# Sorting by hand or searching and inserting

Natan commenting on work the other day:

"You have two lists. One is sorted, the other is not. Every item in one list corresponds to an item in the other. Is it faster to sort the unsorted list and then merge them, or simply go through the unsorted list in order and pair each item with the item you can find in the presorted list?" is a question I asked today in the dressing room of an off-Broadway theater. I've used RegEx at my job before, but I never thought Big O notation would become relevant! Let this be submitted as evidence that a computer science education is valuable no matter what field you go into. That, or I'm just a particular kind of crazy.

There's clearly an unplugged activity here :-)

As a programming exercise, it's pretty easy to analyze. Since one list is sorted, you can use a binary search for each insertion. That's \( n \) searches of \(lgn\) each yields an \( nlogn \) run time. Of course, if the sorted list is an array and you have to shift all the elements down in linear time for each insert, that changes things.

Sorting first is also \(nlogn\). Use an \(nlogn\) sort and then merge the two lists which is linear. Here though, you will also either have to ultimately do the shifting for insertions or use extra space by making an new array or list for the two merged sets.

The interesting discussion points here are the issues like in place vs not and the extra time needed for the shifts.

Unplugged though allows for even richer discussion.

Having humans step through computer algorithms can be popular activities at times and they can also be very instructive but as humans, we're not optimized the same ways computers are. While we might perform some tasks using set algorithms and those algorithms might be the same as on a computer, more often than not, they're different - at least at a higher thought level.

Let's take the insertion solution to our problem. Code wise, we'd use a binary search. Humans however don't use binary search. If we did, it would actually be pretty inefficient. We'd have to locate the exact middle which would probably require specific counting and already we've used linear time. Even if we guessed the first middle correctly, each additional split would require we find a new middle and again, realistically, each time we'd count - more linear components.

In reality, we'd probably use a guided type search. The specific problem involved inserting letters from an unsorted pile into envelopes in a sorted pile so we'd jump down in the envelope pile by a guess amount based on the letter we're looking for - Last name Tucker, we'd jump close to the end. For Davis, near the front. If we ended up far off, we'd jump again, if not, we'd rifle through one by one for the insertion point. There are similarities to a binary search but it's a very different algorithm. We couldn't easily code up our guided search as we have an intuitive instant ability to decide where to jump to and when to jump but on the other hand we're slower if we use the proven \(logn\) binary search.

Sorting is similar. On the computer we'd probably use our langauges built in sort which would be \(nlogn\). By hand, we'd probably use some sort of bucket sort with merging in the individual piles. Even if we were sorting continuous values as opposed to discrete ones like names, we'd probably use buckets.

All of this can lead to very rich discussion - how we do things vs how the computer does and even more, how we decide how to do things. Maybe with a small data set we don't even do a bucket sort but more of an insertion or selection type sort. How do we decide what our jumps our when we're searching by hand or our bucket demarcations when sorting?

Lots of good stuff here, all because somebody at Natan's job forgot to sort the letters before they printed them out.