The primary topics in this part of the specialization are: greedy algorithms (scheduling, minimum spanning trees, clustering, Huffman codes) and dynamic programming (knapsack, sequence alignment, optimal search trees).

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From the course by Stanford University

Greedy Algorithms, Minimum Spanning Trees, and Dynamic Programming

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Stanford University

291 ratings

Course 3 of 4 in the Specialization Algorithms

The primary topics in this part of the specialization are: greedy algorithms (scheduling, minimum spanning trees, clustering, Huffman codes) and dynamic programming (knapsack, sequence alignment, optimal search trees).

From the lesson

Week 3

Huffman codes; introduction to dynamic programming.

- Tim RoughgardenProfessor

Computer Science

So this set of lectures will be our final application of the greedy algorithm

Â design paradigm.

Â It's going to be to an application in compression.

Â Specifically, I'll show you a greedy algorithm for

Â constructing a certain kind of prefix-free binary codes know as Huffman codes.

Â So we're going to spend this video just sort of setting the stage.

Â So let's begin by just defining a binary code.

Â So a binary code is just a way to write down symbols from

Â some general alphabet in a manner that computers can understand.

Â That is, it's just a function mapping each symbol from an alphabet,

Â capital sigma, to a binary string, a sequence of zeroes and ones.

Â So this alphabet capital sigma could be any number of things but, you know,

Â as a simple example, you could imagine it's the letters a through z,

Â say in lowercase, plus maybe the space character and some punctuation.

Â So maybe, for a set of size 32 overall.

Â And if you have 32 symbols you need to encode in binary, well,

Â an obvious way to do it is, there happens to be 32 different binary strings of

Â length five, so why not just use one of each of those for your symbols.

Â So this would be a fixed length code, in the sense we're using the exactly the same

Â number of bits, namely five, to encode each of the symbols of our alphabet.

Â This is pretty similar to what's going on with ASCII codes.

Â And of course, it's a mantra of this class to ask,

Â when can we do better than the obvious solution?

Â So in this context, when can we do better than the fixed length codes?

Â Sometimes we can, in the important case when some symbols are much more likely to

Â appear than others.

Â In that case,

Â we can encode information using fewer bits by deploying variable length codes.

Â And this is, in fact, a very practical idea.

Â Variable length codes are used all the time in practice.

Â One example is in coding MP3 audio files.

Â So if you look up the standard for MP3 encoding,

Â there's this initial phase which you do analog-to-digital conversion.

Â But then, once you're in the digital domain, you do apply Huffman Codes,

Â what I'm going to teach you in these videos,

Â to compress the length of the files even further.

Â And of course as you know, compression, especially lossless compression,

Â like Huffman codes, is a good thing.

Â You want to download a file, you want it to happen as fast as possible.

Â Well, you want to make the file as small as possible.

Â So a new issue arises when you pass from fixed-length codes

Â to variable length codes.

Â So let me illustrate that with a simple example.

Â Suppose our alphabet sigma is just four characters, A, B, C, D.

Â So the obvious fixed length encoding of these characters would

Â just be 00, 01, 10, and 11.

Â Well, suppose you wanted to use fewer bits,

Â and wanted to use a variable length encoding, an obvious idea would be to try

Â to get away with only one bit for a couple of these characters.

Â So, suppose instead of using a double 0 for A, we just use a single 0.

Â And instead of using a double one for D we just use a single one.

Â So that's only fewer bits.

Â So that seems like that can only be better.

Â But now, here's the question.

Â Suppose, someone handed you an encoded transmission consisting of the digits 001.

Â What would have been the original sequence of symbols

Â that led to that encoded version?

Â All right, so the answer is D.

Â There is not enough information to know what 001 was

Â supposed to be an encoding of.

Â The reason is is that having passed to a variable length encoding,

Â there is now ambiguity.

Â There is more than one sequence of original symbols that could have led,

Â under this encoding, to the output 001.

Â Specifically, answers A and C would both lead to 001.

Â The letter A would give you a zero, the letter B would give you a 01.

Â So that would give you 001.

Â On the other hand, AAD would also give you 001.

Â So there's no way to know.

Â Contrast this with fixed-length encoding.

Â If you're given a sequence of bits with a fixed length code,

Â of course you know where one letter ends and the next one begins.

Â For example, if every symbol was encoded with 5 bits, you would just read 5 bits.

Â You would know what symbol it was, you would read the next 5 bits, and so on.

Â With variable length codes, without further precautions,

Â it's unclear where one symbol starts and the next one begins.

Â So that's an additional issue we have to make sure to take care of with variable

Â length codes.

