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like we just have here, you store the compressed text
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by storing your As, Bs, Cs, Ds, and Es and other letters
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using these new encoding, but you somehow
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have to embed in that file in the compressed file the tree itself
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or this cheat sheet of encodings.
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So, with compression-- maybe you're compressing a Microsoft Word
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file, or a dot TXT file, or any other type of file,
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you have to store not just the compressed text using these shorter
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representation-- not 8-bit ASCII, but these shorter representations-- but you
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also somewhere, maybe at the beginning of the file or at the end of the file,
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somewhere where someone else can find it, you need to store this mapping
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or you need to store the tree itself in some digital form.
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And so, it's possible by this logic that you
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might try to compress a really small file,
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and that file could actually become bigger
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because you're storing a tree inside the file to--
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with which to recover the original information.
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Or better yet, most algorithms or most actual
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compression programs will realize, wait a minute, if compressing this file
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is actually going to make it bigger, let's just not compress it at all
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and leave it alone untouched.
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So, what if you take a compressed file and compress it again, and compress it
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again, and compress it again?
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A dangerous assumption to get into is, well, I could just maybe keep
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compressing that video file again, and again, and again, and again,
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and I can maybe compress my big essay, or my big video file,
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or big music file to just maybe one bit.
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Right?
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That's the logical extreme, just keep compressing,
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compressing, compressing, compressing.
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But, of course, that can't possibly make sense,
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because if you compress some file down to just a single bit, 0 or 1,
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you've clearly thrown away information and can't possibly recover it all.
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So, at some point, too, you've hit this lower bound on the size of the file
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until you need to start throwing actual information away.
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At some point, the file just has so much entropy, appears to be so random,
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there really is no pattern to start to leverage to compress it further.
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And so, there generally is some maximum amount of compression
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you can apply to something.
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So, how would we represent this?
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Let's whip out a C struct here.
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So, this time each of the nodes in a Huffman tree
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need a little something different.
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They need, at least in the leaves, some kind of character
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to remember the symbol.
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Now, technically only the leaves need to know what symbols they are,
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so it's a little redundant to have this in every node,
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but we can keep things simple and use the same type of node for everything.
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Float frequency, I could use an integer and treat it exactly as a percentage,
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or I can use a float as the nodes were with 0.1 and 0.45 and so forth,
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and I'll call that frequency.
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And then each of those nodes needs a left child potentially
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and a right child potentially.
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And, again, I'll call these things a node.
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So, again, it's getting a little more involved this node, but it still allows
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me to represent it ultimately in C.
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And now, it's time to pursue lastly the holy grail of data structures,
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if you will.
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Thus far, we've been solving problems, creating new problems,
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trying to solve those again.
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And the problems we've been exploring this week are things like dynamism,
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if we want to be able to grow or shrink our data structure.
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Malloc and pointers give us that flexibility
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but might cost us a bit more time, because we
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have to keep things sorted differently or we
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have to follow all of those pointers.
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And so, a lot of the algorithms we've been discussing today
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at least have-- like linear time, searching, or inserting, or deleting
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potentially like in a linked list.
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Better still would be something logarithmic
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like a balanced binary search tree, so still preserving that nice binary
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aspect from week zero.
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But the holy grail of a data structure for its operations
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is Big O of 1 so to speak, constant time.
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If you are searching, or inserting, or deleting, and somehow changing
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a data structure, wouldn't it be amazing if every darn operation
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takes just one step, or maybe two steps, or three
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steps but a constant number of steps?
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Now, it might be a little naive for us to expect
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that we can store an arbitrary amount of data in some fancy way
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that we get constant time, but maybe just maybe if we're
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clever we can get close to that.
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So, let's introduce a step toward that.
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It turns out there exists in this world things called hash tables.
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And a hash table can be implemented in any number of ways,
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but you can think of it really as just an array.
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So, for instance, this might be a way of representing a hash table called table,
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whose first location is bracket zero and whose last location is bracket
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n minus 1 for however long this is.
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And I just left it as blanks.
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I don't even know what this hash table might want to store.
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It could be numbers, it could be names, it could be letters,
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it could be anything we want.
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But hash table has this nice theoretical property
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that if well-designed and thought through,
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you can maybe just maybe get constant look up time in it.
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And let's do a simple example of a hash table.
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Hash tables are often nicely thought of as buckets,
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so we borrowed these from the loading dock outside just a little moment ago,
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and we've attached thanks to Arturo some of these signs to them.
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