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zzz_vx>4iDx=M--Y~hIL7Z@FcuKCDpBgDB@B8QXz05BWZW9NBofpi{n09JOxt^!*u$^*syePV-A z8!&-3EHecP4;P$(PKgYNISS4(JE6)wOefuW9VhpTh1l8tbAeL#be;IX0F0tG{{Zv% zgu7>p&&B@mft9Je279^y=4KJE-13+IM=%|R_qcJKGb4OGQm9w(eO zNJ_@XTmG`TqoC&$7zMc2LV}~CxqF`STLJhTVe?=viQvMLHBItw_lW~u*n)TOjcXzn zmG6wyhd3z!HO^g}y%}65*E{0?MfzlHquc%X!7*TbVgWb~4l=y8yPVlU^@#RWdN{<* z4eEZeU7^Q0U{kNWMOh8|!+28Doigik~GU{OG$v2mRu6TODa8MGab4BDkf zNr*tIoa-4%<9;)5XR`=N_lzQ6=Kw(;cy1cLv5Yom_eY#a+3y?pCd79 zyDf*NkP;M zDB(0TYKAj<5~UI06)>uvZ|fJ4Re?^LW&uh7uAGzLD2no()c2d8(hu6kz(#J$b^0*O2QuoRNUB2Zh(Xgb$IEUJYC~6lzZ^<1H7M z@Y#z1FTAK49C*OADSY9uy$8-XD8mNZ`ox8f+Up=7hH_}R3wBtOc#$^U9*yHb6q}QO zI3Tu7$If|*op$@l$|Y@gFpxO6r8vDG?c<~cVo5s3K~u_c&ImpsE(jaoJr~w)geKi_ z9e-b(3N5z!-g0XiYeI2;^Q|||nf-sc!LVcpQ6D}$X39yx8WWrVfPmP$_ujh1#{kRb pX#W6j9E8@uBmGR)p0)P+xj&UqmH_`ad|Ji5IJ-Gk? literal 0 HcmV?d00001 diff --git a/src/teaching/cse-250/2022fa/index.erb b/src/teaching/cse-250/2022fa/index.erb index 91a0a059..990ae023 100644 --- a/src/teaching/cse-250/2022fa/index.erb +++ b/src/teaching/cse-250/2022fa/index.erb @@ -347,6 +347,8 @@ schedule: - date: 10/14/22 topic: Order Relations, Priority Queues dow: Fri + section_b: + slides: slide/19b-Orderings.html - week: 8 lectures: - date: 10/17/22 diff --git a/src/teaching/cse-250/2022fa/slide/16b-GraphExploration.erb b/src/teaching/cse-250/2022fa/slide/16b-GraphExploration.erb index 98ac5705..68e0373a 100644 --- a/src/teaching/cse-250/2022fa/slide/16b-GraphExploration.erb +++ b/src/teaching/cse-250/2022fa/slide/16b-GraphExploration.erb @@ -480,7 +480,7 @@ def DFSOne(graph: Graph[…], v: Graph[…]#Vertex) $ = O(\sum_{v \in V} deg(v))$

- $ = O(\frac{1}{2} |E|)$ (by rule) + $ = O(2 |E|)$ (by rule)

$ = O(|E|)$ diff --git a/src/teaching/cse-250/2022fa/slide/19b-Orderings.erb b/src/teaching/cse-250/2022fa/slide/19b-Orderings.erb new file mode 100644 index 00000000..f101e928 --- /dev/null +++ b/src/teaching/cse-250/2022fa/slide/19b-Orderings.erb @@ -0,0 +1,570 @@ +--- +template: templates/cse250_2022_slides.erb +title: Graph Exploration +date: Oct 14, 2022 +textbookTBD: Ch. 15.3 +cat: graphics/19b/cat.png +--- + +

+

Examples

+ +
    +
  • $(B, 10)$, $(D, 3)$, $(E, 40)$
  • +
  • “A+”, “C”, “B-”
  • +
  • Taco Tuesday, Fish Friday, Meatless Monday
  • +
  • The Princess Bride, Crouching Tiger Hidden Dragon, Rob Roy
  • +
  • Aardvark, Baboon, Capybara, Donkey, Echidna, ...
  • +
+
+ +
+

Ordering

+ +

An ordering $(A, \leq)$ (an ordering over type $A$) is ...

+ +
    +
  • A collection (set) of things of type $A$
  • +
  • A "relation" or comparator $\leq$ that relates two things
  • +
+
+ +
+
+
+
$5 \leq 30 \leq 999$
+
Numerical order
+
+ +
+
$(E, 40) \leq (B, 10) \leq (D, 3)$
+
Reverse-numerical order on the 2nd field
+
+ +
+
$C+ \leq B- \leq B \leq B+ \leq A- \leq A$
+
Letter grades
+
+ +
+
$AA \leq AM \leq BZ \leq CA \leq CD$
+
Compare 1st letter, if same compare 2nd, if same compare 3rd, ... ("Lexical Order")
+
+
+
+ +
+

Ordering Properties

+ +
+
+
Crouching Tiger Hidden Dragon $\leq$ Princess Bride
+
Carey Elwes and Mandy Patankin have amazing banter
+
+ +
+
Princess Bride $\leq$ Rob Roy
+
Rob Roy: Best climactic duel of all time
+
+ +
+
Rob Roy $\leq$ Crouching Tiger Hidden Dragon
+
A ton of amazingly technical fights
+
+
+
+ +
+

Ordering Properties

+ +
+ +
+

Ordering Properties

+

An ordering needs to be...

