Merge branch 'master' of git.odin.cse.buffalo.edu:ODIn/Website
This commit is contained in:
commit
1649d73656
GPG key ID:
3E5F9B3ABD3FDB60
|
|
@ -217,7 +217,7 @@ schedule:
|
|||
special: "No Class: Labor Day"
|
||||
dow: Mon
|
||||
- date: 09/07/22
|
||||
topic: Runtime Analysis
|
||||
topic: Runtime Analysis - Intuitions
|
||||
due: AI Quiz
|
||||
dow: Wed
|
||||
section_a:
|
||||
|
|
@ -226,33 +226,44 @@ schedule:
|
|||
section_b:
|
||||
slides: slide/3b-FunctionAnalysis.html
|
||||
- date: 09/09/22
|
||||
topic: Runtime Analysis (contd...)
|
||||
topic: Runtime Analysis - Formalism
|
||||
dow: Fri
|
||||
due: PA0
|
||||
section_a:
|
||||
slides: slide/4b-AsymptoticNotation.html
|
||||
section_b:
|
||||
slides: slide/4b-AsymptoticNotation.html
|
||||
- week: 3
|
||||
lectures:
|
||||
- date: 09/12/22
|
||||
topic: Collections, Sequences, ADTs
|
||||
dow: Mon
|
||||
topic: Runtime Analysis - Formalism (contd...)
|
||||
topic:
|
||||
section_a:
|
||||
slides: slide/4b-AsymptoticNotation.html
|
||||
section_b:
|
||||
slides: slide/4b-AsymptoticNotation.html
|
||||
- date: 09/14/22
|
||||
topic: Array Lists
|
||||
topic: Runtime Analysis - Examples
|
||||
dow: Wed
|
||||
section_b:
|
||||
slides: slide/6b-AsymptoticExamples.html
|
||||
- date: 09/16/22
|
||||
topic: Linked Lists
|
||||
topic: Collections, Sequences, ADTs
|
||||
dow: Fri
|
||||
due: PA1
|
||||
section_b:
|
||||
slides: slide/7b-ADTsandSequences.html
|
||||
- week: 4
|
||||
lectures:
|
||||
- date: 09/19/22
|
||||
topic: Recursion Intro
|
||||
topic: Collections, Sequences, ADTs (contd...)
|
||||
dow: Mon
|
||||
- date: 09/21/22
|
||||
topic: Divide and Conquer, Tail Recursion
|
||||
topic: Collections, Sequences, ADTs (contd...)
|
||||
dow: Wed
|
||||
- date: 09/23/22
|
||||
topic: Recursion (contd...)
|
||||
topic: Recursion Intro
|
||||
dow: Fri
|
||||
due: WA0
|
||||
- week: 5
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
---
|
||||
template: templates/cse250_2022_slides.erb
|
||||
title: Asymptotic Notation
|
||||
date: Sept 7, 2022
|
||||
date: Sept 7 and 12, 2022
|
||||
textbook: "Ch. 7.3-7.4"
|
||||
---
|
||||
|
||||
|
|
@ -342,6 +342,17 @@ textbook: "Ch. 7.3-7.4"
|
|||
<p><b>Problem: </b> What if $g(n)$ doesn't bound $f(n)$ from <b>both</b> above and below?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<pre><code>
|
||||
if input = 1:
|
||||
/* take 1 step */
|
||||
else:
|
||||
/* take n steps */
|
||||
</code></pre>
|
||||
|
||||
<p class="fragment"><b>Schroedinger's Code: </b> Simultaneously like $f_1(n) = 1$ and $f_2(n) = n$</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Upper, Lower Bounds</h4>
|
||||
|
||||
|
|
|
|||
241
src/teaching/cse-250/2022fa/slide/6b-AsymptoticExamples.erb
Normal file
241
src/teaching/cse-250/2022fa/slide/6b-AsymptoticExamples.erb
Normal file
|
|
@ -0,0 +1,241 @@
|
|||
---
|
||||
template: templates/cse250_2022_slides.erb
|
||||
title: Asymptotic Notation Examples
|
||||
date: Sept 14, 2022
|
||||
textbook: "Ch. 7.3-7.4"
|
||||
---
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Big-_ Notation Recap</h4>
|
||||
|
||||
<dl style="font-size: 70%">
|
||||
<dt>Big-ϴ</dt>
|
||||
<dd>Growth Functions in the <b>same</b> complexity class.</dd>
|
||||
<dd>If $f(n) \in \Theta(g(n))$, then an algorithm that takes $f(n)$ steps is exactly as fast as one that takes $g(n)$ steps.</dd>
|
||||
|
||||
<dt>Big-O</dt>
|
||||
<dd>Growth Functions in the <b>same or smaller</b> complexity class.</dd>
|
||||
<dd>If $f(n) \in O(g(n))$, then an algorithm that takes $f(n)$ steps is as fast as or faster than one that takes $g(n)$ steps.</dd>
|
||||
|
||||
<dt>Big-Ω</dt>
|
||||
<dd>Growth Functions in the <b>same or bigger</b> complexity class.</dd>
|
||||
<dd>If $f(n) \in \Omega(g(n))$, then an algorithm that takes $f(n)$ steps is as slow as or slower than one that takes $g(n)$ steps.</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Common Runtimes</h4>
|
||||
|
||||
<dl style="font-size: 70%;">
|
||||
