Pawel Gawrychowski Haim Kaplan Shay Mozes Micha Sharir

Diameter • Compute the longest shortest path • Assume distinct distances, unique shortest paths

Overview of the diameter algorithm • Compute an r-division • Intuitively this is a

Three kinds of shortest paths • Between a vertex and a boundary vertex •

Maximum dist. between vertices in different pieces •

Compute the max from info on the bisectors •

Basic building block • Finding Voronoi vertices • First, in a diagram of 3

Diagram of 3 sites • Lemma: It has at most 2 Voronoi vertices 58

At most 2 Vor. vertices Proof: ? Shortest paths cannot cross. . 59

A diagram of 3 sites • The blue site takes over an interval of

Voronoi diagram • Traversing all triples of sites and finding the associated vertices is

Max distance computation • Each dual vertex computes the max in its triangle •

Slides: 82

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Pawel Gawrychowski, Haim Kaplan, Shay Mozes, Micha Sharir, and Oren Weimann 1

Diameter • Compute the longest shortest path • Assume distinct distances, unique shortest paths 3

Diameter state of the art • 4

Our result • 5

Overview of the diameter algorithm • Compute an r-division • Intuitively this is a recursive application of the planar separator theorem 6

r-division

A piece

Another piece

r-division (Fredrickson 87) •

Three kinds of shortest paths • Between a vertex and a boundary vertex • Between two vertices inside the same piece • Between two vertices in different pieces

Dist. to/from boundary nodes

Distances to/from boundary nodes •

Between vertices in the same piece

Inside pieces •

Vertices in diff. pieces

Between vertices in different pieces •

Voronoi diagram

Additively weighted Voronoi diagram

Voronoi diagram

Maximum dist. between vertices in different pieces •

Compute the furthest in each cell

Voronoi diagram + max in cells •

Between vertices in different pieces •

Total time •

Representing the diagram

Voronoi vertices

Bisectors

Representing the diagram • 34

Compute the max from info on the bisectors •

Computing the bisectors

Computing the bisectors • 37

Computing the Bisectors

Bisectors • 53

Computing the Voronoi diagram • 55

Basic building block • Finding Voronoi vertices • First, in a diagram of 3 sites 56

Diagram of 3 sites 57

Diagram of 3 sites • Lemma: It has at most 2 Voronoi vertices 58

At most 2 Vor. vertices Proof: ? Shortest paths cannot cross. . 59

Computing the diagram of 3 sites

A diagram of 3 sites

A diagram of 3 sites • The blue site takes over an interval of the red/green bisector • The Voronoi vertices are the endpoints of this interval • We can find them by a binary search along the red/green bisector • Generalizes to a situation where instead of a single red/green/blue site we have a contiguous sequence of red/green/blue sites

Voronoi diagram • Traversing all triples of sites and finding the associated vertices is too expensive • Use divide and conquer

Divide and conquer

Divide and conquer

Max distance computation

Max distance computation • Each dual vertex computes the max in its triangle • Vertices of the cotree compute subtree maxima • Gets somewhat more complicated in the presence of holes

Open problems • 88

Thanks 89