Bellman, Dynamic Programming, and Edit Distance



Approaches to the exact edit distance problem



Richard Bellman created the method of dynamic programming in 1953. He was honored with many awards for this and other seminal work, during his lifetime. We probably all know dynamic programming, use it often to design algorithms, and perhaps take it for granted. But, it is one of the great discoveries that changed the theory and practice of computation. One of the surprises–at least to me–is given its wide range of application, it often yields the fastest known algorithm. See our friends at Wikipedia for a nice list of examples of dynamic programming in action.

I never had the honor of meeting Bellman, but he wrote a terrific short autobiographical piece that contained many interesting stories. (I cannot locate it anymore: if you can please let me know.) One story that I recall was how, when he was at The Rand Corporation, he often made extra money playing poker with his doctor friends in the evenings. He usually won, which convinced them that he was using “deep mathematics” against them–perhaps probability theory. Bellman said the truth was much simpler: during the day the doctors were stressed making life and death decisions, at night when they played poker they “let go”. During the day Bellman could just look out at the Pacific Ocean from his office and think about problems. At night he played tight simple poker, and he cleaned up.

One other story from this piece made a long-lasting impression on me. When he was famous he would often be invited to places to give talks, like many scientists. However, he always insisted that he must be picked up at the airport by a driver of a white limo. I understand the driver part, I understand the limo part, but insisting on the color was a very nice touch.

The Edit Distance Problem

The edit distance problem (EDP) is given two strings

determine their edit distance. That is what are the fewest operations that are needed to transform string into . The operations allowed are: insert a character, delete a character, and substitute one character for another. In general their costs can be arbitrary, but for us we will consider only the basic case where insert/delete have unit cost and substitute has cost two. This makes sense, since substitute is the same as one delete and one insert.

This distance is sometimes called the Levenshtein distance, named after Vladimir Levenshtein, who defined it in 1965. The notation of edit distance arises in a multlude of places and contexts, I believe that the notion has been repeatedly discovered. It also has many generalizations where more complex operations are allowed.

People really care about EDP. There are very good heuristic algorithms that are used, for example, in the biological community to solve EDP. One is called BLAST. It is implemented in many languages and runs on everything, there are even special purpose hardware devices that run BLAST. Clearly, there is need for solutions to EDP. Unlike theory algorithms, BLAST has no provable error bounds; unlike theory algorithms, BLAST seems, in practice, to be good enough for the scientists who use it. However, I have no doubt that they would prefer to get the optimal answer–however, the cost of getting the optimal answer cannot be too great.

Upper and Lower Bounds

The dynamic programming algorithm for EDP has been discovered and rediscovered many times. I believe that Robert Wagner and Mike Fischer did it first, but I will not try to be the perfect historian. Their algorithm runs in time and uses the same space: it is convenient to assume that for the rest of this post. With a little bit of work, it is easy to improve the space bound to

There are papers that prove a lower bound of for EDP. These papers assume a restricted model of computation: only input symbols can be compared for equality. Like any restricted lower bound they give you an idea of what not to do: if you wish to beat the the lower bound, then you must do more that compare symbols for equality. Use the value of the symbols, use bit operations, use randomness, do something other than just compare symbols.

Actually, as I recall, there was a bit of an inversion here: there was a better upper bound paper that gave an algorithm for EDP that took time , before the lower bound papers were published. The method used to prove the sub-quadratic algorithm for EDP is based on a general idea called The Four Russian Method. I never liked the name, I do like the method. As Dan Gusfield points out, the four were not even all Russian: perhaps a consequence of the cold war was that information did not always flow easily between east and west.

The Four Russian Method was first, I believe, used to compute boolean matrix products faster than cubic time. The method has been used since then to solve many other problems. If you do not know the method, take a look at the following link.

Still here. Okay, here is an overview of the method: it is based on trading space for time. A typical Four Russian Method algorithm operates in two phases. In the first phase, it examines the input and cleverly pre-computes values to a large number of subproblems, and stores these values in a look-up table. In the second phase, it uses the pre-computed values from the table to make macro steps during the rest of the computation. The result is usually a reduction in time by a logarithmic factor, while the space becomes as large as the time. The method is not generally practical; however, it is a useful method to know.

