Artificial intelligence in games, part 3

As I warned when I started this blog, I've allowed it to lie dormant for a while, but I feel like it's high time to pick up where I left off and finish the series about playing chess by look-ahead.

So to review the posts that led up to this one:

• It's possible, and not even very hard, to construct a tree that describes all games of tic-tac-toe.
• By doing so, you can prune off the branches that lead to a loss (and sometimes that lead to a draw) to write a program that plays a perfect game of tic-tac-toe.

Now we're going to turn our attention to chess.

Although chess is a much more complex game than tic-tac-toe, it's easy to see that you can apply the same tactics to systematically list all possible games of chess, in theory. Of course I'm not going to attempt to diagram them, but there's definitely a finite number of things you can do on each move. If you're the white player, on your first turn you have eight pawns that you can move forward one or two spaces, and then you have two knights that can move to two spots (forward two, left or right one). So the total possible moves on the first turn is (8*2)+(2*2), or 20.

Then it gets a little more complicated. Black has the same choices, but he has the same 20 choices for each of the 20 possible moves that white could make. So after each player has made one move, there are 20*20=400 possible states the board could be in. It only gets worse from there.

For instance, if white performed the traditional P-K4 opening, on the second move he has nearly all the other choices remaining (7 pawns to move one or two squares, plus two knights two ways, 18 moves) but he also has a whole lot of new choices. The king's knight may move into the spot occupied by the pawn (19). If the black king's pawn doesn't block it, the white king's pawn may advance one space (20) or, depending on what black did, capture an enemy pawn.

Also, the white king's bishop is now free to move to any of 5 spaces toward the upper left (25) and the queen can move to any of 4 spaces toward the upper right (29). So the number of possible moves keeps increasing as the board opens up, and so does the complexity of the tree. Now it looks like by the time white has moved twice, there are about (20*20*29) possible states the board could be in. (Some starters obviously make less than 29 moves possible, and there is more than one way to get to the same state; i.e., P-K4 followed by P-Q4 has the same result as P-Q4 followed by P-K4).

As we observed in the tic-tac-toe problem, you can calculate roughly how many possible games there are in two ways: by multiplying out the possibilities for all moves, or by figuring out every possible state of the board. The second method tends to be smaller because it eliminates various ways of GETTING to those states. However, in the case of chess, figuring out how many moves are possible from each state is massively complicated, so will go with the second approach.

Each side has 16 pieces. The pieces are not unique; for instance, the eight pawns are interchangeable with each other and so are the two rooks . However, to simplify the calculations we'll assume that all pieces are unique. So there are 32 pieces on the board, and 64 squares where they could be located. The first piece could be on any of the 64 squares, the second on any of the remaining 63, the third on any of the remaining 62, and so on. So as a rough guess, the number of board configurations is (64*63*62*...*35*34*33) and then the remaining 32 squares are blank. That is to say, the answer is "64 choose 32", or (64!/(32!*32!))

Side note: Cool feature of Google. Type "64 choose 32" into the search bar and it will answer the question for you. It's 1.83*10¹⁸.

Is this number an accurate portrayal of all chess boards? Not exactly. In the first place, many of the moves are not legal. This set of all possible boards includes boards where the two kings are next to each other, or in an impossible multiple check; it includes pawns on the first row, and positions where all eight pawns are on their home squares, but the king and queen are not.

On the other hand, it does not include any position where one or more pieces have been captured, nor does it include positions with two or more queens due to promotion. So there may be more games that are truly possibly, or there may be fewer, but on the order of 10¹⁸ possibilities is a fair guess.

Now, 10¹⁸ is a very large number. In point of fact, it is larger than the number of seconds since the universe began (about 4.4*10¹⁷... although it's only 189 billion if you're a creationist. Har har.) It's close to the number of grains of sand on earth, and much larger than the number of stars in the milky way.

