Parallel Computation
and Algorithms
by
Raymond Greenlaw
Armstrong Atlantic State University
Joint work with:
Pilar de la Torre
Jim Hoover
Department of Computer Science
University of New Hampshire
Computer Science Department
University of Alberta
Magnus Halldorsson
Rossella Petreschi
Decode Genetics
Department of Computer Science
University of Rome La Sapienza
Alex Schaffer
National Institute of Health
Larry Ruzzo
Department of Computer Science
University of Washington
Teresa Przytycka
Department of Biophysics
John Hopkins University Medical School
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
1
Parallel Computation
and Algorithms
by
Raymond Greenlaw
Armstrong Atlantic State University
Part I (a): An Introduction to Parallel Computation
Part I (b): P-Complete Problems
Part II (a): Rankings
Part II (b): Prüfer Codes
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
2
Part I (a): An Introduction to
Parallel Computation
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
3
Part I: Outline
•
•
•
•
•
•
•
Introduction
Parallel Models of Computation
Basic Complexity
Evidence that NC  Ρ
Ρ-Complete Problems
Comparator Circuit Class
Open Problems
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
4
Part I: Outline
•
•
•
•
•
•
•
Introduction
Parallel Models of Computation
Basic Complexity
Evidence that NC  Ρ
Ρ-Complete Problems
Comparator Circuit Class
Open Problems
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
5
Introduction
• Sequential computation:
Feasible ~ n O(1) time
(polynomial time).
• Parallel computation:
Feasible ~ n O(1) time and
~ n O(1) processors.
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
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Introduction
• Goal of parallel computation:
Develop extremely fast ((log n)O(1) )
time algorithms using a reasonable
number of processors.
• Speedup equation:
(best sequential time)
 (parallel time).
(number of processors)
Allow a polynomial number of processors.
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
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Introduction
Roughly,
• A problem is feasible if it can be solved by a
parallel algorithm with worst case time and
processor complexity n O(1).
• A problem is feasible highly parallel if it can be
solved by an algorithm with worst case time
complexity (log n) O(1) and processor complexity
n O(1).
• A problem is inherently sequential if it is
feasible but has no feasible highly parallel
algorithm for its solution.
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
8
Part I: Outline
•
•
•
•
•
•
•
Introduction
Parallel Models of Computation
Basic Complexity
Evidence that NC  Ρ
Ρ-Complete Problems
Comparator Circuit Class
Open Problems
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
9
Parallel Models of Computation
• Parallel Random Access Machine Model
• Boolean Circuit Model
• Circuits and PRAMs
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
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Parallel Random Access Machine
RAM Processors
P0
P1
P2
C0
C1
C2
Global Memory Cells
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
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Boolean Circuit Model
1 0
1 1
AND
OR
OR
AND
0
AND
OR
NOT
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
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Circuits and PRAMS
Theorem:
A function f from {0,1}* to {0,1}* can be computed
by a logarithmic space uniform Boolean circuit family
{n } with depth (n) є (logn)O(1) and size (n ) є n O(1)
if and only if
f can be computed by a CREW-PRAM M on inputs of
length n in time t(n) є (logn)O(1) using p(n) є n O(1).
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
13
Part I: Outline
•
•
•
•
•
•
•
Introduction
Parallel Models of Computation
Basic Complexity
Evidence that NC  Ρ
Ρ-Complete Problems
Comparator Circuit Class
Open Problems
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
14
Basic Complexity
•
•
•
•
Decision, Function, and Search Problems
Complexity Classes
Reducibility
Completeness
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
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Decision, Function, and Search Problems
4
3
Spanning Tree-D
4
6
3
4
6
3
4
Given: An undirected graph G = (V,E) with weights from N labeling edges in E and a
natural number k.
Problem: Is there a spanning tree of G with cost less than or equal to k ?
Spanning Tree-F
Given: Same (no k ).
Problem: Compute the weight of a minimum cost spanning tree.
Spanning Tree-S
Given: Same.
Problem: Find a minimum cost spanning tree.
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
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Complexity Classes
Definitions:
P is the set of all languages L that are decidable in sequential
time n O(1).
NC is the set of all languages L that are decidable in parallel time
(logn)O(1) and processors n O(1).
