slides33
L3-3.2
Copyright ? Radhika Nagpal, 2002.
Partial Order
Two students are related to each other
if one is shorter and younger than
the other
(s1, a1) R(s2, a2)iff(s1≤s2) ∧(a1≤a2)
–Reflexive
–Antisymmetric
–Transitive
L3-3.3
Copyright ? Radhika Nagpal, 2002.
Partial Order
Transitive:
Prove (s
1
, a
1
) R(s
2
, a
2
)and (s
2
, a
2
) R(s
3
, a
3
)
implies (s
1
, a
1
) R(s
3
, a
3
) :
(s
1
≤s
2
≤s
3
)and (a
1
≤a
2
≤a
3
)
Therefore (s
1
≤s
3
)and (a
1
≤a
3
)
If we line up the students in a chain in order of
If we line up the students in an antichain
order of
Older
L3-3.7Copyright ? Radhika Nagpal, 2002.
Normal Person’s Graph x
y
y = f(x)
L3-3.8
Copyright ? Radhika Nagpal, 2002. Computer Scientist’s Graph
a
f
e
c
d
b
edge (e ,a )
L3-3.9
Copyright ? Radhika Nagpal, 2002.
Definition
A Graph is a set of vertices V and a set E of edges, such that
So E is simply a binary relation
on the set V.
V
V E ×?L3-3.10
Copyright ? Radhika Nagpal, 2002.
Relations and Graphs
A = {a,b,c,d}R = {(a,b) (a,c) (c,b)}
a
c
b
d
Properties of Relations
Reflexive
Transitive
Symmetric
Antisymmetric
NO
Types of Relations
Total Order (<)
1
3
2
4
5
6
Equivalence (mod 3)
16
4
3
5
2Partial Order
(a |b)
1
23
5
46
L3-3.13
Copyright ? Radhika Nagpal, 2002.
A Relation on Buildings
a R
b ::= Building a is “physically
adjacent” to Building b
A B C
A R B,
B R A not A R
C A R A
L3-3.14
Copyright ? Radhika Nagpal, 2002.
MIT Building Connections
R
413
10
12
26
8
L3-3.15
Copyright ? Radhika Nagpal, 2002.
Class Problem 1
R
4
13
10
12
26
8
L3-3.16
Copyright ? Radhika Nagpal, 2002.
MIT Building Connections
413
1012
413
1012
413
1012
R 2= {(a,b) | a,b are
connected by path of length 2}
R 3= {(a,b) | a,b are connected by a path of length 3}
Composition and Path Lengths R k is the set of all pairs (a,b ) such
that a and b are connected by a path of length exactly k.
Composition and Path Lengths
If R is not reflexive
If R is reflexive and transitive
R
R 2
R
R 2
L3-3.19
Copyright ? Radhika Nagpal, 2002.
The Same Questions
Question 1:Can you drive from one state to another with at most 5state-boundary crossings ?
C ×C =R 0∪R 1∪???∪R 5?
Question 2:Can you fly on KLM from Boston to
Paramaribo with at most 3stopovers ?
(BOS, PAR ) ∈R 0∪R 1 ∪???∪R 4 ?
Quiz:Paramaribo is the capital of …?
L3-3.20
Copyright ? Radhika Nagpal, 2002.
Connectivity
Is it possible at all to get from bldg a to b ?Is there a path of some length k from a to b ?
(a, b )∈
? =R n-1
Why n-1?
R is reflexive and …
0k k R ∞
=∪L3-3.21Copyright ? Radhika Nagpal, 2002. Connectivity
…the greatest distance between any pair of nodes is n-1:
If longer than n -1
can remove cycle
L3-3.22
Copyright ? Radhika Nagpal, 2002.
Reflexive Transitive Closure
R *::=
k k R ∞
=∪= R 0 ∪R 1∪R 2 ∪???∪R k ∪???
d
I aka the Connectivity Relation
Boolean Matrix Representation
A = {a,b,c,d}
R = {(a,b) (a,c) (c,b)}
a
c
b
d
a b c d
a 0 11 0
b 0 0 0 0
c 0 1 0 0
d 0 0 0 0
Boolean Matrix Operations
e.g R = A×A –R
(all pairs not in R)
a b c d a 0 1 10b 0 0 0 0c 0 1 0 0d 0 0 0 0
a b c d
a 1 0 01
b 1 1 1 1
c 1 0 1 1
d 1 1 1 1
L3-3.25Copyright ? Radhika Nagpal, 2002.
Composition using Matrices
1a 0 1 1 00 0 0 01 0 0 00 0 0 1
R
2a 3a 4
a 1
b 2b 3b 4b 1
c 2c 3c 4c 1b 2b 3b 4
b 00 0 010 0 000 1 000 0 1
S
=
T ::= R )S
10 1 00 0 0 00 0 0 00 0 0 1
1a 2a 3a 1c 2c 3c 4a 4c T (a 1,c 1) = [R (a 1,b 1)∧S (b 1,c 1)] ∨[R (a 1,b 2)∧S (b 2,c 1)] ∨
[R (a 1,b 3)∧S(b 3,c 1)] ∨[R (a 1,b 4)∧S (b 4,c 1)]
L3-3.26
Copyright ? Radhika Nagpal, 2002. Class Problems
L3-3.27Copyright ? Radhika Nagpal, 2002.
MIT Building Connections
812101326
4
4
13
10
12
26
8
812101326
4C
R
C R
L3-3.28
Copyright ? Radhika Nagpal, 2002.
MIT Building Connections
3
R 2
R 413
10
12
26
8
R
413
10
12
26
8
413
1012
26
8