Distributed tree decomposition with privacy Vincent Armant Laurent Simon Philippe Dague
Outline • Introduction • Distributed tree decomposition Preserve network structure Keep local information local • Centralized tree decomp. VS concurrent approaches • Token elimination • Experimental results on small-world graphs • Conclusion / perspectives Vinccent Armant CP2012 2
Introduction Primal graph Pb : f 1 ( l1 , h ) f 2 (l1 , d , e, g) a l3 l2 f 3 (l2 , a , b) b c f 4 (l2 , h ) l4' f 5 (l3 , a , b) h l4 f 6 (l3 , c) e d f 7 (l3 , h) f 8 ( l4 , l4' , c) l1 g l5 f 9 (l4 , e , d) f 10 (l5 , d , g) centralized problem Its primal graph description Vinccent Armant CP2012 3
Introduction Primal graph Pb : f 1 ( l1 , h ) f 2 (l1 , d , e, g) a l3 l2 f 3 (l2 , a , b) b c f 4 (l2 , h ) l4' f 5 (l3 , a , b) h l4 f 6 (l3 , c) e d f 7 (l3 , h) f 8 ( l4 , l4' , c) l1 g l5 f 9 (l4 , e , d) f 10 (l5 , d , g) - Each variable labels exactly one node - All variables contained in the scope of a function in the problem description are neighbors in the primal graph Vinccent Armant CP2012 4
Introduction Tree Decomposition Ct 7 l4' c l1 l4 a l3 l2 Ct 6 b Ct 5 c e d l2 l3 l4' g h l4 l1 c l1 l4 e d Ct 1 Ct 2 Ct 3 Ct 4 b a l1 h e d d g l1 g l5 l2 l3 l2 l3 l1 g l5 Primal graph A tree decomposition 1) is a tree of clusters 2) preserves variables dependency 3) ensures running intersection Vinccent Armant CP2012 5
Introduction Tree Decomposition Ct 7 l4' c l1 l4 a l3 l2 Ct 6 b Ct 5 e d c l2 l3 l4' g h l1 c l4 l1 l4 e d Ct 1 Ct 2 Ct 3 Ct 4 b a l1 h e d d g l1 g l5 l2 l3 l2 l3 l1 g l5 Primal graph A tree decomposition 1) is a tree of clusters 2) preserves variables dependency 3) ensures running intersection Vinccent Armant CP2012 6
Introduction Tree Decomposition Ct 7 l4' c l1 l4 a l3 l2 Ct 6 b Ct 5 e d c l2 l3 l4' g h l1 c l4 l1 l4 e d Ct 1 Ct 2 Ct 3 e d Ct 4 b a l1 h d g l1 g l5 l2 l3 l2 l3 l1 g l5 Primal graph A tree decomposition 1) is a tree of clusters 2) preserves variables dependency 3) ensures running intersection Vinccent Armant CP2012 7
Introduction Why is it useful ? Ct 7 l4' c Pb : f 1 ( l1 , h ) l1 l4 f 2 (l1 , d , e, g) Ct 6 Ct 5 f 8 ( l4 , l4' , c) f 3 (l2 , a , b) e d l3 l2 f 4 (l2 , h ) g l1 c f 5 (l3 , a , b) l1 l4 f 6 (l3 , c) f 6 (l3, c) f 9 (l4, e, d) f 7 (l3 , h) Ct 2 Ct 3 Ct 1 Ct 4 f 8 ( l4 , l4' , c) b a l1 h e d d g f 9 (l4 , e , d) l2 l3 l2 l3 l1 g l5 f 10 (l5 , d , g) f3(l2, a, b) f 1 (l1, h) f 2 (l1, d , e, g ) f 10 (l5, d , g) f 5 (l3 , a , b) f 4 (l2, h) wCmax(init pb)= O(d 13 ) Cmax ( T decomposed pb) = O(d 4 ) f 7 (l3, h) 1) Good points: - divides the initial problem into sub-problems organized in a tree structure - allows concurrent resolution and /or backtrack free search - bounds time and space complexity by the size of the largest cluster ( width ) e.g. allows succinct representation (OBDD, MDD, DNNF, ..) 2) Limitations: - finding an optimal tree-decomposition is NP-Hard Vinccent Armant CP2012 8
Outline • Introduction • Distributed tree decomposition Preserve network structure Keep local information local • Centralized tree decomp. VS concurrent approaches • Token elimination • Experimental results on small-world graphs • Conclusion / perspectives Vinccent Armant CP2012 9
Distributed system p3 Pb(p3) : f 5 (l3 , a , b) Pb : f 6 (l3 , c) f 1 ( l1 , h ) f 7 (l3 , h) f 2 (l1 , d , e, g) p1 f 3 (l2 , a , b) p4 p2 Pb(p2) : f 4 (l2 , h ) p2 Pb(p4) : f 3 (l2 , a , b) f 8 ( l4 , l4' , c) f 5 (l3 , a , b) f 4 (l2 , h ) f 9 (l4 , e , d) f 6 (l3 , c) p3 f 7 (l3 , h) f 8 ( l4 , l4' , c) p4 p1 p5 f 9 (l4 , e , d) Pb(p1) : Pb(p5) : f 10 (l5 , d , g) f 1 ( l1 , h , ) p5 f 10 (l5 , d , g) f 2 (l1 , d , e, g) acquaintance links of p1 Initial problem setting is distributed among a set of peers 1) each peer can only interact with its neighbors by acquaintance links 2) local variables remain local Vinccent Armant CP2012 10