In theory, you could have a computer run through the win/loss state of every one of those games, but not in a reasonable amount of time. If you could analyze one game a second, it would of course take longer than the universe has been around to decide on one move. Computers can do much better than that, of course. Currently the world's fastest supercomputer can perform 596 TeraFLOPs -- 596 trillion floating point operations per second. That means that if you were so incredibly optimistic that you believed that analyzing one board position could be done in a single operation (believe me, it can't), the world's fastest supercomputer would still take 40 minutes to analyze the outcome for a single move. And that level of optimism is ridiculous.

Granted, computers have become a lot more powerful since this topic was discussed in the 1970's, but it's still fairly impractical to play a decent game of perfect chess, unless you have several supercomputers lying around doing parallel computations. Your desktop PC would take years. More realistically, thousands of years. Needless to say, no computer plays chess based on the tic-tac-toe strategy of exploring every possible move.

Next time: What's a heuristic, and how come a computer can beat the human chess champion of the world?

Random numbers aren't random

In a previous post, I mentioned that you can add the appearance of intelligence to a game playing script by throwing in a random component. While the computer can always win tic-tac-toe by playing the same sequence of moves, it looks much less boring if the computer picks different winning moves in each game.

Following up on this, I got a special request to explain how random number generators work. I happen to know that the person who asked for this information is sort of a logical determinist, so she'll be delighted to hear that random numbers are not random at all, but generated by a very specific script.

When we think about random numbers, what we generally want is an unpredictable number in a range, with 0 or 1 at the low end, and some number "m" at the high end. How big m is depends on the application. For instance, to simulate a die roll, we want a random number from 1 to 6. To simulate drawing a card, we need a number from 1 to 52. For a roulette wheel, 1 to 38. For a game with two dice, like monopoly, we need to generate two random numbers and add them together. The rules for a role-playing game such as Dungeons & Dragons will often instruct you to roll 1d20 (roll a 20 sided die, i.e., generate a random from 1-20) or 3d4 (roll three four sided dice) to determine a monster's health, or whether they hit you on a particular round. Games like World of Warcraft have borrowed heavily from tabletop gaming rules in order to come up with event odds in the range that make sense for the game balance.

It's hard to generate a truly random number, since computer programs are fundamentally deterministic in the way they follow algorithms. So they do something slightly easier. Once we have one random number (or pseudo-random), we generate a sequence of randoms after that, where each one is based on the number before it.

There are many functions which can generate this sequence; one common example is this:

• X_n+1 = (a*X_n+b) mod m
  

In this case, a and b are constants that you can pick, often large prime numbers. "m" is a maximum value for the resulting number.

Let's try a few examples. I'll pick arbitrary values a=107, b = 23, and m=101.

• 1 (This comes before the first number. It's called the "seed," as it helps to determine which numbers will come up in the pattern.)
  
• 107*1+23 = 130; 130 % 101 = 29
• 107*29+23 = 3126; 3126 % 101 = 96
• 107*96+23 = 10295; 10295 % 101 = 94
• 107*94+23 = 10081; 10081 % 101 = 82
• 107*82+23 = 8797; 8797 % 101 = 10
• 107*10+23 = 1093; 1093 % 101 = 83

As you can see, this function produces a string of seemingly unconnected numbers from 0 to 100. However, they aren't unpredictable. If you know the formula, and the current number is 29, then you know the next number is always going to be 96.

Because of this, you wouldn't want to use 101 as your "m" value, if you were actually looking for a random number in the range of 0 to 100. You could always predict the next number in the sequence. If you're a company that writes video poker machines, and your random number generator is easy to reverse engineer in this way, you'd be in very big trouble.

So instead, you'd make all your constants, m, a, and b, much larger. Then you'd take the resulting random numbers and mod them by a smaller number to get program data, but you'd still retain the larger number to generate the sequence. That way, if people see the number "3", they have no way of knowing that the real number in your random sequence was generated by 5,495,110 mod 6 plus one, and therefore they can't deduce what the next number is going to be, even if they know the algorithm.

So the random sequence is masked from the program's end user by not telling them what the formula is. Nevertheless, a random number generator is still predictable based on the seed. If you start with a fixed seed number of 1, and you get the sequence of die rolls 4, 2, 6, 1, 3, then you will always get the same sequence after you shut down the program and restart it.