Bit Complexity

Actually there is a logarithmic factor that is hidden in the quadratic dynamic programming algorithm for EDP, since the algorithm must use size numbers. Thus, the bit complexity is, . There is a clever way to remove the factor of that is due to Dan Lopresti:

Theorem 1 There is an algorithm for EDP that uses bit operations and space.

Proof: The key idea is to use only a constant number of bits rather than bits and still run the standard dynamic program.

I will explain this in more detail in a future post. Truth in blogging: Lopresti was once a graduate student of mine, and I worked with him on this result.

Approximation and EDP

Legend has it that when faced with unravelling the famous Gordian Knot, Alexander the Great simply cut the knot with his sword. We do this all the time in theory: when faced by a “knotty” problem we often change the problem. The edit distance problem is any not different: years ago the problem of exact distance was changed to approximate edit distance. The goal is to get a fast algorithm that finds the distance to within a small relative error. For example, in the STOC 2009 conference there will be a paper “Approximating Edit Distance in Near-Linear Time” by Alexandr Andoni and Krzysztof Onak. They show that there is an algorithm (AO) that in time gets a relative error of

This is a huge improvement over the previous work. I will not go into any details today, see their paper for the full proofs.

This is very pretty work. I have started to look at their paper and believe that it has many interesting ideas, they may even help solve other problems. My issue is that we still are no closer to solving the edit distance problem. The AO algorithm still makes too large relative error to be practical–what exactly is the factor for reasonable size ? See my other posts on the use of “ O-notation.”

On the other hand, I am intrigued with their breakthrough. I think that it may be possible to use their new algorithm as a basis of an exact algorithm. The key idea I have in mine is to try to use their algorithm combined with some kind of “amplification trick.”

Amplification

Papers on approximations to the EDP use “relative” error, so let’s take a look at additive error. Suppose that is an algorithm for the EDP that works for any fixed size alphabet. Say that has additive error provided, for all strings of length over the alphabet,

There is a simple amplification trick:

Theorem 1 Suppose there is an algorithm for the EDP that runs in time and has additive error . Then, there is an algorithm for the EDP that runs in time and has an additive error of .

I assume that this is well known, but in case not, here is a sketch of the proof. Suppose that and are strings of length over the alphabet . Consider the following two strings:

The claim is that provided ,

It should be clear that

since one way to change to is to edit the strings and and leave the ‘s alone. The cost of this is twice the cost of editing to .

The opposite inequality follows since any matching of the first with the second (or the other way around: second with first ) must delete all the ‘s and this will cost more than This claim follows since are over an alphabet that does not include . Thus,

and therefore,

Divide by and we see that is within additive error of the edit distance of and .

I conjecture that there is a general construction that can replace by an arbitrary . This would yield an algorithm that runs in time and has additive error of for some absolute constant . Note, we can repeat the above construction; however, the obvious application would require a size alphabet.

If such an amplification were possible, then we could do the following, for example: if there is an algorithm that runs in near linear time, i.e. time , for EDP and has additive error of , then there is a near linear time exact algorithm for EDP. This follows by just setting and doing the calculation.

I have started thinking about the AO algorithm and how to combine it with different amplification methods. Perhaps this could lead, finally, to a fast exact algorithm for EDP. Perhaps. In any event such amplification methods seem potentially useful.

Open Problems

Thus, the main open problem is clear: find an EDP algorithm that runs in time. Or one that runs in time, or near linear time. Or one that runs in I see no reason, even after decades of failure, to believe that there could not be an exact algorithm that runs fast and solves the EDP. This is–in my opinion–an important and exciting open problem. Let’s finally solve the problem.

There are other, perhaps, more approachable open problems. In real applications the inputs are not worst case, nor are they random. For instance, the BLAST program assumes implicitly a kind of randomness to the inputs. Are there reasonable models of the kinds of inputs that are actually encountered? And for such a model can we get provable fast algorithms for EDP?