21 Avoiding Recomputation by Remembering Answers

We have on several instances already referred to a ☛ space-time tradeoff. The most obvious tradeoff is when a computation “remembers” prior results and, instead of recomputing them, looks them up and returns the answers. This is an instance of the tradeoff because it uses space (to remember prior answers) in place of time (recomputing the answer). Let’s see how we can write such computations.

21.1 An Interesting Numeric Sequence

Suppose we want to create properly-parenthesized expressions, and ignore all non-parenthetical symbols. How many ways are there of creating parenthesized expressions given a certain number of opening (equivalently, closing) parentheses?

If we have zero opening parentheses, the only expression we can create is the empty expression. If we have one opening parenthesis, the only one we can construct is “()” (there must be a closing parenthesis since we’re interested only in properly-parenthesized expressions). If we have two opening parentheses, we can construct “(())” and “()()”. Given three, we can construct “((()))”, “(())()”, “()(())”, “()()()”, and “(()())”, for a total of five. And so on. Observe that the solutions at each level use all the possible solutions at one level lower, combined in all the possible ways.

fun catalan(n):
  if n == 0: 1
  else if n > 0:
    for fold(acc from 0, k from range(0, n)):
      acc + (catalan(k) * catalan(n - 1 - k))
    end
  end
end

check:
  catalan(0) is 1
  catalan(1) is 1
  catalan(2) is 2
  catalan(3) is 5
  catalan(4) is 14
  catalan(5) is 42
  catalan(6) is 132
  catalan(7) is 429
  catalan(8) is 1430
  catalan(9) is 4862
  catalan(10) is 16796
  catalan(11) is 58786
end

Do Now!

Check at what value you start to observe a significant slowdown on your machine. Plot the graph of running time against input size. What does this suggest?

The reason the Catalan computation takes so long is precisely because of what we alluded to earlier: at each level, we depend on computing the Catalan number of all the smaller levels; this computation in turn needs the numbers of all of its smaller levels; and so on down the road.

Exercise

Map the subcomputations of catalan to see why the computation time explodes as it does. What is the worst-case time complexity of this function?

Here is a graphical representation of all the sub-computations the Catalan function does for input 3:

Observe the very symmetric computation, reflecting the formula.

21.1.1 Using State to Remember Past Answers

Therefore, this is clearly a case where trading space for time is likely to be of help. How do we do this? We need a notion of memory that records all previous answers and, on subsequent attempts to compute them, checks whether they are already known and, if so, just returns them instead of recomputing them.

Do Now!

What critical assumption is this based on?

Naturally, this assumes that for a given input, the answer will always be the same. As we have seen, functions with state violate this liberally, so typical stateful functions cannot utilize this optimization. Ironically, we will use state to implement this optimization, so we will have a stateful function that always returns the same answer on a given input—and thereby use state in a stateful function to simulate a stateless one. Groovy, dude!

data MemoryCell:
  | mem(in, out)
end

var memory :: List<MemoryCell> = empty

fun catalan(n :: Number) -> Number:
  answer = find(lam(elt): elt.in == n end, memory)
  cases (Option) answer block:
    | none =>
      result =
        if n == 0: 1
        else if n > 0:
          for fold(acc from 0, k from range(0, n)):
            acc + (catalan(k) * catalan(n - 1 - k))
          end
        end
      memory := link(mem(n, result), memory)
      result
    | some(v) => v.out
  end
end

Do Now!

Trace through a call of this revised function and see how many calls it makes.

Here is a revised visualization of computing for input 3:

Observe the asymmetric computation: the early calls perform the computations, while the latter calls simply look up the results.

This process, of converting a function into a version that remembers its past answers, is called memoization.

21.1.2 From a Tree of Computation to a DAG

What we have subtly done is to convert a tree of computation into a DAG over the same computation, with equivalent calls being reused. Whereas previously each call was generating lots of recursive calls, which induced still more recursive calls, now we are reusing previous recursive calls—i.e., sharing the results computed earlier. This, in effect, points the recursive call to one that had occurred earlier. Thus, the shape of computation converts from a tree to a DAG of calls.

This has an important complexity benefit. Whereas previously we were performing a super-exponential number of calls, now we perform only one call per input and share all previous calls—thereby reducing catalan(n) to take a number of fresh calls proportional to n. Looking up the result of a previous call takes time proportional to the size of memory (because we’ve represented it as a list; better representations would improve on that), but that only contributes another linear multiplicative factor, reducing the overall complexity to quadratic in the size of the input. This is a dramatic reduction in overall complexity. In contrast, other uses of memoization may result in much less dramatic improvements, turning the use of this technique into a true engineering trade-off.

