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<!doctype html>
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<title>CS 2150: 06-hashes slide set</title>
<meta name="description" content="A set of slides for a course on Program and Data Representation">
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<div class="slides">
<section data-markdown id="cover"><script type="text/template">
# CS 2150
### Program and Data Representation
<center><small><a href="http://www.cs.virginia.edu/~asb">Aaron Bloomfield</a> (aaron@virginia.edu)<br><a href="http://www.cs.virginia.edu/~nn4pj">Rich Nguyen</a> (nn4pj@virginia.edu)<br><a href="http://www.cs.virginia.edu/~mrf8t">Mark Floryan</a> (mrf8t@virginia.edu)</small></center>
<center><small><a href="http://@github/uva-cs/pdr">@github</a> | <a href="index.html">↑</a> | <a href="./06-hashes.html?print-pdf"><img class="print" width="20" src="../slides/images/print-icon.png"></a></small></center>
## Hash Tables
</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
[ADTs Covered So Far](#/adtssofar)
[Hash Tables](#/hashtables)
[Separate Chaining](#/separatechaining)
[Open Addressing](#/openaddressing)
[Miscellaneous](#/miscellaneous)
</script></section>
<section>
<section id="adtssofar" data-markdown><script type="text/template">
# ADTs Covered So Far
</script></section>
<section data-markdown><script type="text/template">
## Lists
- Operations
- find
- insert
- remove
- findKth
- Implementations
- Array (vector)
- Linked list
</script></section>
<section data-markdown><script type="text/template">
## Lists
| | Array (vector) | Linked List |
|-|-|-|
| find | Θ(*n*) | Θ(*n*) |
| insert | Θ(*n*) worst case,<br>but often Θ(1) | Θ(1) |
| remove | Θ(*n*) | Θ(*n*) |
| findKth | Θ(1) | Θ(*n*) |
<center>The operations are <i>generally</i> linear-time operations</center>
</script></section>
<section data-markdown><script type="text/template">
## Stacks
- List with data handled last-in first-out
- Operations:
- push
- pop
- top
- Implementations
- Array (vector)
- Linked list
</script></section>
<section data-markdown><script type="text/template">
## Stacks
| | Array (vector) | Linked List |
|-|-|-|
| push | Θ(*n*) worst case,<br>but often Θ(1) | Θ(1) |
| pop | Θ(1) | Θ(1) |
| top | Θ(1) | Θ(1) |
<center>The operations are <i>generally</i> constant-time operations</center>
</script></section>
<section data-markdown><script type="text/template">
## Queues
- First-in first-out list
- Operations:
- enqueue
- dequeue
- Implementations
- Array (vector)
- Linked lists
</script></section>
<section data-markdown><script type="text/template">
## Queues
| | Array (vector) | Linked List |
|-|-|-|
| enqueue | Θ(*n*) worst case,<br>but often Θ(1) | Θ(1) |
| dequeue | Θ(1) | Θ(1) |
<center>The operations are <i>generally</i> constant-time operations</center>
</script></section>
<section data-markdown><script type="text/template">
## Trees
- Goal is Θ(log *n*) runtime for most operations
- Binary search trees
- AVL Trees
- Red-black trees
- Splay trees
</script></section>
<section data-markdown><script type="text/template">
## Trees
| | BST | AVL | Red-black | Splay |
|-|-|-|-|-|
| find | Θ(*h*), where<br>log *n* < *h* ≤ *n*-1;<br>worst case is Θ(*n*) | Θ(log *n*) | Θ(log *n*) | Θ(log *n*)<br>amortized |
| insert | Θ(*h*), where<br>log *n* < *h* ≤ *n*-1;<br>worst case is Θ(*n*) | Θ(log *n*) | Θ(log *n*) | Θ(log *n*)<br>amortized |
| remove | Θ(*h*), where<br>log *n* < *h* ≤ *n*-1;<br>worst case is Θ(*n*) | Θ(log *n*) | Θ(log *n*) | Θ(log *n*)<br>amortized |
<center>Balanced trees are <i>generally</i> logarithmic-time operations</center>
</script></section>
<section data-markdown><script type="text/template">
## Is There Anything Faster?
- Fastest possible search using binary comparison: Θ(log *n*)
- Rephrased: binary comparison searches are Ω(log *n*)
- We can do better: (almost) constant (Θ(1)) is possible with *hash tables*
- Hash tables (lookup table)
- Standard set of operations: find, insert, delete
- No ordering property!
