[PPT] - t-spanners for Transmission Graphs Using the Path-Greedy Algorithm PowerPoint Presentation

SLIDE 1

Intro. Transmission Graphs Computing t-Spanner

t-spanners for Transmission Graphs Using the Path-Greedy Algorithm

Stav Ashur and Paz Carmi EuroCG’20 W¨ urzburg

Ashur and Carmi EuroCG20

SLIDE 2

Intro. Transmission Graphs Computing t-Spanner

Overview

1 Introduction

Notations and Definitions Path-Greedy Spanner

2 Transmission Graphs

Definitions Results

3 Computing a t-Spanner for Transmission Graphs

Algorithm Analysis

Ashur and Carmi EuroCG20

SLIDE 3

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Outline

1 Introduction

Notations and Definitions Path-Greedy Spanner

2 Transmission Graphs

Definitions Results

3 Computing a t-Spanner for Transmission Graphs

Algorithm Analysis

Ashur and Carmi EuroCG20

SLIDE 4

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

A t-Spanner for a Directed Graph

Let G = (V , E) be a directed graph. A t-Spanner for Directed Graphs A t-spanner G ′ for G is a sparse subgraph G ′ ⊆ G, s.t. for any two vertices p, q ∈ G, there is a directed path from p to q in G ′ of length at most t times the length of the path from p to q in G.

Ashur and Carmi EuroCG20

SLIDE 5

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

A t-Spanner for a Directed Graph

Let G = (V , E) be a directed graph. A t-Spanner for Directed Graphs A t-spanner G ′ for G is a sparse subgraph G ′ ⊆ G, s.t. for any two vertices p, q ∈ G, there is a directed path from p to q in G ′ of length at most t times the length of the path from p to q in G. Formally: ∀p, q ∈ V : |πG ′(p, q)| ≤ t · |πG(p, q)| Where πG(p, q) is the shortest directed path from p to q in the graph G, and |πG(p, q)| is its length.

Ashur and Carmi EuroCG20

SLIDE 6

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

A t-Spanner for a Directed Graph

Let G = (V , E) be a directed graph.

p q

Formally: ∀p, q ∈ V : |πG ′(p, q)| ≤ t · |πG(p, q)| Where πG(p, q) is the shortest directed path from p to q in the graph G, and |πG(p, q)| is its length.

Ashur and Carmi EuroCG20

SLIDE 7

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

A t-Spanner for a Directed Graph

Let G = (V , E) be a directed graph.

p q

Formally: ∀p, q ∈ V : |πG ′(p, q)| ≤ t · |πG(p, q)| Where πG(p, q) is the shortest directed path from p to q in the graph G, and |πG(p, q)| is its length.

Ashur and Carmi EuroCG20

SLIDE 8

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

A t-Spanner for a Directed Graph

Let G = (V , E) be a directed graph.

p q

Formally: ∀p, q ∈ V : |πG ′(p, q)| ≤ t · |πG(p, q)| Where πG(p, q) is the shortest directed path from p to q in the graph G, and |πG(p, q)| is its length.

Ashur and Carmi EuroCG20

SLIDE 9

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

A t-Spanner for a Directed Graph

Let G = (V , E) be a directed graph.

p q

Formally: ∀p, q ∈ V : |πG ′(p, q)| ≤ t · |πG(p, q)| Where πG(p, q) is the shortest directed path from p to q in the graph G, and |πG(p, q)| is its length.

Ashur and Carmi EuroCG20

SLIDE 10

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Desired Properties

Small stretch factor (spanning ratio)

p q

Ashur and Carmi EuroCG20

SLIDE 11

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Desired Properties

Small stretch factor (spanning ratio) Small number of edges (linear is preferable)

Ashur and Carmi EuroCG20

SLIDE 12

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Desired Properties

Small stretch factor (spanning ratio) Small number of edges (linear is preferable) Bounded degree

Ashur and Carmi EuroCG20

SLIDE 13

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Desired Properties

Small stretch factor (spanning ratio) Small number of edges (linear is preferable) Bounded degree Weight

