Merge branch 'master' of git.odin.cse.buffalo.edu:ODIn/Website
commit
5ee8cf1d87
GPG Key ID:
3E5F9B3ABD3FDB60
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@ -378,25 +378,26 @@ schedule:
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- date: 10/24/22
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topic: Heaps
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dow: Mon
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section_b:
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slides: slide/22b-Heaps.html
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- date: 10/26/22
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topic: Trees
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||||
dow: Wed
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- date: 10/28/22
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topic: Binary Search Trees, Tree Iteration
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due: PA3-Tests
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||||
dow: Fri
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||||
- week: 10
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||||
lectures:
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||||
- date: 10/31/22
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||||
topic: Balanced Trees (AVL)
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dow: Mon
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due: WA2
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||||
- date: 11/02/22
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topic: Balanced Trees (Red-Black)
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dow: Wed
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||||
- date: 11/04/22
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topic: Balanced Trees (contd...)
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||||
dow: Fri
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||||
due: PA3
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||||
- week: 11
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lectures:
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- date: 11/07/22
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@ -412,10 +413,10 @@ schedule:
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- date: 11/11/22
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special: Resign Deadline
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dow: Fri
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due: WA2
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- week: 12
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lectures:
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||||
- date: 11/14/22
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due: PA3-Tests
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topic: Hash Table Analysis
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dow: Mon
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- date: 11/16/22
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@ -428,6 +429,7 @@ schedule:
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- week: 13
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lectures:
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- date: 11/21/22
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due: PA3
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topic: Bloom Filters
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dow: Mon
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- date: 11/23/22
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@ -436,7 +438,6 @@ schedule:
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- date: 11/25/22
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special: "No Class: Fall Recess"
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dow: Fri
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due: PA4-Tests
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- week: 14
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lectures:
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- date: 11/28/22
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@ -446,12 +447,12 @@ schedule:
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topic: Secondary Storage (B+Trees)
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dow: Wed
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- date: 12/02/22
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due: PA4
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topic: Buffer / Advanced Data Structures
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dow: Fri
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- week: 15
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||||
lectures:
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||||
- date: 12/05/22
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due: PA4
|
||||
topic: Wrap-Up / Review
