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.

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

Formally...

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

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

For any two elements $x, y \in A$:

• 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

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

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

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

Can we do better?