From 02f58e72cec1ae07f7df1b2485e42995dd256628 Mon Sep 17 00:00:00 2001 From: Oliver Date: Mon, 24 Oct 2022 00:57:17 -0400 Subject: [PATCH] Heaps slides --- src/teaching/cse-250/2022fa/index.erb | 2 + .../cse-250/2022fa/slide/22b-Heaps.erb | 455 ++++++ .../cse-250/2022fa/slide/graphics/22b/cat.png | Bin 0 -> 482097 bytes .../2022fa/slide/graphics/22b/fix-down.svg | 409 +++++ .../2022fa/slide/graphics/22b/fix-up.svg | 316 ++++ .../2022fa/slide/graphics/22b/heap-array.svg | 634 +++++++ .../2022fa/slide/graphics/22b/heapsort.svg | 1453 +++++++++++++++++ .../slide/graphics/22b/invalid-maxheap.svg | 440 +++++ .../slide/graphics/22b/valid-maxheap.svg | 436 +++++ 9 files changed, 4145 insertions(+) create mode 100644 src/teaching/cse-250/2022fa/slide/22b-Heaps.erb create mode 100644 src/teaching/cse-250/2022fa/slide/graphics/22b/cat.png create mode 100644 src/teaching/cse-250/2022fa/slide/graphics/22b/fix-down.svg create mode 100644 src/teaching/cse-250/2022fa/slide/graphics/22b/fix-up.svg create mode 100644 src/teaching/cse-250/2022fa/slide/graphics/22b/heap-array.svg create mode 100644 src/teaching/cse-250/2022fa/slide/graphics/22b/heapsort.svg create mode 100644 src/teaching/cse-250/2022fa/slide/graphics/22b/invalid-maxheap.svg create mode 100644 src/teaching/cse-250/2022fa/slide/graphics/22b/valid-maxheap.svg diff --git a/src/teaching/cse-250/2022fa/index.erb b/src/teaching/cse-250/2022fa/index.erb index d873966f..291a8d43 100644 --- a/src/teaching/cse-250/2022fa/index.erb +++ b/src/teaching/cse-250/2022fa/index.erb @@ -378,6 +378,8 @@ schedule: - date: 10/24/22 topic: Heaps dow: Mon + section_b: + slides: slide/22b-Heaps.html - date: 10/26/22 topic: Trees dow: Wed diff --git a/src/teaching/cse-250/2022fa/slide/22b-Heaps.erb b/src/teaching/cse-250/2022fa/slide/22b-Heaps.erb new file mode 100644 index 00000000..7664e440 --- /dev/null +++ b/src/teaching/cse-250/2022fa/slide/22b-Heaps.erb @@ -0,0 +1,455 @@ +--- +template: templates/cse250_2022_slides.erb +title: Priority Queues, Heaps +date: Oct 24, 2022 +textbook: Ch. 18 +cat: graphics/22b/cat.png +--- + +
+

Priority Queues

+
+ +
+

PriorityQueue[A <: Ordering]

+ +
+
enqueue(v: A): Unit
+
Insert a value $v$ into the priority queue.
+ +
dequeue: A
+
Remove the greatest element in the priority queue.
+ +
head: A
+
Peek at the greatest element in the priority queue.
+
+
+ +
+

Priority Queues

+ + + + + + + + + + + + + + + + + + + + + + + + + +
OperationLazyProactive
enqueue$O(1)$$O(n)$
dequeue$O(n)$$O(1)$
head$O(n)$$O(1)$
+ +

Can we do better?

+
+ +
+

Priority Queues

+ +
+
Lazy
+
Fast Enqueue, Slow Dequeue
+ +
Proactive
+
Slow Enqueue, Fast Dequeue
+ +
???
+
Fast(-ish) Enqueue, Fast(-ish) Dequeue
+
+
+ +
+

Idea: Keep the priority queue "kinda" sorted.

+ +

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.

+
+ +
+

Binary Heaps

+ +

Challenge: How do we keep track of which elements are still sorted?

+
+ +
+

Binary Heaps

+ +

Idea: Organize the priority queue as a tree

+ +

Directed: A directed edge means "≥"

+
+ +
+

Trees

+ +
+
Child
+
An adjacent node connected by an out-edge
+ +
Leaf
+
A node with no children
+ +
Depth of a node
+
The number of edges from the root to the node
+ +
Depth of a tree
+
The maximum depth of any node in the tree
+ +
Level of a node
+
The depth + 1
+
+
+ +
+

Binary Heaps

+ +

Organize the priority queue as a tree

+ +
+

Directed: A directed edge means "≥"

+

Binary: Max out-degree of 2 (Easy to reason about)

+

Complete: Every "level" except the last is full

    +
  • Balanced: TBD (loosely, all leaves at ~ same level)
  • +
  • Easy to encode in an array (later today)
  • +

+

All nodes in the last level are all-the-way left

+
+
+ +
+

Ordering by "≥" gets us a "Max-Heap"

+
+ +
+

Valid Max-Heaps

+ + +
+ +
+

Invalid Max-Heaps

+ + +
+ +
+

Heaps

+ +

What is the depth of a binary heap containing $n$ items?

