© 2012 Prof. Dr. Franz J. Brandenburg
Stacks, Queues,Deques
and their Representations as Drawings
Franz J. Brandenburg
University of Passau
joint work with
Christopher Auer and Andreas Gleißner
34/1
Basics
© 2012 Prof. Dr. Franz J. Brandenburg
Data Structures are used to store data
by operations which get (read), insert and remove items.
What makes the difference between different data structures? Stack vs queue?
Insertion: anytime, no difference
Removal: When can an item be accessed.
Together: An axiomatic description of the behavior.
Example
Stack: insert (push) and remove (pop) and get (top)
main axiom: s = s.insert(x); s.remove();
The stack is unchanged after insert, remove.
x = s.get(); s.remove(), s.insert(x)
may cause an error if s is empty.
Array: random access
main axiom: NONE, its unpredicable, its chaos
34/2
Stack and Queue
© 2012 Prof. Dr. Franz J. Brandenburg
stack (LIFO)
a linked list with insert and remove to the left (or top)
insert/push
4
3
.....
2
1
remove/pop
4
3
2
1
queue (FIFO)
a linked list with a head (right) for insert
and a tail (left) for get and remove
Axioms: NONE, it is too complex
head
tail
4
3
2
.....
1
34/3
Deques
© 2012 Prof. Dr. Franz J. Brandenburg
deque (double-ended queue)
implemented by the class Arraydeque in Java 6
seen as a doubly linked list with a head (right) and a tail (left)
and insert and remove at either ends.
head
tail
5
3
.....
1
tail
U
5
3
1
6
4
2
2
4
6
head
commuting pipes
34/4
History
© 2012 Prof. Dr. Franz J. Brandenburg
•
Who invented stacks and queues?
•
use of stacks in programming languages
for the formal description of ALGOL (1960)
by F.L. Bauer (TU Munich)
… the axioms for a stack
•
use as data structures
by D.E. Knuth (Stanford) in The Art of Computer Programming (1968)
… stack, queue, deque, input/output restricted deque
deque = double ended queue
This name is misleading.
The queue mode is secondary! But ... (see later)
34/5
Deques
© 2012 Prof. Dr. Franz J. Brandenburg
a deque has a stack mode at either end
has a queue mode from left to right (or right to left)
a deque can simulate 2 stacks
a deque can simulate 1 queue
What happens in queue mode?
Press all items out (6,4,2,1,3,5)
How strong is a deque?
a deque can be simulated by 2 stacks + 1 queue
Can a deque simulate
2 stacks + 1 queue
head
tail
0
5
3
1
.....
2
4
6
34/6
Stack
© 2012 Prof. Dr. Franz J. Brandenburg
What is typical? What is characteristic?
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algorithm
depth first search
structure
LIFO last in, first out
pattern
nested parenthesis ((( )) ()), [ ( ) ]
programming paradigm recursion
storage behavior
destroy information
you cannot save the item you remove
area of influence
local
the closing parenthesis bounds the inner substructure
graphs
nested rainbows
( ( ( (
) ) )
( (
) ) )
34/7
Queue
© 2012 Prof. Dr. Franz J. Brandenburg
What is typical? What is characteristic?
•
•
•
•
•
•
•
algorithm
structure
pattern
breadth first search
FIFO first in, first out
twist
( ( ( ) ) ( ) ), [ ( ] )
equality set p(x) = p‘(x) with projections p, p‘
programming paradigm streams, buffers
storage behavior
copy information
reinsert the removed item
area of influence
global
graphs
twists with very many crossings
34/8
Linear Layouts of Graphs
© 2012 Prof. Dr. Franz J. Brandenburg
Definition
a linear layout of a graph G = (V,E)
place the vertices on a line or number the vertices 1,2,....,n
for every pair of edges (u, v) and (x, y)
nesting
if u < x < y < v
twist or cross if u < x < v < y
e.g. (1,4) (2,3)
e.g. (1,3), (2,4)
A linear layout is
a stack layout if no two edges twist or cross
a queue layout if no two edges nest
This is the complementary view: define by exclusion.