If an enterprising gambler were to discover that his video poker machine was programmed so foolishly, he could take advantage of it in the following way: First, unplug the machine and plug it back in to reset it. Then, bet small amounts of money while writing down the card sequences that you get. Then, reset the machine again, and bet large amounts of money against the cards that you know will turn up.

So predictability is the bane of random number generators. Clearly, it's important not only to have a disguised algorithm, but an unpredictable seed set at the beginning of your program.

How can you set the seed? It doesn't really matter, as long as it's not based on a known quantity. It needs to be a truly random number, but again, there is no such thing as a truly random number to a computer.

One approach is to use the internal clock, including milliseconds. A human can't really control the precise millisecond at which the program starts, so there's no way to predict what the seed will be.

There are wackier ways to handle it. For instance, an algorithm called "lavarand" operated by pointing a camera at a lava lamp, taking a digitized picture, and creating a hash from the bits. This is chaotic enough to provide something that appears to be truly random.

So to recap, in order to generate random numbers, you need

1. An algorithm that is deterministic but unpredictable
2. A random seed that is preferably discovered through some (also unpredictable) aspect of the external world.

There are, of course, also advantages to using predictable seeds. If you always use the same seed, then you can test the same set of data repeatedly, which you sometimes must do if you don't want to tear your hair out trying to nail down a problem with the algorithm that actually does something with the numbers. But in a real world environment, you need some kind of outside information in order to randomize the results and stymie the users.

Artificial intelligence in games, part 2

In the previous post, we discussed the fact that tic-tac-toe is finite game, and you can even calculate all the possible states of the board: it's 19,683, with 362,880 different ways to get to the legal states.  Now we can talk about how to take advantage of this calculation to always win the game.

Let's imagine that we pit two perfect computer opponents against each other.  The first opponent, playing as "X", will want to evaluate each of the nine possible moves on a blank board.  How does it know which is best?

Well, starting in the corner, the question is: "Do I have a sure win on this path?  If so, then I will definitely make this move.  If not, then does my opponent have a sure win?  If so, then I will definitely not make this move."

But how do you know if you or the opponent has a sure win?  Simple: Imagine that you already made this move, and then try to make the best move for the opponent.

You can easily see that this is a recursive process.  It has a recursive step ("Let me try this move") and it definitely has a terminating condition.  After all, if you come to a step where (a) the board is full, (b) X has won, or (c) O has won, you don't need to look ahead to any other moves.

Here's an example of how the computer would evaluate the first possible path.

|  X |   |       X | O |       X | O | X
| ---+---+---   ---+---+---   ---+---+---
|    |   |         |   |         |   |
| ---+---+---   ---+---+---   ---+---+---
|    |   |         |   |         |   |
|
|  X | O | X     X | O | X     X | O | X
| ---+---+---   ---+---+---   ---+---+---
|  O |   |       O | X |       O | X | O
| ---+---+---   ---+---+---   ---+---+---
|    |   |         |   |         |   |
|
|  X | O | X
| ---+---+---
|  O | X | O
| ---+---+---
|  X |   |

This game ends in a win for the X player.  So if you are the X player, you like this game.  If you  are the O player, you do not like this game.  So the computer then backs up by two moves ("waking up," in the dream analogy) and assumes that O would not go there, but would instead do this:

|  X | O | X
| ---+---+---
|  O | X |
| ---+---+---
|  O |   |

Of course, this game will also result in a win for X if you follow it to the conclusion, which is this:

|  X | O | X
| ---+---+---
|  O | X | X
| ---+---+---
|  O | O | X

You get the idea.  The computer will play every game to its ending, at which point it will know exactly which games guarantee a win for one player or the other.

Now the pruning begins.  The X-playing intelligence will try to reach one of the game states that guarantees a win for X.  If it cannot, it will at least avoid reaching any game state that guarantees a win for O.

Writing an unbeatable tic-tac-toe opponent in this way is actually pretty straightforward.  You can write it to pick the first "good" move, where "good" means "First try to pick a move that guarantees a win; then try to pick a move that doesn't guarantee a loss."  If you do this, you'll find that your tic-tac-toe game will always play in the same way.  For instance, in the first move, the upper left corner is as good a move as any (the O play can never force a win).  So your program may well always start in the upper left.  This program will not appear to be intelligent.  But it will never lose.

