minimum ← ∞ for i ← 1 to n do for j ← i+1 to n do distance ← sqrt((x[i] - x[j])² + (y[i] - y[j])²) if distance < minimum then minimum ← distance first ← i second ← j return (first, second)Alas, this is an O(n²) process, so we considered whether we might do better with a divide-and-conquer approach. It did not look promising, though. Divide-and-conquer doesn't let us solve the sub-problems independently. What if the closest pair straddles two partitions? This is a common theme in computing, and problem solving more generally. We try a technique only to find that it doesn't quite work. Something doesn't fit, or a feature of the domain violates a requirement of the technique. It's tempting in such cases to give up and settle for something less. Experienced problem solvers know not to give up too quickly. Many of the great advances in computing came under conditions just like this. Consider Leonard Kleinrock and the theory of packet switching. In a Computing Conversations podcast published last year, Kleinrock talks about his Ph.D. research. He was working on the problem of how to support a lot of bursty network traffic on a shared connection. (You can read a summary of the podcast in an IEEE Computer column also published last year.) His wonderful idea: apply the technique of

His "independence assumption" made it possible for him to complete his analysis and optimize the design of a packet-switching network. But an important question remained: Was his simplifying assumption too big a cheat? Did it skew the theoretical results in such a way that his model was no longer a reasonable approximation of how networks would behave in the real world? Again, Kleinrock didn't give up. He wrote a program instead.I set up this mathematical model and found it was analytically intractable. I had two choices: give up and find another problem to work on, or make an assumption that would allow me to move forward. So I introduced a mathematical assumption that cracked the problem wide open.

High-stakes programming! In the end, Kleinrock was able to demonstrate that his analytical model was sufficiently close to real-world behavior that his design would work. Every one of us reaps the benefit of his persistence every day. Sometimes, a good idea poses obstacles of its own. We should not let those obstacles beat us without a fight. Often, we just have to find a way to make it work. This lesson applies quite nicely to using divide-and-conquer on the closest pairs problem. In this case, we don't make a simplifying assumption; we solve the sub-problem created by our approach: After finding a candidate for the closest pair, we check to see if there is a closer pair straddling our partitions. The distance between the candidate points constrains the area we have to consider quite a bit, which makes the post-processing step feasible. The result is an O(n log n) algorithm that improves significantly on brute force. This algorithm, like packet switching, comes from sticking with a good idea and finding a way to make it work. This is a lesson every computer science student and novice programmer needs to learn. There is a complementary lesson to be learned, of course: knowing when to give up on an idea and move on to something else. Experience helps us tell the two situations apart, though never with perfect accuracy. Sometimes, we just have to follow an idea long enough to know when it's time to move on. -----I had to write a program to simulate these networks with and without the assumption. ... I simulated many networks on the TX-2 computer at Lincoln Laboratories. I spent four months writing the simulation program. It was a 2,500-line assembly language program, and I wrote it all before debugging a single line of it. I knew if I didn't get that simulation right, I wouldn't get my dissertation.