Planar Separator Theorem 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 Disk 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 onep endpoint in V1 and one endpoint in V2, • small if jHj 2 O( n), • balanced if jV1j; jV2j ≤ βn for some constant β Small Balanced Geometric Separators H p 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 p1 n 1 Separator Candidates • choose H0 as smallest square p 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 p1 n 1 p • place n separators H1; H2;:::; Hm at relative p1 distance n 3 Hi Is Balanced Bound number of disks strictly outside Hi : 1 36 jHoutj = n − jH\j − jHinj ≤ n − jHinj ≤ 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 jHoutj = n − jH\j − jHinj ≤ n − jHinj ≤ 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 •j Hinj ≤ 36 · 1 n ≤ 3 i 37 Hi Is Balanced Bound number of disks strictly outside Hi : 1 36 jHoutj = n − jH\j − jHinj ≤ n − jHinj ≤ 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 •j Hi j ≤ 36 · 37 n At Least One Hi Is Small p Hm = H n Hi+1 Hi H0 ≥ n=37 disks p1 n 1 3 At Least One Hi Is Small p1 • if α < n each disk p Hm = H n contributes to at most one Hi+1 separator p H • n contributions across n i H separatorsp ) 9Hi with at 0 most n contributions ≥ n=37 disks p1 n 1 3 At Least One Hi Is Small p1 • if α < n each disk p Hm = H n contributes to at most one Hi+1 separator p H • n contributions across n i H separatorsp ) 9Hi with at 0 most n contributions ≥ n=37 disks p1 n 1 p1 α ≥ n 3 At Least One Hi Is Small p1 Suppose α ≥ n . Hm (and therefore each Hi ) contains at most 32 1 4 · 2 O( ) disks. G is planar, i.e., K5-free π(α=2)2 α2 αp Each disk contributes to at most 1 + 1= n separators, which bounds the sum of all contributions by p 1 α 1 n O( ) · (1 + p ) = O( + ) = O(n + n) = O(n) α2 1= n α2 α p Because we have n separatorp candidates, at least one of these candidates has size O( n). Unit Disk Graphs H p 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 = fd : p ≤ diameter of d < p g: s n n p1 • Disks with diameter ≥ n are partitioned into size classes. p1 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 = fd : p ≤ diameter of d < p g: s n n p1 • Disks with diameter ≥ n are partitioned into size classes. • Hm contains at most 32 n G is planar, i.e., K5-free 4 · 2 O( 2s ) s−1 2 2 upper bound on area of H 2p 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 = fd : p ≤ diameter of d < p g: s n n p1 • 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 p2 upper bound on diameter of disk in Ds n 1 + 2 O(2s ) p1 distance between consecutive separators n separators. At Least One Hi Is Small For s = 1; 2;::: define size class 2s−1 2s D = fd : p ≤ diameter of d < p g: s n n p1 • 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;:::.