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<!doctype html>
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<title>CS 2150: 10-heaps-huffman slide set</title>
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<section data-markdown id="cover"><script type="text/template">
# CS 2150
### Program and Data Representation
<center><small>[Aaron Bloomfield](http://www.cs.virginia.edu/~asb) / [aaron@virginia.edu](mailto:aaron@virginia.edu) / [@bloomfieldaaron](http://twitter.com/bloomfieldaaron)</small></center>
<center><small>Repository: [github.com/aaronbloomfield/pdr](http://github.com/aaronbloomfield/pdr) / [↑](index.html) / <a href="10-heaps-huffman.html?print-pdf"><img class="print" width="20" src="images/print-icon.png"></a></small></center>
## Heaps (Priority Queues) & Huffman Coding
</script></section>
<section>
<h2>CS 2150 Roadmap</h2>
<table class="wide">
<tr><td colspan="3"><p class="center">Data Representation</p></td><td></td><td colspan="3"><p class="center">Program Representation</p></td></tr>
<tr>
<td class="top"><small> <br> <br>string<br> <br> <br> <br>int x[3]<br> <br> <br> <br>char x<br> <br> <br> <br>0x9cd0f0ad<br> <br> <br> <br>01101011</small></td>
<!-- image adapted from http://openclipart.org/detail/3677/arrow-left-right-by-torfnase -->
<td><img class="noborder" src="images/red-double-arrow.png" height="500" alt="vertical red double arrow"></td>
<td class="top"> <br>Objects<br> <br>Arrays<br> <br>Primitive types<br> <br>Addresses<br> <br>bits</td>
<td> </td>
<td class="top"><small> <br> <br>Java code<br> <br> <br>C++ code<br> <br> <br>C code<br> <br> <br>x86 code<br> <br> <br>IBCM<br> <br> <br>hexadecimal</small></td>
<!-- image adapted from http://openclipart.org/detail/3677/arrow-left-right-by-torfnase -->
<td><img class="noborder" src="images/green-double-arrow.png" height="500" alt="vertical green double arrow"></td>
<td class="top"> <br>High-level language<br> <br>Low-level language<br> <br>Assembly language<br> <br>Machine code</td>
</tr>
</table>
</section>
<section data-markdown><script type="text/template">
# Contents
[Priority Queues](#priorityqueues)
[Binary Heaps](#heaps)
[Heap Operations](#heapops)
[File Compression](#filecomp)
[Huffman Coding](#huffman)
[Priority Queue Example](#priorityqueueex)
</script></section>
<section>
<section id="priorityqueues" data-markdown><script type="text/template">
# Priority Queues
</script></section>
<section data-markdown><script type="text/template">
## Motivation
- Multiuser environment
- Operating system must choose which process to run on CPU
- Management of limited resources
- Bandwidth on network router
- Limited bandwidth, but want to give best possible performance
- Send traffic from highest priority queue first
- Example: VoIP
</script></section>
<section data-markdown><script type="text/template">
## Priority Queue ADT - Model
- operations:
- insert
- inserts with a *priority*
- findMin
- finds the minimum element
- deleteMin
- finds, returns, and removes minimum element
</script></section>
<section data-markdown><script type="text/template">
## Implementations of Priority Queue ADT
| Data structure | insert | findMin | deleteMin |
|-|-|-|-|
| Unsorted array | Θ(1) ammortized | Θ(*n*) | Θ(*n*) |
| Unsorted linked list | Θ(1) | Θ(*n*) | Θ(*n*) |
| Sorted array | Θ(*n*) | Θ(1) | Θ(1) |
| Sorted linked list | Θ(*n*) | Θ(1) | Θ(1) |
| Binary search tree | Θ(*n*) | Θ(*n*) | Θ(*n*) |
| AVL or red-black tree | Θ(log *n*) | Θ(log *n*) | Θ(log *n*) |
| Hash table | ideally constant | Θ(*n*) | Θ(*n*) |
- We would like:
- findMin: always constant
- insert: worst case Θ(log *n*), typical case constant
- deleteMin: worst and average case Θ(log *n*)
</script></section>
</section>
<section>
<section id="heaps" data-markdown><script type="text/template">
# Binary Heaps
</script></section>
<section data-markdown><script type="text/template">
## Binary Heap
- A binary heap is a data structure that is one possible implementation of a priority queue
- It's a binary tree (*not* a BST), with a different:
- structure property
- ordering property
</script></section>
<section data-markdown><script type="text/template">
## Definitions
A *perfect* (or *complete*) binary tree is a binary tree with all leaf nodes at the same depth; all internal nodes have 2 children.
