Imagine that you make your living in a somewhat unorthodox but honest way: you live in a tent outside of the castle and every morning, as the king walks into the castle, he hands you a deck of cards. "I've just played cards last night," he says "and I'm going to again tonight, and I'd like these cards ordered and sorted by suite and number by this evening." Then the king goes into his castle and does whatever it is that kings do in castles all day.
So, you take this deck of cards, which is in any old order, as the king just played with it last night, and you set about sorting it. Now, you've been doing this a long time, and your methodology has improved over the years. When you started, you would just lay the cards out in the order the king gave them to you, and then start at the beginning of the line and ask "is this card greater than the next?" and move the two cards in response to the answer so that the two are in order. You would continue this process, repeating from the beginning of the line, until you made a complete pass through the line without making any changes. Sometimes, this took you all day, sometimes you would even make the king wait for you to finish, and that made him very angry, which is very bad.
Your sorting process is compounded by the limitation of your ability to read and compare the value of two cards, because you don't really know how to read or add, it takes you about ten seconds to decide if a card is greater than or less than another card. For a 52 card deck, this means your original sorting method would require 52 times 52 comparisons, or, in real time, seven and a half hours of non-stop work, leaving you no time to do anything else.
Over the years of doing this job for the king you discovered some shortcuts that would take less time but still produce a sorted deck, despite your limited ability to read and understand the values of the cards. While that still takes ten seconds, your new deck sorting approaches require 52 times 1.7 comparisons, or in real time, about fifteen minutes. This is much, much, better than before, but it would of course be better still if it took even less time, as you have discovered that you can now use this extra time to sort cards for all the other members of the court.
One day a traveling wizard observes you sorting cards and says "you know, this method that you have for sorting cards is quite clever but what if I were to tell you that you could sort these cards in 8.6 minutes flat?" This is about half the time it takes you right now, meaning that you could double the number of decks you sort in a day, doubling your income. You are very interested! "Tell me more," you tell the wizard. "Here, read this paper" the wizard replies, and they give you the paper GP linked.
er, given the specific example used, I would expect the wizard to say "look, you know exactly what the cards are, why don't you buy your own sorted deck of cards and staple them to the floor of your tent. Then when the king gives you a deck, just put each card on top of the matching one, then scoop them all up together in order." to which you would of course reply "but my tent is too small!", and the wizard would say "then just put down one of each number of one suit, plus the other 3 aces. Then pile all the cards on the aces first, just by matching suit, and finally take each of those piles in turn and do it by number."
Now, I haven't read the paper, but my guess is that the next objection is "but the king keeps switching the suits! Yesterday, they were diamonds, today they're little squiggly sketches of soft-serve poop! Which he pays me well enough to not think too hard about!" and the wizard would go on to explain that as long as you ask what the suits and numbers are in advance, you can make piles for them. Or something.
Sorry, usually I try to read papers before commenting, but I've gotta run -- I hear that whining noise that usually means my poop dispenser's jammed again.
So, you take this deck of cards, which is in any old order, as the king just played with it last night, and you set about sorting it. Now, you've been doing this a long time, and your methodology has improved over the years. When you started, you would just lay the cards out in the order the king gave them to you, and then start at the beginning of the line and ask "is this card greater than the next?" and move the two cards in response to the answer so that the two are in order. You would continue this process, repeating from the beginning of the line, until you made a complete pass through the line without making any changes. Sometimes, this took you all day, sometimes you would even make the king wait for you to finish, and that made him very angry, which is very bad.
Your sorting process is compounded by the limitation of your ability to read and compare the value of two cards, because you don't really know how to read or add, it takes you about ten seconds to decide if a card is greater than or less than another card. For a 52 card deck, this means your original sorting method would require 52 times 52 comparisons, or, in real time, seven and a half hours of non-stop work, leaving you no time to do anything else.
Over the years of doing this job for the king you discovered some shortcuts that would take less time but still produce a sorted deck, despite your limited ability to read and understand the values of the cards. While that still takes ten seconds, your new deck sorting approaches require 52 times 1.7 comparisons, or in real time, about fifteen minutes. This is much, much, better than before, but it would of course be better still if it took even less time, as you have discovered that you can now use this extra time to sort cards for all the other members of the court.
One day a traveling wizard observes you sorting cards and says "you know, this method that you have for sorting cards is quite clever but what if I were to tell you that you could sort these cards in 8.6 minutes flat?" This is about half the time it takes you right now, meaning that you could double the number of decks you sort in a day, doubling your income. You are very interested! "Tell me more," you tell the wizard. "Here, read this paper" the wizard replies, and they give you the paper GP linked.