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<html>
<head>
<link href="style.css" rel="stylesheet" type="text/css"/>
<title>
Design and Analysis of Algorithms, Lecture 3
</title>
</head>
<body>
<h1>
Design and Analysis of Algorithms, Lecture 3
<a href="#note1">*</a>
</h1>
<h2>Looking at Memoization</h2>
<p>
Last class, I asked you to <a
href="https://en.wikipedia.org/wiki/Memoization">
memoize</a>
the naive, recursive Fibonacci algorithm. Let's take a look at how to do that.
<br>
<a href="https://github.com/gcallah/algorithms/blob/master/python/fibonacci.py">
Here</a>
is the Fibonacci code, now including memo-ization.
</p>
<h2>Dictionaries</h2>
<p>
<img
src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e2/English-English_and_English-Persian_dictionaries.JPG/350px-English-English_and_English-Persian_dictionaries.JPG">
</p>
<h3>
Dictionary ADT.
</h3>
<p>
Operations associated with this data type allow:
<ul>
<li>the addition of a pair to the collection
<li>the removal of a pair from the collection
<li>the modification of an existing pair
<li>the lookup of a value associated with a particular key
</ul>
<p>
(<a href="https://en.wikipedia.org/wiki/Associative_array">Source</a>)
</p>
</p>
<p>
Typical uses:
<ul>
<li> Symbol lookup in a programming language
<li> Counting words in a book
<li> Store colors by name as key and their numeric equivalent as
the value. Then we can write <b>set_text(colors["red"])</b>.
</ul>
</p>
<p>
(Ordered Dictionary ADT next time.)
</p>
<p>
<em>Direct addressing</em> and <em>Hashing</em> are two ways of implementing a
dictionary. Are there others?
</p>
<h3>
Direct addressing.
</h3>
<ul>
<li> <em>O(1)</em> <em>worst</em> case time for lookup.
<li> Uses:
<ul>
<li> Memoization
<li> Bingo
<li> Sieve of Eratosthenes
<li> Mark zipcodes seen
</ul>
<li> Downside: wastes space. If you have no idea how many possible
keys you need, direct addressing is not a good choice.
<br>For instance, if your key is an arbitrary string!
<li><a
href="https://github.com/gcallah/algorithms/blob/master/python/hash.py">
Example code here.
</a>
</ul>
<h3>
Hashing
</h3>
<p>
<img
src="https://upload.wikimedia.org/wikipedia/commons/thumb/5/58/Hash_table_4_1_1_0_0_1_0_LL.svg/300px-Hash_table_4_1_1_0_0_1_0_LL.svg.png">
<br>
<h4>
Basic Hashing:
</h4>
<ul>
<li> <em>O(1)</em> <em>average</em> case time for lookup.
<li> Universe of keys <em>U</em> mapped into slots of a <em>hash table</em>
of size <em>m</em> by hash function <em>h</em>.
<li> Because <em>size(U) > m</em>, collisions are always possible.
<br>Imagine we hash by word length: 'mark' and 'beam' both hash to
4. (Stupid hash function, but it illustrates the idea.) We must
resolve this collision somehow.
<li> Resolve collisions by chaining:
<br> Each slot holds a linked list of values.
<li> <a
href="https://en.wikipedia.org/wiki/Cryptographic_hash_function">
Cryptographic hashing
</a>
<br> Use large hash keys:
<a href="https://en.wikipedia.org/wiki/SHA-1">
SHA-1</a> uses 160 bit keys. <a
href="https://en.wikipedia.org/wiki/SHA-2">SHA-2</a> uses
keys of up to 512 bits.
<br>
<img
src="https://upload.wikimedia.org/wikipedia/commons/thumb/7/7d/SHA-2.svg/400px-SHA-2.svg.png">
<li> <a href="http://www.phash.org">
Perceptual hashing
</a>
</ul>
</p>
<h4>
Introducing probability into an algorithm.
</h4>
<p>
What happens to the usual assumptions?
<br>
<b>Correctness</b>: always, most of the time?
<br>
<b>Termination</b>: always, or almost always?
What does "performance" mean if the running
time/answer/even termination change from one run to the next?
</p>
<h4>Probability Basics</h4>
<p>
<a href="https://gcallah.github.io/algorithms/Probability.html">
Reviewed in this document.
</a>
</p>
<h4>
Simple uniform hashing
</h4>
<p>
This employs <b>chaining</b>. Furthermore, we assume that the
distribution of elements is uniform across hash table slots.
<br>
<img
src="https://upload.wikimedia.org/wikipedia/commons/3/3b/Hasq_hash_chains.png"
height="210" width="240">
<ul>
<li> Hash table <em>T</em> with <em>m</em> slots storing <em>n</em> elements.
<li> <b>Load factor</b>: <em>α = n / m</em>
<br> <em>α</em> is the average number of elements stored in a
chain.
<li> Our analysis is in terms of <em>α</em>, which can be
less than, equal to, or greater than one.
<li><b>Worst case</b> is very bad:
<br>All <em>n</em> keys hash to the same slot.
<br>Worst case for searching becomes <em>Θ(n)</em> plus
time to compute hash function.
<br>We could have just used a linked list directly!
<li><b>Average case</b>:
<br>Assuming any given element is equally likely to hash into
any slot...
<br>We get average case <em>Θ(1 + α)</em> time.
<br>See proofs in our textbook.
</ul>
</p>
<h4>Hash functions</h4>
<ul>
<li> First, convert key to an integer.