FP is the set of all functions from {0,1}* to {0,1}* that are
computable in sequential time n O(1).
FNC is the set of all functions from {0,1}* to {0,1}* that are
computable in parallel time (logn)O(1) and processors n O(1).
NC k, k  1, is the set of all languages L such that L is recognized by
a uniform Boolean circuit family {n } with size(n ) = n O(1) and
depth (n ) = O((logn)k ).
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
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Many-One Reducibility
Definitions:
A language L is many-one reducible to a language L’, written
L mP L’ , if there is a function f such that x є L if and only if
f(x) є L’.
L is P many-one reducible to L’, written L mP L’ , if the function f is
in FP.
k
For k  1, L is NC k many-one reducible to L’, written L mNC L’, if
the function f is in FNC k.
L is NC many-one reducible to L’, written L mNC L’, if the function f
is in FNC.
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
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Many-One Reducibility
Lemma:
The reducibilities
m, mP,
k
NC
m (k
 1), and mNC are transitive.
k
NC L’ then L є P.
– If L’ є P and L mP L’, L mNC L’ (k > 1), or L m
k
NC

L’
– If L’ є NC (k > 1), and L 
m
k
then L є NC k.
– If L’ є NC and L mNC L’ then L є NC .
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
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Turing Reducibility
Definition:
L is NC Turing reducible to L’, written L TNC L’, if and only if the
L’-oracle PRAM on inputs of length n uses time (logn)O(1) and
processors n O(1).
Lemma:
The reducibility TNC is transitive.
– If L mNC L’ then L TNC L’.
– If L 
TNC L’ and L’ є NC then L є NC.
– If L TNC L’ and L’ є P then L є P.
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
20
Completeness
Definitions:
A language L is P-hard under NC reducibility if L’ TNC L for
every L’ є P.
A language L is P-complete under NC reducibility if L є P and
L is P-hard.
Theorem:
If any P-complete problem is in NC then NC equals P.
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
21
Part I: Outline
•
•
•
•
•
•
•
Introduction
Parallel Models of Computation
Basic Complexity
Evidence that NC  Ρ
Ρ-Complete Problems
Comparator Circuit Class
Open Problems
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
22
Evidence That NC  P
• General Simulations Are Not Fast
• Fast Simulations Are Not General
• Natural Approaches Provably Fail
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
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Evidence That NC  P
Gaps in Simulations
Model
DTM
DTM
ATM
PDA
UC
PRAM
Resource
1
Resource
2
Time = n O(1)
Time = n O(1)
2Space = n O(1)
2Space = n O(1)
Size = n O(1)
Procs = n O(1)
Space
Reversals
log(Treesize)
log(Time)
Depth
Time
Max R2
C  NC
log n
(log n)O(1)
(log n)O(1)
(log n)O(1)
(log n)O(1)
(log n)O(1)
Min R2
CP
n O(1)
n O(1)
n O(1)
n O(1)
n O(1)
n O(1)
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
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Part I (b): Ρ-Complete Problem
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
25
Part I: Outline
•
•
•
•
•
•
•
Introduction
Parallel Models of Computation
Basic Complexity
Evidence that NC  Ρ
Ρ-Complete Problems
Comparator Circuit Class
Open Problems
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
26
P-Complete Problems
There are approximately 175 P-complete
problems (500 with variations).
Categories:
–
–
–
–
Circuit complexity
Graph theory
Searching graphs
Combinatorial
optimization and flow
– Local optimality
– Logic
–
–
–
–
–
–
Formal languages
Algebra
Geometry
Real analysis
Games
Miscellaneous
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
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Circuit Value Problem
Given:
An encoding  of a Boolean circuit , inputs x1,…,xn, and
a designated output y.
Problem:
Is output y of  TRUE on input x1,…,xn?
Theorem: [Ladner 75]
The Circuit Value Problem is P-complete under
reductions.
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
1
NC
m
28
P-Complete Variations of CVP
– Topologically Ordered [Folklore]
– Monotone [Goldschlager 77]
– Alternating Monotone Fanin 2, Fanout 2
[Folklore]
– NAND [Folklore]
– Topologically Ordered NOR [Folklore]
– Synchronous Alternating Monotone Fanout 2
CVP [Greenlaw, Hoover, and Ruzzo 87]
– Planar [Goldschlager 77]
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
29
NAND Circuit Value Problem
Given:
An encoding  of a Boolean circuit  that consists solely of NAND
gates, inputs x1,…,xn, and a designated output y.
