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15853:Algorithms in the Real World
Graph Separators
– Introduction
– Applications
– Algorithms
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Edge Separators
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An
edge separator
:
a set of edges E’
⊆
E
which partitions V into
V
1
and V
2
Criteria:
V
1
, V
2
 balanced
E’ is small
V
1
V
2
E’
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Vertex Separators
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An
vertex separator
:
a set of vertices V’
⊆
V
which partitions V into
V
1
and V
2
Criteria:
V
1
, V
2
 balanced
V’ is small
V
1
V
2
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Compared with Mincut
s
t
Mincut
: as in the min
cut, maxflow theorem.
Mincut has no balance
criteria.
Mincut typically has a
source (s) and sink (t).
Will tend to find
unbalanced cuts.
V
1
V
2
E’
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Other names
Sometimes referred to as
–
graph partitioning
(probably more
common than “graph separators”)
– graph bisectors
– graph bifurcators
– balanced
or normalized graph cuts
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Recursive Separation
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What graphs have small separators
Planar graphs
: O(n
1/2
) vertex separators
2d meshes, constant genus, excluded minors
Almost planar graphs
:
the internet, power networks, road networks
Circuits
need to be laid out without too many crossings
Social network graphs
:
phonecall graphs, link structure of the web,
citation graphs, “friends graphs”
3dgrids and meshes
: O(n
2/3
)
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What graphs don’t have small
separatos
Hypercubes
:
O(n) edge separators
O(n/(log n)
1/2
) vertex separators
Butterfly networks
:
O(n/log n) separators ?
Expander graphs:
Graphs such that for any U
∈
V, s.t. U
≤ α
V,

neighbors
(U)
≥ β
U.
(
α
< 1,
β
> 0)
random graphs are expanders, with high probability
It is exactly the fact that they don’t have small
separators that make them useful.