<dt>Constant Time: $\Theta(1)$</dt>
|
||||
<dd>e.g., $T(n) = c$ (runtime is independent of $n$)</dd>
|
||||
|
||||
<dt>Logarithmic Time: $\Theta(\log(n))$</dt>
|
||||
<dd>e.g., $T(n) = c\log(n)$ (for some constant $c$)</dd>
|
||||
|
||||
<dt>Linear Time: $\Theta(n)$</dt>
|
||||
<dd>e.g., $T(n) = c_1n + c_0$ (for some constants $c_0, c_1$)</dd>
|
||||
|
||||
<dt>Quadratic Time: $\Theta(n^2)$</dt>
|
||||
<dd>e.g., $T(n) = c_2n^2 + c_1n + c_0$</dd>
|
||||
|
||||
<dt>Polynomial Time: $\Theta(n^k)$ (for some $k \in \mathbb Z^+$)</dt>
|
||||
<dd>e.g., $T(n) = c_kn^k + \ldots + c_1n + c_0$</dd>
|
||||
|
||||
<dt>Exponential Time: $\Theta(c^n)$ (for some $c \geq 1$)</dt>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h2>Constants vs Asymptotics</h2>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Constant-Factor Speedups</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
for(i ← 0 until n) { /* do work */ }
|
||||
</code></pre>
|
||||
|
||||
<ul>
|
||||
<li class="fragment">Original Body: $10$ steps <span class="fragment">(Total runtime: $10n$ steps)</span></li>
|
||||
<li class="fragment">Optimized Body: $7$ steps <span class="fragment">(Total runtime: $7n$ steps)</span><ul>
|
||||
</ul>
|
||||
|
||||
<p class="fragment"><b>Total Runtime:</b> 7/10 of original (30% faster)</p>
|
||||
<p class="fragment">... but still $\Theta(n)$</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">$c$ and $n_0$</h4>
|
||||
|
||||
<p>Compare $T_1(n) = 100n$ vs $T_2(n) = n^2$</p>
|
||||
<ul>
|
||||
<li class="fragment">$100n = O(n^2)$ ($T_2$ is the slower runtime)</li>
|
||||
<li class="fragment">... but $c_{high} = 1$, $n_0 = 100$</li>
|
||||
<li class="fragment">Until inputs of size 100 or more, $T_2$ is the <b>faster</b> runtime</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">$c$ and $n_0$</h4>
|
||||
|
||||
<p>Asymptotically slower runtimes <i>can</i> be better.</p>
|
||||
<ul>
|
||||
<li>An algorithm with runtime $T_2$ is better on small inputs.</li>
|
||||
<li>An algorithm with runtime $T_2$ might be easier to implement or maintain</li>
|
||||
<li>An algorithm with runtime $T_1$ might not exist. <ul>
|
||||
<li class="fragment">(sometimes we can <i>prove</i> this, see CSE 331)</li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>... but from now on, if $T_2(n)$ is in a bigger complexity class, then $T_1(n)$ is better/faster/stronger.</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h2>Examples</h2>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Bubble Sort</h4>
|
||||
<pre><code class="scala">
|
||||
bubblesort(seq: Seq[Int]):
|
||||
1. n ← seq length
|
||||
2. for i ← n-2 to 0, by -1:
|
||||
3. for j ← i to n-1:
|
||||
4. if seq(j+1) < seq(j):
|
||||
5. swap seq(j) and seq(j+1)
|
||||
</code></pre>
|
||||
<p>What is the runtime complexity class of Bubble Sort?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Summation Rules</h4>
|
||||
|
||||
<ol style="font-size: 50%;">
|
||||
<li>$\sum_{i=j}^{k}c = (k - j + 1)c$</li>
|
||||
|
||||
<li>$\sum_{i=j}^{k}(cf(i)) = c\sum_{i=j}^{k}f(i)$</li>
|
||||
|
||||
<li>$\sum_{i=j}^{k}(f(i) + g(i)) = \left(\sum_{i=j}^{k}f(i)\right) + \left(\sum_{i=j}^{k}g(i)\right)$</li>
|
||||
|
||||
<li>$\sum_{i=j}^{k}(f(i)) = \left(\sum_{i=\ell}^{k}(f(i))\right) - \left(\sum_{i=\ell}^{j-1}(f(i))\right)$ (for any $\ell < j$)</li>
|
||||
|
||||
<li>$\sum_{i=j}^{k}f(i) = f(j) + f(j+1) + \ldots + f(k-1) + f(k)$</li>
|
||||
|
||||
<li>$\sum_{i=j}^{k}f(i) = f(j) + \ldots + f(\ell - 1) + \left(\sum_{i=\ell}^k f(i)\right)$ (for any $j < \ell \leq k$)</li>
|
||||
|
||||
<li>$\sum_{i=j}^{k}f(i) = \left(\sum_{i=j}^{\ell}f(i)\right) + f(\ell+1) + \ldots + f(k)$ (for any $j \leq \ell < k$)</li>
|
||||
|
||||
<li>$\sum_{i=1}^{k}i = \frac{k(k+1)}{2}$</li>
|
||||
|
||||
<li>$\sum_{i=0}^{k}2^i = 2^{k+1}-1$</li>
|
||||
|
||||
<li>$n! \leq c_sn^n$ is a tight upper bound (Sterling: Some constant $c_s$ exists)</li>
|
||||
</ol>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Bubble Sort</h4>
|
