Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh...
Transcript of Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh...
![Page 1: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/1.jpg)
Voronoi Diagrams on Planar Graphs and Computing the
Diameter in Deterministic Õ(n5/3) time
Pawel Gawrychowski, Haim Kaplan, Shay Mozes, Micha Sharir and Oren Weimann
![Page 2: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/2.jpg)
Voronoi diagrams
Voronoi1908
Descartes1644
Dirichlet1850
![Page 3: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/3.jpg)
Voronoi diagrams on planar graphs
![Page 4: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/4.jpg)
Voronoi diagrams on planar graphs
![Page 5: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/5.jpg)
Voronoi diagrams on planar graphs
![Page 6: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/6.jpg)
Voronoi diagrams on planar graphs
![Page 7: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/7.jpg)
A fresh idea in algorithms for planar graph
• Cabello’s breakthrough (best paper in SODA 2017) - • can quickly construct Voronoi diagrams on planar
graphs (using randomized incremental construction of abstract Voronoi diagrams)
• can use Voronoi diagrams to compute the diameter in sub-quadratic Õ(n11/6) randomized time
• Led to exciting developments in distance oracles for planar graphs [Cohen-Addad et al. FOCS17, next talk]
![Page 8: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/8.jpg)
This work
• construction of Voronoi diagrams on planar graphs - faster, deterministic, more general
• leads to a faster O(n5/3)-time algorithm for diameter
![Page 9: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/9.jpg)
Voronoi diagrams on planar graphs
![Page 10: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/10.jpg)
Additively weighted Voronoi diagram
wg
wb
wr
wy
![Page 11: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/11.jpg)
• compute an r-division: O(n/r) pieces, each with O(r) vertices and O(r1/2) boundary vertices
• there are three types of distances:• between a vertex and a boundary vertex• between two vertices in the same piece• between two vertices in different pieces
High-level approach for diameter
![Page 12: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/12.jpg)
High-level approach for diameter
• compute an r-division: O(n/r) pieces, each with O(r) vertices and O(r1/2) boundary vertices
• there are three types of distances:• between a vertex and a boundary vertex• between two vertices in the same piece• between two vertices in different pieces
w
![Page 13: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/13.jpg)
• compute an r-division: O(n/r) pieces, each with O(r) vertices and O(r1/2) boundary vertices
• there are three types of distances:• between a vertex and a boundary vertex• between two vertices in the same piece• between two vertices in different pieces
w
High-level approach for diameter
![Page 14: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/14.jpg)
• compute an r-division: O(n/r) pieces, each with O(r) vertices and O(r1/2) boundary vertices
• there are three types of distances:• between a vertex and a boundary vertex• between two vertices in the same piece• between two vertices in different pieces
w
High-level approach for diameter
![Page 15: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/15.jpg)
Dist. between vertices in different pieces
• already computed distances from v to boundary nodes of the other piece
• compute additively weighted Voronoi diagram for the other piece in Õ(r1/2) time
• use Voronoi diagram to return the node furthest from each boundary site in Õ(1) amortized time per site
• total Õ(n·n/r·r1/2) = Õ(n2/r1/2)
• requires Õ(n/r·r2) = O(nr) preprocessing
wg
wb
wr
wy
# vertices # pieces
![Page 16: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/16.jpg)
• compute an r-division• three types of distance:
• between a vertex and a boundary vertex O(n2/r1/2) time• between two vertices inside the same piece O(nr) time• between two vertices in different pieces Õ(nr + n2/r1/2) time
• setting r = n2/3 yields total running time of Õ(n5/3)
High-level approach for diameter
![Page 17: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/17.jpg)
Remainder of the talk: constructing Voronoi diagrams
![Page 18: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/18.jpg)
Voronoi vertices - adjacent to three different Voronoi cells
![Page 19: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/19.jpg)
Voronoi vertices - adjacent to three different Voronoi cells (= trichromatic faces)
![Page 20: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/20.jpg)
Voronoi diagram with b sites has b cells, O(b) Voronoi vertices, and O(b) Voronoi edges (by Euler’s formula).