- This has height *h*, 2<sup>*h*+1</sup>-1 nodes, 2<sup>*h*</sup>-1 non-leaves, and 2<sup>*h*</sup> leaves
- Note that about half of the nodes are leaves
- Note that this example is a complete binary tree, not a heap
</script></section>
<section data-markdown><script type="text/template">
## Full Binary Tree
- A binary tree in which each node has exactly zero or two children.
- Also known as a proper binary tree
- We will use this later for Huffman trees
</script></section>
<section>
<h2>Heap Structure Property</h2>
<p>A binary heap is an <i>almost complete</i> binary tree, which is a binary tree that is completely filled, with the possible exception of the bottom level, which is filled left to right.</p>
<p> </p>
<p>Examples:</p>
<table class="transparent"><tr><td><img alt="heap 4" src="graphs/heap-4.svg"></td><td style="width:50px"></td><td><img alt="heap 3" src="graphs/heap-3.svg"></td></tr></table>
</section>
<section>
<h2>Almost complete binary tree of height <i>h</i></h2>
<table class="transparent">
<tr><td class="middle" style="text-align:left"><ul><li>For <i>h</i> = 0, just a single node</li></ul></td><td><img alt="heap 5" src="graphs/heap-5.svg"></td><td></td></tr>
<tr><td class="middle" style="text-align:left"><ul><li>For <i>h</i> = 1, left child or two children</li></ul></td><td><img alt="heap 6" src="graphs/heap-6.svg"></td><td><img alt="heap 7" src="graphs/heap-7.svg"></td></tr>
<tr><td colspan="3">
<ul>
<li>For <i>h</i> ≥ 2, either:<ul>
<li>the left subtree of the root is <i>complete</i> with height <i>h</i>-1 and the right is <i>almost complete</i> with height <i>h</i>-1, OR</li>
<li>the left is <i>almost complete</i> with height <i>h</i>-1 and the right is <i>complete</i> with height <i>h</i>-2</li></ul></li>
</ul>
</td></tr></table>
</section>
<section>
<h2>Representing Complete Binary Trees in Arrays</h2>
<table class="transparent">
<tr><td class="top"><img alt="heap 8" src="graphs/heap-8.svg"></td>
<td>
<p> </p>
<p>From node <i>i</i>:</p>
<p> </p>
<p>left child: 2*<i>i</i></p>
<p>right child: (2*<i>i</i>)+1</p>
<p>parent: floor(<i>i</i>/2)</p>
</td></tr></table>
<p>Implicit (array) representation:</p>
<table class="transparent" style="border-collapse:collapse"><tr class="bordercfifty" style="border-bottom:medium solid;"><td> </td><td>A</td><td>B</td><td>C</td><td>D</td><td>E</td><td>F</td><td>G</td><td>H</td><td>I</td><td>J</td><td>K</td><td>L</td><td> </td></tr>
<tr><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td><td>12</td><td>13</td></tr>
</table>
</section>
<section data-markdown><script type="text/template">
## Why better than pointers?
- Saves space
- No need to store parent/child pointers
- Arrays are more compact than dynamic memory
- Saves time
- Arrays work better with cache (we'll see why later this semester)
- \*2, /2, + operations are faster than dereferencing pointer
- Dynamic memory allocation is slow
- Parent easy to locate
</script></section>
<section>
<h2>Heap Ordering Property</h2>
<p>Heap ordering property: For every non-root node <i>X</i>, the key in the parent of <i>X</i> is less than (or equal to) the key in <i>X</i>. Thus, the tree is <i>partially</i> ordered.</p>
<table class="transparent"><tr>
<td class="top"><img alt="heap 9" src="graphs/heap-9.svg"> not a heap</td><td style="width:50px"></td><td><img alt="heap 10" src="graphs/heap-10.svg"> min-heap</td></tr>
</table>
</section>
</section>
<section>
<section id="heapops" data-markdown><script type="text/template">
# Heap Operations
</script></section>
<section data-markdown><script type="text/template">
## Heap Operations
- findMin: just look at the root node
- insert(val): percolate up
- deleteMin: percolate down
See [here](http://people.ksp.sk/~kuko/bak/index.html) for a good heap animation site
</script></section>
<section data-markdown><script type="text/template">
## Heap: Insert(val)
Basic Idea:
1. Put val at "next" leaf position
2. Repeatedly exchange node with its parent if needed
</script></section>
<section data-markdown><script type="text/template">
## insert(int x)
Note that `heap` is an int vector, and `heap_size` is the number of *heap elements* inserted into that vector
Source code: [binary_heap.cpp](code/10-heaps-huffman/binary_heap.cpp.html) ([src](code/10-heaps-huffman/binary_heap.cpp))
```
void binary_heap::insert(int x) {
// a vector push_back will resize as necessary
heap.push_back(x);
// move it up into the right position
percolateUp(++heap_size);
}
```
</script></section>
<section data-markdown><script type="text/template">
## percolateUp(int hole)
Source code: [binary_heap.cpp](code/10-heaps-huffman/binary_heap.cpp.html) ([src](code/10-heaps-huffman/binary_heap.cpp))
```
void binary_heap::percolateUp(int hole) {
// get the value just inserted
int x = heap[hole];
// while we haven't run off the top and while the
// value is less than the parent...
