Communication network for the GPS III system Simon Crevals Prof. - PowerPoint PPT Presentation

Communication network for the GPS III system Simon Crevals Prof. dr. G. Brinkmann N. Van Cleemput Department of Applied Mathematics and Computer Science Ghent University July 2010

Introduction 1 First test 2 Final program 3 Results 4 Conclusion 5 S. Crevals (Ghent University) Communication network for the GPS III system July 2010 2 / 40

Introduction 1 First test 2 Final program 3 Results 4 Conclusion 5 S. Crevals (Ghent University) Communication network for the GPS III system July 2010 3 / 40

S. Crevals (Ghent University) Communication network for the GPS III system July 2010 4 / 40

Advantages of communication continuous telemetry frequent updates S. Crevals (Ghent University) Communication network for the GPS III system July 2010 5 / 40

Restrictions for connections connection possible during the entire orbit 4 connections a satellite S. Crevals (Ghent University) Communication network for the GPS III system July 2010 6 / 40

Definition V GPS is the set of all satellites. S. Crevals (Ghent University) Communication network for the GPS III system July 2010 7 / 40

Definition V GPS is the set of all satellites. Definition G Pos = ( V GPS , E Pos ) , with ∀ v , w ∈ V GPS : { v , w } ∈ E Pos ⇔ communication between satellites v and w is technically possible. S. Crevals (Ghent University) Communication network for the GPS III system July 2010 7 / 40

Definition V GPS is the set of all satellites. Definition G Pos = ( V GPS , E Pos ) , with ∀ v , w ∈ V GPS : { v , w } ∈ E Pos ⇔ communication between satellites v and w is technically possible. Definition A connection graph is a spanning, 4-regular subgraph of G Pos . S. Crevals (Ghent University) Communication network for the GPS III system July 2010 7 / 40

Definition Given a graph G = ( V , E ) and a vertex v ∈ V : the neighbourhood of v in G is N ( v , G ) = { w ∈ V |{ v , w } ∈ E } . S. Crevals (Ghent University) Communication network for the GPS III system July 2010 8 / 40

Definition Given a graph G = ( V , E ) and a vertex v ∈ V : the neighbourhood of v in G is N ( v , G ) = { w ∈ V |{ v , w } ∈ E } . Definition The function uredop of a vertex v with respect to its neighbours is defined as: Let M 4 = { M ⊆ V GPS || M | = 4 } . + � {∞} , so that uredop : V GPS × M 4 → R ∀ v ∈ V GPS : M �⊂ N ( v , G Pos ) ⇒ uredop ( v , M ) = ∞ . S. Crevals (Ghent University) Communication network for the GPS III system July 2010 8 / 40

Definition The uredop value of a vertex v in a connection graph G is defined as: uredop ( v , G ) = uredop ( v , N ( v , G )) . S. Crevals (Ghent University) Communication network for the GPS III system July 2010 9 / 40

Definition The uredop value of a vertex v in a connection graph G is defined as: uredop ( v , G ) = uredop ( v , N ( v , G )) . Definition The uredop value of a connection graph G is defined as: uredop ( G ) = max { uredop ( v , G ) | v ∈ V GPS } . S. Crevals (Ghent University) Communication network for the GPS III system July 2010 9 / 40

Minimum requirements Construct a connection graph G which is 4-regular is a subgraph of G Pos has diameter at most 4 has uredop ( G ) < 3 S. Crevals (Ghent University) Communication network for the GPS III system July 2010 10 / 40

Best connection graph G Diameter 3 Smallest possible value for uredop ( G ) Maximum diameter 4 after one edge removal S. Crevals (Ghent University) Communication network for the GPS III system July 2010 11 / 40

t S. Crevals (Ghent University) Communication network for the GPS III system July 2010 12 / 40

t S. Crevals (Ghent University) Communication network for the GPS III system July 2010 13 / 40

Questions? S. Crevals (Ghent University) Communication network for the GPS III system July 2010 14 / 40

Introduction 1 First test 2 Final program 3 Results 4 Conclusion 5 S. Crevals (Ghent University) Communication network for the GPS III system July 2010 15 / 40

Questions Are there connection graphs with diameter 3? How many? With which uredop value? S. Crevals (Ghent University) Communication network for the GPS III system July 2010 16 / 40

Method Generate 4-regular graphs Filter graphs with diameter 3 Determine whether they are subgraph of G Pos Determine uredop value S. Crevals (Ghent University) Communication network for the GPS III system July 2010 17 / 40

Results Thousands of millions of 4-regular graphs with 27 vertices and diameter 3 Almost all tested graphs were subgraph of G Pos Very different uredop values (also good ones) S. Crevals (Ghent University) Communication network for the GPS III system July 2010 18 / 40

Results Thousands of millions of 4-regular graphs with 27 vertices and diameter 3 Almost all tested graphs were subgraph of G Pos Very different uredop values (also good ones) Conclusion The uredop values will have to provide the biggest restriction. S. Crevals (Ghent University) Communication network for the GPS III system July 2010 18 / 40

