Mathematics for Computer Science
Normal Person’s Graph
MIT 6.042J/18.062J
DAGs,
Partial Orders,
Scheduling
y = f(x)
y
x
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
L4-2.1
September 28, 2005
Computer Scientist’s Graph
a
c
set of vertices V
set of edges E, E ⊆ V × V
f
(Formally the same as
a binary relation on V.)
V={a,b,c,d} E = {(a,b), (a,c), (c,b)}
e
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
L4-2.3
September 28, 2005
Graphical Properties of Relations
Reflexive
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
d
b
c
L4-2.4
September 28, 2005
A
R
Antisymmetric
NO
September 28, 2005
a
Paths
Symmetric
Transitive
L4-2.2
September 28, 2005
Relations and Graphs
b
d
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
L4-2.5
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.6
1
R2 from A to A
A
Path Relations
R= paths
a1Rka2 iff
of length 1
R2 = paths
of length 2
R≤ 2
= paths
of length ≤ 2
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
L4-2.7
September 28, 2005
Reflexive Transitive Closure
a1 and a2 are connected by path of
length exactly k in the graph of R
a1R≤ka2 iff
a1 and a2 are connected by path of
length at most k in the graph of R
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
Prerequisite Relation on Classes
class c is a prerequisite
for class d
a1R*a2 iff
a1 and a2 are connected by path
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
c→d
L4-2.9
September 28, 2005
Some Course 6 Prerequisites
•
•
•
•
•
•
18.01 → 6.042
18.01 → 18.02
18.01 → 18.03
8.01 → 8.02
6.001 → 6.034
6.042 → 6.046
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.10
Indirect Prerequisites
• 18.03, 8.02 → 6.002
• 6.001, 6.002 → 6.004
• 6.001, 6.002 → 6.003
•
6.004 → 6.033
•
6.033 → 6.857
•
6.046 → 6.840
September 28, 2005
L4-2.8
September 28, 2005
L4-2.11
18.01 → 6.042 → 6.046 → 6.840
18.01 is indirect prereq. of 6.840
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.12
2
Classes with no prereqs
18.01
8.01
Minimal elements
6.001
d is minimal for →
"Freshman classes"
there is no c s.t. c→d
d is a Freshman class iff
nothing → d
L4-2.13
September 28, 2005
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
Prerequisite graph
18.01
18.02
6.042
8.01
18.03
6.002
6.840
6.004
September 28, 2005
L4-2.14
minimal not minimum
6.001
8.02
6.046
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
6.034
minimum means "smallest"
-- a prereq. for every class
no minimum in this example
6.003
6.033
6.857
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.15
Team Problem
September 28, 2005
September 28, 2005
L4-2.16
Prerequisite graph
What if there is a cycle in this graph?
-- a path from class c to class d and
back to class c?
No one can graduate!
Comm. on Curricula & Registrar are
supposed to prevent cycles.
Problem 1,2
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
L4-2.17
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.18
3
Directed Acyclic Graph (DAG)
18.01
18.02
6.042
8.01
18.03
6.001
8.02
6.046
6.002
6.840
6.004
6.034
6.003
DAG's = Partial Orders
Theorem:
• The path relation of a DAG is a partial order.
• The graph of a partial order is a DAG.
6.033
6.857
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.19
Constructing the DAG
•
•
•
•
•
•
18.01 → 6.042
18.01 → 18.02
18.01 → 18.03
8.01 → 8.02
6.001 → 6.034
6.042 → 6.046
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
L4-2.20
September 28, 2005
Directed Acyclic Graph (DAG)
• 18.03, 8.02 → 6.002
• 6.001, 6.002 → 6.004
• 6.001, 6.002 → 6.003
•
6.004 → 6.033
•
6.033 → 6.857
•
6.046 → 6.840
18.01
8.01
6.001
Identify Minimal Elements
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.21
Constructing the DAG
•
•
•
•
•
•
18.01 → 6.042
18.01 → 18.02
18.01 → 18.03
8.01 → 8.02
6.001 → 6.034
6.042 → 6.046
• 18.03, 8.02 → 6.002
• 6.001, 6.002 → 6.004
• 6.001, 6.002 → 6.003
•
6.004 → 6.033
•
6.033 → 6.857
•
6.046 → 6.840
September 28, 2005
September 28, 2005
L4-2.22
Constructing the DAG
Remove minimal elements
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
•
6.042
•
18.02
•
18.03
•
8.02
•
6.034
• 6.042 → 6.046
• 18.03, 8.02 → 6.002
•
6.002 → 6.004
•
6.002 → 6.003
•
6.004 → 6.033
•
6.033 → 6.857
•
6.046 → 6.840
Remove minimal elements
L4-2.23
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.24
4
Constructing the DAG
•
6.042
•
18.02
•
18.03
•
8.02
•
6.034
• 6.042 → 6.046
Directed Acyclic Graph (DAG)
• 18.03, 8.02 → 6.002
•
6.002 → 6.004
•
6.002 → 6.003
•
6.004 → 6.033
•
6.033 → 6.857
•
6.046 → 6.840
18.01
18.02
6.042
8.01
18.03
6.001
8.02
6.034
continue in this way…
Identify new minimal elements
L4-2.25
September 28, 2005
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
Prerequisite graph
Directed Acyclic Graph (DAG)
18.01
18.02
6.042
6.046
8.01
18.03
8.02
L4-2.26
September 28, 2005
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
6.001
18.01
6.034
18.02
6.002
6.042
8.01
18.03
6.001
8.02
6.046
6.002
6.840
6.004
6.034
6.003
6.033
6.857
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.27
Topological sort
L4-2.28
September 28, 2005
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
How many terms to graduate?