for ( ; (hole > 1) && (x < heap[hole/2]); hole /= 2 )
heap[hole] = heap[hole/2]; // move parent down
// correct position found! insert at that spot
heap[hole] = x;
}
```
</script></section>
<section>
<h2>Insert: percolate up</h2>
<table class="transparent"><tr><td><img alt="heap 12" src="graphs/heap-12.svg"></td><td class="middle">→</td><td><img alt="heap 13" src="graphs/heap-13.svg"></td></tr></table>
</section>
<section>
<h2>Insert expected running time</h2>
<ul>
<li>How far to move up?<ul>
<li>Half of the nodes are leaves, so half of the inserts will only move up one level</li>
<li>A quarter of the nodes are one level above the leaves, so one quarter of the inserts will move up two levels</li>
<li>One eighth will require moving up 3 levels</li>
<li>One sixteenth will require moving up 4 levels</li>
<li>Etc.</li></ul></li>
<li>Expected running time:<br> </li>
<ul>
\( \frac{1}{2}*1 + \frac{1}{4}*2 + \frac{1}{8}*3 + \ldots = \sum_{i=1}^{n} \frac{1}{2^i}*i = 2 \)
</section>
<section data-markdown><script type="text/template">
## Heap: DeleteMin
Basic Idea:
1. Remove root (that is always the min!)
2. Put "last" leaf node at root
3. Find smallest child (why?)
4. Swap node with smallest child if needed.
5. Repeat steps 3 & 4 until no swaps needed.
</script></section>
<section>
<h2>Which child to swap with</h2>
<table class="transparent">
<tr><td class="middle" style="text-align:left"><ul><li>Consider this min-heap:<ul>
<li>25 needs percolating!</li>
<li>But which way?</li></ul></li>
</ul></td><td><img alt="heap 14" src="graphs/heap-14.svg"></td></tr>
<tr><td class="middle" style="text-align:left"><ul><li>If we swap 25 with the smallest child:<ul>
<li>All's good!</li></ul></li>
</ul></td><td><img alt="heap 15" src="graphs/heap-15.svg"></td></tr>
<tr><td class="middle" style="text-align:left"><ul><li>If we swap 25 with the largest child:<ul>
<li>No longer a min-heap!</li></ul></li>
</ul></td><td><img alt="heap 16" src="graphs/heap-16.svg"></td></tr>
</table>
</section>
<section>
<h2>DeleteMin: percolate down</h2>
<table class="transparent"><tr><td><img alt="heap 17" src="graphs/heap-17.svg"></td><td class="middle">→</td><td><img alt="heap 18" src="graphs/heap-18.svg"></td></tr></table>
</section>
<section data-markdown><script type="text/template">
## deleteMin()
Note that `heap` is an int vector, and `heap_size` is the number of *heap elements* inserted into that vector
Source code: [binary_heap.cpp](code/10-heaps-huffman/binary_heap.cpp.html) ([src](code/10-heaps-huffman/binary_heap.cpp))
```
int binary_heap::deleteMin() {
// make sure the heap is not empty
if ( heap_size == 0 )
throw "deleteMin() called on empty heap";
// save the value to be returned
int ret = heap[1];
// move the last inserted position into the root
heap[1] = heap[heap_size--];
// make sure the vector knows that there is one less
// element
heap.pop_back();
// percolate the root down to the proper position
percolateDown(1);
// return the old root node
return ret;
}
```
</script></section>
<section data-markdown><script type="text/template">
## percolateDown()
Source code: [binary_heap.cpp](code/10-heaps-huffman/binary_heap.cpp.html) ([src](code/10-heaps-huffman/binary_heap.cpp))
```
void binary_heap::percolateDown(int hole) {
// get the value to percolate down
int x = heap[hole];
// while there is a left child...