Â So to solve this problem, that with variable length codes and without further

Â precautions, it's unclear where one symbol ends and where the next one begins,

Â we're going to insist that our variable length codes are prefix free.

Â So what this means is when we encode a bunch of symbols,

Â we're going to make sure that for each pair of symbols i and

Â j from the original alphabet sigma, the corresponding encodings will have

Â the property that neither one is a prefix of the other.

Â So going back to the previous slide,

Â you'll see that that example was not prefix free.

Â For example, we used zero, was a prefix of zero one, that led to ambiguity.

Â Similarly, one, the encoding for d, was a prefix of 10, the encoding for

Â c, and that also leads to an ambiguity.

Â So if you don't have prefixes for each other, and

Â we'll develop this more shortly, then there's no ambiguity.

Â Then there's a unique way to decode, to reconstruct what the original sequence of

Â symbols was, given just the zeros and ones.

Â So lest you think this is too strong a property, certainly interesting and

Â useful variable length codes exist that satisfy the prefix-free property.

Â So one simple example, again just to encode the letters A, B, C, D.

Â We can get away with encoding the symbol A just using a single bit,

Â just using a zero.

Â Now, of course, to be prefix free,

Â it better be the case that our encodings of B and C and D all start with the bit 1.

Â Otherwise we're not prefix free.

Â But we can get away with that, so let's encode a B with a one and

Â then a zero, and now both C and

Â D better have the property that they start neither with 0 nor with 10.

Â That is, they better start with 11, but let's just encode c using 110 and

Â D using 111.

Â So that would be a variable length code.

Â The number of bits varies between one and three, but it is prefix free.

Â And again, the reason we might want to do this,

Â the reason we might want to use a variable-length encoding, is to take

Â advantage of non-uniform frequencies of symbols from a given alphabet.

Â So let me show you a concrete example of the benefits you can get from these kinds

Â of codes on the next slide.

Â So let's continue with just our four-symbol alphabet, A, B, C, and D.

Â And let's suppose we have good statistics in our application domain about

Â exactly how frequent each of these symbols are.

Â So, in particular, let's assume we know that A is by far the most likely symbol.

Â Let's say 60% of the symbols are going to be As,

Â whereas 25% are Bs, 10% are Cs, and 5% are Ds.

Â So why would you know the statistics?

Â Well, in some domains you're just going to have a lot of expertise.

Â In genomics you're going to know the usual frequencies of As, Cs, Gs and Ts.

Â For something like an mp3 file, well, you can literally just take an intermediate

Â version of the file after you've done the analog to digital transformation, and

Â just count the number of occurrences of each of the symbols.

Â And then you know exact frequencies, and then you're good to go.

Â So let's compare the performance of the obvious fixed length code,

Â where we used 2 bits for each of the 4 characters, with that of the variable

Â length code that's also prefix-free that we mentioned on the previous slide.

Â And we're going to measure the performance of these codes by looking, on average,

Â how many bits do you need to encode a character.

Â Where the average is over the frequencies of the four different symbols.

Â So for the fixed-length encoding, of course, it's two bits per symbol.

Â We don't even need the average.

Â Just whatever the symbol is, it uses exactly two bits.

Â So what about the variable length encoding that's shown on the right in pink?

Â How many bits, on average, for an average character, given these frequencies of

Â the different symbols, are needed to encode a character of the alphabet sigma?

Â Okay, so the correct answer is the second one.

Â It's, on average, 1.55 bits per character.

Â So what's the computation?

Â Well, 60% of the time, it's going to use only 1 bit, and

Â that's where the big savings comes from.

Â 1 bit is all that's needed whenever we see an A, and most of the characters are As.

Â We don't do too bad when we see a B either, which is 25% of the time.

Â We're only using 2 bits for each B.

Â Now it is true that Cs and Ds, we're paying a price.

Â We've having to use 3 bits for each of those, but there aren't very many.

Â Only 10% of the time is it a C and 5% of the time is it a D.

Â And if you add up the result,

Â that's taking the average over the simple frequencies, we get the result of 1.55.

Â So this example draws our attention to a very neat algorithmic opportunity.

Â So namely, given a alphabet and frequencies of the symbols,

Â which in general are not uniform,

Â we now know that the obvious solution fixed length codes need not be optimal.

Â We can improve upon them using variable length prefix-free codes.

Â So the computation you probably want to solve is which one is the best?

Â How do we get optimal compression?

Â Which variable length code gives us the minimum average encoding

Â length of a symbol from this alphabet?

Â So Huffman codes are the solution to that problem.

Â We'll start developing them in the next video.

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