Ashur and Carmi EuroCG20

SLIDE 14

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Desired Properties

Small stretch factor (spanning ratio) Small number of edges (linear is preferable) Bounded degree Weight Easy construction

Ashur and Carmi EuroCG20

SLIDE 15

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Outline

1 Introduction

Notations and Definitions Path-Greedy Spanner

2 Transmission Graphs

Definitions Results

3 Computing a t-Spanner for Transmission Graphs

Algorithm Analysis

Ashur and Carmi EuroCG20

SLIDE 16

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Algorithm

Path-Greedy, O(n3 log n)

I. Alth¨
fer, G. Das, D. Dobkin, D. Joseph, J. Soares, (1993)

Input: Given a graph G = (P, E), where P ⊂ Rd, E are edges with Euclidean weights, and a real number t > 1. Output: The Path-Greedy t-spanner G ′ = (P, E ′) for G. E ← E sorted in non-decreasing order of length E ′ := ∅ G ′ := (P, E ′) ForEach (u, v) ∈ E (in sorted order) If πG ′(u, v) > t · |uv| E ′ := E ′ ∪ {(u, v)} Return: G = (P, E ′)

Ashur and Carmi EuroCG20

SLIDE 17

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Algorithm

Path-Greedy, O(n3 log n)

I. Alth¨
fer, G. Das, D. Dobkin, D. Joseph, J. Soares, (1993)

The Path-Greedy spanner has the best properties

Ashur and Carmi EuroCG20

SLIDE 18

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Algorithm

Path-Greedy, O(n3 log n)

I. Alth¨
fer, G. Das, D. Dobkin, D. Joseph, J. Soares, (1993)

The Path-Greedy spanner has the best properties

O(n) edges

Ashur and Carmi EuroCG20

SLIDE 19

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Algorithm

Path-Greedy, O(n3 log n)

I. Alth¨
fer, G. Das, D. Dobkin, D. Joseph, J. Soares, (1993)

The Path-Greedy spanner has the best properties

O(n) edges Bounded degree

Ashur and Carmi EuroCG20

SLIDE 20

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Algorithm

Path-Greedy, O(n3 log n)

I. Alth¨
fer, G. Das, D. Dobkin, D. Joseph, J. Soares, (1993)

The Path-Greedy spanner has the best properties

O(n) edges Bounded degree Total weight O(wt(MST(G)))

Ashur and Carmi EuroCG20

SLIDE 21

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Algorithm

Path-Greedy, O(n3 log n)

I. Alth¨
fer, G. Das, D. Dobkin, D. Joseph, J. Soares, (1993)

The Path-Greedy spanner has the best properties

O(n) edges Bounded degree Total weight O(wt(MST(G))) Very simple

Ashur and Carmi EuroCG20

SLIDE 22

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Algorithm

Path-Greedy, O(n3 log n)

I. Alth¨
fer, G. Das, D. Dobkin, D. Joseph, J. Soares, (1993)

The Path-Greedy spanner has the best properties

O(n) edges Bounded degree Total weight O(wt(MST(G))) Very simple

Its main weakness is its time complexity

Ashur and Carmi EuroCG20

SLIDE 23

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

≈Greedy

Approximate Greedy, O(n log2 n)

G. Das and G. Narasimhan, (1997)

Ashur and Carmi EuroCG20

SLIDE 24

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

≈Greedy

Approximate Greedy, O(n log2 n)

G. Das and G. Narasimhan, (1997)

Approximating Dijkstra’s algorithm by querying a cluster graph

Ashur and Carmi EuroCG20

SLIDE 25

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

≈Greedy

Approximate Greedy, O(n log2 n)

G. Das and G. Narasimhan, (1997)

Approximating Dijkstra’s algorithm by querying a cluster graph Calculating a t-spanner in O(n log2 n)

Ashur and Carmi EuroCG20

SLIDE 26

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

≈Greedy

Approximate Greedy, O(n log2 n)

G. Das and G. Narasimhan, (1997)

Approximating Dijkstra’s algorithm by querying a cluster graph Calculating a t-spanner in O(n log2 n) Theoretically has good properties as the Path-Greedy spanner