- Thus, no findMin or findMax
</script></section>
</section>
<section>
<section id="hashtables" data-markdown><script type="text/template">
# Hash Tables
</script></section>
<section data-markdown><script type="text/template">
## Key-value pairs
- Hash tables store key-value pairs
- Each value has a specific key associated with it
- Keys and values need not be the same type!
- Examples
- Definitions: "set", "v.tr. 1. To put in a specified position..."
- Uva e-mail redirects: "aaron@", "asb2t@cms .virginia.edu"
- Anything that can be stored in a tree
- Userid / IDnum pairs
- Userid / lots_of_info_about_them_in_an_object pairs
</script></section>
<section>
<h2>Hash Tables</h2>
<table class="transparent"><tr><td>
<table class="transparent"><tr><td>
<div style="font-size:130%;line-height:110%">
<ul>
<li>Hash table<ul>
<li>fixed size <i>array</i> of some size, usually a prime number<ul>
<li>Should be larger than the number of elements</li></ul></li></ul></li>
<li>Given a key space:</li>
</ul>
</div>
</td></tr>
<tr><td>
<table class="transparent"><tr><td><img alt="blob" src="images/06-hashes/blob.png"></td><td class="middle">
<table class="transparent"><tr><td><div style="font-size:130%;line-height:110%">hash function</div></td></tr><tr><td><div style="font-size:130%;line-height:110%"><i>hash</i>(<i>k</i>)</div></td></tr><tr><td><div style="font-size:200%">→</div></td></tr></table>
</td></tr></table>
</td></tr></table>
</td><td class="middle">
<table class="transparent">
<tr><td> </td><td></td></tr>
<tr><td> </td><td></td></tr>
<tr><td style="text-align:right;">hash</td><td style="text-align:left;">table</td></tr>
<tr><td> </td><td></td></tr>
<tr><td>0</td><td class="border" style="width:100px"></td></tr>
<tr><td>1</td><td class="border" style="width:100px"></td></tr>
<tr><td>2</td><td class="border" style="width:100px"></td></tr>
<tr><td> </td><td class="border" style="width:100px"></td></tr>
<tr><td>...</td><td class="border" style="width:100px"></td></tr>
<tr><td> </td><td class="border" style="width:100px"></td></tr>
<tr><td>tablesize‑1</td><td class="border" style="border-bottom:medium solid;"></td></tr>
</table>
</td></tr></table>
</section>
<section data-markdown><script type="text/template">
## Hash function
- A hash function takes in a "thing"...
- string, int, object, etc.
- and returns an *unsigned* integer value
- which is then mod'ed by the size of the hash table to yield a spot within the bounds of the hash table array
- Three *required* properties
- Must be *deterministic*
- Meaning it must return the same value each time for the same "thing"
- Must be *fast*
- Must be *evenly distributed*
- Technically, only the first is required for *correctness*, but the other two are required for fast running times
</script></section>
<section>
<h2>Hash functions KLA</h2>
<ul>
<li>I'm going hash all of you into 10 buckets (0-9) by your birthday<ul>
<li>(you are welcome to make up another birthday, as long as you are consistent)</li></ul></li>
<li>The hash functions:<ul>
<li class="fragment" data-fragment-index="1">By the decade of your birth year<ul>
<li class="fragment" data-fragment-index="1"><i>hash</i>(<i>birthday</i>) = (<i>year</i>/10) % 10</li></ul></li>
<li class="fragment" data-fragment-index="2">By the last digit of your birth year<ul>
<li class="fragment" data-fragment-index="2"><i>hash</i>(<i>birthday</i>) = <i>year</i> % 10</li></ul></li>
<li class="fragment" data-fragment-index="3">By the last digit of your birth month<ul>