If you want your program to be more interesting, throw in a randomizer.  If there are multiple moves that guarantee a win, then pick one at random.  Otherwise, if there are multiple moves that don't guarantee a loss, pick one of those.  Then you'll have a program that never loses, and also it will make moves that appear unpredictable.

Even though tic-tac-toe is an unwinnable game with two perfect players, some moves are better than others.  Against novice opponents, there are tricks you can use to moderately increase the likelihood that your opponent will make a mistake and lose.  This program doesn't know that.  It doesn't evaluate different positions as tactically stronger.  It just knows whether or not one player is guaranteed to win.

By the way, needless to say, this tic-tac-toe program does not have sufficient reasoning capacity to prevent it from launching a global thermonuclear war.  ;)  The computer doesn't have abstract thoughts; it's just following instructions.  Completely deterministic instructions for winning if it can.

Next time: Chess as a deterministic game.

Separating form from content

In a previous post I highlighted a video showcasing some experimental wearable Technology. One of the features in this video was that a person picked up a product from the shelf, scanned it, and received some information (green, yellow, or right lights) indicating whether the product was environmentally friendly.

A friend asked:

The thing that interests me the most about this is the impact it will claim to have on shoppers' ability to make more eco-friendly purchasing decisions. Where will the database be kept? Does it already exist? Also wondering how companies would be held to objective standards about the environmentally friendly nature of their products. Are there standards already in place? Will the database include packaging environmental impacts?

The important thing to recognize about the gizmo in the video is that it is not an information creator. It is simply a display device. In the current web 2.0 world, it's important to recognize the separation of form and content.

Essentially, the device consists of a wireless receiver, a camera, and a projected screen. All computers and computer-like devices require three basic components: input, output, and data. They've gotten more complicated over time, but they can still be best understood as variations on those basic components.

As a kid in 1982, I had an IBM PC 8086. For output, it had a screen that would display only four colors (red, green, blue, and of course black), and it had a speaker that could generate various pitches of annoying beeps. For input, it had a keyboard that made loud solid clicky noises; no mouse. It also had no hard drive. All data was stored to 5-1/4" floppy disks, which would turn on a red light when writing and produce an audible "kachunk kachunk" noise.

Many of my friends had Atari 2600s. Ataris didn't come with their own screen or speakers of course; instead, they had a dongle thingy that would plug into a standard TV and provide output. Input was limited to a joystick that had one button and four directions, which are not sensitive to the level of force you apply. Data was in the form of interchangeable cartridges. You could not save the state of your games; your progress only existed in memory while the console was on.

Now the components for a high end gaming system are getting more and more elaborate, especially with high end games: output includes complex 3D graphics and digitized sound, while input may include mouse, keyboard, camera, gamepad, and a microphone. Game files (data) take up several gigabytes, sometimes read from your hard disk and sometimes directly from a CD. Saved games may take several megabytes.

But something else is happening too.

The most important aspect of this is the internet. Thanks to the internet, the data component has shifted away from individual computers and onto servers that can be accessed from anywhere in the world. I don't have to be actually at my home computer anymore to look up some important information that's saved in an email. I don't have to be holding a physical copy of "Consumer Reports" in my hand to check out a product rating.

At the same time, wireless devices -- basically a fancy method of receiving additional input -- are infiltrating everything. We've got wi-fi hubs popping up all over the place, in coffee shops and restaurants and airports and malls. If you're not near a hub and you're willing to pay extra (I'm not yet), your cell phone will pull down an internet connection from satellite. And they're coming up with new methods to connect people, like Mobile Ad Hoc Networks, which can turn a room full of people with wireless devices into a grid of amplified signals.

So, how does the device in the video generate the information about environmental products? It doesn't. It doesn't take up space in the device's memory. It comes from the internet, from sites that are maintained by someone else.

