Recursion
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@ -284,11 +284,10 @@ schedule:
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- date: 09/28/22
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topic: Recursive Analysis (contd...)
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dow: Wed
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due: WA0
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due: WA1
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- date: 09/30/22
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topic: Data-Bounded Analysis
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dow: Fri
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due: WA1
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- week: 6
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lectures:
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- date: 10/03/22
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@ -10,10 +10,30 @@ textbook: Ch. 15
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<attribution><a href="https://www.etsy.com/listing/916447505/ukrainian-nesting-doll-nesting-dolls">Ukranian Nesting Doll Nesting Dolls Handmade Dolls | Etsy</a>
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</section>
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<section>
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<p>Computational problems can be recursive too!</p>
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</section>
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<section>
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<h4 class="slide_title">Factorial</h4>
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<pre><code>
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<p class="fragment">
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$439!$ <span class="fragment">$= 439 \cdot 438!$</span>
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</p>
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</section>
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<section>
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<h4 class="slide_title">Factorial</h4>
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<p><b>Recursive Case: </b> $n! = n \cdot (n-1)!$</p>
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<p class="fragment"><b>Base Case: </b> $1! = 1$</p>
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</section>
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<section>
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<h4 class="slide_title">Factorial</h4>
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<pre><code class="scala">
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def factorial(n: Int): Long =
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if(n <= 1){ 1 }
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else { n * factorial(n-1) }
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@ -24,12 +44,12 @@ textbook: Ch. 15
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<h4 class="slide_title">Factorial</h4>
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<b>Base Case:</b>
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<pre><code>
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<pre><code class="scala">
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if(n <= 1){ 1 }
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</code></pre>
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<b>Recursive Case:</b>
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<pre><code>
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<pre><code class="scala">
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else { n * factorial(n-1) }
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</code></pre>
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</section>
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@ -51,11 +71,18 @@ textbook: Ch. 15
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<section>
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<h4 class="slide_title">Fibonacci</h4>
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<p><b>Base Cases:</b> $\texttt{fib}(1) = 1$, $\texttt{fib}(2) = 1$</p>
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<p><b>Recursive Cases:</b> $\texttt{fib}(n) = \texttt{fib}(n-1) + \texttt{fib}(n-2)$</p>
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</section>
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<pre><code>
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def fibb(n: Int): Long =
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<section>
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<h4 class="slide_title">Fibonacci</h4>
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<pre><code class="scala">
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def fib(n: Int): Long =
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if(n < 2){ 1 }
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else { fibb(n-1) + fibb(n-2) }
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else { fib(n-1) + fib(n-2) }
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</code></pre>
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</section>
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@ -63,13 +90,13 @@ textbook: Ch. 15
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<h4 class="slide_title">Fibonacci</h4>
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<b>Base Case:</b>
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<pre><code>
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<pre><code class="scala">
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if(n < 2){ 1 }
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</code></pre>
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<b>Recursive Case:</b>
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<pre><code>
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else { fibb(n-1) + fibb(n-2) }
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<pre><code class="scala">
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else { fib(n-1) + fib(n-2) }
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</code></pre>
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</section>
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@ -82,32 +109,71 @@ textbook: Ch. 15
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<section>
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<h4 class="slide_title">Towers of Hanoi</h4>
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<pre><code>
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<p><b>Task:</b> Move $n$ blocks from <b>A</b> to <b>C</b></p>
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<p><b><u>Base Case</u></b> ($n=1$)
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<ol>
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<li>Move the block from <b>A</b> to <b>C</b></li>
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</ol>
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</p>
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<p><b><u>Recursive Case</u></b> ($n \geq 2$)
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<ol>
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<li>Move $n-1$ blocks from <b>A</b> to <b>B</b></li>
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<li>Move the bottom block from <b>A</b> to <b>C</b></li>
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<li>Move $n-1$ blocks from <b>B</b> to <b>C</b></li>
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</ol>
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</p>
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</section>
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<section>
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<h4 class="slide_title">Towers of Hanoi</h4>
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<pre><code class="scala">
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var towers = Array(new Stack(), new Stack(), new Stack())
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def moveFrom(fromTower: Int, toTower: Int, numDisks: Int): Unit =
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def move(fromTower: Int, toTower: Int, numDisks: Int): Unit =
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{
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val otherTower = (Set(0, 1, 2) - fromTower - toTower).head
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if(numDisks < 0){
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return
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} else if(numDisks == 1){
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moveTopDisk(from = fromTower, to = toTower)
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if(numDisks == 1){
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moveOne(from = fromTower, to = toTower)