<li class="fragment" data-fragment-index="3"><i>hash</i>(<i>birthday</i>) = <i>month</i> % 10</li></ul></li>
<li class="fragment" data-fragment-index="4">By the last digit of your birth day<ul>
<li class="fragment" data-fragment-index="4"><i>hash</i>(<i>birthday</i>) = <i>day</i> % 10</li></ul></li>
</ul></li></ul>
</section>
<section>
<h2>Keys</h2>
<table class="wide">
<tr>
<td class="top">
<div style="width:400px;font-size:130%;line-height:110%">
<ul>
<li>How can we hash the keys if the keys can be anything?</li>
<li>Best one binary comparison can do is eliminate one half of the elements Θ(log <i>n</i>)</li>
<li>We want Θ(1)</li>
<li>The keys must be bits, so we can do better!</li>
</ul></div>
</td>
<td style="width:75px"></td>
<td class="top">
<br><small>"Hello"</small><br> <br><small>['H','i',\0]</small><br> <br><small>3.14</small><br> <br><small>'x'</small><br> <br><small>0x42381a</small><br> <br><small>01001010</small></td>
<!-- image adapted from http://openclipart.org/detail/3677/arrow-left-right-by-torfnase -->
<td style="width:150px;vertical-align:top"><img class="noborder" src="images/red-double-arrow.png" height="500" alt="vertical red double arrow"></td>
<td class="top"> <br> <br><small>Objects</small><br> <br><small>Arrays</small><br> <br><small>Primitive types</small><br> <br><small>Addresses</small><br> <br><small>bits</small></td>
<td> </td>
</tr>
</table>
</section>
<section data-markdown><script type="text/template">
## Lookup Table
| hash(key) | key |
|-|-|
| 000000 | "red" |
| 000001 | "orange" |
| 000010 | "blue" |
| 000011 | `null` |
| 000100 | "green" |
| 000101 | ... |
This can work, unless the key space is sparse, or we don't know the keys ahead of time. But it's slow to look up a value in a table!
</script></section>
<section>
<h2>Example</h2>
<table class="transparent"><tr><td>
<div style="font-size:120%">
<ul>
<li>Key space: integers<br> </li>
<li>Table size: 10<br> </li>
<li><i>hash</i>(<i>k</i>) = <i>k</i> mod 10<ul>
<li>Technically, <i>hash</i>(<i>k</i>) = <i>k</i>,<br>which is <i>then</i> mod'ed by<br>the table size of 10<br> </li>
</ul></li>
<li>Insert: 7, 18, 41, 34<br> </li>
<li>How do we find them?</li>
</ul></div>
</td><td style="width:100px"></td><td class="top">
<table class="transparent">
<tr><td>0</td><td class="border" style="width:100px"></td></tr>
<tr><td>1</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="3">41</span></td></tr>
<tr><td>2</td><td class="border" style="width:100px"></td></tr>
<tr><td>3</td><td class="border" style="width:100px"></td></tr>
<tr><td>4</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="4">34</span></td></tr>
<tr><td>5</td><td class="border" style="width:100px"></td></tr>
<tr><td>6</td><td class="border" style="width:100px"></td></tr>
<tr><td>7</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="1">7</span></td></tr>
<tr><td>8</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="2">18</span></td></tr>
<tr><td>9</td><td class="border" style="border-bottom:medium solid;"></td></tr>
</table>
</td></tr></table>
</section>
<section data-markdown><script type="text/template">
## Table size issues...
- Why not just have a table of size 100
- And map them directly to the location corresponding to their key?
- We assume that the key space is too large
- Example: mapping social security numbers for students at UVa
- There are not 999,999,999 students at UVa, even if taken across all time
- Do you see why find max and find min are not easy?