34/9
Book Embeddings
© 2012 Prof. Dr. Franz J. Brandenburg
Definition (Bernhard, Kainen, 1979)
A book-embedding of a (undirected) graph G = (V,E) is
– a linear ordering of the vertices 1,2,...n
on the spine of the book
– a planar drawing of the edges (u,v) on the pages
as nested arches (LIFO)
1 2 3
4
5 6
7 8 9
10 11 12
Add some more edges, (2,8) and (2,11), (2,12)
but (5,9) were illegal
34/10
Book Embeddings
© 2012 Prof. Dr. Franz J. Brandenburg
Theorem (Bernhard, Kainen, 1979)
A graph G has a book embedding (with 1 page)
if and only if
G is a subgraph of an outerplanar graph
if and only if
G is a stack graph
visit the vertices from left to right and push/pop the edges
At 1: push (1,12), push (1,2)
At 2: pop (1,2), push (2,12), (2,11), (2,8), (2,7)
top
1 2 3
4
5 6
7 8 9
10 11 12
(2,7)
(2,8)
(2,11)
(2,12)
(1,12)
34/11
Graph and Data Structures
© 2012 Prof. Dr. Franz J. Brandenburg
Definiton
Let DS be a structure with insert and remove operations
e.g., stack, queue, deque,
2-stacks, 1-stack + 1-queue, ....
with insert/remove at the head/tail (1st, 2nd stack....)
A graph G = (V,E) is a DS graph (e.g. stack/queue/deque/2-stack/...)
if there is a linear order of the vertices, say 1,2,....,n
such that the edges at vertex v can be correctly processed by DS
(u,v) is removed from DS if u < v
(v,w) is inserted into DS if v < w.
The local order at a vertex is chosen appropriately.
For the correctness:
an edge (u,v) must be directly accessible if it is removed at v.
The order of the edges must meet the data structure.
34/12
Example: K4
© 2012 Prof. Dr. Franz J. Brandenburg
Process on the stack in the order 1,2,3,4
1: push (1,4), (1,3), (1,2)
2: pop (1,2)
top
push (2,4), (2,3)
(2,4)
3: pop (2,3), pop(1,3)
(1,3)
(1,4)
Process on a queue in the order 1,2,3,4
1: insert (1,2), (1,3), 1,4)
2: remove (1,2)
insert (2,3), (2,4)
3: remove (1,3), (2,3)
4
3
1
tail (insert)
2
head (remove)
(1,4) (1,3) (1,2)
(2,4) (2,3) (1,4) (1,3)
(2,4) (2,3) (1,4)
34/13
K4 and Deque
© 2012 Prof. Dr. Franz J. Brandenburg
Process the vertices 1,2,3,4
4
1: insert-left (1,2), insert-right (1,4), (1,3)
2: remove-left (1,2)
insert-left (2,4), (2,3)
3: remove-right (1,3), remove-left (2,3)
insert-left (3,4)
4: remove-right (1,4), (2,4), (3,4),
3
1
(2,4)
(2,3) (1,2)
2
(1,3)
(1,4)
34/14
Stack Graphs
© 2012 Prof. Dr. Franz J. Brandenburg
Theorem (Bernhard, Kainen (1979))
1-stack = outerplanar (subgraph of)
Test: in O(n)
2-stack = subgraph of a planar graph with a Hamilton cycle
Test: NP-complete
Theorem (Yannakakis (1989))
Every planar graph is a 4-stack graph
Claim: some planar graphs need 4 stacks.
stack-number = min # stacks
Test: NP-hard.
34/15
Queue Graphs
© 2012 Prof. Dr. Franz J. Brandenburg
Theorem (Heath, Rosenberg (1992))
1-queue = arch leveled planar
Test: NP-hard
Definition
A graph G = (V,E) is arch leveled planar if
there is a leveling of the vertices
- proper level edges
- arches from the leftmost vertex
to the „open“ right vertices
on the same level
and planar
34/16
Simulations
© 2012 Prof. Dr. Franz J. Brandenburg
Theorem (Heath, Leighton, Rosenberg (1993))
Every 1-queue graph is a 2-stack graph
namely: a planar sub-Hamiltonian graph
Every 1-stack graph is a 2-queue graph
namely: outerplanar
1-queue < 2-stack
and 1-stack < 2-queue
Hence stack-thickness < 2 queue-thickness
queue-thickness < 2 stack-thickness
But this does not scale to stack-number and queue-number
since you must choose one linear odering of the vertices
OPEN
stack-number vs queue-number
34/17
Linear Cylindric Drawing
© 2012 Prof. Dr. Franz J. Brandenburg
Definition (Auer, Brandenburg, et al GD2010)
A linear cylindric drawing is a planar drawing
on a rolling cylinder
with the vertices in a linear ordering on the front line
with edges on the front or winding over the backside
34/18
2D-drawing
© 2012 Prof. Dr. Franz J. Brandenburg
cut along the front line
split or copy the vertices on the front line and unfold
34/19
Linear Cylindric Drawings
© 2012 Prof. Dr. Franz J. Brandenburg
•
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take two horizontal levels
or the upper and lower borders of a square