||||
<pre><code class="scala">
|
||||
bubblesort(seq: Seq[Int]):
|
||||
1. n ← seq length
|
||||
2. for i ← n-2 to 0, by -1:
|
||||
3. for j ← i to n-1:
|
||||
4. if seq(j+1) < seq(j):
|
||||
5. swap seq(j) and seq(j+1)
|
||||
</code></pre>
|
||||
<p><b>Note:</b> We can ignore the exact number of steps required by any step in the algorithm, as long as we know its complexity</p>
|
||||
<p class="fragment"><b>Can we safely say this algorithm is $\Theta(n^2)$?</b></p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Bubble Sort (for Mutable Sequences)</h4>
|
||||
<pre><code class="scala">
|
||||
def sort(seq: mutable.Seq[Int]): Unit =
|
||||
{
|
||||
val n = seq.length
|
||||
for(i <- n - 2 to 0 by -1; j <- i to n)
|
||||
{
|
||||
if(seq(n) < seq(j))
|
||||
{
|
||||
val temp = seq(j+1)
|
||||
seq(j+1) = seq(j)
|
||||
seq(j) = temp
|
||||
}
|
||||
}
|
||||
}
|
||||
</code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Bubble Sort (for <b>Im</b>mutable Sequences)</h4>
|
||||
<pre><code class="scala">
|
||||
def sort(seq: Seq[Int]): Seq[Int] =
|
||||
{
|
||||
val newSeq = seq.toArray
|
||||
val n = seq.length
|
||||
for(i <- n - 2 to 0 by -1; j <- i to n)
|
||||
{
|
||||
if(newSeq(n) < newSeq(j))
|
||||
{
|
||||
val temp = newSeq(j+1)
|
||||
newSeq(j+1) = newSeq(j)
|
||||
newSeq(j) = temp
|
||||
}
|
||||
}
|
||||
return newSeq.toList
|
||||
}
|
||||
</code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Searching Sequences</h4>
|
||||
<pre><code class="scala">
|
||||
def indexOf[T](seq: Seq[T], value: T, from: Int): Int =
|
||||
{
|
||||
for(i <- from until seq.length)
|
||||
{
|
||||
if(seq(i).equals(value)) { return i }
|
||||
}
|
||||
return -1
|
||||
}
|
||||
</code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Searching Sequences</h4>
|
||||
<pre><code class="scala">
|
||||
def count[T](seq: Seq[T], value: T): Int =
|
||||
{
|
||||
var count = 0;
|
||||
var i = indexOf(seq, value, 0)
|
||||
while(i != -1)
|
||||
{
|
||||
count += 1;
|
||||
i = indexOf(seq, value, i+1)
|
||||
}
|
||||
return count
|
||||
}
|
||||
</code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Searching Sorted Sequences</h4>
|
||||
|
||||
<p>... with $O(1)$ access to elements ('random access')</p>
|
||||
|
||||
<ul style="font-size: 80%">
|
||||
<li>To search $[begin, end)$:<ul>
|
||||
<li>compare $target$ to $seq[middle]$ ($middle = \frac{end+begin}{2}$</li>
|
||||
<li>If $seq[middle] = target$ return $middle$</li>
|
||||
<li>If $target < seq[middle]$ search $[begin, middle)$</li>
|
||||
<li>If $seq[middle] < target$ search $[middle+1, end)$</li>
|
||||
<li>If $begin = end$, the value doesn't exist.</li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
|
||||
<p class="fragment">What if no random access is available?</p>
|
||||
</section>
|
||||
196
src/teaching/cse-250/2022fa/slide/7b-ADTsandSequences.erb
Normal file
196
src/teaching/cse-250/2022fa/slide/7b-ADTsandSequences.erb
Normal file
|
|
@ -0,0 +1,196 @@
|
|||
---
|
||||
template: templates/cse250_2022_slides.erb
|
||||
title: Asymptotic Notation Examples
|
||||
date: Sept 16, 2022
|
||||
textbook: "Ch. 7.1, 1.7.2"
|
||||
---
|
||||
|
||||
<!--
|
||||
<section>
|
||||
<h4 class="slide_title">Types of Collections</h4>
|
||||
|
||||
<dl>
|
||||
<dt>Iterable</dt>
|
||||
<dd>Any collection of items.</dd>
|
||||
|
||||
<dt>Seq</dt>
|
||||
<dd>A collection of items arranged in a specific order.</dd>
|
||||
|
||||
<dt>IndexedSeq</dt>
|
||||
<dd>Like Seq, but $O(1)$ access to individual items by index.</dd>
|
||||
|
||||
<dt>Set</dt>
|
||||
<dd>A collection of unique items.</dd>
|
||||
|
||||
<dt>Map</dt>
|
||||
<dd>A collection of items identified by a key.</dd>
|
||||
</dl>
|
||||
</section>
|
||||
<section>
|
||||
<h4 class="slide_title">Types of Collections</h4>
|
||||
|
||||
<dl>
|
||||
<dt>mutable.Seq</dt>
|
||||
<dd>Like Seq, but can be changed.</dd>
|
||||
|
||||
<dt>mutable.Buffer</dt>
|
||||
<dd>Like mutable.Seq, but "efficient" append.</dd>
|
||||
|
||||