- We have not preserved any ordering info
</script></section>
<section>
<h2>Another Example</h2>
<table class="transparent"><tr><td>
<div style="font-size:120%">
<ul>
<li>Key space: integers<br> </li>
<li>Table size: 6<br> </li>
<li><i>hash</i>(<i>k</i>) = <i>k</i> mod 6<br> </li>
<li>Insert: 7, 18, 41, 34, <span class='red'>12</span><br> </li>
<li>How do we find them?</li>
</ul></div>
</td><td style="width:100px"></td><td class="top">
<table class="transparent">
<tr><td>0</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="2">18</span><span class="fragment" data-fragment-index="5"><span class="red"> 12</span></span></td></tr>
<tr><td>1</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="1">7</span></td></tr>
<tr><td>2</td><td class="border" style="width:100px"></td></tr>
<tr><td>3</td><td class="border" style="width:100px"></td></tr>
<tr><td>4</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="4">34</span></td></tr>
<tr><td>5</td><td class="border" style="border-bottom:medium solid;"><span class="fragment" data-fragment-index="3">41</span></td></tr>
</table>
</td></tr></table>
</section>
<section data-markdown><script type="text/template">
## Hash Table
- Hash function: *hash*: *key* → [0, *m*-1]
- Really to any unsigned integer, which is then mod'ed by *m*, the table size
- Here, *hash*(*key*) = `firstletter`(*key*)<br>
| Location | Key | Value |
|-|-|-|
| 0 | "Alice" | "red" |
| 1 | "Bob" | "orange" |
| 2 | "Colleen" | "blue" |
| 3 | `null` | `null` |
| 4 | "Eve" | "green" |
| ... | ... | ... |
| *m*-1 | "Zeus" | "purple" |
</script></section>
<section data-markdown><script type="text/template">
## Hash Functions
- Required properties described earlier
- Must be deterministic
- Must be fast
- Must be evenly distributed
- This implies avoiding of collisions
- A perfect hash function has:
- No blanks (i.e., no empty cells)
- No collisions
</script></section>
<section>
<h2>Sample String Hash Functions</h2>
<ul>
<li>Key space: strings</li>
<li>A string <i>s</i> is made up of characters <i>s<sub>i</sub></i></li>
<li>\( s = s_0s_1s_2s_3\ldots s_{k-1} \)</li>
</ul>
<p> </p>
<ol>
<li class="fragment">\( hash(s) = s_0 \mod table\_size \)<br> </li>
<li class="fragment">\( hash(s) = \left( \sum_{i=0}^{k-1}s_i \right) \mod table\_size \)<br> </li>
<li class="fragment">\( hash(s) = \left( \sum_{i=0}^{k-1}s_i*37^i \right) \mod table\_size \)<br> </li>
</ol>
</section>
<section data-markdown><script type="text/template">
## Hash function notes
- They should always return an *unsigned* int
- Otherwise your program will be trying to find a negative array index
- Integer overflow is fine, as long as it overflows *deterministically*
- Meaning the same way each time
- This will especially be true with the last of the string hash functions presented on the previous slide
</script></section>
<section data-markdown><script type="text/template">
## Collision Resolution
- Collision: when two keys map to the same location in the hash table
- Two primary ways to resolve collisions:
1. Separate Chaining (make each spot in the table a 'bucket' or a collection)
2. Open Addressing, of which there are 3 types:
- Linear probing
- Quadratic probing
- Double hashing
</script></section>
</section>
<section>
<section id="separatechaining" data-markdown><script type="text/template">
# Separate Chaining
</script></section>
<section>
<h2>Separate Chaining</h2>
<table class="transparent"><tr><td class="top">
<table class="transparent">
<tr><td>0</td><td class="border" style="width:100px"></td></tr>
<tr><td>1</td><td class="border" style="width:100px"></td></tr>
<tr><td>2</td><td class="border" style="width:100px"></td></tr>
<tr><td>3</td><td class="border" style="width:100px"></td></tr>
<tr><td>4</td><td class="border" style="width:100px"></td></tr>
<tr><td>5</td><td class="border" style="width:100px"></td></tr>
<tr><td>6</td><td class="border" style="width:100px"></td></tr>
<tr><td>7</td><td class="border" style="width:100px"></td></tr>
<tr><td>8</td><td class="border" style="width:100px"></td></tr>
<tr><td>9</td><td class="border" style="border-bottom:medium solid;width:100px"></td></tr>
</table>
</td><td style="width:200px"></td><td class="top">
<div style="font-size:120%;line-height:110%">
<ul>
<li>All keys that map to the same hash value are kept in a "bucket"<ul>
<li>This "bucket" is another data structure, typically a linked list</li></ul><br> </li>
<li><i>hash</i>(<i>k</i>) = <i>k</i> mod 10<br> </li>
<li>Insert: 10, 22, 107, 12, 42</li>
</ul></div>
</td></tr></table>
<script type="text/javascript">insertCanvas();</script>
</section>
<section data-markdown><script type="text/template">
## Analysis of find
- Definition: The *load factor*, λ, of a hash table is the ratio of the number of elements divided by the table size
- For separate chaining, λ is the average number of elements in a bucket
- Average time on unsuccessful find: λ
- Average length of a list at *hash*(*k*)
- Average time on successful find: 1 + (λ/2)
- One node, plus half the average length of a list (not including the item)
</script></section>
<section data-markdown><script type="text/template">
## Load factor
- How big should we make the hash table?