Problem:
Is output y of  TRUE on input x1,…,xn?
Theorem:
The NAND Circuit Value Problem is P-complete.
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
30
NAND Circuit Value Problem
Proof:
Reduce AM2CVP to NAND CVP. Complement all inputs. Relabel all gates as
NAND.
1
0
0
OR
1
OR
1
1
0
0
1
1
NAND
1
NAND
1
AND
0
1
NAND
1
0
OR
1
NAND
1
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
31
Graph Theory
– Lexicographically First Maximal Independent Set
[Cook 85]
– Lexicographically First ( + 1)-Vertex Coloring
[Luby 84]
– High Degree Subgraph
[Anderson and Mayr 84]
– Nearest Neighbor Traveling Salesman Heuristic
[Kindervater, Lenstra, and Shmoys 89]
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
32
Lexicographically First Maximal Independent Set
Theorem: [Cook 85]
LFMIS is P-complete.
Proof:
Reduce TopNOR CVP to LFMIS. Add new vertex 0. Connect to all false
inputs.
1
1
1
2
0
0
3
0
4
NOR
5
NOR
6
X
X
X
1
2
0
3
4
5
7
0
6
1
NOR
8
X
7
8
0
NOR
9
0
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
9
33
Searching Graphs
– Lexicographically First Depth-First Search Ordering
[Reif 85]
– Stack Breadth-First Search
[Greenlaw 92]
– Breadth-Depth Search
[Greenlaw 93]
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
34
Context-Free Grammar Empty
Given: A context-free grammar G = (N,T,P,S).
Problem: Is L(G) empty?
Theorem: [Jones and Laaser 76], [Goldschlager 81],
[Tompa 91]
CFGempty is P-complete.
Proof: Reduce Monotone CVP to CFGempty. Given 
construct G = (N,T,P,S) with N, T, S, and P as follows:
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
35
Context-Free Grammar Empty
N = {i | vi is a vertex in }
T = {a }
S = n, where vn is the output of .
P as follows:
1. For input vi, i  a if value of vi is 1,
2. i  jk if vi  vj Λ vk, and
3. i  j | k if vi  vj V vk .
* , where  є {a }+.
Then the value of vi is 1 if and only if i 
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
36
CFGempty Example
x1 = 0, x2 = 0, x3 = 1, and x4 = 1.
1
x1
x3 3
2 x2
5 OR
x4
4
AND 6
7 AND
G = (N, T, S, P ), where
N = {1, 2, 3, 4, 5, 6, 7 }
T = {a }
S=7
P = {3  a, 4  a, 5  1 | 2, 6  34, 7  56}
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
37
Part I: Outline
•
•
•
•
•
•
•
Introduction
Parallel Models of Computation
Basic Complexity
Evidence that NC  Ρ
Ρ-Complete Problems
Comparator Circuit Class
Open Problems
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
38
Comparator Circuit Class
Definition: [Cook 82]
Comparator Circuit Value Problem (CCVP)
Given: An encoding  of a circuit  composed of
comparator gates, plus inputs x1,…,xn, and a
designated output y.
Problem: Is output y of  TRUE on input x1,…,xn?
Definition:
CC is the class of languages NC reducible to CCVP.
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
39
CC-Complete Problems
– Lexicographically First Maximal Matching
[Cook 82]
– Telephone Connection
[Ramachandran and Wang 91]
– Stable Marriage
[Mayr and Subramanian 92]
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
40
Part I: Outline
•
•
•
•
•
•
•
Introduction
Parallel Models of Computation
Basic Complexity
Evidence that NC  Ρ
Ρ-Complete Problems
Comparator Circuit Class
Open Problems
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
41
Open Problems
Find an NC algorithm or classify
as P-complete.