<br> E.g., we can interpret characters in a string by their ASCII values.
<br> Then treat each value as a digit in a radix-128 integer.
<br>
(<a
href="http://www.drdobbs.com/database/generating-sequential-keys-in-an-arbitra/184409688">
See this article for more on radices.</a>)
<li> Keys could be many other things besides strings.
<br> E.g., genomes:
<br>
<img
src="https://upload.wikimedia.org/wikipedia/commons/6/63/Part_of_DNA_sequence_prototypification_of_complete_genome_of_virus_5418_nucleotides.gif"
height="320" width="340">
<li> Division method:
<br><em>h(k) = k mod P</em>, where <em>P</em> is a
suitably-chosen prime number.
<li> Multiplication method:
<br><em>h(k) = [m (k A mod 1)]</em>,
where <em>0 < A < 1</em>.
</ul>
<h4>
Universal hashing
</h4>
<ul>
<li>Establish a <em>family</em> of hash functions.
<li>Choose so that <em>Prob[h(x) = h(y)] <= 1/m</em>, where m is
the size of our hash table.
<br>In other words, the hash functions have no more chance of
collision than simply randomly choosing to slots between 1 and m.
<li>Choose one at random each execution.
<br>Tricky: what if we store hash values?
<li>Good average case behavior
<br>If a "bad" function handles some
data once, a "good" one will handle it another time.
<br>So a "bad" set of programming variable names one run will turn
into a good set the next run.
</ul>
<h4>Open addressing</h4>
<ul>
<li>All elements are stored directly in the table; no chaining.
<li>Linear probing
<br>Easy: just move along array indices!
<br>Prone to clustering
<li>Quadratic probing
<li><a href="https://en.wikipedia.org/wiki/Double_hashing">
Double hashing
</a>
<br>Uses two hash functions to search array for key.
<li><a
href="https://github.com/gcallah/algorithms/blob/master/hash.py">
Source code here</a>.
</ul>
<h4>
Perfect hashing
</h4>
<h3>Homework</h3>
<p>
</p>
<ul>
<li> Devise three possible hash functions. Analyze each for size of hash
table necessary and possibility of collisions.
<li> Let's say our hash function is <em>h(k) = k mod 11</em> and we resolve
conflicts with chaining. Analyze what happens if our data values are
23, 34, 12, 8, 19, 33, 2, 5, 15, 4, 31, 9, 3, 6, 18, 8, 19 and 22.
<br>What will happen when we look up 19?
<li>Write (pseudo) code to resolve collisions with chaining.
<li>Consider the following hashing scheme:
<br>
<em>h(k) = k mod m</em>
<br>
<em>m = 7</em>
<br>We convert a string into a hashable key by treating it as a
base-8 number.
<br>So 'abc', where a = 1, b = 2, and c = 3, is converted to a
key as follows: 1 * 8<sup>2</sup> + 2 * 8 + 3 = 83.
<br>In this hashing scheme, what do the strings 'cba' and 'bac'
hash to?
<br>Can you write a more general statement about a pattern we
can detect here? Something along the lines of, "If the table
size is 2<sup>P</sup> - 1, and strings are interpreted in radix
2<sup>P</sup>..."
<br>
<br>
<b>Answer:</b>
<br>
<br>
If <em>h(k) = k mod m</em>, where <em>m = 2<sup>P</sup> -
1</em>, and <em>k</em> is a character string interpreted in
radix 2<sup>P</sup>, then all permutations of a given string
will hash to the same value. So in the example above, 'abc',
'cba', and 'bac' all hash to the same value.
<br>
<br>
<b>Proof</b>:
<br>
<br>
Assumed (could be proven, but we won't do it here):
<ol>
<li> (x + y) mod z ==
(x mod z + y mod z) mod z
<br><b>Example:</b> (10 + 12) mod 7 ==
(10 mod 7 + 12 mod 7) mod 7
<li> (x * y) mod z ==
(x mod z) * (y mod z)
mod z
<br><b>Example:</b> (10 * 12) mod 7 ==
(10 mod 7) * (12 mod 7) mod 7
<br>(7 * 17 = 119)
<li>if x mod z == 1, then x<sup>n</sup> mod z == 1
<br><b>Example:</b> 8 mod 7 == 1, and
8<sup>2</sup> mod 7 == 1
<br>This is a special case of 2!
</ol>
<br>
So, we have:
<br>
<br>
<img src="https://raw.githubusercontent.com/gcallah/algorithms/master/graphics/H3Eq1.gif">
<br>
<br>
<img src="https://raw.githubusercontent.com/gcallah/algorithms/master/graphics/H3Eq2.gif">
<br>
<br>
<img src="https://raw.githubusercontent.com/gcallah/algorithms/master/graphics/H3Eq3.gif">
<br>
<br>
<img
src="https://raw.githubusercontent.com/gcallah/algorithms/master/graphics/H3Eq4.gif">
(By 1)
<br>
<br>
<img
src="https://raw.githubusercontent.com/gcallah/algorithms/master/graphics/H3Eq5.gif">
(By 2)
<br>
<br>
<img
src="https://raw.githubusercontent.com/gcallah/algorithms/master/graphics/H3Eq6.gif">
(By 3)
<br>
<br>
<img
src="https://raw.githubusercontent.com/gcallah/algorithms/master/graphics/H3Eq7.gif">
</ol>
</ul>
<br>
<hr>
<a name="note1">* Based on Prof. Boris Aronov's lecture notes. </a>
<br>
</body>
</html>