+
+
Reflexive
+
$x \leq x$
+ +
Antisymmetric
+
If $x \leq y$ and $y \leq x$ then $x = y$
+ +
Transitive
+
If $x \leq y$ and $y \leq z$ then $x \leq z$
+
+
+ +
+

Ordering Properties

+ +

Course 1 $\leq$ Course 2 iff Course 1 is a prereq of Course 2

+
    +
  • CSE 115 $\leq$ CSE 116
  • +
  • CSE 116 $\leq$ CSE 250
  • +
  • CSE 115 $\leq$ CSE 191
  • +
  • CSE 191 $\leq$ CSE 250
  • +
+
+ +
+

Ordering Properties

+ + +
+ +
+

Is this a valid ordering?

+ +

Yes!

+ +

(Partial order, as opposed to Total order)

+
+ +
+

(Partial)Ordering Properties

+

A partial ordering needs to be...

+
+
Reflexive
+
$x \leq x$
+ +
Antisymmetric
+
If $x \leq y$ and $y \leq x$ then $x = y$
+ +
Transitive
+
If $x \leq y$ and $y \leq z$ then $x \leq z$
+
+
+ +
+

(Total)Ordering Properties

+

A total ordering needs to be...

+
+
Reflexive
+
$x \leq x$
+ +
Antisymmetric
+
If $x \leq y$ and $y \leq x$ then $x = y$
+ +
Transitive
+
If $x \leq y$ and $y \leq z$ then $x \leq z$
+ +
Complete
+
Either $x \leq y$ or $y \leq x$ for any $x, y \in A$.
+
+
+ +
+

Formally...

+ +

For a sort order $(A, \leq)$

+ +
+
Greatest
+
An element $x \in A$ s.t. there is no $y \in A$, where $x \leq y$
+ +
Least
+
An element $x \in A$ s.t. there is no $y \in A$, where $y \leq x$
+
+ +

A partial order may not have a unique greatest/least element

+
+ +
+

Formally...

+ +

$\leq$ can be described explicitly, by a set of tuples.

+ + $$\{\; (a, a), (a, b), (a, c), \ldots, (b, b), \ldots, (z, z) \;\}$$ + +

If $(x, y)$ is in the set, $x \leq y$

+
+ +
+

Formally...

+ +

$\leq$ can be described by a mathematical rule.

+ + $$\{\; (x, y) \;|\; x, y \in \mathbb Z, \exists a \in \mathbb Z_0^{+} : x + a = y \;\}$$ + +

$x \leq y$ iff $x, y$ are integers, and there is a non-negative integer $a$ s.t. $x + a = y$

+
+ +
+

Formally...

+ +

Multiple orderings can be defined for the same set.

+
    +
  • RottenTomatoes vs Metacritic vs Box Office Gross.
  • +
  • "Best Movie" first vs "Worst Movie" first.
  • +
  • Rank by number of air ducts crawled through.
  • +
+ +

We use a subscripts to separate orderings (e.g., $\leq_1$, $\leq_2$, $\leq_3$)

+
+ +
+

Formally...

+ +

We can transform orderings.

+ +

Reverse: If $x \leq_1 y$, then define $y \leq_r x$

+ +

Lexical: Given $\leq_1, \leq_2, \leq_3, \ldots$ +

    +
  • If $x \leq_1 y$ then $x \leq_\ell y$
  • +
  • If $x =_1 y$ and $x \leq_2 y$ then $x \leq_\ell y$
  • +
  • If $x =_2 y$ and $x \leq_3 y$ then $x \leq_\ell y$
  • +
  • ....
  • +
+

+
+ +
+

Examples of Lexical Ordering

+ +
    +
  • Names: First letter, then second, then third...
  • +
  • Movies: Average of reviews, then number of reviews...
  • +
  • Tuples: First field, then second field, then third...
  • +
+
+ +
+

Formally...

+ +

$\leq$ can be described an ordering over a key derived from the element.

+ +

$x \leq_{edge} y$ iff $weight(x) \leq weight(y)$

+

$x \leq_{student} y$ iff $name(x) \leq_{lex} name(y)$

+ +

We say that the weight (resp., name) is a key.