21.1.3 The Complexity of Numbers

As we start to run larger computations, however, we may start to notice that our computations are starting to take longer than linear growth. This is because our numbers are growing arbitrarily large—for instance, catalan(100) is 896519947090131496687170070074100632420837521538745909320—and computations on numbers can no longer be constant time, contrary to what we said earlier [The Size of the Input]. Indeed, when working on cryptographic problems, the fact that operations on numbers do not take constant time are absolutely critical to fundamental complexity results (and, for instance, the presumed unbreakability of contemporary cryptography). (See also Factoring Numbers.)

21.1.4 Abstracting Memoization

Now we’ve achieved the desired complexity improvement, but there is still something unsatisfactory about the structure of our revised definition of catalan: the act of memoization is deeply intertwined with the definition of a Catalan number, even though these should be intellectually distinct. Let’s do that next.

In effect, we want to separate our program into two parts. One part defines a general notion of memoization, while the other defines catalan in terms of this general notion.

data MemoryCell:
  | mem(in, out)
end

fun memoize-1<T, U>(f :: (T -> U)) -> (T -> U):

  var memory :: List<MemoryCell> = empty

  lam(n):
    answer = find(lam(elt): elt.in == n end, memory)
    cases (Option) answer block:
      | none =>
        result = f(n)
        memory := link(mem(n, result), memory)
        result
      | some(v) => v.out
    end
  end
end

rec catalan :: (Number -> Number) =
  memoize-1(
    lam(n):
      if n == 0: 1
      else if n > 0:
        for fold(acc from 0, k from range(0, n)):
          acc + (catalan(k) * catalan(n - 1 - k))
        end
      end
    end)

Exercise

Why is sharing memoization tables a bad idea? Be concrete.

21.2 Edit-Distance for Spelling Correction

Text editors, word processors, mobile phones, and various other devices now routinely implement spelling correction or offer suggestions on (mis-)spellings. How do they do this? Doing so requires two capabilities: computing the distance between words, and finding words that are nearby according to this metric. In this section we will study the first of these questions. (For the purposes of this discussion, we will not dwell on the exact definition of what a “word” is, and just deal with strings instead. A real system would need to focus on this definition in considerable detail.)

Do Now!

Think about how you might define the “distance between two words”. Does it define a metric space?

Exercise

Will the definition we give below define a metric space over the set of words?

Exercise

Observe that we have not included the triangle inequality relative to the properties of a metric. Why not? If we don’t need the triangle inequality, does this let us define more interesting distance functions that are not metrics?

There are several variations of this definition possible. For now, we will consider the simplest one, which assumes that each of these errors has equal cost. For certain input devices, we may want to assign different costs to these mistakes; we might also assign different costs depending on what wrong character was typed (two characters adjacent on a keyboard are much more likely to be a legitimate error than two that are far apart). We will return briefly to some of these considerations later [Nature as a Fat-Fingered Typist].

check:
  levenshtein(empty, empty) is 0
  levenshtein([list: "x"], [list: "x"]) is 0
  levenshtein([list: "x"], [list: "y"]) is 1
  # one of about 600
  levenshtein(
    [list: "b", "r", "i", "t", "n", "e", "y"],
    [list: "b", "r", "i", "t", "t", "a", "n", "y"])
    is 3
  # http://en.wikipedia.org/wiki/Levenshtein_distance
  levenshtein(
    [list: "k", "i", "t", "t", "e", "n"],
    [list: "s", "i", "t", "t", "i", "n", "g"])
    is 3
  levenshtein(
    [list: "k", "i", "t", "t", "e", "n"],
    [list: "k", "i", "t", "t", "e", "n"])
    is 0
  # http://en.wikipedia.org/wiki/Levenshtein_distance
  levenshtein(
    [list: "S", "u", "n", "d", "a", "y"],
    [list: "S", "a", "t", "u", "r", "d", "a", "y"])
    is 3
  # http://www.merriampark.com/ld.htm
  levenshtein(
    [list: "g", "u", "m", "b", "o"],
    [list: "g", "a", "m", "b", "o", "l"])
    is 2
  # http://www.csse.monash.edu.au/~lloyd/tildeStrings/Alignment/92.IPL.html
  levenshtein(
    [list: "a", "c", "g", "t", "a", "c", "g", "t", "a", "c", "g", "t"],
    [list: "a", "c", "a", "t", "a", "c", "t", "t", "g", "t", "a", "c", "t"])
    is 4
  levenshtein(
    [list: "s", "u", "p", "e", "r", "c", "a", "l", "i",
      "f", "r", "a", "g", "i", "l", "i", "s", "t" ],
    [list: "s", "u", "p", "e", "r", "c", "a", "l", "y",
      "f", "r", "a", "g", "i", "l", "e", "s", "t" ])
    is 2
end