Introduction 1 First test 2 Final program 3 Results 4 Conclusion 5 S. Crevals (Ghent University) Communication network for the GPS III system July 2010 19 / 40

Definition Given a graph G = ( V , E ) . A vertex set IS ⊆ V is an independent set in G ⇔ ∀ v , w ∈ IS : { v , w } / ∈ E. Definition Given a graph G = ( V , E ) . A vertex set C ⊆ V is a clique in G ⇔ ∀ v , w ∈ C : { v , w } ∈ E. If there doesn’t exist a clique C ′ with | C ′ | > | C | , then C is a maximum clique in G. S. Crevals (Ghent University) Communication network for the GPS III system July 2010 20 / 40

Definition G C ( u ) = ( V C ( u ) , E C ( u )) , with V C ( u ) = { ( s , N ) | s ∈ V GPS , N ⊂ N ( s , G Pos ) , | N | = 4 and uredop ( s , N ) < u } and E C ( u ) = {{ v , w }| v = ( s , N s ) ∈ V C ( u ) , w = ( t , N t ) ∈ V C ( u ) , s � = t and s ∈ N t ⇔ t ∈ N s } . 1 1 3 15 13 13 7 7 9 3 2 2 5 5 5 2 2 12 21 9 9 1 3 3 7 S. Crevals (Ghent University) Communication network for the GPS III system July 2010 21 / 40

Definition G C ( u ) = ( V C ( u ) , E C ( u )) , with V C ( u ) = { ( s , N ) | s ∈ V GPS , N ⊂ N ( s , G Pos ) , | N | = 4 and uredop ( s , N ) < u } and E C ( u ) = {{ v , w }| v = ( s , N s ) ∈ V C ( u ) , w = ( t , N t ) ∈ V C ( u ) , s � = t and s ∈ N t ⇔ t ∈ N s } . Theorem The largest cliques in G C ( u ) have at most 27 vertices. S. Crevals (Ghent University) Communication network for the GPS III system July 2010 22 / 40

Definition G C ( u ) = ( V C ( u ) , E C ( u )) , with V C ( u ) = { ( s , N ) | s ∈ V GPS , N ⊂ N ( s , G Pos ) , | N | = 4 and uredop ( s , N ) < u } and E C ( u ) = {{ v , w }| v = ( s , N s ) ∈ V C ( u ) , w = ( t , N t ) ∈ V C ( u ) , s � = t and s ∈ N t ⇔ t ∈ N s } . Theorem The largest cliques in G C ( u ) have at most 27 vertices. Theorem Every connection graph G = ( V GPS , E ) with uredop values smaller than u, corresponds to a maximum clique C in G C ( u ) , with | C | = 27 . S. Crevals (Ghent University) Communication network for the GPS III system July 2010 22 / 40

Definition G C ( u ) = ( V C ( u ) , E C ( u )) , with V C ( u ) = { ( s , N ) | s ∈ V GPS , N ⊂ N ( s , G Pos ) , | N | = 4 and uredop ( s , N ) < u } and E C ( u ) = {{ v , w }| v = ( s , N s ) ∈ V C ( u ) , w = ( t , N t ) ∈ V C ( u ) , s � = t and s ∈ N t ⇔ t ∈ N s } . Theorem Every maximum clique C in G C ( u ) , with | C | = 27 , corresponds to a connection graph with uredop smaller than u. S. Crevals (Ghent University) Communication network for the GPS III system July 2010 23 / 40

Definition The language Uredop is the set of all strings with the structure g ′ # c # u, with g ′ represents a graph G ′ c represents all possible sets of 4 neighbours with corresponding value for each vertex u represents a natural number U and for which there exists a subgraph G of G ′ , such that C ( G ) < U. S. Crevals (Ghent University) Communication network for the GPS III system July 2010 24 / 40

Theorem The language Uredop is NP-complete. Proof It is easy to see that Uredop ∈ NP. We still have to prove that each problem in NP can be reduced to Uredop in polynomial time. To prove this, we reduce the language HC (graphs containing a Hamiltonian cycle) to Uredop. S. Crevals (Ghent University) Communication network for the GPS III system July 2010 25 / 40

S. Crevals (Ghent University) Communication network for the GPS III system July 2010 26 / 40

General clique solver Too slow for this specific problem S. Crevals (Ghent University) Communication network for the GPS III system July 2010 27 / 40

Pseudocode recursion Determine central vertex with smallest number of possible neighbourhoods Amount = 0: backtrack For each possible neighbourhood: ◮ choose neighbourhood ◮ adjust lists with possible neighbourhoods ◮ found connection graph or continue in recursion S. Crevals (Ghent University) Communication network for the GPS III system July 2010 28 / 40

Some details Diameter as bounding criterium Use of bitvectors Reduce iteration length Efficient adjustment of lists with possible neighbourhoods S. Crevals (Ghent University) Communication network for the GPS III system July 2010 29 / 40