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dow: Mon
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- date: 12/07/22
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@ -0,0 +1,455 @@
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---
|
||||
template: templates/cse250_2022_slides.erb
|
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title: Priority Queues, Heaps
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||||
date: Oct 24, 2022
|
||||
textbook: Ch. 18
|
||||
cat: graphics/22b/cat.png
|
||||
---
|
||||
|
||||
<section>
|
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<h2>Priority Queues</h2>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">PriorityQueue[A <: Ordering]</h4>
|
||||
|
||||
<dl>
|
||||
<dt>enqueue(v: A): Unit</dt>
|
||||
<dd>Insert a value $v$ into the priority queue.</dd>
|
||||
|
||||
<dt>dequeue: A</dt>
|
||||
<dd>Remove the greatest element in the priority queue.</dd>
|
||||
|
||||
<dt>head: A</dt>
|
||||
<dd>Peek at the greatest element in the priority queue.</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Priority Queues</h4>
|
||||
|
||||
<table>
|
||||
<tr>
|
||||
<th>Operation</th>
|
||||
<th>Lazy</th>
|
||||
<th>Proactive</th>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td>enqueue</td>
|
||||
<td>$O(1)$</td>
|
||||
<td>$O(n)$</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td>dequeue</td>
|
||||
<td>$O(n)$</td>
|
||||
<td>$O(1)$</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td>head</td>
|
||||
<td>$O(n)$</td>
|
||||
<td>$O(1)$</td>
|
||||
</tr>
|
||||
</table>
|
||||
|
||||
<p class="fragment">Can we do better?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Priority Queues</h4>
|
||||
|
||||
<dl>
|
||||
<dt>Lazy</dt>
|
||||
<dd>Fast Enqueue, Slow Dequeue</dd>
|
||||
|
||||
<dt>Proactive</dt>
|
||||
<dd>Slow Enqueue, Fast Dequeue</dd>
|
||||
|
||||
<dt>???</dt>
|
||||
<dd>Fast(-ish) Enqueue, Fast(-ish) Dequeue</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p><b>Idea: </b> Keep the priority queue "kinda" sorted.</p>
|
||||
|
||||
<p class="fragment">Larger items tend to be closer to the front of the list. The closer we are to the front of the list, the more sorted it gets.</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Binary Heaps</h4>
|
||||
|
||||
<p><b>Challenge: </b> How do we keep track of which elements are still sorted?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Binary Heaps</h4>
|
||||
|
||||
<p><b>Idea: </b> Organize the priority queue as a tree</p>
|
||||
|
||||
<p class="fragment"><b>Directed: </b> A directed edge means "≥"</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Trees</h4>
|
||||
|
||||
<dl style="font-size: 80%">
|
||||
<dt><u>Child</u></dt>
|
||||
<dd>An adjacent node connected by an out-edge</dd>
|
||||
|
||||
<dt><u>Leaf</u></dt>
|
||||
<dd>A node with no children</dd>
|
||||
|
||||
<dt><u>Depth</u> of a node</dt>
|
||||
<dd>The number of edges from the root to the node</dd>
|
||||
|
||||
<dt><u>Depth</u> of a tree</dt>
|
||||
<dd>The maximum depth of any node in the tree</dd>
|
||||
|
||||
<dt><u>Level</u> of a node</dt>
|
||||
<dd>The depth + 1</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Binary Heaps</h4>
|
||||
|
||||
<p>Organize the priority queue as a tree</p>
|
||||
|
||||
<div style="text-align: left;">
|
||||
<p><b>Directed: </b> A directed edge means "≥"</p>
|
||||
<p class="fragment"><b>Binary: </b> Max out-degree of 2 (Easy to reason about)</p>
|
||||
<p class="fragment"><b>Complete: </b> Every "level" except the last is full<ul>
|
||||
<li class="fragment"><b>Balanced: </b> TBD (loosely, all leaves at ~ same level)</li>
|
||||
<li class="fragment">Easy to encode in an array (later today)</li>
|
||||
</ul></p>
|
||||
<p class="fragment">All nodes in the last level are all-the-way left</p>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>Ordering by "≥" gets us a "Max-Heap"</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Valid Max-Heaps</h4>
|
||||
|
||||
<svg data-src="graphics/22b/valid-maxheap.svg" class="stretch" />
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Invalid Max-Heaps</h4>
|
||||
|
||||
<svg data-src="graphics/22b/invalid-maxheap.svg" class="stretch" />
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Heaps</h4>
|
||||
|
||||
<p>What is the depth of a binary heap containing $n$ items?</p>
|
||||
|
||||
<ul style="font-size: 70%">
|
||||
<li class="fragment">Level 1: Up to 1 item</li>
|
||||
<li class="fragment">Level 2: Up to 2 items</li>
|
||||
<li class="fragment">Level 3: Up to 4 items</li>
|
||||
<li class="fragment">Level 4: Up to 8 items</li>
|
||||
<li class="fragment">Level 5: Up to 16 items</li>
|
||||
<li class="fragment">Level $i$: Up to $O(2^i)$ items</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Heaps</h4>
|
||||
|
||||
<p>What is the depth of a binary heap containing $n$ items?</p>
|
||||
|
||||
