And Or Evaluating this expression at runtime is best modeled by a stack. Operations like IntConst, VariableRead, This push a value onto the stack; operations like Sub, Less, And pop two operands off the stack and push the result of the operation onto the stack; and so on.
By Alex Allain The Huffman encoding algorithm is an optimal compression algorithm when only the frequency of individual letters are used to compress the data.
There are better algorithms that can use more structure of the file than just letter frequencies. The idea behind the algorithm is that if you have some letters that are more frequent than others, it makes sense to use fewer bits to encode those letters than to encode the less frequent letters.
For instance, take the following phrase: There are a few ways that might be reasonable ways of encoding this phrase using letter frequencies. First, notice that there are only a very few letters that show up here. It would be silly to use chars, with eights bits apiece, to encode each character of the string.
In fact, given that there are only seven characters, we could get away with using three bits for each character!
That's not too bad! But we can do even better if we consider that if one character shows up many times, and several characters show up only a few times, then using one bit to encode one character and many bits writing a compiler in c tutorial point encode another might actually be useful if the character that uses many bits only shows up a small number of times!
Prefix Property To get away with doing this, we now need a way of knowing how to tell which encoding matches which letter. For instance, before, we knew that every three or eight bits was a boundary for a letter.
Now, with different length encodings for different letters, we need to have some way of separating the words out. For instance, given the stringif we know that every letter is encoded with three bits, it's easy to break apart into If some letters are encoded with one bit, and another with four, it's not as easy to know how to do it.
But we can use a trick commonly referred to as the "prefix property". The idea is that the encoding for any one character isn't a prefix for any other character. For instance, if A is encoded with 0, then no other character will be encoded with a zero at the front.
That way, if we start reading a string of bits and the first bit is a zero, we know that we can stop reading, and we know that bit encodes an A because no other character encoding begins with a 0.
In general, the idea is that if we have a full encoding for a character, then that encoding won't show up at the beginning of the encoding for any other character.
This means that once we actually read a string of bits that match a particular character, we know that it must mean that that's the next character and we can start fresh from the next bit, looking for a new character encoding match.
Note that it is perfectly fine for the encoding for a character to show up in the middle of the encoding for another character because there's no way we'd mistake that as the encoding for another character so long as we start decoding from the first bit in the compressed file.
On the other hand, if we get off by a bit, this might cause some headaches. Let's take a look at how this might actually work using some simple encodings that all have the property that the encoding for one character doesn't show up at the beginning of the encoding for another character.
Some of the encodings share the same prefix, but that's perfectly fine because we can always tell them apart at some point by reading in more bits.
For instance, the original string we had could be encoded by 39 bits which we could break apart as 0 0 0 10 0 10 A D A Space A T E Space A P P L E Notice that even using this somewhat simple approach to generating encodings that satisfy the prefix property, we managed to save 4 bits over the approach of using 3 bits per character.
With even more unbalanced word frequencies, we could do even better. A nice way of visualizing the process of decoding a file compressed with Huffman encoding is to think about the encoding as a binary treewhere each leaf node corresponds to a single character.
At each inner node of the tree, if the next bit is a 0, move to the left node, otherwise move to the right node. For instance, the prefix encoding used above would have a binary tree representation that looks like this -- the X's indicate inner nodes.
Once a leaf node is reached, we output the character stored at the leaf and go back up to the root of the tree. The Huffman Algorithm So far, we've gone over the basic principles we'll need for the Huffman algorithm, both for encoding and decoding, but we've had to guess at what would be the best way of actually encoding the characters.
For our simple text string, it wasn't too hard to figure out a decent encoding that saved a few bits.
But in the general case, it might be hard to figure out a good solution, let alone the best possible solution. The Huffman algorithm is a so-called "greedy" approach to solving this problem in the sense that at each step, the algorithm chooses the best available option.
It turns out that this is sufficient for finding the best encoding. The basic idea behind the algorithm is to build the tree bottom-up.
First, every letter starts off as part of its own tree and the trees are ordered by the frequency of the letters in the original string. Then the two least-frequently used letters are combined into a single tree, and the frequency of that tree is set to be the combined frequency of the two trees that it links together.
For instance, if we started out with two characters that showed up once, L and T, in our sample string, they would be recombined into a new tree that has a "supernode" that links to both L and T, and has a frequency of 2: The process is then repeated, treating trees with more than one element the same as any other trees except that their frequencies are the sum of the frequencies of all of the letters at the leaves.
This is just the sum of the left and right children of any node because each node stores the frequency information about its own children.
The process completes when all of the trees have been combined into a single tree -- this tree will describe a Huffman compression encoding. Essentially, a tree is built from the bottom up -- we start out with trees for an ASCII file -- and end up with a single tree with leaves along with internal nodes one for each merging of two trees, which takes place times.by Tom Niemann.
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