<dt>Queue</dt>
|
||||
<dd>Like Seq, but "efficient" append and remove first.</dd>
|
||||
|
||||
<dt>Stack</dt>
|
||||
<dd>Like Seq, but "efficient" prepend and remove first.</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
-->
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Sequences</h4>
|
||||
|
||||
<dl class="fragment">
|
||||
<dt>The Fibonacci Sequence</dt>
|
||||
<dd>1, 1, 2, 3, 5, 8, 13, 21, 34, 55</dd>
|
||||
|
||||
<dt>Characters of a String</dt>
|
||||
<dd>'H', 'e', 'l', 'l', 'o', ' ', 'W', 'o', 'r', 'l', 'd'</dd>
|
||||
|
||||
<dt>Lines of a file.</dt>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">What can you do with a Sequence?</h4>
|
||||
|
||||
<ul>
|
||||
<li>Enumerate every element of the sequence.</li>
|
||||
<li>Get the 'n'th element.</li>
|
||||
<li>Modify the 'n'th element (mutable only).</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Abstract Data Types (ADTs)</h4>
|
||||
|
||||
<svg data-src="graphics/07b/adt_opaque_box.svg"/>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">The <tt>Seq</tt> ADT</h4>
|
||||
|
||||
<dl style="font-size: 70%;">
|
||||
<dt><tt>apply(<b>idx</b>: Int): [A]</tt></dt>
|
||||
<dd>Get the element (of type A) at position <b>idx</b>.</dd>
|
||||
|
||||
<dt><tt>iterator: Iterator[A]</tt></dt>
|
||||
<dd>Get access to <ul>view</ul> all elements in the seq, in order, once.</dd>
|
||||
|
||||
<dt><tt>length: Int</tt></dt>
|
||||
<dd>Count the number of elements in the seq.</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">The <tt>mutable.Seq</tt> ADT</h4>
|
||||
|
||||
<dl style="font-size: 70%;">
|
||||
<dt><tt>apply(<b>idx</b>: Int): [A]</tt></dt>
|
||||
<dd>Get the element (of type A) at position <b>idx</b>.</dd>
|
||||
|
||||
<dt><tt>iterator: Iterator[A]</tt></dt>
|
||||
<dd>Get access to <ul>view</ul> all elements in the seq, in order, once.</dd>
|
||||
|
||||
<dt><tt>length: Int</tt></dt>
|
||||
<dd>Count the number of elements in the seq.</dd>
|
||||
|
||||
<dt><tt>insert(<b>idx</b>: Int, <b>elem</b>: A): Unit</tt></dt>
|
||||
<dd>Insert an element at position <b>idx</b> with value <b>elem</b>.</dd>
|
||||
|
||||
<dt><tt>remove(<b>idx</b>: Int): A</tt></dt>
|
||||
<dd>Remove the element at position <b>idx</b> and return the removed value.</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>... so how do we implement it?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">RAM (CSE 220 crossover)</h4>
|
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|
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<svg data-src="graphics/07b/ram.svg"/>
|
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</section>
|
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|
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<section>
|
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<h4 class="slide_title">RAM</h4>
|
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|
||||
<dl>
|
||||
<dt><tt>new T()</tt></dt>
|
||||
<dd>Find some unused part of memory big enough to fit a <tt>T</tt>, mark it used, and return the <i>address</i> of that location.</dd>
|
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</dl>
|
||||
|
||||
<div class="fragment">
|
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<pre><code>
|
||||
var arr = new Array[Int](50)
|
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</code></pre>
|
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allocates $50 * 4 = 200$ bytes of memory (Java <tt>Int</tt> is 4 bytes).
|
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</div>
|
||||
|
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<p class="fragment">If <tt>arr</tt> is at address $a$, where should you look for <tt>arr(19)</tt>? <span class="fragment"><tt>arr(55)</tt>?</span></p>
|
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</section>
|
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|
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<section>
|
