Chapter 5 Guillotine Cut

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Chapter 5 Guillotine Cut

• (2) Portals

Ding-Zhu Du
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Rectilinear Steiner Tree

• Given a set of points in the rectilinear plane,
  find a minimum length tree interconnecting them.
• Those given points are called terminals.

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Initially
Edge length lt RSMT
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Initially
L
Total moving Length
n of terminals
2
2
n x n grid
If PTAS exists for grid points, then it exists
for general case.
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(1/3-2/3)-cut
Longer edge
1/3
2/3
Shorter edge
gt 1/3
Longer edge
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Cut line position
L
Cut line always passes through the center of a
cell.
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2
n x n grid
1 ( assume)
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Depth of (1/3-2/3)-cut
Note that every two parallel cut lines has
distance at least one. Therefore, the smallest
rectangle has area 1.
After one cut, each resulting rectangle has area
Within a factor of 2/3 from the original one.
Hence, depth of cuts lt (4 log n)/(log (3/2))
O(log n) since
depth
4
(2/3) n gt 1
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(1/3-2/3)-Partition
O(log n)
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Portals
m portals divide a cut segment equally.
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Restriction
A rectilinear Steiner tree T is restricted if
there exists a (1/3-2/3)-partition such that If a
segment of T passes through a cut Line, it passes
at a portal.
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Minimum Restricted RST can be computed in time n
2 by dynamic programming
O(m)
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2
Choices of each cut line O(n )
O(m)
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of subproblems n 2
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of subproblem
Each subproblems can be described by three facts
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O(n )
1. Position of for edges of a rectangle.
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O(n )
2. Position of portals at each edge.
O(m)
3. Set of using portals.
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4. Partition of using portals on the boundary.
(In each part of the partition, all portals
are connected and every terminal inside of
the rectangle is connected to some tree
containing a portal. )
O(m)
2
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Position of portals
2
2
O(n )
O(n )
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of partitions
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N(k) of partitions
N(0)1
N(k) N(k-1) N(k-2)N(1) N(1)N(k-2)
N(k-1) N(k-1)N(0) N(k-2)N(1)
N(0)N(k-1)
2
k
f(x) N(0) N(1)x N(2)x N(k)x
1
k
2
xf(x) f(x) - 1
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Analysis (idea)

• Consider a MRST T.
• Choose a (1/3-2/3)-partition.
• Modify it into a restricted RST by moving
  cross-points to portals.
• Estimate the total cost of moving cross-points.

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Choice of (1/3-2/3)-partition
1/3
2/3
Each cut is chosen to minimize of cross-points.
( of cross-points) x (1/3 longer edge length) lt
(length of T lying in rectangle).
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Moving cross-points to portals
Cost ( of cross-points) x ( edge
length/(m1)) lt (3/(m1)) x (length of
T lying in rectangle)
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Moving cost at each level of (1/3-2/3)-Partition
lt (3/(m1)) x (length of T )
O(log n)
Total cost lt O(log n)(3 / (m1)) x (length of T)
O(m)
O(1/e)
Choose m (1/e) O(log n). Then 2 n
.
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RSMT has (1e)-approximation with running Time
n .
O(1/e)
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Thanks, End