while ( hole*2 <= heap_size ) {
int child = hole*2; // the left child
// is there a right child? if so, is it lesser?
if ( (child+1 <= heap_size) &&
(heap[child+1] < heap[child]) )
child++;
// if the child is greater than the node...
if ( x > heap[child] ) {
heap[hole] = heap[child]; // move child up
hole = child; // move hole down
} else
break;
}
// correct position found! insert at that spot
heap[hole] = x;
}
```
</script></section>
<section data-markdown><script type="text/template">
## Other Possible Heap Operations
- `decreaseKey(processID, amount)`: "raise" the priority of a process, percolate up
- `increaseKey(processID, amount)`: "lower" the priority of a process, percolate down
- `remove(processID)`: remove a process, move to top, then delete.
1. `decreaseKey(processID, -infinity)`
2. `deleteMin()`
- Worst case running time for all of these: Θ(*n*), because of the find() required; without the find(), it's Θ(log *n*)
- What about FindMax?
- ExpandHeap: when heap fills, copy into new space.
</script></section>
<section data-markdown><script type="text/template">
## Heaps (Summary)
- insert: percolate up; Θ(log *n*) time worst case, but constant expected time
- deleteMin: percolate down; Θ(log *n*) time worst case; also logarithmic expected time
- findMin: Θ(1) time
</script></section>
<section data-markdown><script type="text/template">
## Heapsort
- Insert *n* elements, then remove n elements
- Each one has an insertion time of log *n*
- And then a removal time of log *n*
- Hence Θ(*n* log *n*)
- But it's not a *stable* sort, so it's not used as often as mergesort
</script></section>
<section>
<h2>An xkcd about heaps...</h2>
<img class="stretch" src="images/10-heaps-huffman/tree.png" title="Not only is that terrible in general, but you just KNOW Billy's going to open the root present first, and then everyone will have to wait while the heap is rebuilt." alt="Tree" />
<p class="center"><a href="http://xkcd.com/835/">xkcd # 835</a></p>
</section>
</section>
<section>
<section id="filecomp" data-markdown><script type="text/template">
# File Compression
</script></section>
<section data-markdown><script type="text/template">
## Why compress files?
- Disk space is limited
- File transfer
- Bigger files take longer to transfer
- Smaller file might fit in memory more easily
</script></section>
<section data-markdown><script type="text/template">
## What is a file?
- Named collection of information
- C++ program
- Application executables
- Word documents
- Email
- Web pages
- Pictures, audio, video
- Which of these needs to be exactly the same when we use them again?
</script></section>
<section data-markdown><script type="text/template">
## Data Compression
- Lossless compression: X = X'
- Lossy compression: X != X'
- Information is lost (irreversible)
- Compression ratio: |X|/|Y|
- Where |X| is the number of bits (i.e., file size) of X
</script></section>
<section data-markdown><script type="text/template">
## Lossy Compression
- Some data is lost, but not "noticable" data
- Standards:
- JPEG (Joint Photographic Experts Group)
- Still images (pictures)
- MP3 (MPEG-1, Layer 3), Ogg vorbis, AAC
- Audio
- MPEG (Motion Picture Experts Group), as well as most video codecs (standards)
- Audio and video
- Compression ratios of 10:1 are possible
</script></section>
<section>
<table class="transparent">
<tr><td colspan="2" class="top"><h2> <br>JPEG image quality<br>comparison</h2></td><td style="width:30%"><a href="http://en.wikipedia.org/wiki/Jpg#Sample_photographs"><img class="fragment" data-fragment-index="1" alt="jpeg @ 25%" src="images/10-heaps-huffman/jpeg-025.jpg"></a></td></tr>
<tr>
<td class="top">
<ul>
<li style="text-align:left" class="fragment" data-fragment-index="1">Quality = 100; image size: 83,261 (100%)</li>
<li style="text-align:left" class="fragment" data-fragment-index="2">Quality = 50; image size: 15,138 (18%)</li>
<li style="text-align:left" class="fragment" data-fragment-index="3">Quality = 25; image size: 9,553 (11%)</li>
<li style="text-align:left" class="fragment" data-fragment-index="4">Quality = 10; image size: 4,787 (6%)</li>
<li style="text-align:left" class="fragment" data-fragment-index="5">Quality = 1; image size: 1,523 (2%)</li>
</ul>
</td>
<td style="width:30%"><a href="http://en.wikipedia.org/wiki/Jpg#Sample_photographs"><img class="fragment" data-fragment-index="2" alt="jpeg @ 100%" src="images/10-heaps-huffman/jpeg-100.jpg"></a>
<a href="http://en.wikipedia.org/wiki/Jpg#Sample_photographs"><img class="fragment" data-fragment-index="3" alt="jpeg @ 50%" src="images/10-heaps-huffman/jpeg-050.jpg"></a></td>
<td><a href="http://en.wikipedia.org/wiki/Jpg#Sample_photographs"><img class="fragment" data-fragment-index="4" alt="jpeg @ 10%" src="images/10-heaps-huffman/jpeg-010.jpg"></a>
<a href="http://en.wikipedia.org/wiki/Jpg#Sample_photographs"><img class="fragment" data-fragment-index="5" alt="jpeg @ 1%" src="images/10-heaps-huffman/jpeg-001.jpg"></a></td></tr>
</table>
</section>
<section data-markdown><script type="text/template">
## Lossless Compression
- No data is lost.