Ashur and Carmi EuroCG20

SLIDE 27

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path Greedy superiority

Experimental Study of Geometric t-Spanners

M. Farshi, & J. Gudmundsson, (2009)

Algorithm Edges Degree

Weight wt(MST)

Path-Greedy 36K 17 11 θ-Graph 370K 144 327 ≈-Greedy 852K 403 WSPD spanner 11,119K 5,192 70,470

Table: Results for 8000 random uniformly distributed points with t = 1.1

Ashur and Carmi EuroCG20

SLIDE 28

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Improvements

Fast Path-Greedy, ˜ O(n2)

P. Bose, P. Carmi, M. Farshi, A. Maheshwari, M. Smid, (2010)

δ-Greedy, ˜ O(n2)

G. Bar-On & P. Carmi, (2017)

Ashur and Carmi EuroCG20

SLIDE 29

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Improvements

Fast Path-Greedy, ˜ O(n2)

P. Bose, P. Carmi, M. Farshi, A. Maheshwari, M. Smid, (2010)

Keeping a stack of actions preformed by Dijkstra’s algorithm δ-Greedy, ˜ O(n2)

G. Bar-On & P. Carmi, (2017)

Ashur and Carmi EuroCG20

SLIDE 30

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Improvements

Fast Path-Greedy, ˜ O(n2)

P. Bose, P. Carmi, M. Farshi, A. Maheshwari, M. Smid, (2010)

Keeping a stack of actions preformed by Dijkstra’s algorithm The running time is proportional to running Dijkstra from each point O(n2 log n) δ-Greedy, ˜ O(n2)

G. Bar-On & P. Carmi, (2017)

Ashur and Carmi EuroCG20

SLIDE 31

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Improvements

Fast Path-Greedy, ˜ O(n2)

P. Bose, P. Carmi, M. Farshi, A. Maheshwari, M. Smid, (2010)

Keeping a stack of actions preformed by Dijkstra’s algorithm The running time is proportional to running Dijkstra from each point O(n2 log n) Drawback - complex and difficult to implement δ-Greedy, ˜ O(n2)

G. Bar-On & P. Carmi, (2017)

Ashur and Carmi EuroCG20

SLIDE 32

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Improvements

Fast Path-Greedy, ˜ O(n2)

P. Bose, P. Carmi, M. Farshi, A. Maheshwari, M. Smid, (2010)

Keeping a stack of actions preformed by Dijkstra’s algorithm The running time is proportional to running Dijkstra from each point O(n2 log n) Drawback - complex and difficult to implement δ-Greedy, ˜ O(n2)

G. Bar-On & P. Carmi, (2017)

Maintain cones in directions where t-spanning paths are already guaranteed

Ashur and Carmi EuroCG20

SLIDE 33

Intro. Transmission Graphs Computing t-Spanner Definitions Path-Greedy Spanner

Path-Greedy Improvements

Fast Path-Greedy, ˜ O(n2)

P. Bose, P. Carmi, M. Farshi, A. Maheshwari, M. Smid, (2010)

Keeping a stack of actions preformed by Dijkstra’s algorithm The running time is proportional to running Dijkstra from each point O(n2 log n) Drawback - complex and difficult to implement δ-Greedy, ˜ O(n2)

G. Bar-On & P. Carmi, (2017)

Maintain cones in directions where t-spanning paths are already guaranteed Simpler than Fast-Greedy

Ashur and Carmi EuroCG20

SLIDE 34

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Outline

1 Introduction

Notations and Definitions Path-Greedy Spanner

2 Transmission Graphs

Definitions Results

3 Computing a t-Spanner for Transmission Graphs

Algorithm Analysis

Ashur and Carmi EuroCG20

SLIDE 35

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Transmission Graphs

D = {d1, ..., dn} a set of disks in Rd

Ashur and Carmi EuroCG20

SLIDE 36

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Transmission Graphs

D = {d1, ..., dn} a set of disks in Rd C = {c1, ..., cn} the centers of the disks

