C++ Hash Table Memoization: Simplifying Dynamic Programming

DP Problems

From my perspective, the traditional approach to DP suffers from a couple of problems.

First, it seems to me that people feel more comfortable attacking problems in a top-down fashion as opposed to bottoms-up. This is certainly a matter of psychology, not computer science, but it is relevant to consider human factors when designing for maintenance, testability, and review. In many cases the expression of the algorithm seems more natural when given in the original top-down mode, and requires fewer mental gymnastics on the part of whoever is studying the algorithm.

The second problem is the traditional use of array structures to hold subproblems. The problem starts to creep into play when you look at algorithms like Matrix Chain Multiplication, in which half of the storage space is not even used. Things get even worse when you use a problem like that described later in this article, in which subproblems just don't fit into a row/column organizational format.

Both of these problems can be addressed effectively by using standard C++ hash tables. The procedure is as follows:

1. Implement your algorithm using the basic, inefficient recursive implementation.
2. Create a C++ hash table with global scope that will hold your subproblem results. The key for the hash table should be a concatenation of the input parameters to your function, and the value will be the return type from your function.
3. At the entry point to your function, check the hash table for the presence of an existing value for your subproblem, using the input variables to form the key. If the value is present, instead of executing your function, return immediately.
4. At the exit for your function, store the value you are about to return in the hash table, using the input parameters as the key.

With these minor changes, which can often be accomplished in change to as few as three lines of code, you can convert your inefficient recursive algorithm to use dynamic programming, without having to refactor to a bottoms-up implementation, and without having to shoehorn your results into tabular format.

A Simple Example

Applying this technique to our Fibonacci example, we get a routine that looks like this:

std::map<int,int> results; 
int fib( int n ) 
{ 
  if ( n == 0 || n == 1 ) 
    return 1; 
  std::hash_map<int,int>::iterator ii = results.find( n ); 
  if ( ii != results.end() ) 
    return ii.second; 
  else 
    return results[ n ] = fib( n -1 ) + fib( n - 2 ); 
} 

So the first time we encounter fib(4), we have to make the recursive calls to fib(3) and fib(2). But the second time around, the result has been stored in the map, and it is returned without any additional calculations taking place.