Another Look at Dynamic Programming

Whilst on the tube today, I overheard a mother teaching her child how to count, using a method likely to be extremely familiar to many – fingers. The child counted correctly from one to ten, and then the mother added her hands too and asked the child to count how many fingers there were now.

“One, two, three -“

And so on, till twenty. The mother then attempted to explain that it would have been faster if the child continued from ten, rather than starting again. Although it wasn’t really an example of the concept, the words dynamic programming immediately shot to the front of my mind. I initially found this to be a rather confusing concept to grasp (let’s say that up to high school programming contests, if a problem wasn’t solvable by exhaustive search or greedy algorithms I’d likely have struggled), so I figured a post on it might be worthwhile.

(This isn’t really an example of DP; I’d say it’s closer to divide and conquer plus the use of a cache. We’ve cached the answer that the child has ten fingers, and identified the problem as being taking the sum of the child’s and parent’s fingers. Note that because of the possibility of amputation or polydactyly, the subproblems are not the same – and, specifically, saying 2 * 10 = 20 isn’t generally correct.)

Essentially, the key idea behind dynamic programming (DP) is that we save time by not re-doing work that we’ve already done, by remembering the results to intermediate steps. Of course, this tends to mean that there’s a space overhead. This is generally useful in cases where a problem is too large to solve, yet it can be decomposed into smaller pieces, and importantly we can combine optimal solutions to these smaller pieces, to get a solution that is optimal for the original problem. (More formally, this is known as optimal substructure.)

Furthermore, we want to get some significant benefit out of actually remembering the answers (in practice, we want to use our previous work multiple times; this manifests in the form of overlapping subproblems). This is what would distinguish an approach as being a DP-based one, as opposed to divide and conquer.

Of course, the fingers example is trivial. There are many other natural examples (the ones that come to mind first for me include knapsack problems and route-planning), though I’m not sure I directly apply DP that much in a natural context (although quite a few days have tasklists that could be done solving an ordered constrained TSP, the last time I used the Held-Karp algorithm was probably for my third year project). It certainly does see many applications that are relevant to daily life (error correction in search queries / autocorrect via Levenshtein distance; not sure how they are actually implemented but routing applications like Citymapper and Google Maps are likely to involve such algorithms as well).

In terms of implementation, the cache-based “top-down” solution was what I learned first, and to me at least was intuitively easier to understand. When you encounter a subproblem, you check a lookup table to see if you’ve done the problem before; if you have, you just take the answer from that. If you haven’t, solve the problem the hard way (this may involve more subproblems – when solving these, it’s important to look at the table again), and then (important!) store the answer you obtained back in the table.

The alternative “bottom-up” method involves generating solutions to smaller subproblems, and using these to build up the solution to a bigger problem. I’d probably first actually used a method along these lines when introduced to the Fibonacci sequence (probably in year 4 or so) – I remember being asked to compute F_{13} and did something like “1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, uh… 144, 233”. (This is linear time. It’s possible to do better via matrix exponentiation, or Binet’s formula – bonus points if you pair the exponentiation with a fancy multiplication algorithm like Karatsuba or even Schonhage-Strassen.)

From a computing point of view there can be both benefits and disadvantages to this versus the “top-down” method.

  • Ease of understanding and/or code readability are likely to depend on the problem; for Fibonacci I would prefer bottom-up, but I usually find the top-down case to be more approachable (it’s more intuitive to me at least to think “here’s how I decompose this problem” as opposed to “here’s how I build a solution from smaller solutions”).
  • The top-down approach might be able to solve some problems without necessarily computing all smaller subproblems that a bottom-up solution counting up from 0 or 1 might deal with. You can, of course, implement this in a bottom-up solution… provided you know how to compute the required subproblems in a way that isn’t itself too costly. With a top-down approach you get this “avoidance” for free.
  • As an extension of the previous point: for bottom-up you’ll need to figure out a safe ordering to go through the subproblems (you can’t have a solution depending on something that hasn’t been computed yet). This is easy in most cases (*cough* Fibonacci), but can be extremely difficult in others (chess transposition tables come to mind; problems with online input, many base cases and a massive domain).
  • Recursive implementations (which tend to be top-down, though could plausibly be in either direction; it’s possible to maintain your own call stack on the heap, or pass some kind of lookup table around) incur the overhead of function calls, and can cause stack overflows for large problems.
  • Handling limited memory (there are many 2D array problems for which only the last row of results needs to be kept; alternatively with Fibonacci we only need the last two results) tends to be more naturally expressed with the bottom up method (though of course, you can clean the top-down cache). This is probably because you’ll have defined an order for solving the subproblems, which may not be as immediately clear with the top-down method.

Note that although this is a powerful tool, there are quite a number of cases where you don’t actually need to consider all of the ways of decomposing a problem into subproblems. A well-known example would be the activity selection problem; given a set of mutually exclusive activities with start and end times, find the largest set of activities I can participate in. I can solve this optimally by sorting events by their ending time, and aggressively picking events to fill my schedule where feasible. The key differentiator here is what’s known as the greedy choice property; that making an optimal choice at each step gives us the overall optimal solution.

In practice anyway it’s highly unlikely that I’d weight my activities equally, so we then get to the weighted activity selection problem, and the greedy method no longer works (but we can still use dynamic programming – as before, sort the activities by their ending time E, and for each activity, pick the better of not attending it, or attending it and behaving optimally before the start time of said activity).

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