- Standards:
- Gzip, Unix compress, zip, Morse code
- PNG image file formats
- Run-length encoding (RLE)
- Can get compression ratios of 4:1
</script></section>
<section data-markdown><script type="text/template">
## Lossless Compression of Text
- ASCII = fixed 8 bits per character
- Example: "hello there"
- 11 characters * 8 bits = 88 bits
- Can we encode this message using fewer bits?
</script></section>
</section>
<section>
<section id="huffman" data-markdown><script type="text/template">
# Huffman Coding
</script></section>
<section>
<h2>Huffman Coding</h2>
<table>
<tr style="background-color: transparent"><td style="width:60%">
<ul><li>Uses <i>frequencies</i> of symbols in a string to build a <i>prefix code</i></li>
<li>The more frequent a character is, the fewer bits we'll use to represent it</li>
<li>Prefix code: no code in our encoding is a prefix of another code</li>
</ul></td><td class="top">
<table>
<tr><th>Letter</th><th>Code</th></tr>
<tr><td>a</td><td>0</td></tr>
<tr><td>b</td><td>100</td></tr>
<tr><td>c</td><td>101</td></tr>
<tr><td>d</td><td>11</td></tr>
</table>
</td></tr></table>
</section>
<section data-markdown><script type="text/template">
## Decoding a Prefix Code
- Create the Huffman coding tree
- Loop
- start at root of tree
- loop
- if bit read = 1 then go right
- else, go left
- until node is a leaf
- Report character found!
- Until end of the message
</script></section>
<section id="lab10tree">
<h2>Decode: 1110001010011</h2>
<table>
<tr style="background-color: transparent"><td class="top">
<table>
<tr><th>Letter</th><th>Code</th></tr>
<tr><td>a</td><td>0</td></tr>
<tr><td>b</td><td>100</td></tr>
<tr><td>c</td><td>101</td></tr>
<tr><td>d</td><td>11</td></tr>
</table>
</td><td style="width:50px"></td><td class="top"><img alt="huffman 13" src="graphs/huffman-13.svg"></td><td style="width:50px"></td><td class="middle">This is a full<br>binary tree!</td></tr></table>
<p> </p>
<p class="center">11 100 0 101 0 0 11 = dbacaad</p>
</section>
<section>
<h2>Huffman Trees</h2>
<p>Cost of a file encoded via a Huffman Tree containing <i>n</i> symbols:</p>
<p> </p>
\( C(T) = p_1 * r_1 + p_2 * r_2 + p_3 * r_3 + \ldots + p_n * r_n \)
<p> </p>
<p>Where:</p>
<ul>
<li><i>p<sub>i</sub></i> = the frequency (or probability) that a symbol occurs</li>
<li><i>r<sub>i</sub></i> = the length of the path from the root to the node</li>
</ul>
</section>
<section>
<h2>Huffman encoding costs</h2>
<table>
<tr style="background-color: transparent"><td class="top">
<table><tr style="background-color: transparent"><td>This is the example<br> from 2 slides ago</td></tr>
<tr style="background-color: transparent"><td> </td></tr>
<tr style="background-color: transparent"><td> </td></tr>
<tr style="background-color: transparent"><td>
<table>
<tr><th>Letter</th><th>Frequency</th><th>Code</th></tr>
<tr><td>a</td><td>3/7</td><td>0</td></tr>
<tr><td>b</td><td>1/7</td><td>100</td></tr>
<tr><td>c</td><td>1/7</td><td>101</td></tr>
<tr><td>d</td><td>2/7</td><td>11</td></tr>
</table>
</td></tr></table>
</td><td style="width:50px"></td><td style="text-align:left">
<ul>
<li>a: 3/7 * 1 = 3/7</li>
<li>b: 1/7 * 3 = 3/7</li>
<li>c: 1/7 * 3 = 3/7</li>
<li>d: 2/7 * 2 = 4/7</li>
</ul>
<p> </p>
<ul>
