Localiser une cible dans un graphe

Transcription

1 Julien Bensmail 1, Dorian Mazauric 2, Fionn Mc Inerney 1, Nicolas Nisse 1, Stéphane Pérennes 1 1 Université Côte d Azur, Inria, CNRS, I3S, France 2 Université Côte d Azur, Inria, France May 30, 2018, Roscoff, France AlgoTel /15

2 Metric Dimension [Slater, 1975 ; Harary and Melter, 1976] A facility, city or some other location is modelled by a graph G. A target is hidden at a vertex of G (i.e., a building). Detector at each vertex ; when probed, returns its distance to the target. B (1,1,1,1) (0) (1) (2) (3) (4) (0,1) (1,0) (2,1) A A (1,2) (2,2) A B C D (0,2,2,2) (2,0,2,2) (2,2,0,2) (2,2,2,0) (2,2,2,2) Metric Dimension of G Min. # of vertices needed to be probed all at once to locate the target in G. 2/15

3 Sequential Locating Game on Graphs [Seager, 2013] & Game of Guess Who? SL(G ) : Sequential location number of G Min. # of turns of probing one vertex per turn, to locate a target hidden in G. 3/15 Bensmail, Mazauric, Mc Inerney, Nisse, Pe rennes

4 Sequential Locating Game on Graphs [Seager, 2013] & Game of Guess Who? SL(G ) : Sequential location number of G Min. # of turns of probing one vertex per turn, to locate a target hidden in G. 3/15 Bensmail, Mazauric, Mc Inerney, Nisse, Pe rennes

5 Sequential Locating Game on Graphs Philip Bill Barbu Anne Richard Max Moustache Couettes Lunettes Roux Blond Charles Alex Alfred Cheveux Bruns/Noirs Eric Joe David Chauve & Game of Guess Who? [Seager, 2013] Anita Ch. blanc Femme Herman Sam Boucles d'oreilles Paul Claire Tom Chapeau Gros nez Maria George Bernard Susan Frans Robert Peter Yeux bleus/ clairs 4/15 Bensmail, Mazauric, Mc Inerney, Nisse, Pe rennes

6 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

7 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. Blue vertex probed MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

8 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. Blue vertex probed Answer : d = 0 MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

9 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. Blue vertex probed Answer : d = 1 MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

10 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. Blue vertex probed Answer : d = 2 MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

11 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. Blue vertex probed Answer : d = 3 MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

12 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. Blue vertex probed Answer : d = 4 MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

13 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. Two vertices in blue NOT probed MD(G) > 18 MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

14 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. Two vertices in blue NOT probed MD(G) > 18 Vertices at d = 1 MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

15 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. Two vertices in blue NOT probed MD(G) > 18 Vertices at d = 2 MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

16 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. Two vertices in blue NOT probed MD(G) > 18 Vertices at d = 3 MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

17 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. All vertices in blue probed MD(G) = 19 MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

18 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. A B Now sequential probing All vertices in blue probed Locate in 2 turns (FIRST turn) D C MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

19 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. A B Now sequential probing All vertices in blue probed Answer : (0, 4, 4, 4) Locate in 2 turns (FIRST turn) D C MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

20 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. A B Now sequential probing All vertices in blue probed Answer : (2, 2, 2, 2) Locate in 2 turns (FIRST turn) D C MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

21 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. A B Now sequential probing All vertices in blue probed Answer : (1, 3, 3, 3) Locate in 2 turns (FIRST turn) D C MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

22 Sequential Metric Dimension [Bensmail, Mazauric, M., Nisse, Pérennes, 2018] Sequential Metric Dimension of G Given k, l, G, is it possible to locate the immobile target in G in at most l turns by probing at most k vertices each turn. Related : Locate moving target with k? [Bosek et al, 2017] ; k = 1 [Seager, 2012]. MD(G) k turns suffice to locate target. But it can be located faster. A B Now sequential probing All vertices in blue probed Locate in 2 turns (SECOND turn) D C MD(G) = 19 and for k = 4, the target can be located in 2 turns. 5/15

23 Sequential Metric Dimension - Our Results Complexity NP-complete when either number of vertices to be probed or number of turns is fixed. Trees NP-complete in trees. Difficulty only comes from first turn. polynomial-time (+1)-approximation algorithm for trees. 6/15

24 Summary of results in trees λ k (T ) : min. # turns to locate target in T. (+1)-approximation algorithm computes strategy that locates target in T in at most λ k (T ) + 1 steps. Time complexity : O(n log n). Exact algorithm computes strategy that locates target in T in at most λ k (T ) steps. Time complexity : O(n k+2 log n). 7/15

25 Trees : why problem is easy after 1 st turn Probe any 1 vertex on 1 st turn. 8/15

26 Trees : why problem is easy after 1 st turn Probe any 1 vertex on 1 st turn. r 8/15

27 Trees : why problem is easy after 1 st turn Probe any 1 vertex on 1 st turn. Result : Tree T rooted in r where all leaves are the same distance from r and the target is known to occupy a leaf. r 8/15

28 Trees : why problem is easy after 1 st turn Probe any 1 vertex on 1 st turn. Result : Tree T rooted in r where all leaves are the same distance from r and the target is known to occupy a leaf. r 9/15

29 Second key argument for easiness T v : subtree rooted in v of T rooted in r, v is a child of r. Probing 1 vertex in T v allows to know if target is in T v or in T \ T v. r T v1 v 1 v 2 10/15

30 Two parameters needed for algorithm Assume target is known to occupy a leaf of T i. Then, λ k (T i ) : min. # of turns of probings to locate target in T i. π k (T i ) : min. # of vertices to probe in T i during first turn of probing vertices in T i to locate target in λ k (T i ) turns. k = 3 λ 3 (T ) = 3 π 3 (T ) = 2 11/15

31 Need for tradeoff between probing 1 or π(t vi ) vertices in T vi k = 3 λ(t ) = 5 r v 1 v 2 v 3 v 4 v 5 v (5,2) (3,2) (3,2) (3,2) (3,2) (3,2) (λ 3 (T vi ), π 3 (T vi )) 12/15

32 Need for tradeoff between probing 1 or π(t vi ) vertices in T vi k = 3 λ(t ) = 5 r v 1 v 2 v 3 v 4 v 5 v (5,2) (3,2) (3,2) (3,2) (3,2) (3,2) (λ 3 (T vi ), π 3 (T vi )) 12/15

33 Need for tradeoff between probing 1 or π(t vi ) vertices in T vi k = 3 λ(t ) = 5 r v 1 v 2 v 3 v 4 v 5 v (5,2) (3,2) (3,2) (3,2) (3,2) (3,2) (λ 3 (T vi ), π 3 (T vi )) 12/15

34 Variant with relative distances Relative distances returned instead of exact distances. Target v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 Relative dist. vector from probing blue vertices : (v 7 < v 4 = v 8 < v 1 ). Complexity NP-complete when either number of vertices to be probed or number of turns is fixed. 13/15

35 Future work Study the problem with exact distances in other graph classes such as planar and interval graphs. Try to get exact values for the problem with relative distances in paths [Foucaud et al, 2014]. Study the problem with relative distances in trees. 14/15

36 Thanks! 15/15