- Well, we want "constant" time for find and insert...
- Possible sizes for hash table with separate chaining
- λ = 1
- Make hash table be the number of elements expected; average bucket size is 1
- Also make it a prime number
- λ = 0.75
- Java's [Hashtable](http://docs.oracle.com/javase/7/docs/api/java/util/Hashtable.html) but can be set to another value
- Table will always be bigger than number of elements
- This reduces the chance of a collision!
- Good trade-off between memory use and time
- λ = 0.5
- Uses more memory, but fewer collisions
</script></section>
<section data-markdown><script type="text/template">
## Separate Chaining: find()
- Note that we now have to keep each key in the chain, as well as the value!
- What is the worst case?
- Hint: [Wikipedia](http://en.wikipedia.org/wiki/Hash_table) is wrong on this one...
- In the worst case, every key could hash to the same spot!
- As nobody uses anything other than a linked list as the secondary data structure, this means it will be a Θ(*n*) algorithm to perform a find!
- What is the "hopeful" case?
</script></section>
<section data-markdown><script type="text/template">
## What data structure to use for the buckets?
- AVL & red-black trees will give the best running time
- But that's a lot of overhead!
- Vectors are easier and simpler, but take up a *lot* of space
- All those extra, unused, cells
- Don't *ever* use vectors for this!
- Linked lists are quick and easy, and take up very little extra space
- That's Θ(*n*)!
- Still faster *in practice* than trees due to having a very small number of items in the bucket
</script></section>
<section data-markdown><script type="text/template">
## Requirements for "Hopeful" Case
- Our ideal hash function and hash table:
- Function *hash*(*k*) is well distributed for key space
- For a randomly selected *k* ∈ *K*,
- probability(*hash*(*k*) = i) = 1/*table_size*
- Size of table scales linearly with number of elements
- Expected bucket size is Θ(*num_elements* / *table_size*)
- Finding a good hash function can be tough
</script></section>
<section data-markdown><script type="text/template">
## Separate chaining insert is Θ(1)
- In an unsorted linked list, you can just put it on the front
- So all inserts into a seprate chained hash table, that uses linked lists, are actually in constant time
- If you were to *sort* the linked list, that would be linear time
- And finds (and thus deletes) are still linear time
</script></section>
</section>
<section>
<section id="openaddressing" data-markdown><script type="text/template">
# Open Addressing
</script></section>
<section data-markdown><script type="text/template">
## Saving Memory
<center>Can we avoid the overhead of all those linked lists?</center>
</script></section>
<section data-markdown><script type="text/template">
## Three Types of Probing Strategies
- The three types:
- Linear
- Quadratic
- Double hashing
- The general idea with all of them is that, if a spot is occupied, to 'probe', or try, other spots in the table to use
- How we determine where else to probe depends on which strategy we are using
</script></section>
<section>
<h2>Linear Probing</h2>
<table class="transparent"><tr><td class="top">
<table class="transparent">
<tr><td>0</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="3">37</span></td></tr>
<tr><td>1</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="4">14</span></td></tr>
<tr><td>2</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="5">21</span></td></tr>
<tr><td>3</td><td class="border" style="width:100px"></td></tr>
<tr><td>4</td><td class="border" style="width:100px"></td></tr>
<tr><td>5</td><td class="border" style="width:100px"></td></tr>
<tr><td>6</td><td class="border" style="width:100px"></td></tr>
<tr><td>7</td><td class="border" style="width:100px"></td></tr>
<tr><td>8</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="2">27</span></td></tr>
<tr><td>9</td><td class="border" style="border-bottom:medium solid;width:100px"><span class="fragment" data-fragment-index="1">4</span></td></tr>
</table>
</td><td style="width:200px"></td><td class="top">
<ul>
<li>Check spots in this order:<ul>
<li><i>hash</i>(<i>k</i>)</li>
<li><i>hash</i>(<i>k</i>)+1</li>
<li><i>hash</i>(<i>k</i>)+2</li>
<li><i>hash</i>(<i>k</i>)+3</li>
<li>etc.</li>
</ul> </li>
<li><i>hash</i>(<i>k</i>) = 3<i>k</i>+7<ul><li>Which is then mod'ed by the table size (10)</li><li>Result: <i>hash</i>(<i>k</i>) = (3<i>k</i>+7) mod 10</li></ul> </li>
<li>Insert: 4, 27, 37, 14, 21<ul>
<li><i>hash</i>(<i>k</i>) values: 19, 88, 118, 49, 70, respectively</li>
</ul></li>
</ul>
</td></tr></table>
</section>
<section data-markdown><script type="text/template">
## Linear Probing
- With all open addressing schemes, we examine ('probe') the cells in the order:
- *p*<sub>0</sub>(*k*), *p*<sub>1</sub>(*k*), *p*<sub>2</sub>(*k*), ...