<li>Cost is 3/7 + 3/7 + 3/7 + 4/7 = 13/7 = 1.85 bits per character</li>
<li>ASCII is 8 bits per char</li>
<li>"Straight" encoding is 2 bits per char</li>
</ul></td></tr></table>
</section>
<section data-markdown><script type="text/template">
## Compression Phase
1. Determine the frequencies of the characters stored in the source file
- Read the source file
- Then store the character frequencies in a *min-heap*
2. Build a *tree* of prefix codes (a Huffman code) that determines the unique bit codes for each character
3. Write the prefix codes or code tree to the output file
4. Re-read the source file and for each character read, write its prefix code into the output file
- You must WRITE the prefix code/tree and the encoded file to the SAME output file
</script></section>
<section data-markdown><script type="text/template">
## Huffman coding example
- For the quote "if it is to be, it is up to me"
- Total of 12 different characters; string length is 30
- Normal ASCII encoding is 8 bits per character
- 30\*8 = 240 bits
- Which is 30 bytes, obviously
- "Straight" encoding is 4 bits per character
- 30\*4 = 120 bits
- Which is 15 bytes
</script></section>
<section>
<h2>Compression step 1 (a)</h2>
<p>Determine frequencies of letters</p>
<table style="font-size:80%">
<tr><th>Character</th><th>Frequency</th></tr>
<tr><td>b</td><td>1</td></tr>
<tr><td>e</td><td>2</td></tr>
<tr><td>f</td><td>1</td></tr>
<tr><td>i</td><td>5</td></tr>
<tr><td>m</td><td>1</td></tr>
<tr><td>o</td><td>2</td></tr>
<tr><td>p</td><td>1</td></tr>
<tr><td>s</td><td>2</td></tr>
<tr><td>t</td><td>4</td></tr>
<tr><td>u</td><td>1</td></tr>
<tr><td>, (comma)</td><td>1</td></tr>
<tr><td>(space)</td><td>9</td></tr>
</table>
</section>
<section>
<h2>Compression step 1 (b)</h2>
<table><tr style="background-color: transparent"><td class="top"><p>Build a min-heap, sorted by frequency</p><img alt="huffman-14" src="graphs/huffman-14.svg"></td><td class="top">
<table>
<tr><th>Character</th><th>Frequency</th></tr>
<tr><td>b</td><td>1</td></tr>
<tr><td>e</td><td>2</td></tr>
<tr><td>f</td><td>1</td></tr>
<tr><td>i</td><td>5</td></tr>
<tr><td>m</td><td>1</td></tr>
<tr><td>o</td><td>2</td></tr>
<tr><td>p</td><td>1</td></tr>
<tr><td>s</td><td>2</td></tr>
<tr><td>t</td><td>4</td></tr>
<tr><td>u</td><td>1</td></tr>
<tr><td>, (comma)</td><td>1</td></tr>
<tr><td>(space)</td><td>9</td></tr>
</table>
</td></table>
</section>
<section data-markdown><script type="text/template">
## Compression step 2
- Build the tree, starting with a "forest" of trees:
- Repeat
- Take the two trees that have the lowest frequency
- The next two removals from the heap
- Make them children of a new node
- Keep track of the total frequency of that node
- And stick that tree back into the heap
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## The final Huffman coding tree
</script></section>
<section>
<h2>The Prefix codes</h2>
<table><tr style="background-color: transparent"><td class="top"><img alt="huffman-12" src="graphs/huffman-12.svg"></td><td class="top">
<table>
<tr><th>Character</th><th>Prefix code</th></tr>
<tr><td>b</td><td>00000</td></tr>
<tr><td>e</td><td>0011</td></tr>
<tr><td>f</td><td>00001</td></tr>
<tr><td>i</td><td>11</td></tr>
<tr><td>m</td><td>00010</td></tr>