<p class="fragment">$$n = O\left(\sum_{i = 1}^{\ell_{max}} 2^i\right) = O\left(2^{\ell_{max}}\right)$$</p>
|
||||
<p class="fragment">$$\ell_{max} = O\left(\log(n)\right)$$</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">The Heap ADT</h4>
|
||||
|
||||
<dl>
|
||||
<dt>enqueue(elem: A) [pushHeap]</dt>
|
||||
<dd>Place an item into the heap</dd>
|
||||
|
||||
<dt>dequeue: A [popHeap]</dt>
|
||||
<dd>Remove and return the maximal element from the heap</dd>
|
||||
|
||||
<dt>head: A</dt>
|
||||
<dd>The maximal element in the heap</dd>
|
||||
|
||||
<dt>length: Int</dt>
|
||||
<dd>The number of elements in the heap.</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">enqueue</h4>
|
||||
|
||||
<p class="fragment"><b>Idea:</b> Insert element at next available spot, then fix.</p>
|
||||
|
||||
<ul>
|
||||
<li class="fragment">Call the insertion point <tt>current</tt></li>
|
||||
<li class="fragment">While <tt>current != root</tt> and <tt>current > parent</tt>
|
||||
<ul>
|
||||
<li class="fragment">Swap <tt>current</tt> with <tt>parent</tt></li>
|
||||
<li class="fragment">Repeat with <tt>current</tt> ← <tt>parent</tt></li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">enqueue</h4>
|
||||
|
||||
<svg data-src="graphics/22b/fix-up.svg" class="stretch" />
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">enqueue</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
def fixUp[A: Ordering](current: Vertex[A]): Unit =
|
||||
{
|
||||
if(current.parent.isDefined){
|
||||
val parent = current.parent.get
|
||||
if( Ordering[A].lt( parent.value, current.value ) ){
|
||||
swap(current.value, parent.value)
|
||||
fixUp(parent)
|
||||
}
|
||||
}
|
||||
}
|
||||
</code></pre>
|
||||
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
<p class="fragment">(How many swaps are required?)</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">dequeue</h4>
|
||||
|
||||
<p class="fragment"><b>Idea:</b> Replace root with last element, then fix.</p>
|
||||
|
||||
<ul>
|
||||
<li class="fragment">Call start with <tt>current</tt> ← <tt>root</tt></li>
|
||||
<li class="fragment">While <tt>current</tt> has a <tt>child > current</tt>
|
||||
<ul>
|
||||
<li class="fragment">Swap <tt>current</tt> with larger <tt>child</tt></li>
|
||||
<li class="fragment">Repeat with <tt>current</tt> ← <tt>child</tt></li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">dequeue</h4>
|
||||
|
||||
<svg data-src="graphics/22b/fix-down.svg" height="600px" />
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">enqueue</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
def fixDown[A: Ordering](current: Vertex[A]): Unit =
|
||||
{
|
||||
if(current.leftChild.isDefined){
|
||||
val left = current.leftChild.get
|
||||
if( Ordering[A].lt( current.value, left.value ) ){
|
||||
if(current.rightChild.isDefined){
|
||||
val right = current.rightChild.get
|
||||
if( Ordering[A].lt( right.value, left.value ) ){
|
||||
swap(current.value, left.value); fixDown(left)
|
||||
} else {
|
||||
swap(current.value, right.value); fixDown(right)
|
||||
}
|
||||
} else {
|
||||
swap(current.value, left.value); fixDown(left)
|
||||
}
|
||||
} else if(current.rightChild.isDefined) {
|
||||
val right = current.rightChild.get
|
||||
if( Ordering[A].lt( current.value, right.value ) ){
|
||||
swap(current.value, right.value); fixDown(right)
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
</code></pre>
|
||||
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Priority Queues</h4>
|
||||
<table>
|
||||
<tr>
|
||||
<th>Operation</th>
|
||||
<th>Lazy</th>
|
||||
<th>Proactive</th>
|
||||
<th>Heap</th>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td>enqueue</td>
|
||||
<td>$O(1)$</td>
|
||||
<td>$O(n)$</td>
|
||||
<td>$O(\log(n))$</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td>dequeue</td>
|
||||
<td>$O(n)$</td>
|
||||
<td>$O(1)$</td>
|
||||
<td>$O(\log(n))$</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td>head</td>
|
||||
<td>$O(n)$</td>
|
||||
<td>$O(1)$</td>
|
||||
<td>$O(1)$</td>
|
||||
</tr>
|
||||
</table>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4>Storing Heaps</h4>
|
||||
|
||||
<ul>
|
||||
<li>Each layer has a maximum size</li>
|
||||
<li>Each layer grows left-to-right</li>
|
||||
<li>Only the last layer grows</li>
|
||||
</ul>
|
||||
|
||||
<p class="fragment"><b>Idea: </b> Use an <tt>ArrayBuffer</tt> to store the heap.</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Storing Heaps</h4>
|
||||
|
||||
<svg data-src="graphics/22b/heap-array.svg" width="500px"/>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Analysis</h4>
|
||||
|
||||
<dl>
|
||||
<dt>Enqueue</dt>
|
||||
<dd class="fragment"><b>Append to ArrayBuffer: </b> <span class="fragment">$O(n)$, Amortized $O(1)$</span></dd>
|
||||
<dd class="fragment"><b>fixUp: </b> <span class="fragment">$O(\log(n))$ fixes, each $O(1)$ = $O(\log(n))$</span></dd>
|
||||
<dd class="fragment"><b>Total: </b> Amortized $O(\log(n))$, Worst-case $O(n)$</dd>
