6.889: Algorithms for Planar Graphs and Beyond
October 26, 2011
Problem Set 7
This problem set is due Wednesday, 11/2/2011 at noon.
Problem:
Give an
O
(
n
log
n
)–time algorithm to compute an
r
–division (
O
(
n/r
) pieces of
size
O
(
r
) and boundary
O
(
√
r
)) with the additional property that the boundary nodes of
each piece lie on a constant number of faces (called “holes”). Note that a face of a piece is
not necessarily a face of the graph.
For simplicity, you may assume that the cycle separator theorem achieves perfect balance
(meaning that, whenever we apply the separator theorem partitioning
V
into
A, B, S
each
of the two components
A
∪
S, B
∪
S
has weight exactly
w
(
V
)
/
2).
Solution:
The following algorithm was extracted from a preprint of Christian WulffNilsen
(
arXiv:1007.3609v2
), wherein the algorithm is analyzed in great detail.
Regard the entire graph
G
as a piece with no boundary nodes, split it recursively into two
subpieces using a cycle separator, retriangulate each piece, and recurse. The recursion stops
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 Fall '11
 ErikDemaine
 Algorithms, Graph Theory, Recursion, Planar graph, boundary nodes

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