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} else {
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moveFrom(fromTower, otherTower, numDisks-1)
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moveTopDisk(from = fromTower, to = toTower)
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moveFrom(otherTower, toTower, numDisks-1)
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move(fromTower, otherTower, numDisks-1)
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moveOne(from = fromTower, to = toTower)
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move(otherTower, toTower, numDisks-1)
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}
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}
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</code></pre>
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</section>
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<section>
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<p>How do we get the complexity of recursive algorithms?</p>
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</section>
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<section>
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<h4 class="slide_title">Factorial</h4>
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<p>What's the complexity? (in terms of <tt>n</tt>)</p>
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<pre><code>
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<pre><code class="scala">
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def factorial(n: Int): Long =
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if(n <= 1){ 1 }
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else { n * factorial(n-1) }
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</code></pre>
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</section>
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<section>
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<p><b>Idea:</b> Write down a recursive runtime growth function</p>
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</section>
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<section>
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<h4 class="slide_title">Factorial</h4>
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<p>What's the complexity? (in terms of <tt>n</tt>)</p>
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<pre><code class="scala">
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def factorial(n: Int): Long =
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if(n <= 1){ 1 }
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else { n * factorial(n-1) }
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@ -124,13 +190,16 @@ textbook: Ch. 15
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</section>
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<section>
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<h4 class="slide_title">Fibonacci</h4>
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<h4 class="slide_title">Factorial</h4>
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<p>
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<b><u>Base Case</u></b> ($n \leq 0$)
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$$\Theta(1)$$
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</p>
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<b>Base Case:</b>
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$$\Theta(1) \textbf{ if } n < 2$$
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<b>Recursive Case"</b>
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$$T(n-1) + \Theta(1) \textbf{ otherwise}$$
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<p>
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<b><u>Recursive Case</u></b> ($n > 0$)
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$$T(n-1) + \Theta(1)$$
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</p>
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</section>
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<section>
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@ -152,30 +221,30 @@ textbook: Ch. 15
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<p><b>Approach:</b>
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<ol>
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<li>Generate a Hypothesis (🔮)</li>
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<li>Prove the Hypothesis for the Base Case (🏡)</li>
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<li>Prove the Hypothesis Inductively (🏢)</li>
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<li>Prove the Hypothesis for the Base Case</li>
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<li>Prove the Hypothesis Inductively</li>
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</ol>
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</p>
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</section>
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<section>
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<h4 class="slide_title">Generate a Hypothesis (🔮)</h4>
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<h4 class="slide_title">Generate a Hypothesis</h4>
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<p><b>Approach:</b> Write out the rule for increasing $n$</p>
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<p>
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$\Theta(1)$,
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<span class="fragment">$2\Theta(1)$ </span>
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<span class="fragment">$3\Theta(1)$ </span>
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<span class="fragment">$4\Theta(1)$ </span>
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<span class="fragment">$5\Theta(1)$ </span>
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<span class="fragment">$6\Theta(1)$ </span>
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<span class="fragment">$7\Theta(1)$ </span>
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<span class="fragment">$2\Theta(1)$, </span>
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<span class="fragment">$3\Theta(1)$, </span>
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<span class="fragment">$4\Theta(1)$, </span>
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<span class="fragment">$5\Theta(1)$, </span>
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<span class="fragment">$6\Theta(1)$, </span>
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<span class="fragment">$7\Theta(1)$, $\ldots$ </span>
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</p>
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<p class="fragment">What's the pattern?</p>
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<p class="fragment"><b>Hypothesis: </b> $T(n) \in O(n)$ (there is some $c > 0$ such that $T(n) \leq c \cdot n$)</p>
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<p class="fragment"><b>Hypothesis: </b> $T(n) \in O(n)$<br> (there is some $c > 0$ such that $T(n) \leq c \cdot n$)</p>
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</section>
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<section>
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@ -192,16 +261,16 @@ textbook: Ch. 15
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</section>
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<section>
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<h4 class="slide_title">Prove the Hypothesis for the Base Case (🏡)</h4>
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<h4 class="slide_title">Prove the Hypothesis for the Base Case</h4>
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<p>$T(1) \leq c \cdot 1$</p>
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<p class="fragment">$T(1) \leq c$</p>
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<p class="fragment">$c_0 \leq c$</p>
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<p class="fragment">True for any $c \geq c_0</p>
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<p class="fragment">True for any $c \geq c_0$</p>
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</section>
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<section>
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<h4 class="slide_title">Prove the Hypothesis for the Base Case+1 (🏡)</h4>
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<h4 class="slide_title">Prove the Hypothesis for the Base Case+1</h4>
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<p>$T(2) \leq c \cdot 2$</p>
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<p class="fragment">$T(1) + c_1 \leq 2c$</p>
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@ -211,21 +280,21 @@ textbook: Ch. 15
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</section>