+
+ +
+

Topological Sort

+ +

A Topological Sort $(A, \leq_2)$ is any total order over $x$ that "agrees" with a partial order $(A, \leq_1)$

+ +

For any two elements $x, y \in X$: +

    +
  • If $x \leq_1 y$ then $x \leq_2 y$
  • +
  • If $y \leq_1 x$ then $y \leq_2 x$
  • +
  • Otherwise, either $x \leq_2 y$ or $y \leq_2 x$
  • +
+

+
+ +
+

Topological Sort

+ +

The following are all topological sorts over our partial order example:

+
    +
  • CSE 115, CSE 116, CSE 191, CSE 241, CSE 250
  • +
  • CSE 241, CSE 115, CSE 116, CSE 191, CSE 250
  • +
  • CSE 115, CSE 191, CSE 116, CSE 250, CSE 241
  • +
+ +

(If the partial order is a schedule requirement, each topological sort is a possible schedule)

+
+ +
+

And now, for an ordering-based ADT...

+
+ +
+

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.
+
+
+ +
+
    +
  • enqueue(5)
  • +
  • enqueue(9)
  • +
  • enqueue(2)
  • +
  • enqueue(7)
  • +
  • head → 9
  • +
  • dequeue → 9
  • +
  • size → 3
  • +
  • head → 7
  • +
  • dequeue → 7
  • +
  • dequeue → 5
  • +
  • dequeue → 2
  • +
  • isEmpty → true
  • +
+
+

How do we store these items?

+

5, 9, 2, 7?

+

9, 7, 5, 2?

+

2, 5, 7, 9?

+
+
+ +
+

Two mentalities...

+ +
+
Lazy: Keep everything in a mess
+
"Selection Sort"
+
Proactive: Keep everything organized
+
"Insertion Sort"
+
+
+ +
+

Lazy Priority Queue

+ +

Base Data Structure: Linked List

+ +
+
enqueue(t:A)
+
Append $t$ to the end of the linked list. $O(1)$
+ +
head/dequeue
+
Traverse the list to find the largest value. $O(n)$
+
+
+ +
+

Sort

+ +

+  def pqueueSort[A](items: Seq[A], pqueue: PriorityQueue[A]): Seq[A] =
+  {
+    val out = new Array[A](items.size)
+    for(item <- items){ pqueue.enqueue(item) }
+    i = out.size - 1
+    while(!pqueue.isEmpty) { buffer(i) = pqueue.dequeue; i-- }
+    return out.toSeq
+  }
+  
+
+ +
+

Selection Sort

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Seq/BufferPQueue
Input(7, 4, 8, 2, 5, 3, 9)()
Step 1(4, 8, 2, 5, 3, 9)(7)
Step 2(8, 2, 5, 3, 9)(7, 4)
፧ ፧ ፧
Step N[_, _, _, _, _, _, _](7, 4, 8, 2, 5, 3, 9)
Step N+1[_, _, _, _, _, _, 9](7, 4, 8, 2, 5, 3)
Step N+2[_, _, _, _, _, 8, 9](7, 4, 2, 5, 3)
Step N+3[_, _, _, _, 7, 8, 9](4, 2, 5, 3)
Step N+4[_, _, _, 5, 7, 8, 9](4, 2, 3)
Step N+5[_, _, 4, 5, 7, 8, 9](2, 3)
Step N+6[_, 3, 4, 5, 7, 8, 9](2)
Step 2N[2, 3, 4, 5, 7, 8, 9]()
+
+ +
+

Selection Sort

+ +

+  def pqueueSort[A](items: Seq[A], pqueue: PriorityQueue[A]): Seq[A] =
+  {
+    val out = new Array[A](items.size)
+    for(item <- items){ pqueue.enqueue(item) }
+    i = out.size - 1
+    while(!pqueue.isEmpty) { buffer(i) = pqueue.dequeue; i-- }
+    return out.toSeq
+  }
+  
+ +

What's the complexity?

+
+ +
+

Proactive Priority Queue

+ +

Base Data Structure: Linked List

+ +
+
enqueue(t:A)
+
Find $t$'s position in reverse sorted order. $O(n)$
+ +
head/dequeue
+
Refer to the head item in the list. $O(1)$
+
+
+ +
+ +
+

Insertion Sort

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Seq/BufferPQueue
Input(7, 4, 8, 2, 5, 3, 9)()
Step 1(4, 8, 2, 5, 3, 9)(7)
Step 2(8, 2, 5, 3, 9)(7, 4)
Step 3(2, 5, 3, 9)(8, 7, 4)
Step 4(5, 3, 9)(8, 7, 4, 2)
Step 5(3, 9)(8, 7, 5, 4, 2)
Step 6(9)(8, 7, 5, 4, 3, 2)
Step N[_, _, _, _, _, _, _](9, 8, 7, 5, 4, 3, 2)
Step N+1[_, _, _, _, _, _, 9](8, 7, 5, 4, 3, 2)
፧ ፧ ፧
Step 2N[2, 3, 4, 5, 7, 8, 9]()
+
+ +
+

Insertion Sort

+ +

+  def pqueueSort[A](items: Seq[A], pqueue: PriorityQueue[A]): Seq[A] =
+  {
+    val out = new Array[A](items.size)
+    for(item <- items){ pqueue.enqueue(item) }
+    i = out.size - 1
+    while(!pqueue.isEmpty) { buffer(i) = pqueue.dequeue; i-- }
+    return out.toSeq
+  }
+  
+ +

What's the complexity?

+
+ +
+

Priority Queues

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

Can we do better?

+
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