Ashur and Carmi EuroCG20

SLIDE 37

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Transmission Graphs

D = {d1, ..., dn} a set of disks in Rd C = {c1, ..., cn} the centers of the disks Transmission Graph The transmission graph G = (C, E) of D is a directed graph over the centers of the disks

Ashur and Carmi EuroCG20

SLIDE 38

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Transmission Graphs

D = {d1, ..., dn} a set of disks in Rd C = {c1, ..., cn} the centers of the disks Transmission Graph The transmission graph G = (C, E) of D is a directed graph over the centers of the disks, where (ci, cj) is a directed edge if ci lies inside Di.

ci cj Di

Ashur and Carmi EuroCG20

SLIDE 39

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Transmission Graphs

D = {d1, ..., dn} a set of disks in Rd C = {c1, ..., cn} the centers of the disks Transmission Graph The transmission graph G = (C, E) of D is a directed graph over the centers of the disks, where (ci, cj) is a directed edge if ci lies inside Di.

cj Di

Ashur and Carmi EuroCG20

SLIDE 40

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Transmission Graphs

D = {d1, ..., dn} a set of disks in Rd C = {c1, ..., cn} the centers of the disks Transmission Graph The transmission graph G = (C, E) of D is a directed graph over the centers of the disks, where (ci, cj) is a directed edge if ci lies inside Di.

ci cj Di

Ashur and Carmi EuroCG20

SLIDE 41

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Radius Ratio

D = {d1, ..., dn} a set of disks in Rd

Ashur and Carmi EuroCG20

SLIDE 42

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Radius Ratio

D = {d1, ..., dn} a set of disks in Rd R = {r1, ..., rn} the radii of the disks

Ashur and Carmi EuroCG20

SLIDE 43

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Radius Ratio

D = {d1, ..., dn} a set of disks in Rd R = {r1, ..., rn} the radii of the disks Radius Ratio The radius ratio Ψ of D is the ratio between the largest and smallest disk radii.

Ashur and Carmi EuroCG20

SLIDE 44

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Radius Ratio

D = {d1, ..., dn} a set of disks in Rd R = {r1, ..., rn} the radii of the disks Radius Ratio The radius ratio Ψ of D is the ratio between the largest and smallest disk radii. rmax = max{ri}, rmin = min{ri}, Ψ = rmax rmin

Ashur and Carmi EuroCG20

SLIDE 45

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Outline

1 Introduction

Notations and Definitions Path-Greedy Spanner

2 Transmission Graphs

Definitions Results

3 Computing a t-Spanner for Transmission Graphs

Algorithm Analysis

Ashur and Carmi EuroCG20

SLIDE 46

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Transmission graph Spanners

All results are of a t-spanner for any t > 1. O(n · Ψ) edges, O(m log n) time

D. Peleg, L. Roditty, (2010)

O(n) edges, O(n log n + n log Ψ) or O(n log5 n) time

H. Kaplan, W. Mulzer, L. Roditty, and P. Seiferth, (2015, 2018)

Ashur and Carmi EuroCG20

SLIDE 47

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Transmission graph Spanners

All results are of a t-spanner for any t > 1. O(n · Ψ) edges, O(m log n) time

D. Peleg, L. Roditty, (2010)

Relatively big (many edges) O(n) edges, O(n log n + n log Ψ) or O(n log5 n) time

H. Kaplan, W. Mulzer, L. Roditty, and P. Seiferth, (2015, 2018)

Ashur and Carmi EuroCG20

SLIDE 48

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Transmission graph Spanners

All results are of a t-spanner for any t > 1. O(n · Ψ) edges, O(m log n) time

D. Peleg, L. Roditty, (2010)

Relatively big (many edges) O(n) edges, O(n log n + n log Ψ) or O(n log5 n) time

H. Kaplan, W. Mulzer, L. Roditty, and P. Seiferth, (2015, 2018)

Difficult construction and analysis

Ashur and Carmi EuroCG20

SLIDE 49

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Transmission graph Spanners

All results are of a t-spanner for any t > 1. O(n · Ψ) edges, O(m log n) time

D. Peleg, L. Roditty, (2010)

Relatively big (many edges) O(n) edges, O(n log n + n log Ψ) or O(n log5 n) time

H. Kaplan, W. Mulzer, L. Roditty, and P. Seiferth, (2015, 2018)

Difficult construction and analysis No previous result of a t-spanner with bounded weight.