– Edge Ranking
– Edge-Weighted Matching
– Restricted Lexicographically First Maximal
Independent Set
– Integer Greatest Common Divisor
– Modular Powering
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
42
Part II (a): Rankings
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
43
Part II: Outline
•
•
•
•
•
•
Introduction
Parallel Preliminaries
Node Ranking
Edge Ranking
Prüfer Codes
Open Problems
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
44
Part II: Outline
•
•
•
•
•
•
Introduction
Parallel Preliminaries
Node Ranking
Edge Ranking
Prüfer Codes
Open Problems
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
45
Trees
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
46
Part II: Outline
•
•
•
•
•
•
Introduction
Parallel Preliminaries
Node Ranking
Edge Ranking
Prüfer Codes
Open Problems
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
47
Parallel Random Access Machine
RAM Processors
P0
P1
P2
C0
C1
C2
Global Memory Cells
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
48
Preliminary Parallel Algorithms
•
•
•
•
•
Brent’s Scheduling Principle
Parallel Prefix Computation
Euler Tour
List Ranking
Parallel Tree Contraction
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
49
Brent’s Scheduling Principle
(Brent 1974)
If processor allocation is not a problem
then a t(n) time parallel algorithm that
requires w(n) computational operations can
be simulated using w(n)/p(n) + t(n) time and
p(n) processors.
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
50
Example of Brent’s
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
51
Parallel Prefix Computation
(Ladner & Fisher 1980)
The Parallel Prefix Problem can be
solved in O(log n) time using n/log n
processors on an EREW-PRAM.
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
52
Example of Parallel Prefix
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
53
Euler Tour
(Tarjan & Vishkin 1985)
An Euler tour of an n-node tree can be
computed in O(log n) time using n/log n
processors on an EREW-PRAM.
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
54
Example of Euler Tour
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
55
List Ranking
(Anderson & Miller 1988)
Given a list with n nodes, the List
Ranking Problem can be solved in O(log n)
time using n/log n processors on an
EREW-PRAM.
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
56
Example of List Ranking
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
57
Parallel Tree Contraction
(He 1986; Miller & Teng 1987; Abrahamson,
Dadoun, Kirkpatrick & Przytycka 1989)
Let T be an n-leaf regular binary expression
tree. Then all of the algebraic expressions
associated with the internal nodes (one per
node) of T can be evaluated in O(log n) time
using n/log n processors on an EREW-PRAM.
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
58
Example of Tree Contraction
Scrunched tree
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
59
Part II: Outline
•
•
•
•
•
•
Introduction
Parallel Preliminaries
Node Ranking
Edge Ranking
Prüfer Codes
Open Problems
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
60
Node Ranking
• A node ranking is a labeling of the nodes of
a tree with natural numbers such that if
nodes u and v have the same label then
there exists another node with a greater
label on the path between them.
• An optimal node ranking is a node ranking in
which the largest label assigned to any
node is as small as possible among all node
rankings.
• The node ranking problem is to compute an
optimal node ranking.
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
61
Node Ranking Example
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
62
Node Ranking: Sequential Results
• Node Ranking Problem
• O(n log n) time
(Iyer, Ratliff & Vijayan 1988)
• O(n) time
(Schaffer 1989)
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
63
Example of Sequential Node Ranking
Algorithm
• Critical list at * is {3,4}
• Label 3 at ** covers values 1 and 2
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
64
Example of Sequential Node Ranking
Algorithm
• Critical list at * is {3,4}
• Label 3 at ** covers values 1 and 2
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
65
Node Ranking: Parallel Results
• Approximate Optimal Tree Ranking
• O(log2 n) time, n processors EREW-PRAM
(Liang, Dhall & Lakshmivarahan 1990)
• Optimal Tree Ranking
• O(log n) time, n2/log n procs CREW-PRAM
(de la Torre & Greenlaw 1991)
(Przytycka 1991)
• Super Critical Tree Numbering
O(log n) time, n2/log n procs CREW-PRAM
(de la Torre, Greenlaw & Przytycka 1992)
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
66
Part II: Outline
•
•
•
•
•
•
Introduction
Parallel Preliminaries
Node Ranking
Edge Ranking
Prüfer Codes
Open Problems
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
67
Edge Ranking
• An edge ranking is a labeling of the edges
of a tree with natural numbers such that if
edges u and v have the same label then
there exists another edge with a greater
label on the path between them.