take identical copies of the order of the vertices
draw „stack“ edges / rainbows below the upper level
draw „stack“ edge / rainbows above the lower level
draw „queue“ edges between the levels
PLANAR
34/20
Deque Graphs
© 2012 Prof. Dr. Franz J. Brandenburg
THEOREM 1 (Auer, Bachmaier, Brandenburg, Brunner, Gleißner, GD2010)
A graph G is a deque graph
if and only if
G has a linear cylindric drawing
Proof:
Take the order of the vertices, 1,2,...,n
Classify the edges:
lower stack iff insert and remove at the head
upper stack iff insert and remove at the tail
queue from bottom left to top right iff insert at head and remove at tail
queue from top left to bottom right iff insert at tail and remove from head
34/21
Queue Graphs
© 2012 Prof. Dr. Franz J. Brandenburg
Corollary
A graph G is a queue graph
if and only if
G is arch leveled planar (Heath, Rosenberg 1992)
if and only if
G is linear cylindric with only queue arcs
with only straight lines from NW to SE
between the two sides
34/22
2-Stack vs Deque
© 2012 Prof. Dr. Franz J. Brandenburg
Theorem
G is a 2-stack graph (Bernhard, Kainen (1979))
iff G is a subgraph of a planar graph with a Hamilton cycle
G is a deque graph
iff G is a subgraph of a planar graph with a Hamilton path
... and there are planar graphs
which have no Hamilton path
which have a Hamilton path and not a Hamilton cycle
Di Giacomo et al Comput Geom 30 (2005),
34/23
Deque & Hamilton Path
© 2012 Prof. Dr. Franz J. Brandenburg
Proof: Take the linear order.
Add edges for a Hamilton path.
Draw the stack edges
Draw the queue edges as arches over the north pole (right)
34/24
Computability
© 2012 Prof. Dr. Franz J. Brandenburg
Computability is defined via Turing machines with a finite state control
a read-only input tape
and (one or more) read-write working tapes.
pushdown automaton = Turing machine with 1 stack
... the working tape is/operates like a stack and nondeterminism.
... these are the context-free languages with an O(n3) membership test
queue machine = Turing machine with 1 queue (Post (1936), Brandenburg (1982))
... the working tapes are queues with two heads (head, tail)
34/25
Universality
© 2012 Prof. Dr. Franz J. Brandenburg
Universal Machines
• Turing machines (with one or more Turing tapes)
• machines with 2-pushdown stacks with O(1) slowdown per operation
with O(n) to simulate a queue
• machines with 1 queue with Q(n2) slowdown
The Power of a Queue
the Post Correspondence Problem
Given: a finite set of pairs of strings (xi, yi)
Problem: Is there a correspondence?
xi1 xi2 ...xiq = yi1 yi2 ...yiq
Solution: Use a queue to buffer the difference between
xi1 xi2 ... xij and yi1 yi2 ...yij
34/26
PCP
© 2012 Prof. Dr. Franz J. Brandenburg
Example 1:
1.
2.
Solution:
1
10
011
101
00
11
1 3 2 3 = 1 011 10 011 = 101 11 00 11
Example 2:
1:
011
01
2:
0
011
Solution (with 66 entries)
01
101
10
011
24344 21243 43443 44214 42134 11344 42121 11343 41214 42141 13411 31131 21411 3
34/27
Typical Languages
© 2012 Prof. Dr. Franz J. Brandenburg
Formal Language Theory
abstract families of languages, AFL (here trios)
generators of language families
initiators:
investigators:
N. Chomsky, G. Schützenberger
S.A. Greibach, S. Ginsburg, J. Berstel .. in the 60-80th
Which languages characterizes the context-free languages?
the Dyck set of well nested parenthesis ( ) and brackets [ ]
generated by the rules S  SS, S  ( S ), S  [ S ], S  e
[[()()] ()()]
generator: use homomorphisms g and h-1 over alphabets/strings S*
and intersection with regular sets R
such that CFL = g(h-1(Dyck)  R)
known as the Chomsky-Schützenberger Theorem
34/28
Queue Languages
© 2012 Prof. Dr. Franz J. Brandenburg
equality sets of two homomorphisms g, h
Eg(g,h) = {w  S* | g(w) = h(w)}
Example:
let S = (, ), [, ]
let g preserve ( and ] and g erases ) ]
let h preserve and rename (  ) and ]  [
Then Eg(g,h) = all FIFO nested strings of parenthesis and brackets
( [ [ ) ] ( ] [ ) ]
Theorem (Book, Brandenburg SICOMP80)
A language L is a queue language
iff L is recognized by a nondeterministic real-time queue machine
iff L = g(h-1(FIFO)  R), where FIFO are all FIFO nestings.