+ + +
+ +
+

Heaps

+ +

What is the depth of a binary heap containing $n$ items?

+ +

$$n = O\left(\sum_{i = 1}^{\ell_{max}} 2^i\right) = O\left(2^{\ell_{max}}\right)$$

+

$$\ell_{max} = O\left(\log(n)\right)$$

+
+ +
+

The Heap ADT

+ +
+
enqueue(elem: A) [pushHeap]
+
Place an item into the heap
+ +
dequeue: A [popHeap]
+
Remove and return the maximal element from the heap
+ +
head: A
+
The maximal element in the heap
+ +
length: Int
+
The number of elements in the heap.
+
+
+ +
+

enqueue

+ +

Idea: Insert element at next available spot, then fix.

+ + +
+ +
+

enqueue

+ + +
+ +
+

enqueue

+ +

+    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)
+        }
+      }
+    }
+  
+ +

What's the complexity?

+

(How many swaps are required?)

+
+ +
+

dequeue

+ +

Idea: Replace root with last element, then fix.

+ + +
+ +
+

dequeue

+ + +
+ +
+

enqueue

+ +

+    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)
+          }
+        }
+      }
+    }
+  
+ +

What's the complexity?

+
+ +
+

Priority Queues

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
OperationLazyProactiveHeap
enqueue$O(1)$$O(n)$$O(\log(n))$
dequeue$O(n)$$O(1)$$O(\log(n))$
head$O(n)$$O(1)$$O(1)$
+
+ +
+

Storing Heaps

+ + + +

Idea: Use an ArrayBuffer to store the heap.

+
+ +
+

Storing Heaps

+ + +
+ +
+

Analysis

+ +
+
Enqueue
+
Append to ArrayBuffer: $O(n)$, Amortized $O(1)$
+
fixUp: $O(\log(n))$ fixes, each $O(1)$ = $O(\log(n))$
+
Total: Amortized $O(\log(n))$, Worst-case $O(n)$
+ +
Dequeue
+
Remove End of ArrayBuffer: $O(1)$
+
fixDown: $O(\log(n))$ fixes, each $O(1)$ = $O(\log(n))$
+
Total: Worst-case $O(\log(n))$
+
+
+ +
+

Heap Sort

+ + +
+ +
+

Heap Sort

+ + +
+ +
+

Heap Sort

+ + + +

+ $$\left(\sum_{i=1}^n O(\log(i))\right) + \left(\sum_{i=1}^n O(\log(n-i))\right)$$ +

+

+ $$< O(n\log(n))$$ +

+
+ +
+

Updating Heap Elements

+ +

+ After updating current, fixUp or fixDown. +

+ +

+ If new current > parent, current moves up.
+ If new current < parent, current moves down. +

+ +

+ How do we know where the value appears in the heap? +

+
+ +
+

Heapify

+ +

Input: Array

+

Output: Array reorderd to be a heap

+
+ +
+

Heapify

+ +
+
+
Level $\log(n)$
+
fixDown all $\frac{n}{2}$ nodes at this level (max $0$ swaps each)
+
+ +
+
Level $\log(n)-1$
+
fixDown all $\frac{n}{4}$ nodes at this level (max $1$ swaps each)
+
+ +
+
Level $\log(n)-2$
+
fixDown all $\frac{n}{8}$ nodes at this level (max $2$ swaps each)
+
+
+
+ +
+

Heapify

+ +
+

$$O\left(\sum_{i = 1}^{\log(n)} \frac{n}{2^{i}} \cdot (i+1) \right)$$

+

$$O\left(n \sum_{i = 1}^{\log(n)} \frac{i}{2^{i}} + \frac{1}{2^i}\right)$$

+

$$O\left(n \sum_{i = 1}^{\log(n)} \frac{i}{2^{i}}\right)$$

+

$$O\left(n \sum_{i = 1}^{\infty} \frac{i}{2^{i}}\right)$$

+

$\sum_{i = 1}^{\infty} \frac{i}{2^{i}}$ is known to converge to a constant.

+

$$O\left(n\right)$$

+
+
+ +
+

Heapify

+ +

We can go from an unsorted array to a heap in $O(n)$

+ +

(but heap sort still requires $n \log(n)$ for dequeueing)

+
+ +
+

Next Time...

+ +

Sets, Bags, and Binary Search Trees

+
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