Regarding problems which can be solved by dynamic programming techniques
Jan 22, 2015

I have recently watched quite a few videos on dynamic programming. What is dynamic programming? It’s a method for constructing efficient solutions to some specific problems. In order to apply the method, there are some criteria needed to be met.

• The problem should have a brute-force solution/exhaustive search solution.
• The problem should be dividable into sub-problems with optimal solutions (explained further down)
• The problem should have a recursive definition.

NOTE: I’m not doing this in a formal manner, I simply think out loud in order to understand the subject better.

Let’s use these criteria to make up a problem which we could solve.

Definition of optimal

I’m thinking of the second criteria, that the problem should be dividable into sub-problems with optimal solutions. What do I mean by that? Well, being optimal means that if the original problem is split into parts, and you find a solution for a part, you know that the solution is the best possible solution for that part. Simple as that.

A recursive thought

You can use these “smaller” solutions to construct a “bigger” solution for the original problem. This is obviously a recursive thought.

• “I start with a lot of small simple problems, and I use them to construct a bigger solution.”

It is however a lot more common to state (or think about) recursion the other way around.

• “In order to find a solution to this problem, I need to build it up from smaller solutions by dividing the problem into sub-problems. When the sub-problems become small enough, the solutions will be trivial.”

The second statement is a lot longer when expressed in text, but when expressed with math, it becomes a lot more concise as well as more beautiful.

Brute force?

Well, this part I am not so sure about. Something in the back of my mind tells me that the problems in question should be solvable by performing some kind of exhaustive search (trying all variants of solutions which could possibly be generated) in a (what computer scientists would call a) non-efficient manner (referring to that it would take A LOT of time if the problem is big - usually exponential compared to the size of input).

Do you have a problem?

I think before I dwell any further into what the dynamic programming method is all about, I’d like to try and see if I can come up with a good problem. I’ve recently come to realize that constructing problems can be just as efficient as a learning tool as actually solving them. Conversation with myself follows:

• A1: Let’s pick a subject!
• A2: Sports!
• A1: Okay, what now… Where’s the recursion?
• A2: I’m thinking a football league… NO! Ice-hockey!
• A1: Okay, keep going.
• A2: Well, let’s say there are 12 teams like in the Swedish Hockey League. They will all play against each other twice per season.

$$\mathbf{I}$$: This would result in $$nPr(12,2) = 132\, matches$$.

$$\mathbf{II}$$: It could probably also be expressed like $$nCr(12,2)*2 = 132$$.

In fact, $$\mathbf{I}$$ is a special case of permutations when we only have two matches per season. It makes more sense to use $$\mathbf{II}$$ if we for any reason would like to increase the number of matches per season. Hmm… Interesting!

• A1: Enough with the combinatorics, go on!
• A2: Okay, let’s think of a property of the matches we can use. The first one I can think of is the total number of goals made per match.

That’s enough. I’ll continue this some other time…