- where: *p*<sub>i</sub>(*k*) = (*hash*(*k*) + *f*(*i*)) mod *table_size*
- With *linear probing*, <span class="red">*f*(*i*) = *i*</span>
- After searching spot *hash*(*k*) in the array, look in:
- *hash*(*k*) + 1
- *hash*(*k*) + 2
- *hash*(*k*) + 3
- etc.
</script></section>
<section data-markdown><script type="text/template">
## Problems with Linear Probing
- Primary clustering
- Large blocks of occupied cells
- As table fills, increased number of attempts required to solve collision
- And thus slower lookup times
- "Holes" when an element is removed
- We'll see how to solve this later
- When to stop looking?
</script></section>
<section data-markdown><script type="text/template">
## Quadratic Probing
- With all open addressing schemes, we examine ('probe') the cells in the order:
- *p*<sub>0</sub>(*k*), *p*<sub>1</sub>(*k*), *p*<sub>2</sub>(*k*), ...
- where: *p*<sub>i</sub>(*k*) = (*hash*(*k*) + *f*(*i*)) mod *table_size*
- With *quadratic probing*, <span class="red">*f*(*i*) = *i*<sup>2</sup></span>
- After searching spot *hash*(*k*) in the array, look in:
- *hash*(*k*) + 1
- *hash*(*k*) + 4
- *hash*(*k*) + 9
- etc.
</script></section>
<section>
<h2>Quadratic Probing</h2>
<table class="transparent"><tr><td class="top">
<table class="transparent">
<tr><td>0</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="3">14 </span></td></tr>
<tr><td>1</td><td class="border" style="width:100px"></td></tr>
<tr><td>2</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="4">37</span></td></tr>
<tr><td>3</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="5">22</span></td></tr>
<tr><td>4</td><td class="border" style="width:100px"></td></tr>
<tr><td>5</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="6">34</span></td></tr>
<tr><td>6</td><td class="border" style="width:100px"></td></tr>
<tr><td>7</td><td class="border" style="width:100px"></td></tr>
<tr><td>8</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="2">27</span></td></tr>
<tr><td>9</td><td class="border" style="border-bottom:medium solid;width:100px"><span class="fragment" data-fragment-index="1">4</span></td></tr>
</table>
</td><td style="width:200px"></td><td class="top">
<ul>
<li>Check spots in this order:<ul>
<li><i>hash</i>(<i>k</i>)</li>
<li><i>hash</i>(<i>k</i>)+1<sup>2</sup> = <i>hash</i>(<i>k</i>)+1</li>
<li><i>hash</i>(<i>k</i>)+2<sup>2</sup> = <i>hash</i>(<i>k</i>)+4</li>
<li><i>hash</i>(<i>k</i>)+3<sup>2</sup> = <i>hash</i>(<i>k</i>)+9</li>
<li>etc.</li>
</ul> </li>
<li><i>hash</i>(<i>k</i>) = 3<i>k</i>+7<ul><li>Which is then mod'ed by the table size (10)</li><li>Result: <i>hash</i>(<i>k</i>) = (3<i>k</i>+7) mod 10</li></ul> </li>
<li>Insert: 4, 27, 14, 37, 22, 34<ul>
<li><i>hash</i>(<i>k</i>) values: 19, 88, 49, 118, 73, 109, respectively</li>
</ul></li>
</ul>
</td></tr></table>
</section>
<section data-markdown><script type="text/template">
## Double Hashing
- With all open addressing schemes, we examine ('probe') the cells in the order:
- *p*<sub>0</sub>(*k*), *p*<sub>1</sub>(*k*), *p*<sub>2</sub>(*k*), ...