rec levenshtein :: (List<String>, List<String> -> Number) =
  <levenshtein-body>

lam(s, t):
  <levenshtein-both-empty>
  <levenshtein-one-empty>
  <levenshtein-neither-empty>
end

if is-empty(s) and is-empty(t): 0

else if is-empty(s): t.length()
else if is-empty(t): s.length()

else:
  if s.first == t.first:
    levenshtein(s.rest, t.rest)
  else:
    min3(
      1 + levenshtein(s.rest, t),
      1 + levenshtein(s, t.rest),
      1 + levenshtein(s.rest, t.rest))
  end
end

fun min3(a :: Number, b :: Number, c :: Number):
  num-min(a, num-min(b, c))
end

This algorithm will indeed pass all the tests we have written above, but with a problem: the running time grows exponentially. That is because, each time we find a mismatch, we recur on three subproblems. In principle, therefore, the algorithm takes time proportional to three to the power of the length of the shorter word. In practice, any prefix that matches causes no branching, so it is mismatches that incur branching (thus, confirming that the distance of a word with itself is zero only takes time linear in the size of the word).

Observe, however, that many of these subproblems are the same. For instance, given “kitten” and “sitting”, the mismatch on the initial character will cause the algorithm to compute the distance of “itten” from “itting” but also “itten” from “sitting” and “kitten” from “itting”. Those latter two distance computations will also involve matching “itten” against “itting”. Thus, again, we want the computation tree to turn into a DAG of expressions that are actually evaluated.

data MemoryCell2<T, U, V>:
  | mem(in-1 :: T, in-2 :: U, out :: V)
end

fun memoize-2<T, U, V>(f :: (T, U -> V)) -> (T, U -> V):

  var memory :: List<MemoryCell2<T, U, V>> = empty

  lam(p, q):
    answer = find(
      lam(elt): (elt.in-1 == p) and (elt.in-2 == q) end,
      memory)
    cases (Option) answer block:
      | none =>
        result = f(p, q)
        memory :=
        link(mem(p, q, result), memory)
        result
      | some(v) => v.out
    end
  end
end

rec levenshtein :: (List<String>, List<String> -> Number) =
  memoize-2(
    lam(s, t):
      if is-empty(s) and is-empty(t): 0
      else if is-empty(s): t.length()
      else if is-empty(t): s.length()
      else:
        if s.first == t.first:
          levenshtein(s.rest, t.rest)
        else:
          min3(
            1 + levenshtein(s.rest, t),
            1 + levenshtein(s, t.rest),
            1 + levenshtein(s.rest, t.rest))
        end
      end
    end)

The complexity of this algorithm is still non-trivial. First, let’s introduce the term suffix: the suffix of a string is the rest of the string starting from any point in the string. (Thus “kitten”, “itten”, “ten”, “n”, and “” are all suffixes of “kitten”.) Now, observe that in the worst case, starting with every suffix in the first word, we may need to perform a comparison against every suffix in the second word. Fortunately, for each of these suffixes we perform a constant computation relative to the recursion. Therefore, the overall time complexity of computing the distance between strings of length \(m\) and \(n\) is \(O([m, n \rightarrow m \cdot n])\). (We will return to space consumption later [Contrasting Memoization and Dynamic Programming].)

Exercise

Modify the above algorithm to produce an actual (optimal) sequence of edit operations. This is sometimes known as the traceback.

21.3 Nature as a Fat-Fingered Typist

We have talked about how to address mistakes made by humans. However, humans are not the only bad typists: nature is one, too!