34/29
2-stacks
© 2012 Prof. Dr. Franz J. Brandenburg
Theorem (Ginsburg, Greibach 1968)
L is a 2-stack language
iff L is recognized in time n by a nondeterministic TM with two stacks
iff L = g(h-1(S2)  R)
where S2 is the shuffle / interleavings of all strings
from two disjoint Dyck sets
i.e. take well nested parenthesis over (), [ ]
and over < >, { }
and then consider all strings which are pairwise well-nested
e.g. < { [ ( > ] } )
34/30
Deque Languages
© 2012 Prof. Dr. Franz J. Brandenburg
Consider the equality set FIFO = Eq(g,h) which characterizes a queue,
say over the alphabet a, b and a, b
Define a (context-free) substitution by
(i) a  Dyck a Dyck and b  Dyck b Dyck
(ii) a  Dyck a Dyck and b  Dyck b Dyck
where underscoring means a new alphabet
Thus attach an arbitrary well nested string to the left and right.
Theorem
A language L is a deque language
iff L is recognized in time T(n)=n by a nondeterminstic TM with a deque
iff L = g(h-1( X  R),
The language X is the above context-free substitution on FIFO
such that the nesting is complete between any pair of a,b or underscored
34/31
Interpretation
© 2012 Prof. Dr. Franz J. Brandenburg
Queues – at Turingmachines are universal
since they can copy information
since they exactly describe the Post Correspondence Problem
Deques – in formal language theory
behave like queues (FIFO) with context-free substitition
.... if this were true, the name deque = double ended queue
should be
queue with free stack mode
34/32
Deque ?
© 2012 Prof. Dr. Franz J. Brandenburg
The behavior of a deque:
a stack at the tail
a stack at the head
queue mode: an item x moves from the head/tail to the tail/head
tail
3
1
.....
U
2
4
head
commuting pipes
34/33
Deque ?
© 2012 Prof. Dr. Franz J. Brandenburg
The behavior of a deque:
a stack at the tail
a stack at the head
queue mode: an item x moves from the head/tail to the tail/head
tail
3
1
.....
U
2
4
head
remove red to the right
if blue has been removed before
The right stack must be empty.
If empty
the right stack steels the red item
34/34
Deque
? Stack
Steeling
Double
© 2012 Prof. Dr. Franz J. Brandenburg
The behavior of a deque:
a stack at the tail
a stack at the head
queue mode: if one stack is empty, it steels the deepest item
34/35
Generalization
© 2012 Prof. Dr. Franz J. Brandenburg
A tripod:
take three stacks
which meet at their bottom and
if a stack is empty, it takes the next item from the bottom of
the next nonempty stack in circular order
34/36
Queue Graphs
© 2012 Prof. Dr. Franz J. Brandenburg
Corollary
The dual G* of a queue graph has an Eulerian path.
Proof: Sweep the drawing from left to right.
Theorem
Tests are NP-hard.
Is G a deque (queue, 2-stack) graph?
Theorem
G is a bipartite queue graph
if and only if
G is proper arch leveled planar (no arches)
34/37
Arches
© 2012 Prof. Dr. Franz J. Brandenburg
When do we get the arches?
... at the ends of the full queue phases
...where can we add edges?
... (3,7) or (4,6)
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
7
8
9
4
5
6
1
2
3
8
9
4
5
6
1
2
3
7
7
8
9
4
5
6
1
2
3
34/38
Stacks
© 2012 Prof. Dr. Franz J. Brandenburg
4-stacks (Yannakakis (1989))
Every planar graph is a 4-stack graph.
and there are planar graphs which need 4 stacks
3-stacks (Dujmovic, Wood (2005))
Every graph has a 3-stack embedding with O(g(n)) subdivisions
g(n) is minimum of stack-number and queue-number
2-stacks (Di Giacomo et al. (2005), Kaufmann,Wiese (1999) Dujmovic, Wood (2005))
Every planar graph has a 2-stack embedding with 1 subdivision
34/39
1 Stack + 1 Queue
© 2012 Prof. Dr. Franz J. Brandenburg
Drawing style
Example
K5 and K3,3 are 1-stack + 1-queue graphs
K6 and K4,4 are not
Conjecture of Heath and Rosenberg (1992)
Every planar graph is a 1 stack + 1 queue graph.
I say NO!
but my first counterexamples were refuted by A. Gleißner
using a SAT solver
34/40
Perspectives and Open Problems
© 2012 Prof. Dr. Franz J. Brandenburg
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a new drawing style for queues
Dujmovic, Wood (2005)
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planar graphs and 3-stacks
planar graphs and 1-stack+ 1-queue (Heath, Rosenberg conjecture)
stack and queue number: are they related
deque number and its relation to stack (queue) number
a sophisticated data structure to capture all planar graphs
34/41