A Geometric Approach

based on

A Framework for ETH-Tight Algorithms and Lower Bounds in Geometric Intersection Graphs

by

Mark de Berg, Hans L. Bodlaender, S´andor Kisfaludi-Bak, D´anielMarx and Tom C. van der Zanden Unit Graphs

• represent vertices as unit disks, i.e., disks with diameter 1 • disks intersect iff corresponding vertices are adjacent

a d a b d b f ↔ e f e g g h h c c Unit Disk Graphs

• represent vertices as unit disks, i.e., disks with diameter 1 • disks intersect iff corresponding vertices are adjacent • some unit disk graphs are planar

a d a b d b f ↔ e f e g g h h c c Unit Disk Graphs

• represent vertices as unit disks, i.e., disks with diameter 1 • disks intersect iff corresponding vertices are adjacent • some unit disk graphs are planar • some planar graphs are not unit disk graphs

c c d b a d a 6↔ b

e f e f Geometric Separators Geometric Separators Geometric Separators Geometric Separators

H

Observation: Disks intersecting the boundary of H separate disks strictly inside H from disks strictly outside H . Small Balanced Separators

Let G = (V , E) be a graph. H ⊆ V is • a separator if there is a partition H, V1, V2 of V so that no edge in E has one√ endpoint in V1 and one endpoint in V2, • small if |H| ∈ O( n), • balanced if |V1|, |V2| ≤ βn for some constant β Small Balanced Geometric Separators

H

√ Claim: There exists an H intersecting O( n) disks with ≤ 36/37n disks strictly inside H and ≤ 36/37n disks strictly outside H , i.e., H is a small balanced separator. Separator Candidates

• choose H0 as smallest square that contains ≥ n/37 disks • resize disk representation so that H0 has diameter 1

• all disks now have diameter α H0

≥ n/37 disks √1 n 1 Separator Candidates

• choose H0 as smallest square √ Hm = H n that contains ≥ n/37 disks Hi+1 • resize disk representation so H that H0 has diameter 1 i

• all disks now have diameter α H0

≥ n/37 disks √1 n 1 √ • place n separators H1, H2,..., Hm at relative √1 distance n 3 Hi Is Balanced

Bound number of disks strictly outside Hi : 1 36 |Hout| = n − |H∩| − |Hin| ≤ n − |Hin| ≤ n − n = n i i i i 37 37

Hi contains H0 and H0 contains ≥ n/37 disks Hi Is Balanced

Bound number of disks strictly outside Hi : 1 36 |Hout| = n − |H∩| − |Hin| ≤ n − |Hin| ≤ n − n = n i i i i 37 37

H0 Bound number of disks strictly inside H for α < 1/4: 1 i • H0 is smallest square that 1 contains ≥ 37 n disks H 2 i • subdivide Hi into 6 = 36 subsquares with edge length 1 ≤ 1/2 ≤ 2 • every subsquare Hsub of Hi touches < 1/37n disks •| Hin| ≤ 36 · 1 n ≤ 3 i 37 Hi Is Balanced

Bound number of disks strictly outside Hi : 1 36 |Hout| = n − |H∩| − |Hin| ≤ n − |Hin| ≤ n − n = n i i i i 37 37

Bound number of disks strictly Hsub < 1/4 inside Hi for α < 1/4: • H0 is smallest square that 1 contains ≥ 37 n disks • subdivide H into 62 = 36 ≤ 1/2 i subsquares with edge length < 1 ≤ 1/2 • every subsquare Hsub of Hi disks that touch Hsub are touches < 1/37n disks contained inside the dashed in 1 square with edge length < 1 •| Hi | ≤ 36 · 37 n At Least One Hi Is Small

√ Hm = H n Hi+1

Hi

H0

≥ n/37 disks √1 n 1

3 At Least One Hi Is Small

√1 • if α < n each disk √ Hm = H n contributes to at most one Hi+1 separator √ H • n contributions across n i H separators√ ⇒ ∃Hi with at 0 most n contributions

≥ n/37 disks √1 n 1

3 At Least One Hi Is Small

√1 • if α < n each disk √ Hm = H n contributes to at most one Hi+1 separator √ H • n contributions across n i H separators√ ⇒ ∃Hi with at 0 most n contributions

≥ n/37 disks √1 n 1

√1 α ≥ n

3 At Least One Hi Is Small

√1 Suppose α ≥ n . Hm (and therefore each Hi ) contains at most 32 1 4 · ∈ O( ) disks. G is planar, i.e., K5-free π(α/2)2 α2

α√ Each disk contributes to at most 1 + 1/ n separators, which bounds the sum of all contributions by √ 1 α 1 n O( ) · (1 + √ ) = O( + ) = O(n + n) = O(n) α2 1/ n α2 α √ Because we have n separator√ candidates, at least one of these candidates has size O( n). Unit Disk Graphs

H

√ Theorem: There exists an H intersecting O( n) disks with ≤ 36/37n disks strictly inside H and ≤ 36/37n disks strictly outside H , i.e., H is a small balanced separator. (Non-Unit) Disk Graphs

• represent vertices as disks with arbitrary diameter • disks intersect iff corresponding vertices are adjacent

c c d b a d b ↔ a

f e e f

Circle Packing Theorem: The intersection graphs of non-crossing circles are exactly the planar graphs. [Koebe 1936] At Least One Hi Is Small

For s = 1, 2,... define size class

2s−1 2s D = {d : √ ≤ diameter of d < √ }. s n n

√1 • Disks with diameter ≥ n are partitioned into size classes.

√1 Disks with diameter < n contribute to at most one separator. At Least One Hi Is Small

For s = 1, 2,... define size class

2s−1 2s D = {d : √ ≤ diameter of d < √ }. s n n

√1 • Disks with diameter ≥ n are partitioned into size classes.

• Hm contains at most

32 n G is planar, i.e., K5-free 4 · ∈ O( 2s ) s−1 2 2 upper bound on area of H  2√  i n π 2 lower bound on area of disk in Ds

disks in Ds . At Least One Hi Is Small

For s = 1, 2,... define size class

2s−1 2s D = {d : √ ≤ diameter of d < √ }. s n n

√1 • Disks with diameter ≥ n are partitioned into size classes. n • Hm contains at most O( 22s ) disks in Ds .

• One disk in Ds contributes to at most

s √2 upper bound on diameter of disk in Ds n 1 + ∈ O(2s ) √1 distance between consecutive separators n

separators. At Least One Hi Is Small

For s = 1, 2,... define size class

2s−1 2s D = {d : √ ≤ diameter of d < √ }. s n n

√1 • Disks with diameter ≥ n are partitioned into size classes. n • Hm contains at most O( 22s ) disks in Ds . s • One disk in Ds contributes to at most O(2 ) separators.

Bound the number of contributions: X X 1 O( n ) · O(2s )= O(n · ) = O(n) 22s 2s s=1,2,... s=1,2,...