a tested program; you then “mutate” some part of your program that you feel must change the output, and see whether any tests break. If no tests break, then either you’ve misunderstood your program or, more likely, your test suite is not good enough. Improve your test suite to catch the error in your program, or convince yourself the change didn’t matter.

    There are mutation testing tools that will randomly try to alter your program using “mutant” strategies—e.g., replacing a + with a -—and re-run your suites, and then report back on how many potential mutants the suites actually caught. But we can’t and shouldn’t only rely on tools; we can also apply the principle of mutation testing by hand, as we have above. At the very least, it will help us understand our program better!

15.2.7 More on Value Printing: An Aside from Racket

Earlier, we talked about how the standard recursive size can still be thought of as a “size of printed value” computation. However, that actually depends on your language’s value printer.

In Racket, you can turn on (it’s slightly more expensive, so off by default) a value-printer that shows value sharing: Language | Choose Language … | Show Details | Show sharing in values. So if we take the data definition above and translate it into Racket structures

(struct mt () #:transparent)

(struct nd (v l r) #:transparent)

and then construct (almost) the same data as in the first example:

(define n1 (nd 1 (mt) (mt)))

(define n2 (nd 2 (mt) n1))

(define n3 (nd 3 n1 (mt)))

(define n4 (nd 4 n2 n3))

and then ask Racket to print it, we get:

> n4

(nd 4 (nd 2 (mt) #0=(nd 1 (mt) (mt))) (nd 3 #0# (mt)))

The #0= notation is the moral equivalent of saying, “I’m going to refer to this value again later, so let’s call it the 0th value” and #0# is saying “Here I’m referring to the aforementioned 0th value”.

(Yes, there can be more than one shared value in an output, so each is given a different “name”. We’ll see that in a moment.)

The later example above translates to

(define n13 (nd 13 (mt) (mt)))

(define n11 (nd 11 n13 n13))

(define n12 (nd 12 n13 (mt)))

(define n10 (nd 10 n11 n12))

which prints as

> n10

(nd 10 (nd 11 #0=(nd 13 (mt) (mt)) #0#) (nd 12 #0# (mt)))

So it is possible for a language to reflect the sharing in its output. It’s just that most programming languages choose to not do that, even optionally.

Remember the “almost” above? What was that about?

In Racket, we’ve made a new instance of mt over and over. We can more accurately reflect what is happening in Pyret by instantiating it only once:

(struct mt () #:transparent)

(define the-mt (mt))

(struct nd (v l r) #:transparent)

We then rewrite the earlier example to use that one instance only:

(define n1 (nd 1 the-mt the-mt))

(define n2 (nd 2 the-mt n1))

(define n3 (nd 3 n1 the-mt))

(define n4 (nd 4 n2 n3))

And now when we print it:

> n4

(nd 4 (nd 2 #0=(mt) #1=(nd 1 #0# #0#)) (nd 3 #1# #0#))

And now you can see there are two different shared values, one is the single instance of mt, the other is the nd with 1 in it. Thus, Racket uses both #0= / #0# and #1= / #1#. Notice how all the leaves are sharing the same mt instance. (The numbering is picked in the order in which nodes are encountered while traversing, which is why the nd instance was #0 the previous time and is #1 this time.)

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