As I envision the guts of this device, you probably install a minimal amount of middleware -- software which "knows" where to get raw data from and what is the best way to display it. The database that keeps track of individual products would probably be very large, but that needn't concern you, any more than the inner workings of your favorite web pages concerns you. You ask for a specific piece of information -- what is THIS product? -- and that gets interpreted into a web request, and the server sends it back as a stream of formatted information, to which the middleware then adds some graphical bells and whistles, for the display.

So then the question is, where does the environmental information come from? Does it already exist now, or is it just a fantasy from the video?

Well, yes it does, I think. I'm not real up on this stuff, but I googled around and came up with this and this. The data may not yet be formatted in a way that is useful to a mobile device, but the information exists in some format. If nothing else, somebody could easily write a program to scan the web page and throw out most of the HTML, pulling out only relevant information and turning it into a graphical display.

So the remaining question is: what is the standard for this information? Are these sites reliable? Well, that's up to you to decide. The main thing that has been accomplished by web 2.0 is that many competing organizations can generate this sort of content in a standardized format. Whether you trust the information depends on whether you trust the organization that generated it.

I can understand the suspicion if it is, for instance, a government agency that is known to have been strongly influenced by a political agenda. Likewise, if it's a corporate site, and the corporation is influenced by advertising dollars. But there are also consumer advocacy groups with a reputation to uphold. And while I am not a pure capitalist by any means, this really is a case where if there's a strong enough demand, the market provides what you want. "The market" in this case may be profit driven, or it may just be a bunch of activists who want to make this data available to further their agenda. Whatever the case, it's likely that you can find the data you're looking for.

The bottom line is that technology is a tool for getting at the information that already exists, or can be created. It gives you quick access to the same data from anywhere in the world.

I don't think the device shown in the video is perfect. After having some discussions about it, I think that waving your arms around is a clunky and tiring way to interact. I bet the screen is hard to read in a brightly lit area. That's not the point, though. The point is that it's becoming easier to apply technological solutions to pull in information about the world around you, and get back useful background information that's not immediately obvious to your senses.

Wearable technology

I want one of these setups. NOW.

Artificial intelligence in games, part 1

"Chess is not a game. Chess is a well-defined form of computation. You may not be able to work out the answers, but in theory there must be a solution."

- John von Neumann

I'd like to talk about some basic concepts of Artificial Intelligence, by talking about games. In a few posts I'm going to tell you how to play a perfect game of chess. Not just a good strategy, but a perfect strategy. With this strategy, you will always beat a non-perfect opponent. If no win is possible, you will most likely be able to end the game in a draw.

Unfortunately, once I've explained this strategy, you will realize that you can't actually use it. So before I talk about that, I'm going to try something less ambitious. I'll tell you how to teach your computer to play a perfect game of tic-tac-toe, also known as "noughts and crosses" in England.

Here's some quick background for those who don't know how to play. Tic-tac-toe is played on a three-by-three grid. Two players mark their squares, one with the letter "X", the other with the letter "O". Traditionally the X player goes first, and may place a mark on any open square. The players continue to alternate marks until one of two conditions is met. A player can win by placing three of his or her marks in a row -- either horizontally, vertically, or on a diagonal. Much more commonly if the players are experienced, the board will fill up, no player will get three in a row, and the game ends in a draw.

Complexity of tic-tac-toe

How many possible games of tic-tac-toe are there? One way to do it might be to count all possible sequences of moves. The first player has nine possible moves to make. The second player can only go in an unoccupied square, so that's eight moves left. The first player can then go in any of seven remaining squares. And so on, so the number of possible games is at most 9*8*7*6*5*4*3*2*1.

This is a common mathematical operation called a factorial. "9 factorial" is abbreviated as "9!" Factorials are very big. Some algorithms have a running time of O(n!), which can be extremely dangerous -- but I'll get back to that in another post. Luckily, in this case n is only 9, and 9! is a fairly manageable number: It's 362,880.

That seems like a big number to people who are good at thinking in the abstract but bad at storing large amounts of information. However, computers are the opposite, and 362,880 is a very small, manageable number of objects to explore.

