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Chapter 4 Partition (3) Double Partition Ding-Zhu Du

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Partition a big thing (hard to deal) into small ones (easy to deal). It is a natural idea.

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Partition is also an important technique in design of approximation algorithms. Example: To find a dominating set, we may find a dominating set in each small area.

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Weighted Dominating Set in unit disk graphs Given a unit disk graph G=(D,E) with node weight c:DR, find a dominating set with minimum total weight. + <1<1

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Backgroud 72-approximation (Ambuhl, et al. 2006).

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(6+ε)-approximation (Gao, et al. 2008).

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Partition into big cells General Case

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Partition No node lies on a cut-line.

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Problem A(i,j) 1 Dominating area

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Lemma Problem A(i,j) has 2-approximation. Theorem

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16 ? 14 !

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2-approximation for A(I,j) Case 1 Minimum weight of node in D ij

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2-approximation for A(I,j) Case 2. nodes in N(D ij ) dominate nodes in D ij ? ALAM AR CL CR BL BM BR

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A problem on strip: outside disks cover inside points p1p1 p2p2 pipi Ti(D,D) : minimum weight set with D, D, dominating p 1, …, p i such that (1) D (lowest intersection point on L) among disks above the strip (2) D(highest intersection point on L) among disks below the strip L

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Dynamic Programming

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p1p1 p2p2 p i-1 D 1 (lowest intersection point on L) among disks above the strip, in T i (D,D) D 2 (highest intersection point on L) among disks below the strip, in T i (D,D) L

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pjpj pipi p i-1 D D1

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2-approximation for A(I,j) Case 2. nodes in N(D ij ) dominate nodes in D ij ?

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LM Lemma If p is dominated by u in LM area, then every point in is dominated by u.

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p u v p u v

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p p Lemma If p and p can be dominated by nodes in BM but not nodes in CL and CR, then every node in can be dominated in nodes in A and B. A B CLCR

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is the leftmost one for p dominated by a node in BM, but not any node in CL and CR is the rightmost one for p dominated by a node in LM, but not any node in CL and CR contains all nodes dominated by nodes in BM but not nodes in CL and CR. pp Consider OPT

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is the leftmost one for p dominated by a node in UM, but not any node in CL and CR is the rightmost one for p dominated by a node in UM, but not any node in CL and CR contains all nodes dominated by nodes in UM but not nodes in CL and CR. qq Consider OPT

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L R U R

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How do we find p, p, q, q? Try all possibilities. How many possibilities?

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Idea: Combine cells into a strip Each strip contains m cells.

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6-approximation for a special case: For every subset C of cells, 1.every cell e in C is in case 1; 2.every cell e not in C is in case 2.

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6! 1 2 3 4 56

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Partition into big cells General Case

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(6+ε)-approximation in general case Shafting to minimize # of disks on boundaries

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(9.875+ε)-approximation for minimum weight connected dominating set in unit disk graph. Connecting a dominating set into a cds needs to add at most 3.875 opt nodes. (Zou et al, 2008) (improved 17opt)

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