Ashur and Carmi EuroCG20

SLIDE 50

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Our results

O(n · Ψ) edges, O(m log n) time

D. Peleg, L. Roditty, (2010)

O(n) edges, O(n log n + n log Ψ) or O(n log5 n) time

H. Kaplan, W. Mulzer, L. Roditty, and P. Seiferth, (2015, 2018)

Our results: O(n) edges, O(n3 log n) time

S. Ashur, P. Carmi, (2020)

Ashur and Carmi EuroCG20

SLIDE 51

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Our results

O(n · Ψ) edges, O(m log n) time

D. Peleg, L. Roditty, (2010)

O(n) edges, O(n log n + n log Ψ) or O(n log5 n) time

H. Kaplan, W. Mulzer, L. Roditty, and P. Seiferth, (2015, 2018)

Our results: O(n) edges, O(n3 log n) time

S. Ashur, P. Carmi, (2020)

Very simple (implementation and analysis)

Ashur and Carmi EuroCG20

SLIDE 52

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Our results

O(n · Ψ) edges, O(m log n) time

D. Peleg, L. Roditty, (2010)

O(n) edges, O(n log n + n log Ψ) or O(n log5 n) time

H. Kaplan, W. Mulzer, L. Roditty, and P. Seiferth, (2015, 2018)

Our results: O(n) edges, O(n3 log n) time

S. Ashur, P. Carmi, (2020)

Very simple (implementation and analysis) Weight bounded by a function of n and Ψ

Ashur and Carmi EuroCG20

SLIDE 53

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Our results

O(n · Ψ) edges, O(m log n) time

D. Peleg, L. Roditty, (2010)

O(n) edges, O(n log n + n log Ψ) or O(n log5 n) time

H. Kaplan, W. Mulzer, L. Roditty, and P. Seiferth, (2015, 2018)

Our results: O(n) edges, O(n3 log n) time

S. Ashur, P. Carmi, (2020)

Very simple (implementation and analysis) Weight bounded by a function of n and Ψ Disadvantage - runtime

Ashur and Carmi EuroCG20

SLIDE 54

Intro. Transmission Graphs Computing t-Spanner Definitions Results

Our results

O(n · Ψ) edges, O(m log n) time

D. Peleg, L. Roditty, (2010)

O(n) edges, O(n log n + n log Ψ) or O(n log5 n) time

H. Kaplan, W. Mulzer, L. Roditty, and P. Seiferth, (2015, 2018)

Our results: O(n) edges, O(n2 log n) time

S. Ashur, P. Carmi, (2020)

Very simple (implementation and analysis) Weight bounded by a function of n and Ψ Disadvantage - runtime, can be reduced by using the δ-Greedy

Ashur and Carmi EuroCG20

SLIDE 55

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Outline

1 Introduction

Notations and Definitions Path-Greedy Spanner

2 Transmission Graphs

Definitions Results

3 Computing a t-Spanner for Transmission Graphs

Algorithm Analysis

Ashur and Carmi EuroCG20

SLIDE 56

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Algorithm

Let: D = {d1, ..., dn} a set of disks in R2 1 < t ∈ R Algorithm for computing a t-spanner for the disk graph G of D:

Ashur and Carmi EuroCG20

SLIDE 57

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Algorithm

Let: D = {d1, ..., dn} a set of disks in R2 1 < t ∈ R Algorithm for computing a t-spanner for the disk graph G of D: G ′ ← PathGreedy(G)

Ashur and Carmi EuroCG20

SLIDE 58

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Algorithm

Let: D = {d1, ..., dn} a set of disks in R2 1 < t ∈ R Algorithm for computing a t-spanner for the disk graph G of D: G ′ ← PathGreedy(G) Return G ′