It is also an upper bound. We know that there are no MORE than 362,880 possible games of tic-tac-toe, but most of the games resemble each other. Another trick we could apply is to calculate possible states of the board without thinking of moves. Each square can be in one of three states. Either it is empty, or it has an X, or it has an O. Thus, the total number of possible states of the board (legal or not) is 3^9, which is 19,683. So actually, there are no more than this many different ways to set up the board.

In fact, there are even fewer possible states, because many of them aren't legal. For instance, the board could contain all X's, but that's not the state of a real game, because the "O" player never got to move. So there are actually far less than 19,683 states, but all we need is a rough order of magnitude estimate to see how big a problem we have ahead of us.

Calculating all possible games of tic-tac-toe

If you go first in a game of tic-tac-toe, there are nine moves that you can make. The possibilities break down like this:

|  X |   |         | X |         |   | X
| ---+---+---   ---+---+---   ---+---+---
|    |   |         |   |         |   |
| ---+---+---   ---+---+---   ---+---+---
|    |   |         |   |         |   |
|
|    |   |         |   |         |   |
| ---+---+---   ---+---+---   ---+---+---
|  X |   |         | X |         |   | X
| ---+---+---   ---+---+---   ---+---+---
|    |   |         |   |         |   |

|    |   |         |   |         |   |
| ---+---+---   ---+---+---   ---+---+---
|    |   |         |   |         |   |
| ---+---+---   ---+---+---   ---+---+---
|  X |   |         | X |         |   | X 

Actually, these nine positions do not really represent unique games. Really, there are three unique moves that you can make: Go in the center, go in a corner, and go on an edge. The rest of the moves simply lead to rotated or mirrored versions of the same games. Some algorithms might take advantage of this similarity and reduce the search space, but for our purposes, it still doesn't take too long to treat all combinations as unique.

Now, if you go in the upper left corner, there are eight moves that the "O" player could make. Here are a few.

|  X | O |       X |   | O     X |   |
| ---+---+---   ---+---+---   ---+---+---
|    |   |         |   |       O |   |
| ---+---+---   ---+---+---   ---+---+---
|    |   |         |   |         |   |

Again, these do not represent unique games. For all practical purposes, the first move and the third move are the same game. But it's easier to program (though slower to run) if you don't bother trying to analyze the games.

In the next post, I'll explain how we can use this system for categorizing every possible game to write an unbeatable opponent.

Recursive sorting 2: QuickSort

I got good feedback from my last post about recursive sorting, some of it by email and in person.  I also heard some serious concerns about my world domination scheme. You can all relax, because my Evil Generals are all away on extended vacation for the foreseeable future. This post probably won't be as amusing as the last one, as I've already explained the basic groundwork. But I'll try to give you the basics so you can do QuickSorting yourself.

Here are the fundamentals of how QuickSort works. In Merge Sort, we split up two decks of cards by size, and then sorted each half. How exactly you "sort each half" is up to you. If the remaining divided up decks are still large, you can just divide them in half again and run two more Merge Sorts. But if the decks are small, you can sort them in another way. Even Selection Sort might be a good choice if you have no more than, let's say 10 cards. OR, you can just keep dividing and dividing until you get down to two or three cards, and then perform a couple of swaps to get the cards in order.

QuickSort is similar, but a little different. We still split up the cards, but in this case we split them around a card value instead of a specific deck size. Here's an example. Start with unsorted card:

• 8, 4, K, A, J, 5, 9
  

In QuickSort our plan is to start with the first card (8) and put it in the correct place.  You remember that in Merge Sort we did the recursive step -- the splitting up into smaller problems -- first, and then the action of actually moving cards around was saved until we reached the lowest level (merging the cards).  That's when we actually do something constructive... like telling an evil private "Go kill that guy."  In this one, we do the reverse.  The action step comes first, and then the recursive step second.  After the 8 is in the right place, we'll sort the cards around it, using another call to QuickSort.

When I say we want the 8 in "the right place,"  what I mean is that we want all the cards that are less than 8 to be on the left side of the 8, and we want all the cards that are greater than 8 on the right side.