|
||||
|
||||
<dt class="fragment">Dequeue</dt>
|
||||
<dd class="fragment"><b>Remove End of ArrayBuffer: </b> <span class="fragment">$O(1)$</span></dd>
|
||||
<dd class="fragment"><b>fixDown: </b> <span class="fragment">$O(\log(n))$ fixes, each $O(1)$ = $O(\log(n))$</span></dd>
|
||||
<dd class="fragment"><b>Total: </b> Worst-case $O(\log(n))$</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Heap Sort</h4>
|
||||
|
||||
<ul>
|
||||
<li>Insert items into priority queue with enqueue</li>
|
||||
<li>Reconstruct sequence (in reverse order) with dequeue</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Heap Sort</h4>
|
||||
|
||||
<svg data-src="graphics/22b/heapsort.svg" />
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Heap Sort</h4>
|
||||
|
||||
<ul>
|
||||
<dt>Enqueue Element $i$</dt>
|
||||
<dd class="fragment">$O(\log(i))$</dd>
|
||||
|
||||
<dt class="fragment">Dequeue Element $i$</dt>
|
||||
<dd class="fragment">$O(\log(n-i))$</dd>
|
||||
</ul>
|
||||
|
||||
<p class="fragment">
|
||||
$$\left(\sum_{i=1}^n O(\log(i))\right) + \left(\sum_{i=1}^n O(\log(n-i))\right)$$
|
||||
</p>
|
||||
<p class="fragment">
|
||||
$$< O(n\log(n))$$
|
||||
</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Updating Heap Elements</h4>
|
||||
|
||||
<p>
|
||||
After updating <tt>current</tt>, <tt>fixUp</tt> or <tt>fixDown</tt>.
|
||||
</p>
|
||||
|
||||
<p class="fragment">
|
||||
If new <tt>current > parent</tt>, <tt>current</tt> moves up.<br/>
|
||||
If new <tt>current < parent</tt>, <tt>current</tt> moves down.
|
||||
</p>
|
||||
|
||||
<p class="fragment">
|
||||
How do we know where the value appears in the heap?
|
||||
</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Heapify</h4>
|
||||
|
||||
<p><b>Input: </b> Array</p>
|
||||
<p><b>Output: </b> Array reorderd to be a heap</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Heapify</h4>
|
||||
|
||||
<dl>
|
||||
<div class="fragment">
|
||||
<dt>Level $\log(n)$</dt>
|
||||
<dd><tt>fixDown</tt> all $\frac{n}{2}$ nodes at this level (max $0$ swaps each)</dd>
|
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</div>
|
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<div class="fragment">
|
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<dt>Level $\log(n)-1$</dt>
|
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<dd><tt>fixDown</tt> all $\frac{n}{4}$ nodes at this level (max $1$ swaps each)</dd>
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</div>
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<div class="fragment">
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<dt>Level $\log(n)-2$</dt>
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<dd><tt>fixDown</tt> all $\frac{n}{8}$ nodes at this level (max $2$ swaps each)</dd>
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</div>
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</dl>
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</section>
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<section>
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<h4 class="slide_title">Heapify</h4>
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<div style="font-size: 70%">
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<p>$$O\left(\sum_{i = 1}^{\log(n)} \frac{n}{2^{i}} \cdot (i+1) \right)$$</p>
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<p class="fragment">$$O\left(n \sum_{i = 1}^{\log(n)} \frac{i}{2^{i}} + \frac{1}{2^i}\right)$$</p>
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<p class="fragment">$$O\left(n \sum_{i = 1}^{\log(n)} \frac{i}{2^{i}}\right)$$</p>
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<p class="fragment">$$O\left(n \sum_{i = 1}^{\infty} \frac{i}{2^{i}}\right)$$</p>
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<p class="fragment">$\sum_{i = 1}^{\infty} \frac{i}{2^{i}}$ is known to converge to a constant.</p>
|
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<p class="fragment">$$O\left(n\right)$$</p>
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</div>
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</section>
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<section>
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<h4 class="slide_title">Heapify</h4>
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<p>We can go from an unsorted array to a heap in $O(n)$</p>
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<p class="fragment">(but heap sort still requires $n \log(n)$ for dequeueing)</p>
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<section>
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<h4 class="slide_title">Next Time...</h4>
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<p>Sets, Bags, and Binary Search Trees</p>
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After Width: | Height: | Size: 19 KiB |
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Reference in New Issue