When studying living matter we obtain sequences of amino acids and other such chemicals that comprise molecules, such as DNA, that hold important and potentially determinative information about the organism. These sequences consist of similar fragments that we wish to identify because they represent relationships in the organism’s behavior or evolution.This section may need to be skipped in some states and countries. Unfortunately, these sequences are never identical: like all low-level programmers, nature slips up and sometimes makes mistakes in copying (called—wait for it—mutations). Therefore, looking for strict equality would rule out too many sequences that are almost certainly equivalent. Instead, we must perform an alignment step to find these equivalent sequences. As you might have guessed, this process is very much a process of computing an edit distance, and using some threshold to determine whether the edit is small enough.To be precise, we are performing local sequence alignment. This algorithm is named, after its creators, Smith-Waterman, and because it is essentially identical, has the same complexity as the Levenshtein algorithm.

The only difference between traditional presentations of Levenshtein and Smith-Waterman is something we alluded to earlier: why is every edit given a distance of one? Instead, in the Smith-Waterman presentation, we assume that we have a function that gives us the gap score, i.e., the value to assign every character’s alignment, i.e., scores for both matches and edits, with scores driven by biological considerations. Of course, as we have already noted, this need is not peculiar to biology; we could just as well use a “gap score” to reflect the likelihood of a substitution based on keyboard characteristics.

21.4 Dynamic Programming

We have used memoization as our canonical means of saving the values of past computations to reuse later. There is another popular technique for doing this called dynamic programming. This technique is closely related to memoization; indeed, it can be viewed as the dual method for achieving the same end. First we will see dynamic programming at work, then discuss how it differs from memoization.

Dynamic programming also proceeds by building up a memory of answers, and looking them up instead of recomputing them. As such, it too is a process for turning a computation’s shape from a tree to a DAG of actual calls. The key difference is that instead of starting with the largest computation and recurring to smaller ones, it starts with the smallest computations and builds outward to larger ones.

We will revisit our previous examples in light of this approach.

21.4.1 Catalan Numbers with Dynamic Programming

MAX-CAT = 11

answers :: Array<Option<Number>> = array-of(none, MAX-CAT + 1)

fun catalan(n):
  cases (Option) array-get-now(answers, n):
    | none => raise("looking at uninitialized value")
    | some(v) => v
  end
end

fun fill-catalan(upper) block:
  array-set-now(answers, 0, some(1))
  when upper > 0:
    for each(n from range(1, upper + 1)):
      block:
        cat-at-n =
          for fold(acc from 0, k from range(0, n)):
            acc + (catalan(k) * catalan(n - 1 - k))
          end
        array-set-now(answers, n, some(cat-at-n))
      end
    end
  end
end

fill-catalan(MAX-CAT)

Notice that we have had to undo the natural recursive definition—which proceeds from bigger values to smaller ones—to instead use a loop that goes from smaller values to larger ones. In principle, the program has the danger that when we apply catalan to some value, that index of answers will have not yet been initialized, resultingin an error. In fact, however, we know that because we fill all smaller indices in answers before computing the next larger one, we will never actually encounter this error. Note that this requires careful reasoning about our program, which we did not need to perform when using memoization because there we made precisely the recursive call we needed, which either looked up the value or computed it afresh.

21.4.2 Levenshtein Distance and Dynamic Programming

fun levenshtein(s1 :: List<String>, s2 :: List<String>) block:
  <levenshtein-dp/1>
end

s1-len = s1.length()
s2-len = s2.length()
answers = array2d(s1-len + 1, s2-len + 1, none)
<levenshtein-dp/2>

Exercise

Define the functions

array2d :: Number, Number, A -> Array<A>
set-answer :: Array<A>, Number, Number, A -> Nothing
get-answer :: Array<A>, Number, Number -> A

fun put(s1-idx :: Number, s2-idx :: Number, n :: Number):
  set-answer(answers, s1-idx, s2-idx, some(n))
end
fun lookup(s1-idx :: Number, s2-idx :: Number) -> Number block:
  a = get-answer(answers, s1-idx, s2-idx)
  cases (Option) a:
    | none => raise("looking at uninitialized value")
    | some(v) => v
  end
end

for each(s1i from range(0, s1-len + 1)):
  put(s1i, 0, s1i)
end
for each(s2i from range(0, s2-len + 1)):
  put(0, s2i, s2i)
end
<levenshtein-dp/4>

for each(s1i from range(0, s1-len)):
  for each(s2i from range(0, s2-len)):
  <levenshtein-dp/compute-dist>
  end
end
<levenshtein-dp/get-result>

Do Now!

Is this strictly true?