![Page 21: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/21.jpg)
Bisectors - Voronoi diagram with just two sites
![Page 22: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/22.jpg)
Every Voronoi edge is a subpath of a bisector
![Page 23: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/23.jpg)
• Every Voronoi edge is a subpath of a bisector
• Every Voronoi vertex is the intersection of two bisectors
Strategy:
![Page 24: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/24.jpg)
• Every Voronoi edge is a subpath of a bisector
• Every Voronoi vertex is the intersection of two bisectors
• Precompute and store all possible bisectors (for all pairs of sites and all possible weights…)
Strategy:
![Page 25: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/25.jpg)
• Every Voronoi edge is a subpath of a bisector
• Every Voronoi vertex is the intersection of two bisectors
• Precompute and store all possible bisectors (for all pairs of sites and all possible weights…)
• Represent Voronoi edges as subpaths of bisectors
Strategy:
![Page 26: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/26.jpg)
• Every Voronoi edge is a subpath of a bisector
• Every Voronoi vertex is the intersection of two bisectors
• Precompute and store all possible bisectors (for all pairs of sites and all possible weights…)
• Represent Voronoi edges as subpaths of bisectors• Construct Voronoi diagram with b sites in Õ(b) time
using divide and conquer by intersecting bisectors
Strategy:
![Page 27: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/27.jpg)
Computing bisectors
![Page 28: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/28.jpg)
Computing bisectors• only depends on the
difference wb-wg
reminiscent of MSSP[Cabello, Chambers, Erickson]
wg
wb
![Page 29: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/29.jpg)
• only depends on the difference wb-wg
• as we increase wb-wg the bisector sweeps the graph
• changes occur at discrete critical values, where blue vertices become green
Computing bisectors
wg
wb
reminiscent of MSSP[Cabello, Chambers, Erickson]
![Page 30: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/30.jpg)
Computing bisectors
wg
wb• only depends on the
difference wb-wg • as we increase wb-wg the
bisector sweeps the graph• changes occur at discrete
critical values, where blue vertices become green reminiscent of MSSP
[Cabello, Chambers, Erickson]
![Page 31: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/31.jpg)
Computing bisectors
wg
wb• only depends on the
difference wb-wg • as we increase wb-wg the
bisector sweeps the graph• changes occur at discrete
critical values, where blue vertices become green reminiscent of MSSP
[Cabello, Chambers, Erickson]
![Page 32: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/32.jpg)
Computing bisectors
wg
wb• only depends on the
difference wb-wg • as we increase wb-wg the
bisector sweeps the graph• changes occur at discrete
critical values, where blue vertices become green reminiscent of MSSP
[Cabello, Chambers, Erickson]
![Page 33: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/33.jpg)
Computing bisectors
wg
wb• only depends on the
difference wb-wg • as we increase wb-wg the
bisector sweeps the graph• changes occur at discrete
critical values, where blue vertices become green reminiscent of MSSP
[Cabello, Chambers, Erickson]
![Page 34: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/34.jpg)
• as we increase wb-wg the bisector sweeps the graph
• changes occur at discrete critical values, where blue vertices become green
➡ In a graph with O(r) vertices there are only O(r) bisectors (for each pair of sites). can be computed in Õ(r) time
• for each pair of sites, all bisectors stored in Õ(r) space and time using persistent binary search trees
• for r1/2 sites total preprocessing Õ(r2) time and space
Computing bisectors
wg
wb
• compared to Õ(r3) time and space in Cabello’s
![Page 35: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/35.jpg)
Intersecting bisectors = finding trichromatic faces
• three sites with weights wb, wg, wg.
• want to find a face with one vertex in each of the Voronoi cells.
wg
wb
wr
wg
wb
wr
![Page 36: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/36.jpg)
Dynamics of trichromatic faces
wg
wb
wr=∞
• fix wb and wg, and gradually decrease wg.
![Page 37: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/37.jpg)
wg
wb
• fix wb and wg, and gradually decrease wg.
wr
Dynamics of trichromatic faces
![Page 38: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/38.jpg)
wg
wb
wr
wg
wb
wr
• fix wb and wg, and gradually decrease wg.
• as we decrease wr, the trichromatic face moves monotonically along the (g,b)-bisector.
Dynamics of trichromatic faces
![Page 39: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/39.jpg)
• fix wb and wg, and gradually decrease wg.
• as we decrease wr, the trichromatic face moves monotonically along the (g,b)-bisector.
wg
wb
wr
wg
wb
wr
Dynamics of trichromatic faces
![Page 40: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/40.jpg)
• fix wb and wg, and gradually decrease wg.
• as we decrease wr, the trichromatic face moves monotonically along the (g,b)-bisector.
wg
wb
wr
wg
wb
wr
Dynamics of trichromatic faces
![Page 41: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/41.jpg)
Finding trichromatic faces• as we decrease wr, the
trichromatic face moves monotonically along the (g,b)-bisector.
wg
wb
wr
wg
wb
wr
• given wb, wg, wr can determine in constant timewhether an edge has a red endpoint.
• binary search for the last edge on the (g,b)-bisector that has a red endpoint
• takes Õ(1) time• need to extend to groups of sites. Becomes much
more complicated
![Page 42: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/42.jpg)
• we have precomputed bisectors in Õ(r2) • we know how to find a trichromatic face in Õ(1)
time • we compute the weighted Voronoi diagram of r1/2
sites in Õ(r1/2) time using divide and conquer.
Constructing a Voronoi diagram
![Page 43: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/43.jpg)
Constructing diagram via divide and conquer
![Page 44: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/44.jpg)
Constructing diagram via divide and conquer
![Page 45: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/45.jpg)
Constructing diagram via divide and conquer
![Page 46: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/46.jpg)
rg
![Page 47: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/47.jpg)
rg g
b
r
b
![Page 48: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/48.jpg)
Combined diagram
![Page 49: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/49.jpg)
• since we can compute trichromatic vertices in Õ(1) time, we can merge two Voronoi diagrams with b1 and b2 sites in Õ(b1 + b2) time
• so constructing a Voronoi diagram with r1/2 sites takes Õ(r1/2) time
Constructing a Voronoi diagram
![Page 50: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/50.jpg)
• handling sites on more than one face (holes)
• mechanism for finding furthest vertex in a Voronoi cell (similar to Cabello’s)
Things I swept under the rug
![Page 51: Voronoi Diagrams on Planar Graphs and Computing the ...oren/Publications/diameterSlides.pdfA fresh idea in algorithms for planar graph • Cabello’s breakthrough (best paper in SODA](https://reader033.fdocuments.us/reader033/viewer/2022060502/5f1bbc714f42af5bff7122bc/html5/thumbnails/51.jpg)
Conclusion• efficient deterministic construction of Voronoi diagrams on
planar graphs • diameter of a planar graph in deterministic Õ(n5/3) time
• can we get below for Õ(n5/3) time diameter?• nearly linear time / (conditional) lower bounds? • what other problems can benefit from these techniques?