<tr><td>o</td><td>1000</td></tr>
<tr><td>p</td><td>00011</td></tr>
<tr><td>s</td><td>1001</td></tr>
<tr><td>t</td><td>101</td></tr>
<tr><td>u</td><td>00100</td></tr>
<tr><td>, (comma)</td><td>00101</td></tr>
<tr><td>(space)</td><td>01</td></tr>
</table>
</td></table>
</section>
<section>
<h2>Resulting Encoding Table</h2>
<table style="font-size:90%">
<tr><th>Character</th><th>Frequency</th><th>Prefix code</th><th>Total bits</th></tr>
<tr><td>b</td><td>1</td><td>00000</td><td>5</td></tr>
<tr><td>e</td><td>2</td><td>0011</td><td>8</td></tr>
<tr><td>f</td><td>1</td><td>00001</td><td>5</td></tr>
<tr><td>i</td><td>5</td><td>11</td><td>10</td></tr>
<tr><td>m</td><td>1</td><td>00010</td><td>5</td></tr>
<tr><td>o</td><td>2</td><td>1000</td><td>8</td></tr>
<tr><td>p</td><td>1</td><td>00011</td><td>5</td></tr>
<tr><td>s</td><td>2</td><td>1001</td><td>8</td></tr>
<tr><td>t</td><td>4</td><td>101</td><td>12</td></tr>
<tr><td>u</td><td>1</td><td>00100</td><td>5</td></tr>
<tr><td>, (comma)</td><td>1</td><td>00101</td><td>5</td></tr>
<tr><td>(space)</td><td>9</td><td>01</td><td>18</td></tr>
</table>
<p>Total is 94 bits</p>
</section>
<section data-markdown><script type="text/template">
## Huffman Encoding Statistics
- Total encoding is 94 bits for 30 characeters
- Or an average of 3.13 bits per character
- ASCII was 240 bits: compression ratio of 2.6
- "Straight" encoding was 120 bits: compression ratio of 1.3
</script></section>
<section data-markdown><script type="text/template">
## Compression step 3
- Write the prefix codes to a file
- We'll use regular ASCII characters
- Does this actually do any compression, then?
```
b 00000
e 0011
f 00001
i 11
m 00010
o 1000
p 00011
s 1001
t 101
u 00100
, 00101
space 01
```
(note the double-space to indicate a space!)
</script></section>
<section>
<h2>The Prefix codes</h2>
<table><tr style="background-color: transparent">
<td class="top">
<ul>
<li>Write the text to the file using the Huffman encoding<br> </li>
<li>"be" becomes 00000 0011</li>
<li>"set" becomes 1001 0011 101</li>
<li>"stumps" becomes 1001 101 00100 00010 00011 1001</li>
</ul>
</td><td style="width:50px"></td><td class="top">
<table>
<tr><th>Character</th><th>Prefix code</th></tr>
<tr><td>b</td><td>00000</td></tr>
<tr><td>e</td><td>0011</td></tr>
<tr><td>f</td><td>00001</td></tr>
<tr><td>i</td><td>11</td></tr>
<tr><td>m</td><td>00010</td></tr>
<tr><td>o</td><td>1000</td></tr>
<tr><td>p</td><td>00011</td></tr>
<tr><td>s</td><td>1001</td></tr>
<tr><td>t</td><td>101</td></tr>
<tr><td>u</td><td>00100</td></tr>
<tr><td>, (comma)</td><td>00101</td></tr>
<tr><td>(space)</td><td>01</td></tr>
</table>
</td></table>
</section>
<section data-markdown><script type="text/template">
## Compression Phase Review
1. Determine the frequencies of the characters stored in the source file
- Read the source file
- Then store the character frequencies in a *min-heap*
2. Build a *tree* of prefix codes (a Huffman code) that determines the unique bit codes for each character
3. Write the prefix codes or code tree to the output file
4. Re-read the source file and for each character read, write its prefix code into the output file
- You must WRITE the prefix code/tree and the encoded file to the SAME output file
