I LOVE meaningless questions.
The match algorithm is public knowledge and posted on the NRMP website. Basically, the algorithm starts by choosing any applicant and looking at their #1 rank, and "matching" them to that program if they are on the rank list. Then it takes the next applicant and does the same thing -- if when the algorithm tries to put someone into a program there is already someone on that list, they are placed on the list in the order of the program's rank list. If a program is "full", then the lowest person is bumped off the list, and the algorithm looks at their next rank, and tries to match them. This continues until all applicants are completed. Of note, it does not matter whom the algorithm starts with -- no matter what order it goes in, the final result is the same.
So, how long does this take?
The best way to guess is to count the number of iterations of the algorithm, with an "iteration" being defined as "trying to put someone on a rank list". Every time the system tries to place someone on a rank list, it could "bump" someone off the bottom of the rank list, which would generate another iteration of the algorithm. Everytime this happens, it takes more time to find a solution.
So, imagine that every applicant ranked as their #1 choice a program that listed them in the top N spots (where N=number of slots that program has). This would be a "perfect" match -- everyone gets exactly what they want. This would be the fastest option, and the number of iterations would = the number of applicants in the match.
That gives you the shortest time it takes to run the match, but is not very realistic. To better answer the question, we should look at the other extreme -- assuming the worst possible match solution, how long does it take? The "worst" solution (as far as computer computaional time is concerned) is that everyone matches into their lowest ranked position.
Rounding the numbers:
Total active applicants in 2007 = 28,000
Average length of rank list = 8 (Table 2, Charting Outcomes, "worst case")
Therefore, in a worst-case match scenario, there would be 28000 x 8 = 224,000 iterations of the match algorithm.
So, how long does it take a computer to do one iteration of the algorithm. Well, it would depend on the speed of the computer and the software being run. However, think of how fast a plain laptop can recalculate an excel spreadsheet. It would be very easy for a computer to run 10-20 iterations of the algoritm per second, and that's a very lowball estimate.
244,000 iterations / 10 per second = 24,400 seconds = 373 minutes = 6.2 hours.
And remember, that's worst case and a good computer can probably process 100's of iterations per second.
Most of the time between now and match day is double checking the data, printing, sorting, and delivering the results. Running the actual match is peanuts.