• An optimal edge ranking is an edge ranking
in which the largest label assigned to any
edge is as small as possible among all edge
rankings.
• The edge ranking problem is to compute an
optimal edge ranking.
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
68
Edge Ranking Example
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
69
Edge Ranking: Sequential Results
• Approximate Edge Ranking
• O(n log n) time
(Iyer, Ratliff & Vijayan 1991)
• Optimal Edge Ranking
• O(n3 log n) time
(de la Torre, Greenlaw & Schaffer 1995)
• O(n2 log n) time
(Zhou, Kashem & Nishizeki 1996)
• O(n log n) time, O(n) time
(Lam & Ling 1996, 1997)
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
70
From Local to Global Optimality
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
71
Greedy Cover Labeling
• Lc > Lc-1 > … > L1
is a greedy cover labeling if and only if
cover(Li, crit(vj)) lex cover(Li, crit(vi))
for all i and for all j < i.
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
72
Example of Edge
Ranking Algorithm
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
73
Example of Edge
Ranking Algorithm
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
74
Example of Edge
Ranking Algorithm
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
75
Edge Ranking: Parallel Results
• Approximate Edge Ranking
• O(log2 n) time, n2/log n processors
CREW-PRAM
(de la Torre, Greenlaw & Schaffer 1995)
• Optimal Edge Ranking of Constant Degree
Trees
NC
(de la Torre, Greenlaw & Schaffer 1995)
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
76
Motivation for Studying Rankings
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
77
Motivation for Studying Rankings
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
78
Part II (b): Prüfer Codes
Parallel Computation and Algorithms, Part I: An Introduction to Parallel Computation and Ρ-Completeness Theory
Raymond Greenlaw
79
Part II: Outline
•
•
•
•
•
•
Introduction
Parallel Preliminaries
Node Ranking
Edge Ranking
Prüfer Codes
Open Problems
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
80
Definition of Prüfer Code
A Prüfer code of a labeled free tree with n
nodes is a sequence of length n – 2
constructed by the following sequential
process:
for i ranging from 1 to n – 2 do
insert the label of the neighbor of the
smallest remaining leaf into the i-th
position of the sequence;
delete the leaf;
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
81
Example of a Prüfer Code
(9,6,5,6,1,6,1,1)
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
82
Related Work
• Prüfer code to tree
EREW-PRAM O(log n) time, n processors
proposed as an open problem the reverse
direction
(Wang, Chen & Liu 1997)
• Tree to Prüfer code
EREW-PRAM O(log n) time, n processors
(Chen & Wang 1998, Greenlaw & Petreschi
1998)
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
83
Related Work Continued
• Chain to Prüfer code
EREW-PRAM O(log n) time, n/log n
(Greenlaw & Petreschi 1998)
• Used Prüfer codes for random
tree generation
(Kumar, Deo & Kumar 1998)
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
84
Can Sequential Algorithm
Prüfer Code Be Parallelized?
• Initial intuition suggests no.
• What type of running time do we want?
• How do we proceed?