18.01
• Is there a way of graduating?
(in any number of semesters?)
18.02
• Yes - take a minimal remaining course each semester
6.042
8.01
18.03
6.001
8.02
6.046
6.002
6.840
6.004
6.033
6.034
6.003
maximum chain
6.857
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.29
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.30
5
Parallel Task Scheduling
and Sufficient…
• 6 terms are necessary to complete the curriculum
18.01
18.02
• and sufficient (if you can take unlimited courses per term...)
6.042
8.01
18.03
6.001
8.02
6.046
6.002
6.840
6.004
6.034
6.003
6.033
6.857
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.31
Antichains
Parallel Task Scheduling
Set of courses that can be taken in any order: Theorem: If the longest chain has size t, then the elements can be partitioned into
Any two courses in set are incomparable
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
t successive antichains,
with no element in any block
preceding anything in a preceding
block
L4-2.34
and Sufficient…
18.01
Take
Bi = {a | largest chain ending
in a is of size i}
18.02
If there is a y in Bi such that xÆ y and
x not in B1….Bi-1 then there is a chain
of size >i ending in y
September 28, 2005
L4-2.35
September 28, 2005
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
Why sufficient?
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
L4-2.32
September 28, 2005
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
L4-2.36
6.042
8.01
18.03
6.001
8.02
6.046
6.002
6.840
6.004
6.034
6.003
6.033
6.857
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.37
6
Minimum “Parallel” Time
Minimum “Parallel” Time
parallel time = max chain size.
required # processors
≤ max antichain size.
L4-2.38
September 28, 2005
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
but 5-course term not necessary.
Possible that
min-time #processors
< max antichain size
Min-time Schedule
18.01
18.02
6.042
Another Min-time Schedule
8.01
18.03
6.001
8.02
6.046
6.002
6.840
6.004
18.01
6.034
6.042
18.02
6.003
18.03
6.001
8.02
6.002
6.840
6.004
6.034
6.003
6.033
6.857
6.857
September 28, 2005
L4-2.40
3 Subjects per Term Possible
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
8.01
6.046
6.033
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
L4-2.39
September 28, 2005
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
18.01
8.01
6.001
18.03
8.02
6.034
6.042
6.002
6.003
6.046
6.004
6.840
6.033
18.02
6.857
September 28, 2005
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.41
A 3-course term is necessary
• 15 subjects
• max chain size = 6
• size of some block must be
≥ ⎡15/6⎤ = 3.
∴ to finish in 6 terms, must
take ≥3 classes some term
L4-2.43
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.44
7
Dilworth’s Lemma
Height/Birthday Partial Order
A partial order on n items has
• a chain of size ≥ t, or
⎡n⎤
• or an antichain of size ≥ ⎢ ⎥
⎢t⎥
for all 1 ≤ t ≤ n.
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
Two students are related to each other iff
one is shorter and younger than the other
(s1, a1) Y (s2, a2) iff
(s1 ≤ s2) and (a1 ≤ a2)
L4-2.45
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.46
Dilworth Demo
Height/Birthday Partial Order
Chain of students:
get older as they get taller.
AntiChain of students:
get younger as they get taller.
Older
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.47
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.48
Team Problem
Problem 4
Copyright © Albert R. Meyer and Ronitt Rubinfeld 2005.
September 28, 2005
L4-2.49
8