Ashur and Carmi EuroCG20

SLIDE 59

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Algorithm

Let: D = {d1, ..., dn} a set of disks in R2 1 < t ∈ R Algorithm for computing a t-spanner for the disk graph G of D: G ′ ← PathGreedy(G) Return G ′ Claim 1 G ′ is a t-spanner of G

Ashur and Carmi EuroCG20

SLIDE 60

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Outline

1 Introduction

Notations and Definitions Path-Greedy Spanner

2 Transmission Graphs

Definitions Results

3 Computing a t-Spanner for Transmission Graphs

Algorithm Analysis

Ashur and Carmi EuroCG20

SLIDE 61

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t.

Ashur and Carmi EuroCG20

SLIDE 62

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof:

θ p q r

Ashur and Carmi EuroCG20

SLIDE 63

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof:

θ p q r r′

Ashur and Carmi EuroCG20

SLIDE 64

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof:

θ p q r r′

sin θ · |rp| cos θ · |rp|

Ashur and Carmi EuroCG20

SLIDE 65

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof: |qr| ≤ |rr′| + |r′q|

θ p q r r′

Ashur and Carmi EuroCG20

SLIDE 66

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof: |qr| ≤ |rr′| + |r′q| = |rr′| + (|pq| − |pr′|)

θ p q r r′

Ashur and Carmi EuroCG20

SLIDE 67

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof: |qr| ≤ |rr′| + |r′q| = |rr′| + (|pq| − |pr′|) = sin θ|rp| + (|pq| − cos θ|rp|)

θ p q r r′

sin θ · |rp| cos θ · |rp|

Ashur and Carmi EuroCG20

SLIDE 68

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof: |qr| ≤ |rr′| + |r′q| = |rr′| + (|pq| − |pr′|) = sin θ|rp| + (|pq| − cos θ|rp|)= |pq| − |rp|(cos θ − sin θ)

θ p q r r′

Ashur and Carmi EuroCG20

SLIDE 69

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof: |qr| ≤ |rr′| + |r′q| = |rr′| + (|pq| − |pr′|) = sin θ|rp| + (|pq| − cos θ|rp|)= |pq| − |rp|(cos θ − sin θ) |qr| + |rp|(cos θ − sin θ) ≤ |qp|

θ p q r r′

Ashur and Carmi EuroCG20

SLIDE 70

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof: |qr| ≤ |rr′| + |r′q| = |rr′| + (|pq| − |pr′|) = sin θ|rp| + (|pq| − cos θ|rp|)= |pq| − |rp|(cos θ − sin θ) 1 cos θ − sin θ|qr| + |rp| ≤ 1 cos θ − sin θ|qp|

θ p q r r′

Ashur and Carmi EuroCG20

SLIDE 71

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof: |qr| ≤ |rr′| + |r′q| = |rr′| + (|pq| − |pr′|) = sin θ|rp| + (|pq| − cos θ|rp|)= |pq| − |rp|(cos θ − sin θ) 1 cos θ − sin θ|qr| + |rp| ≤ 1 cos θ − sin θ|qp|

θ p q r r′

Ashur and Carmi EuroCG20

SLIDE 72

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof: |qr| ≤ |rr′| + |r′q| = |rr′| + (|pq| − |pr′|) = sin θ|rp| + (|pq| − cos θ|rp|)= |pq| − |rp|(cos θ − sin θ) t|qr| + |rp| ≤ t|qp|

θ p q r r′

Ashur and Carmi EuroCG20

SLIDE 73

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof: |qr| ≤ |rr′| + |r′q| = |rr′| + (|pq| − |pr′|) = sin θ|rp| + (|pq| − cos θ|rp|)= |pq| − |rp|(cos θ − sin θ) t|qr| + |rp| ≤ t|qp|

θ p q r

Ashur and Carmi EuroCG20

SLIDE 74

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof: |qr| ≤ |rr′| + |r′q| = |rr′| + (|pq| − |pr′|) = sin θ|rp| + (|pq| − cos θ|rp|)= |pq| − |rp|(cos θ − sin θ) 1 cos θ − sin θ ≤ t

