As soon as we settled on the problem to be solved, I ran to my nearest copy of Knuth's _Seminumerical Algorithms_ (having copies of Knuth's three volumes both at home and at work has been well worth the investment). Because I had studied the book carefully a decade earlier, I vaguely recalled that it contained several algorithms for problems like this. After spending a few minutes considering several possible designs that we'll study shortly, I realized that Algorithm S in Knuth's Section 3.4.2 was the ideal solution to my problem.
If Bentley needed to go look up an algorithm that he vaguely recalled studying in the past, just how awful of programmers are the rest of us that look things up, really?