Chapter 4. Partition

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Chapter 4. Partition

• (2) Multi-layer Partition

Ding-Zhu Du
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Intersection Disk Graph
Consider n points in the Euclidean plane, each
is associated with a disk. An edge exists between
two points if and only if their associated disks
have nonempty intersection.
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Maximum Independent Set in Intersection Disk
Graph
Given a intersection disk graph D, find a
maximum Independent set opt(D).
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Multi-layer
Suppose the largest disk has diameter 1-e. Let
dmin be The diameter of smallest disk. Fix an
integer k gt 0. Let
Put all disks into m1 layers. For 0 lt j lt m,
layer j consists of all disks with diameter di,

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Partition P(0,0) in layer j
(0,0)
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Partition P(0) in layer j and layer j1

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Partition P(a,b) in layer j
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Layer j
Layer j1
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A disk hits a cut line.
At each layer, a disk can hit at most one among
Parallel lines apart each other with distance
.
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D(a,b)
In partition P(a,b), delete all disks each
hitting a cut line in the same layer. The
remaining disks form a collection D(a,b).
Maximum Independent set opt(D(a,b)) can be
computed in time
Why use it?
by dynamic programming.
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Dynamic Programming
j-cell is a cell in layer j.
For any j-cell S and a set I of independent disks
in layers lt j, intersecting S,
Table(S,I) maximum independent set of disks
layers gt j, contained in S, and
disjoint from I.
opt(D(a, b)) US Table(S, ?) where S is over all
cells in layer 0.
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Recursive Relation
For j-cell S and I,
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of Table(S,I)
of S too large
How do we overcome this difficulty?
Relevant cell
A j-cell is relevant if it contains a disk in
layer j.
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Dynamic Programming
j-cell is a cell in layer j.
For any relevant j-cell S and a set I of
independent disks in layers lt j, intersecting S,

Table(S,I) maximum independent set of disks
layers gt j, contained in S, and
disjoint from I.
opt(D(a, b)) USTable(S, ?) where S is over all
maximal relevant cells.
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Children of a relevant cell

S
S
S
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Maximal relevant cell
A relevant cell is maximal if it is not contained
by Another relevant cell.
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Recursive Relation
For j-cell S,
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of Table(S,I)
of relevant S n.
of I
of Table(S,I)
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of I
S
of Is
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Computation Time of Recursion
of S
of J
Time
Running Time of dynamic programming
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of J
S
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(1e)-Approximation
Choose k ?.
Compute opt(D(0,0)), opt(D(1,1)), ,
opt(D(k-1,k-1).
Choose maximum one among them.
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Analysis

• Consider an optimal solution D.
• For each partition P(a,b), let H(a,b) be the
  collection of all disks hitting cut line in the
  same layer.
• Estimate H(0,0)H(1,1)H(k-1,k-1).

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H(0,0)H(1,1)H(k-1,k-1)
Each disk appears in at most two terms in this
sum.
There exists i such that H(2i,2i) lt 2D/k.
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Performance ratio
Opt/approx 1/(1-2/k) 1 2/(k-4)
Choose
We obtain a (1e)-approximation With time
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Thanks, End