How do we get there? The 8 starts in the low position, so put your left hand on the 8, then put your right hand on the right side of the list. That's 9. 9 is bigger than 8. No switch is necessary yet, so move your right hand one space to the left, toward the 8. You're now on 5. 5 is smaller than 8, so it's in the wrong position.  Do a swap and you get this:

• 5, 4, K, A, J, 8, 9
  

So now the 8 is on the high side. It is correctly positioned with respect to some of the cards: it is between 5 and 9, which is where it should be. But you're not done yet. Now you have your left hand on 5, right hand on 8.  Move your left hand up one space, to the 4.

4 is less than 8. Keep going.

King is more than 8. It's in the wrong place.  Time to swap.

• 5, 4, 8, A, J, K, 9
  

Now the 8 is correctly sorted with respect to the cards you already looked at. It is more than 5 and 4, but less than king and 9. Nearly done.

The 8 is now on the low side. With your right hand on the king, move backwards. The jack is in the right place. The ace is smaller than the 8. Swap.

• 5, 4, A, 8, J, K, 9
  

...your hands now meet in the middle, and you're done. The 8 is in the precise position where it should be.

Now what? Here's the recursive step. You have the 8 in the right place, but you have three cards on the left that are totally scrambled, and three cards on the right that are also scrambled. How do you unscramble them? You QuickSort them, of course.

First you sort this array:

• 5, 4, A

Then you sort this array:

• J, K, 9

Do it in the same way as before. For the first set of three cards, put your left hand on the 5, and move it to the proper place (the third position). For the second set, put your left hand on the jack, and move it to the proper place (second position, between J and K). By the time you've finished, we discover that all the cards are sorted. Quick, wasn't it?

Seriously, get your own deck and try this on 20 cards. It's easy and fun once you get the hang of it. Lay them out in a line. As you work, you'll put a card at a time in its final location. When you do this, push the card up slightly so it will stand out from the others. Then work on the unsorted cards from left to right until you're finished.

Complexity of QuickSort

Starting with the first card, you made one complete pass through the array in order to put it in the right place. So this algorithm is at least O(n). Then you performed this pass through the array multiple times. How many?

In the best case, QuickSort does the same thing as Merge Sort: it splits the array almost exactly in half. (In the above case, our first number, 8, landed smack in the middle of the array, with three numbers on each side. That's a perfect split.) So basically, you're dividing by two each time, and after (log₂ n) divisions, you're down to sorting one card -- just like what we did in a binary search. So we simplify this and say we examined "n" cards approximately "log n" times, so as I indicated earlier, this algorithm is O(n*log n).

If you're lucky!

But what happens if you start with an array that's already sorted?

• A, 4, 5, 8, 9, J, K
  

Well, the first card is already the lowest. So you'll start from the king and start moving left, but eventually you'll get to the ace, having examined all the cards but not moved any.

Then you'll "split" around the ace... but there are no cards at all on the left, and (n-1) cards on the right. So you'll do almost as many comparisons as before against the 4. Then against the 5. And so on, until you've compared every card to every other card. This takes O(n²) time. Needless to say, that's bad.

So what we see here is that there's a difference between the complexity of a typical execution of an algorithm and the worst case scenario. Knowing that, we can do things to plan for the worst case.

Unfortunately, sooner or later it's highly likely that you will accidentally try to sort an array that is already sorted. So although it is counter-intuitive, we can guard against this by scrambling the array first and then sorting it. Scrambling is easy... all you do is go through the entire array (n operations), pick a random number (picking random numbers takes constant time, so it's O(1)), and swap the current number with the one at the random location.

This takes 1*n moves, so it's O(n) time. Then "scramble + sort" would take O(n log n + n) time But n is of lower complexity than n log n, and at high numbers it will be trivially small in comparison. So the "scramble + sort" routine is simply expressed as O(n log n).

Of course, you could be super unlucky and wind up with a scrambled array that happens to be perfectly sorted.  But as you get a very large number for n, the chances of that happening become extremely small.

Another way to randomize this thing is by picking random pivot point.  Instead of always trying to position the first card, pick a RANDOM card, move it to the leftmost location, and put that one in position.  This way you fix the problem of potentially starting with the smallest card every time.