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<h4 class="slide_title">Array[T] : Seq[T]</h4>
|
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|
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<p>An Array of $n$ items of type <tt>T</tt>:</p>
|
||||
|
||||
<ul>
|
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<li>4 bytes for $n$ (optional)</li>
|
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<li>4 bytes for $\textt{sizeof}(\texttt{T})$) (optional).</li>
|
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|
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</ul>
|
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<svg data-src="graphics/07b/array.svg"/>
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<section>
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<h4 class="slide_title">Array[T] : Seq[T]</h4>
|
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|
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<p>How do we implement...</p>
|
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<dl style="font-size: 70%;">
|
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<dt><tt>apply(<b>idx</b>: Int): [A]</tt></dt>
|
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<dd>Get the element (of type A) at position <b>idx</b>.</dd>
|
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|
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<dt><tt>length: Int</tt></dt>
|
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<dd>Count the number of elements in the seq.</dd>
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|
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<dt><tt>insert(<b>idx</b>: Int, <b>elem</b>: A): Unit</tt></dt>
|
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<dd>Insert an element at position <b>idx</b> with value <b>elem</b>.</dd>
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<dt><tt>remove(<b>idx</b>: Int): A</tt></dt>
|
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<dd>Remove the element at position <b>idx</b> and return the removed value.</dd>
|
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</dl>
|
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</section>
|
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|
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<section>
|
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<p>Insert and remove don't make sense on arrays...</p>
|
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<p class="fragment"><b>Idea</b> Reserve extra space in the array!</p>
|
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</section>
|
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|
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<section>
|
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<h4 class="slide_title">ArrayBuffer[T] : Buffer[T]</h4>
|
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|
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<p>An ArrayBuffer of $n$ items of type <tt>T</tt>:</p>
|
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|
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<ul>
|
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<li>4 bytes for $n$ (optional)</li>
|
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<li>4 bytes for $\textt{sizeof}(\texttt{T})$) (optional).</li>
|
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<li style="color: red">4 bytes for the number of fields $used$</li>
|
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<li>$n \times \texttt{sizeof}(\texttt{T})$ bytes of memory.</li>
|
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</ul>
|
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<svg data-src="graphics/07b/arraybuffer.svg"/>
|
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</section>
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|
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<section>
|
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Reference in a new issue