Recursion slides
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@ -144,6 +144,158 @@ textbook: tbd
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<p>Solve for $T(n)$</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>Solve for $T(n)$</p>
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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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</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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<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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</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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</section>
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<section>
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<h4 class="slide_title">Factorial</h4>
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<p>Let's make the constants explicit</p>
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$$T(n) = \begin{cases}
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c_0 & \textbf{if } n \leq 0\\
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T(n-1) + c_1 & \textbf{otherwise}
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\end{cases}$$
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<p>Solve for $T(n)$</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 (🏡)</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 $c \geq c_0</p>
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</section>
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<section>
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<h4 class="slide_title">Prove the Hypothesis for the Base Case+1 (🏡)</h4>
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<p>$T(2) \leq c \cdot 2$</p>
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<p class="fragment">$T(1) + c_1 \leq 2c$</p>
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<p class="fragment">$c_0 + c_1\leq 2c$</p>
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<p class="fragment">We know there's a $c \geq c_0$, so... $c_1 \leq c$</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+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">$c_1 \leq c$</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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<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>$c_1 \leq c$</p>
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<p>True for any $c \geq c_1$</p>
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</section>
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<section>
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<p>Hey... this looks like a pattern!</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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<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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</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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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>How much space is used?</p>
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</section>
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<section>
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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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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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def factorial(n: Int): Long =
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{
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var total = 1l
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for(i <- 1 until n){ total *= i }
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return total
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}
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</code></pre>
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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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<p>
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<b>If the last action in the function is a recursive call</b>, the compiler will turn it into a loop.
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</p>
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<p><b>Scala:</b> Add <tt>@tailrec</tt> to a function to get the compiler to yell at you if it can't convert the function.</p>
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</section>
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<section>
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<p>Time permitting...</p>
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</section>
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<section>
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<h4 class="slide_title">Fibonacci</h4>
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@ -162,4 +314,12 @@ textbook: tbd
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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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<p>Solve for $T(n)$</p>
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</section>
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<section>
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<h4 class="slide_title">Next time...</h4>
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<p>Recursion Trees</p>
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</section>
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@ -0,0 +1,191 @@
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