</script></section>
<section data-markdown><script type="text/template">
## Decompression Phase Review
1. Read in the prefix code structure (tree or array) from the compressed file.
- Build a new Huffman tree for performing decompression
2. Read in one bit at a time from the compressed file and move through the prefix code tree until a leaf node is reached
3. Write the character stored at the leaf node into the decompressed file
4. While there is still input, repeat
</script></section>
<section data-markdown><script type="text/template">
## Huffman Coding Lab
- Compression (pre-lab)
- Decompression (in-lab)
- Report (post-lab)
</script></section>
<section data-markdown><script type="text/template">
## Huffman Encoding Prelab
- Write a program that reads input from a file (see class website for test files)
- Build a binary heap (priority queue) that calculates the letter frequencies
- Build the huffman tree
- Print the letter-encoding mapping to the file
- There is a specific format, as specified in the lab
- Encode the message, printing this also to the file
</script></section>
<section data-markdown><script type="text/template">
## Tips/Hints
- You may use the heap code included here, or create your own heap code, but you may not use it from other sources (including the STL)
- File input/output reference material:
- The [fileio.cpp](../labs/lab10/fileio.cpp.html) ([src](../labs/lab10/fileio.cpp)) file from [lab 10 (Huffman)](../labs/lab10/index.html)
- The [Input/output with files article](http://www.cplusplus.com/doc/tutorial/files/) on [cplusplus.com](http://www.cplusplus.com/)
- The [getWordInTable.cpp](../labs/lab06/code/getWordInTable.cpp.html) ([src](../labs/lab06/code/getWordInTable.cpp)) file from [lab 6 (hashes)](../labs/lab06/index.html)
</script></section>
<section data-markdown><script type="text/template">
## ASCII
- American Standard Code for Information Interchange (ASCII)
- Character encoding standard
- Specifies correspondence between digital bit patterns and English alphabet
- 7-bit encoding with 1 bit for parity (error correction)
- 128 characters
- 95 printable characters
- 33 non-printable characters
- More details in the [Wikipedia ASCII article](http://en.wikipedia.org/wiki/ASCII)
</script></section>
<section id="asciiset">
<h2>ASCII Character Codes in Hexadecimal</h2>
<table class="transparent">
<tr>
<td class="width:45%"><img alt="ascii" src="images/10-heaps-huffman/ascii.png"></td>
<td style="width:10%"></td>
<td class="top" style="width:45%"><p>For the lab, you only need to account for the printable characters (0x20 to 0x7e)</p><p> </p>
<p>Character codes:</p>
<ul>
<li>32<sub>10</sub> (0x20) = space</li>
<li>33<sub>10</sub> (0x21) = !</li>
<li>126<sub>10</sub>(0x7e) = ~</li>
</ul></td></tr></table>
</section>
</section>
<section>
<section id="priorityqueueex" data-markdown><script type="text/template">
# Priority Queue<br>Example
</script></section>
<section data-markdown><script type="text/template">
## Priority Queue Operations
- insert (x)
- deleteMin()
- findMin()
- isEmpty()
- makeEmpty()
- size()
</script></section>
<section data-markdown><script type="text/template">
## Binary Heap Class
Source code: [binary_heap.h](code/10-heaps-huffman/binary_heap.h.html) ([src](code/10-heaps-huffman/binary_heap.h))
```
class binary_heap {
public:
binary_heap();
binary_heap(vector<int> vec);
~binary_heap();
void insert(int x);
int findMin();
int deleteMin();
unsigned int size();
void makeEmpty();
bool isEmpty();
void print();
private:
vector<int> heap;
unsigned int heap_size;
void percolateUp(int hole);
void percolateDown(int hole);
};
```
</script></section>
<section data-markdown><script type="text/template">
## Heap Data Members
- `int heap_size`
- Size of the heap
- This holds the number of *heap elements*, which is always one less than the number of elements in the vector
- `vector<int> heap`
- The heap implemented as an array (vector)
</script></section>
<section data-markdown><script type="text/template">
## Methods seen already
- `insert(int x)`
- `deleteMin()`
- `percolateUp(int hole)`
- `percolateDown(int hole)`
</script></section>
<section data-markdown><script type="text/template">
## Constructors & Destructors
Source code: [binary_heap.cpp](code/10-heaps-huffman/binary_heap.cpp.html) ([src](code/10-heaps-huffman/binary_heap.cpp))
```
// default constructor
binary_heap::binary_heap() : heap_size(0) {
heap.push_back(0);
}
// builds a heap from an unsorted vector
binary_heap::binary_heap(vector<int> vec) :
heap_size(vec.size()) {
heap = vec;
heap.push_back(heap[0]);
heap[0] = 0;
for ( int i = heap_size/2; i > 0; i-- )
percolateDown(i);
}
// the destructor doesn't need to do much
binary_heap::~binary_heap() {