- where: *p*<sub>i</sub>(*k*) = (*hash*(*k*) + *f*(*i*)) mod *table_size*
- With *double hashing*, <span class="red">*f*(*i*) = *i* \* hash<sub>2</sub>(*k*)</span>
- Which means we have to define a *secondary* hash function!
- After searching spot *hash*(*k*) in the array, look in:
- *hash*(*k*) + 1 \* *hash*<sub>2</sub>(*k*)
- *hash*(*k*) + 2 \* *hash*<sub>2</sub>(*k*)
- *hash*(*k*) + 3 \* *hash*<sub>2</sub>(*k*)
- etc.
</script></section>
<section>
<h2>Double Hashing</h2>
<table class="transparent"><tr><td class="top">
<table class="transparent">
<tr><td>0</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="5">69</span></td></tr>
<tr><td>1</td><td class="border" style="width:100px"></td></tr>
<tr><td>2</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="6">60</span></td></tr>
<tr><td>3</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="3">58</span></td></tr>
<tr><td>4</td><td class="border" style="width:100px"></td></tr>
<tr><td>5</td><td class="border" style="width:100px"></td></tr>
<tr><td>6</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="4">49</span></td></tr>
<tr><td>7</td><td class="border" style="width:100px"></td></tr>
<tr><td>8</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="2">18</span></td></tr>
<tr><td>9</td><td class="border" style="border-bottom:medium solid;width:100px"><span class="fragment" data-fragment-index="1">89</span></td></tr>
</table>
</td><td style="width:200px"></td><td class="top">
<ul>
<li>Check spots in this order:<ul>
<li><i>hash</i>(<i>k</i>)</li>
<li><i>hash</i>(<i>k</i>) + 1 * <i>hash</i><sub>2</sub>(<i>k</i>)</li>
<li><i>hash</i>(<i>k</i>) + 2 * <i>hash</i><sub>2</sub>(<i>k</i>)</li>
<li><i>hash</i>(<i>k</i>) + 3 * <i>hash</i><sub>2</sub>(<i>k</i>)</li>
<li>etc.</li>
</ul> </li>
<li><i>hash</i>(<i>k</i>) = <i>k</i><ul>
<li>The hash function was made simpler for this example...</li>
<li>Which is then mod'ed by the table size (10)</li>
<li>Result: <i>hash</i>(<i>k</i>) = <i>k</i> mod 10</li></ul></li>
<li><i>hash</i><sub>2</sub>(<i>k</i>) = 7 - (<i>k</i> mod 7)<br> </li>
<li>Insert: 89, 18, 58, 49, 69, 60</li>
</ul>
</td></tr></table>
</section>
<section>
<h2>Double Hashing Thrashing</h2>
<table class="transparent"><tr><td class="top">
<table class="transparent">
<tr><td>0</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="1">10</span></td></tr>
<tr><td>1</td><td class="border" style="width:100px"></td></tr>
<tr><td>2</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="2">12</span></td></tr>
<tr><td>3</td><td class="border" style="width:100px"></td></tr>
<tr><td>4</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="3">14</span></td></tr>
<tr><td>5</td><td class="border" style="width:100px"></td></tr>
<tr><td>6</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="4">16</span></td></tr>
<tr><td>7</td><td class="border" style="width:100px"></td></tr>
<tr><td>8</td><td class="border" style="width:100px"><span class="fragment" data-fragment-index="5">18</span></td></tr>
<tr><td>9</td><td class="border" style="border-bottom:medium solid;width:100px"></td></tr>