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
85
Parallel Algorithm Prüfer Chain
The Prüfer code of this chain is
(2,8,3,7,5,6)
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
86
Parallel Algorithm Prüfer Chain
Step 1:
/* Compute the position of each node in the
chain. */
1. Use parallel list ranking to construct the array
Position such that Position[i] = v, 1  i  n,
where node v has a ranking of i in T
4
1
3
2
7
3
5
4
6
5
8
6
2
7
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
1
8
87
Parallel Algorithm Prüfer Chain
Step 2:
/* Compute the maximum nodes encountered thus far in left-to-right
and right-to-left traversals over the chain. */
2. Use parallel prefix computation to construct the arrays LRMax and
RLMax
LRMax[i] = max{Position[j] | 1  j  i}
RLMax[i] = max{Position[j] | n – i + 1  j  n}
0
0
4
1
4
2
7
8
7
8
7
8
8
8
8
8
8
8
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
9
9
88
Parallel Algorithm Prüfer Chain
Step 3:
/* Compute when a node becomes a maximum (if it does) for both leftto-right and right-to-left traversals. */
3. For 1  i  n in parallel do
if LRMax[i-1]  LRMax[i] then
LRStart[LRMax[i]] = i;
if RLMax[i-1]  RLMax[i] then
RLStart[RLMax[i]] = i;
0
1
0
2
0
0
1
0
0
0
0
0
3
0
6
3
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
89
Parallel Algorithm Prüfer Chain
Step 4:
/* Compute when a node is no longer a maximum (if it was) for both
left-to-right and right-to-left traversals. */
4. For 1  i  n in parallel do
if LRMax[i]  LRMax[i+1] then
LREnd[LRMax[i]] = i;
if RLMax[i]  RLMax[i+1] then
RLEnd[RLMax[i]] = i;
0
1
0
2
0
0
2
0
0
0
0
0
5
0
8
8
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
90
Parallel Algorithm Prüfer Chain
Step 5:
/* Compute how many positions a node was maximum for. */
5. For 1  i  n in parallel do
if LRStart[i]  0 then
LRSpan[i] = LREnd[i] - LRStart[i] + 1;
if RLStart[i]  0 then
RLSpan[i] = RLEnd[i] - RLStart[i] + 1;
0
0
0
1
0
1
0
0
2
0
0
0
0
0
3
0
3
6
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
91
Parallel Algorithm Prüfer Chain
Step 6:
/* Compute how many nodes a given node is greater than from the left.
Similiarly for the right. */
6. Use parallel prefix to construct the array
LeftClear and RightClear, where 1  i  n
LeftClear[i] = LRSpan[0] + … + LRSpan[i-1];
RightClear[i] = RLSpan[0] + … + RLSpan[i-1];
0
0
0
0
2
2
2
5
0
1
2
2
2
2
2
2
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
92
Parallel Algorithm Prüfer Chain
Step 7:
/* Removal[i] denotes when the node in Position[i]
is removed. */
7. For 1  i  n in parallel do
if RLMax[n-i] > LRMax[i-1]
then k = LRMax[i];
Removal[i] = i + RightClear[k];
else k = RLMax[n-i+1];
Removal[i] = (n – i) + 1 + LeftClear[k];
3
4
5
6
7
8
2
1
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
93
Theorem on Chains
(Greenlaw & Petreschi 1998)
The Prüfer code of an n-node labeled chain
can be computed in O(log n) time using
n/log n processors on an
EREW-PRAM.
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
94
Parallel Algorithm Prüfer Tree
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
95
Parallel Algorithm Prüfer Tree
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
96
Theorem on Trees
(Greenlaw, Halldorsson & Petreschi 1999)
The Prüfer code of an n-node labeled
free tree can be computed in O(log n) time
using n/log n processors on an
EREW-PRAM.
In the other direction we show
Given the Prüfer code of an n-node labeled
chain we can output the corresponding
chain in O(log n) time using n/log n
processors on an EREW-PRAM.
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
97
Prüfer Code Summary
Tree to P-code
P-code to Tree
sequential
O(n) folklore
O(n) (K 78)
parallel
O(log n) time, n processors
EREW-PRAM (CW 98, GP 98)
O(log n) time, n processors
EREW-PRAM (WCL 97)
For a chain
O(log n) time,
n/log n processors
EREW-PRAM (GHP 99)
For a chain
O(log n) time, n/log n
processors
EREW-PRAM (GHP 99)
For a tree
O(log n) time, n/log n processors
EREW-PRAM (GHP 99)
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
98
Part II: Outline
•
•
•
•
•
•
Introduction
Parallel Preliminaries
Node Ranking
Edge Ranking
Prüfer Codes
Open Problems
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
99
Open Problems
• Can an optimal parallel algorithm be
developed for building a tree given its
Prüfer code?
• Can the Element Distinctness Problem be
solved on a CREW-PRAM in O(log n) time
using n/log n processors?
• Is the edge ranking problem on trees
P-complete?
• Can all n-node trees be labeled with the
values 1 through n such that neighors
receive pairwise relatively prime labels?
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
100
References
See
www.cs.armstrong.edu/greenlaw
Parallel Computation and Algorithms, Part II:
I: An
Sequential
Introduction
and Parallel
to Parallel
Algorithms
Computation
for Tree
and Ρ-Completeness
Problems
Theory
Raymond Greenlaw
101