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<section>
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<h4 class="slide_title">Prove the Hypothesis for the Base Case+2 (🏡)</h4>
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<h4 class="slide_title">Prove the Hypothesis for the Base Case+2</h4>
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<p>$T(3) \leq c \cdot 3$</p>
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<p class="fragment">$T(2) + c_1 \leq 3c$</p>
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<p class="fragment">We know there's a $c$ s.t. $T(2) \leq 2c$, so if we show that $2c + c_1 \leq 3c$, then $T(2) + c_1 \leq 2c + c_1 \leq 3c$</p>
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<p class="fragment">We know there's a $c$ s.t. $T(2) \leq 2c$,<br>... so if we show that $2c + c_1 \leq 3c$, then $T(2) + c_1 \leq 2c + c_1 \leq 3c$</p>
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<p class="fragment">$c_1 \leq c$</p>
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<p class="fragment">True for any $c \geq c_1$</p>
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</section>
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<section>
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<h4 class="slide_title">Prove the Hypothesis for the Base Case+3 (🏡)</h4>
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<h4 class="slide_title">Prove the Hypothesis for the Base Case+3</h4>
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<p>$T(4) \leq c \cdot 4$</p>
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<p>$T(3) + c_1 \leq 4c$</p>
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<p>We know there's a $c$ s.t. $T(3) \leq 3c$, so if we show that $3c + c_1 \leq 4c$, then $T(3) + c_1 \leq 3c + c_1 \leq 4c$</p>
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<p>We know there's a $c$ s.t. $T(3) \leq 3c$, <br>...so if we show that $3c + c_1 \leq 4c$, then $T(3) + c_1 \leq 3c + c_1 \leq 4c$</p>
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<p>$c_1 \leq c$</p>
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<p>True for any $c \geq c_1$</p>
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</section>
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@ -235,22 +304,29 @@ textbook: Ch. 15
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</section>
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<section>
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<h4 class="slide_title">Prove the Hypothesis Inductively (🏢)</h4>
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<h4 class="slide_title">Prove the Hypothesis Inductively</h4>
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<p><b>Approach:</b> Assume the hypothesis is true for any $n' < n$; use that to prove for $n$</p>
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<p class="fragment"><b>Assume: </b>There is a $c > 0$ s.t. $T(n-1) \leq c\cdot (n-1)$</p>
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<p class="fragment"><b>Prove: </b>There is a $c > 0$ s.t. $T(n) \leq c\cdot n$</p>
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<p class="fragment">$T(n-1) + c_1 \leq c \cdot n$</p>
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<p class="fragment">By the inductive assumption, there is a $c$ s.t. $T(n-1) \leq (n-1)c$, so if we show that $(n-1)c + c_1 \leq nc$, then $T(n-1) + c_1 \leq (n-1)c + c_1 \leq nc$</p>
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<p class="fragment">$c_1 \leq c$</p>
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<p class="fragment">True for any $c \geq c_1$</p>
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<p><b>Approach:</b> Assume the hypothesis is true for any $n' < n$;<br> use that to prove for $n$</p>
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</section>
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<section>
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<h4 class="slide_title">Prove the Hypothesis Inductively</h4>
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<div style="font-size: 90%">
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<p><b>Assume: </b>There is a $c > 0$ s.t. $T(n-1) \leq c\cdot (n-1)$</p>
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<p><b>Prove: </b>There is a $c > 0$ s.t. $T(n) \leq c\cdot n$</p>
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<p class="fragment">$T(n-1) + c_1 \leq c \cdot n$</p>
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<p class="fragment" style="font-size: 80%">By the inductive assumption, there is a $c$ s.t. $T(n-1) \leq (n-1)c$, so if we show that $(n-1)c + c_1 \leq nc$, then $T(n-1) + c_1 \leq (n-1)c + c_1 \leq nc$</p>
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<p class="fragment">$(n-1)c + c_1 - (n-1)c \leq nc - (n-1)c$</p>
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<p class="fragment">$c_1 \leq c $</p>
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<p class="fragment">True for any $c \geq c_1$</p>
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</div>
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</section>
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<section>
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<h4 class="slide_title">Factorial</h4>
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<pre><code>
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<pre><code class="scala">
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def factorial(n: Int): Long =
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if(n <= 1){ 1 }
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else { n * factorial(n-1) }
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@ -260,19 +336,20 @@ textbook: Ch. 15
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</section>
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<section>
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<h4 class="slide_title">The Call Stack</h4>
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<svg data-src="graphics/10b/callstack.svg"/>
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</section>
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<section>
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<h4 class="slide_title">Tail Recursion</h4>
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<pre><code>
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<pre><code class="scala">
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def factorial(n: Int): Long =
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if(n <= 1){ 1 }
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else { n * factorial(n-1) }
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</code></pre>
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<p class="fragment">↳ the compiler can (sometimes) figure this out on its own ↴</p>
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<pre><code>
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<p class="fragment" style="font-size: 80%">↳ the compiler can (sometimes) figure this out on its own ↴</p>
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<pre><code class="scala">
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def factorial(n: Int): Long =
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{
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var total = 1l
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@ -300,10 +377,10 @@ textbook: Ch. 15
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<h4 class="slide_title">Fibonacci</h4>
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<p>What's the complexity? (in terms of <tt>n</tt>)</p>
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<pre><code>
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def fibb(n: Int): Long =
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<pre><code class="scala">
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def fib(n: Int): Long =
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if(n < 2){ 1 }
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else { fibb(n-1) + fibb(n-2) }
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else { fib(n-1) + fib(n-2) }
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</code></pre>
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</section>
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<h4 class="slide_title">Fibonacci</h4>
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$$T(n) = \begin{cases}
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\Theta(1) & \textbf{if } n < 2
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\Theta(1) & \textbf{if } n < 2\\
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T(n-1) + T(n-2) + \Theta(1) & \textbf{otherwise}
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\end{cases}$$
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