</table>
</td><td style="width:200px"></td><td class="top">
<ul>
<li><i>hash</i>(<i>k</i>) = <i>k</i> mod 10 <ul>
<li>Same as the previous slide</li>
<li>Result: <i>hash</i>(<i>k</i>) = <i>k</i> mod 10</li></ul> </li>
<li><i>hash</i><sub>2</sub>(<i>k</i>) = (<i>k</i> mod 5) +1<br> </li>
<li>Insert: 10, 12, 14, 16, 18, <span class='red'>36</span></li>
</ul>
</td></tr></table>
</section>
<section data-markdown><script type="text/template">
## Table size must be prime!
- The table size must always be a prime number
- It will prevent the thrashing from the previous slide
- Thrashing will only occur when the double hash value is a *factor* of the table size
- The only factors of a prime number *p* are 1 and *p*
- 1 is effectively linear probing, which is fine
- *p* will mod to zero, which is an invalid return value for a secondary hash function
- It will provide better distribution of the hash keys into the table
- Less clustering, etc.
- A prime number table size does not remove the need for a good hash function!
</script></section>
</section>
<section>
<section id="miscellaneous" data-markdown><script type="text/template">
# Miscellaneous
</script></section>
<section data-markdown><script type="text/template">
## Rehashing
- Problem: when the table gets too full, running time for operations increases
- Solution: create a bigger table and hash all the items from the original table into the new table
- The position in a table is dependent on the table size, which means we have to *rehash* each value
- This means we have to re-compute the hash value for *each* element, and insert it into the new table!
</script></section>
<section data-markdown><script type="text/template">
## Rehashing
- When to rehash?
- When half full (λ = 0.5)
- When mostly full (λ = 0.75)
- Java's hashtable does this by default
- When an insertion fails
- Some other threshold
- Cost of rehashing
- Let's assume that the hash function computation is constant
- We have to do *n* inserts, and if each key hashes to the same spot, then it will be a Θ(*n*<sup>2</sup>) operation!
- Although it is not likely to ever run that slow
</script></section>
<section data-markdown><script type="text/template">
## Removing an element
- How to handle this?
- You could:
- Rehash upon each delete, which is *very* expensive
- Put in a 'placeholder' or 'sentinel' value
- But the table gets filled with these rather fast
- Perhaps rehashing after a certain number of deletes
- Disallow deletes entirely; you can do this for [lab 6](../labs/lab06/index.html)
- Hash tables are not an ideal data structure if you need to perform a lot of deletions
</script></section>
<section data-markdown><script type="text/template">
## Hashing: MD5
- [MD5: Message Digest 5](http://en.wikipedia.org/wiki/Md5)
- Given a string (or file contents, etc.) generate a 128-bit hash
- 2<sup>128</sup> = 3.4*10<sup>38</sup> (coincidentally, this is is also the [maximum finite value](03-numbers.html#/maxfloatvalue) of a `float`)
- Typically an MD5 is always written in hex: 16e28b7986fd74f65b061de89dc8b78e
- This could then be used as the key
- Obviously having to mod it by the table size
- (Was) good for checking if a download completed successfully
</script></section>
<section data-markdown><script type="text/template">
## Can you reverse an MD5 hash?
- Technically, no
- A 129-bit file has 2<sup>129</sup> possibilities, and if you were to hash each one, it would go into 2<sup>128</sup> buckets
- By the pigeonhole principle, there would be at least one hash value (pigeonhole) with multiple keys (pigeons), and you don't know which one
- In reality, *many* (and eventually *all*) would have multiple keys
- But if a password is stored by it's MD5 hash...
- ... then there are enough online hash libraries that you can find at least one password that hashes to that value
- Try Googling for [3858f62230ac3c915f300c664312c63f](https://www.google.com/search?q=3858f62230ac3c915f300c664312c63f)
- Plus there are lots of weaknesses in MD5...
</script></section>
<section data-markdown><script type="text/template">
## More hashing: SHA
- MD5 has been "broken"
- One can generate two files that have the same hash; this is called a [collision attack](https://en.wikipedia.org/wiki/Collision_attack)
- In fact, I have the students do something similar when I teach Defense Against the Dark Arts (albeit with a [weaker hashing algorithm](https://en.wikipedia.org/wiki/Crc32))
- So it is useless for any security-related purposes
- [SHA (Secure Hash Algorithm)](http://en.wikipedia.org/wiki/Secure_Hash_Algorithm) is a family of algorithms that (the more recent ones) are much more secure
- Same overall idea: it generates a hash value up to 512 bits
- SHA-1 has been broken also, but more recent SHAs are secure
</script></section>
</section>
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