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6.889: Algorithms for Planar Graphs and Beyond
September 28, 2011
Problem Set 3
This problem set is due Wednesday, October 5 at noon.
1. Let
P
:=
{
p
1
,...,p
k
}
be points in the plane and
{
Q
1
,...,Q
t
}
be a partition of
P
into
t
sets. Argue (informally) that there exist disjoint curves in the plane,
C
1
,...,C
t
, such
that for
i
= 1
,...,t
,
Q
i
⊆
C
i
.
Deduce (informally) that there is a suitable function
g
, such that for every
k
, the
planar grid graph
G
of size
g
(
k
)
×
g
(
k
) has the following property: if
u
1
,...,u
k
∈
V
(
G
)
are suﬃciently far apart from each other and from the boundary (i.e. their pairwise
distance and distance to the boundary is at least
f
(
k
) for a suitable function
f
) and
{
Q
1
,...,Q
t
}
is a partition of
{
u
1
,...,u
k
}
, then there exist disjoint trees
T
1
,...,T
t
in
G
such that for
i
= 1
,...,t
,
Q
i
⊆
T
i
.
Solution:
Let Π be the plane and observe that if
F
is a ﬁnite set of points in Π then
Π

F
is still connected. So for any two given points
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 Fall '11
 ErikDemaine
 Algorithms

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