θ p q r

Ashur and Carmi EuroCG20

SLIDE 75

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Bounded In-Degree

Claim 2 The in-degree of every vertex in G is bounded by a constant that depends on t. Proof: |qr| ≤ |rr′| + |r′q| = |rr′| + (|pq| − |pr′|) = sin θ|rp| + (|pq| − cos θ|rp|)= |pq| − |rp|(cos θ − sin θ) θ = O( 1 t − 1)

θ p q r

Ashur and Carmi EuroCG20

SLIDE 76

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Weight Bound

Claim 2 The weight of G ′, denoted by wt(G ′), is bounded by O((1 + Ψ) · log n · wt(MST(G))), where wt(MST(G)) is the weight of the MST of the disk centers.

Ashur and Carmi EuroCG20

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Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Weight Bound

Claim 2 The weight of G ′, denoted by wt(G ′), is bounded by O((1 + Ψ) · log n · wt(MST(G))), where wt(MST(G)) is the weight of the MST of the disk centers. w-Gap Property A set of directed edges E has the w-gap property for w ∈ R+, if for any to edges pq and rs, |pr|> w · min{|pq|, |rs|}

Ashur and Carmi EuroCG20

SLIDE 78

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Weight Bound

w-Gap Property A set of directed edges E has the w-gap property for w ∈ R+, if for any to edges pq and rs, |pr|> w · min{|pq|, |rs|}

Ashur and Carmi EuroCG20

SLIDE 79

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Weight Bound

w-Gap Property A set of directed edges E has the w-gap property for w ∈ R+, if for any to edges pq and rs, |pr|> w · min{|pq|, |rs|}

Ashur and Carmi EuroCG20

SLIDE 80

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Weight Bound

w-Gap Property A set of directed edges E has the w-gap property for w ∈ R+, if for any to edges pq and rs, |pr|> w · min{|pq|, |rs|}

Ashur and Carmi EuroCG20

SLIDE 81

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Weight Bound

w-Gap Property A set of directed edges E has the w-gap property for w ∈ R+, if for any to edges pq and rs, |pr|> w · min{|pq|, |rs|}

Ashur and Carmi EuroCG20

SLIDE 82

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Weight Bound

w-Gap Property A set of directed edges E has the w-gap property for w ∈ R+, if for any to edges pq and rs, |pr|> w · min{|pq|, |rs|}

Ashur and Carmi EuroCG20

SLIDE 83

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Weight Bound

w-Gap Property A set of directed edges E has the w-gap property for w ∈ R+, if for any to edges pq and rs, |qs|> w · min{|pq|, |rs|}

Ashur and Carmi EuroCG20

SLIDE 84

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Weight Bound

w-Gap Property A set of directed edges E has the w-gap property for w ∈ R+, if for any to edges pq and rs, |qs|> w · min{|pq|, |rs|} Lemma [G. Narasimhan, M.Smid 2007] A set of directed edges E that admit the w-gap property for w ∈ R+, then the total weight of E is less than (1 + 2 w ) · log |P| · wt(MST(P)), where P is the set of the end-points of E.

Ashur and Carmi EuroCG20

SLIDE 85

Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Weight Bound

w-Gap Property A set of directed edges E has the w-gap property for w ∈ R+, if for any to edges pq and rs, |qs|> w · min{|pq|, |rs|} Lemma [G. Narasimhan, M.Smid 2007] A set of directed edges E that admit the w-gap property for w ∈ R+, then the total weight of E is less than (1 + 2 w ) · log |P| · wt(MST(P)), where P is the set of the end-points of E. We divide E(G ′) into a constant number of sets that admit the

1 Ψ-gap property.

Ashur and Carmi EuroCG20

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Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Conclusion

1 A simple t-spanner for transmission-graphs 2 Bounded in-degree 3 Bounded weight 4 O(n2 log n) runtime Ashur and Carmi EuroCG20

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Intro. Transmission Graphs Computing t-Spanner Algorithm Analysis

Thank You

Ashur and Carmi EuroCG20