No, it isn’t. We did first fill in values for the “borders” of the table. This is because doing so in the midst of <levenshtein-dp/compute-dist> would be much more annoying. By initializing all the known values, we keep the core computation cleaner. But it does mean the order in which we fill in the table is fairly complex.

dist =
  if get(s1, s1i) == get(s2, s2i):
    lookup(s1i, s2i)
  else:
    min3(
      1 + lookup(s1i, s2i + 1),
      1 + lookup(s1i + 1, s2i),
      1 + lookup(s1i, s2i))
  end
put(s1i + 1, s2i + 1, dist)

lookup(s1-len, s2-len)

fun levenshtein(s1 :: List<String>, s2 :: List<String>) block:
  s1-len = s1.length()
  s2-len = s2.length()
  answers = array2d(s1-len + 1, s2-len + 1, none)

  fun put(s1-idx :: Number, s2-idx :: Number, n :: Number):
    set-answer(answers, s1-idx, s2-idx, some(n))
  end
  fun lookup(s1-idx :: Number, s2-idx :: Number) -> Number block:
    a = get-answer(answers, s1-idx, s2-idx)
    cases (Option) a:
      | none => raise("looking at uninitialized value")
      | some(v) => v
    end
  end

  for each(s1i from range(0, s1-len + 1)):
    put(s1i, 0, s1i)
  end
  for each(s2i from range(0, s2-len + 1)):
    put(0, s2i, s2i)
  end

  for each(s1i from range(0, s1-len)):
    for each(s2i from range(0, s2-len)):
      dist =
        if get(s1, s1i) == get(s2, s2i):
          lookup(s1i, s2i)
        else:
          min3(
            1 + lookup(s1i, s2i + 1),
            1 + lookup(s1i + 1, s2i),
            1 + lookup(s1i, s2i))
        end
      put(s1i + 1, s2i + 1, dist)
    end
  end

  lookup(s1-len, s2-len)
end

21.5 Contrasting Memoization and Dynamic Programming

Now that we’ve seen two very different techniques for avoiding recomputation, it’s worth contrasting them. The important thing to note is that memoization is a much simpler technique: write the natural recursive definition; determine its time complexity; decide whether this is problematic enough to warrant a space-time trade-off; and if it is, apply memoization. The code remains clean, and subsequent readers and maintainers will be grateful for that. In contrast, dynamic programming requires a reorganization of the algorithm to work bottom-up, which can often make the code harder to follow and full of subtle invariants about boundary conditions and computation order.

That said, the dynamic programming solution can sometimes be more computationally efficient. For instance, in the Levenshtein case, observe that at each table element, we (at most) only ever use the ones that are from the previous row and column. That means we never need to store the entire table; we can retain just the fringe of the table, which reduces space to being proportional to the sum, rather than product, of the length of the words. In a computational biology setting (when using Smith-Waterman), for instance, this saving can be substantial. This optimization is essentially impossible for memoization.

From a software design perspective, there are two more considerations.

First, the performance of a memoized solution can trail that of dynamic programming when the memoized solution uses a generic data structure to store the memo table, whereas a dynamic programming solution will invariably use a custom data structure (since the code needs to be rewritten against it anyway). Therefore, before switching to dynamic programming for performance reasons, it makes sense to try to create a custom memoizer for the problem: the same knowledge embodied in the dynamic programming version can often be encoded in this custom memoizer (e.g., using an array instead of list to improve access times). This way, the program can enjoy speed comparable to that of dynamic programming while retaining readability and maintainability.

Second, suppose space is an important consideration and the dynamic programming version can make use of significantly less space. Then it does make sense to employ dynamic programming instead. Does this mean the memoized version is useless?

Do Now!

What do you think? Do we still have use for the memoized version?

Yes, of course we do! It can serve as an oracle [Oracles for Testing] for the dynamic programming version, since the two are supposed to produce identical answers anyway—and the memoized version would be a much more efficient oracle than the purely recursive implemenation, and can therefore be used to test the dynamic programming version on much larger inputs.

In short, always first produce the memoized version. If you need more performance, consider customizing the memoizer’s data structure. If you need to also save space, and can arrive at a more space-efficient dynamic programming solution, then keep both versions around, using the former to test the latter (the person who inherits your code and needs to alter it will thank you!).

Exercise

We have characterized the fundamental difference between memoization and dynamic programming as that between top-down, depth-first and bottom-up, breadth-first computation. This should naturally raise the question, what about:

• top-down, breadth-first

• bottom-up, depth-first

orders of computation. Do they also have special names that we just happen to not know? Are they uninteresting? Or do they not get discussed for a reason?