Distributed system p3 Pb(p3) : f 5 ( l3 , a , b) Pb : f 6 ( l3 , c) f 1 ( l1 , h ) f 7 ( l3 , h) f 2 (l1 , d , e, g) p1 f 3 (l2 , a , b) p4 p2 Pb(p2) : f 4 (l2 , h ) p2 Pb(p4) : f 3 ( l2 , a , b) f 8 ( l4 , l4' , c) f 5 (l3 , a , b) f 4 ( l2 , h ) f 9 ( l4, e , d) f 6 (l3 , c) p3 f 7 (l3 , h) f 8 ( l4 , l4' , c) p4 p1 p5 f 9 (l4 , e , d) Pb(p1) : Pb(p5) : f 10 (l5 , d , g) f 1 ( l1 , h , ) p5 f 10 ( l5 , d , g) f 2 ( l1 , d , e, g) each « li » represents a local variable of pi Initial setting is distributed among a set of peers 1) each peer can only interact with neighbors by acquaintance links 2) local variables remain local Vinccent Armant CP2012 11
How to decompose a distributed system respecting privacy and the peer acquaintances ? Ct 7 l4' c l1 l4 a l3 l2 Ct 5 Ct 6 b e d c l3 l2 l4' g l1 c h l4 l1 l4 e d Ct 2 Ct 3 Ct 4 Ct 1 b a h d g l1 e d l1 g l5 l1 g l2 l3 l2 l3 l5 a primal graph its tree decomposition The classical notion of tree decomposition is not sufficient it does not respect the privacy of local variables it does not preserve the peer acquaintances Vinccent Armant CP2012 12
Distributed Tree Decomposition Acquaintance Graph p3 p3 Pb(p3) : a l3 f 5 (l3 , a , b) h f 6 (l3 , c) b c f 7 (l3 , h) p2 p4 p4 p2 e c Pb(p2) : Pb(p4) : l2 a f 3 (l2 , a , b) l4 f 8 ( l4 , l4' , c) h b f 4 (l2 , h ) f 9 (l4 , e , d) d l4' p1 p5 p1 p5 Pb(p1) : h e Pb(p5) : d f 1 ( l1 , h , ) d f 10 (l5 , d , g) g l5 f 2 (l1 , d , e, g) l1 g Distributed system Acquaintance Graph G((P,V), ACQ) 1) P represents the set of peers 2) V labels each peer by its set of variables 3) ACQ P x P represents is acquaintance links Vinccent Armant CP2012 13
Distributed Tree Decomposition ct3 p3 a l3 a l3 b h h c b c p2 ct4 ct2 p4 ct6 e c a h l2 l2 a e c l4 c h b h b d l4 d l4' l4' p1 p5 ct5 h e d ct1 h e d g l5 d d g l1 g g l5 l1 c Acquaintance Graph Distributed Tree Decomposition 1) is a tree of clusters 2) preserves the variables dependencies 3) respects the running intersection property 4) preserves the peers acquaintance 5) respectis the privacy of local variables Vinccent Armant CP2012 14
Distributed Tree Decomposition ct3 p3 a l3 a l3 b h h c b c p2 ct4 ct2 p4 ct6 e c a l2 l2 a h e c l4 c h b h b d l4 d l4' l4' p1 p5 ct5 h e d ct1 h e d g l5 d d g l1 g g l5 c l1 Acquaintance Graph Distributed Tree Decomposition 1) is a tree of clusters 2) preserves the variables dependencies 3) respects the running intersection property 4) preserves the peers acquaintance 5) respectis the privacy of local variables Vinccent Armant CP2012 15
Distributed Tree Decomposition ct3 p3 a l3 a l3 b h h c b c ct4 ct2 p4 ct6 e c a l2 l2 a h e c l4 h b c h b d l4 d l4' l4' p5 ct5 h e d ct1 h e d g l5 d d g l1 g g l5 l1 c Acquaintance Graph Distributed Tree Decomposition 1) is a tree of clusters 2) preserves the variables dependencies 3) respects the running intersection property 4) preserves the peers acquaintance 5) respectis the privacy of local variables Vinccent Armant CP2012 16
Distributed Tree Decomposition p3 ct3 p3 a l3 a l3 b h h c b c p2 p4 p2 ct4 ct2 p4 ct6 e c c a l2 a l2 e c l4 h b h h b d l4 d l4' l4' p1 p1 p1 p5 p1 ct5 h e d ct1 h e d g l5 d d g l1 g g l5 l1 c Acquaintance Graph Distributed Tree Decomposition 1) is a tree of clusters -a cluster is created by one peer 2) preserves the variables dependencies -2 neighboring clusters come from: 3) respects the running intersection property - the same peer 4) preserves the peers acquaintance - neighboring peers 5) respects the privacy of local variables Vinccent Armant CP2012 17
Distributed Tree Decomposition p3 ct3 p3 a l3 a l3 b h h c b c p2 p4 p2 ct4 ct2 p4 ct6 e c a c a l2 l2 e c l4 h b h h b d l4 d l4' l4' p1 p1 p1 p5 p1 ct5 h e d ct1 h e d g l5 d d g l1 g g l5 c l1 Acquaintance Graph Distributed Tree Decomposition 1) is a tree of clusters 2) preserves the variables dependencies A local variable from pi can only 3) respects the running intersection property appear in a cluster created by pi 4) preserves the peers acquaintance 5) respects the privacy of local variables Vinccent Armant CP2012 18
Outline • Introduction • Distributed tree decomposition Preserve network structure Keep local information local • Centralized tree decomp. VS concurrent approaches • Token elimination • Experimental results on small-world graphs • Conclusion / perspectives Vinccent Armant CP2012 19
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