Follo lowin ing g De Derek ek's 's foots tstep eps s Feodor Dragan May, 2014
Derek’s Primary Universe The talk is not about this Derek These footsteps are hard to follow
Derek’s Parallel Universe domination decomposition minors LBFS Interval LDFS Tree-width AT-free Co-comp Search Clique-width Partial k-tree Path cover NP-hard Linear cographs Cliques Dom. path Dom. pair Tree spanner Perfect graphs Efficient Tree power Not a complete picture I didn’t want to block the Sun
Tal alk k ou outl tlin ine collaborating with Derek o fast estimation of diameters o representing approximately graph distances with few tree distances following Derek's footsteps o tree- and path-decompositions and new graph parameters o Approximating tree t-spanner problem using tree-breadth o Approximating bandwidth using path-length o Approximating line-distortion using path-length 4
Tal alk k ou outl tlin ine collaborating with Derek o fast estimation of diameters 5
The e Di Diam ameter er Pr Prob oblem lem o The eccentricity ecc 𝑤 = 𝑒𝑗𝑏𝑛 𝐻 of a vertex 𝑤 is the maximum distance from 𝑤 to a vertex in 𝐻 y o The diameter 𝑒𝑗𝑏𝑛 𝐻 is the maximum eccentricity of a vertex of 𝐻 x o The diameter problem (find a longest shortest path in a graph) : find 𝑒𝑗𝑏𝑛 𝐻 and 𝑦, 𝑧 such that 𝑒 𝑦, 𝑧 = 𝑒𝑗𝑏𝑛 𝐻 (in other words, find a vertex of maximum diam ( G ) d ( x , y ) 20 eccentricity) 6
Ou Our App pproach oach Examine the naïve algorithm of o choosing a vertex o performing some version of BFS from this vertex and then o showing a nontrivial bound on the eccentricity of the last vertex visited in this search. This approach has already received considerable attention o (classical result [Handler’ 73]) for trees this method produces a vertex of maximum eccentricity o [Dragan et al’ 97] if LexBFS is used for chordal graphs, then ecc(v) ≥ 𝑒𝑗𝑏𝑛 𝐻 − 1 whereas for interval graphs and Ptolemaic graphs ecc v = 𝑒𝑗𝑏𝑛 𝐻 o [Corneil et al’ 99] if LexBFS is used on AT-free graphs, then ecc(v) ≥ 𝑒𝑗𝑏𝑛 𝐻 − 1 o [Dragan’ 99] if LexBFS is used, then ecc(v) ≥ 𝑒𝑗𝑏𝑛 𝐻 − 2 for HH-free graphs, ecc v ≥ 𝑒𝑗𝑏𝑛 𝐻 − 1 for HHD-free graphs and ecc v = 𝑒𝑗𝑏𝑛 𝐻 for HHD-free and AT-free graphs o [Corneil et al’ 01] considered multi sweep LexBFSs … 7
Variants riants of BF BFS us used ed Can be implemented to run in linear time ( ) ( ) ( 7 , 6 ) ( 7 , 6 ) ( 6 ) ( 7 ) 6 7 8 u 8
Ou Our Res esult ults s on Res estr trict icted ed Fami milies lies of Gr Graph aphs No induced cycles of length >3 No asteroidal triples No asteroidal triples and The intersection graph of intervals of a line No induced cycles of length >4 c b a asteroidal triple a,b,c 9
Arbitrar itrary y k-Chor ordal dal gr graphs phs a graph is 𝑙 - chordal if it has no induced cycles of length greater than 𝑙 . o if 𝑀𝑀 is used for 𝑙 -chordal graphs ( 𝑙 > 3 ), then ecc(v) ≥ 𝑒𝑗𝑏𝑛 𝐻 − 𝑙/2 o 𝑙 = 4𝑚 o 𝑒𝑗𝑏𝑛 𝐻 = 4𝑚 = 𝑙 = 𝑒(𝑏, 𝑐) o 𝑓𝑑𝑑 𝑤 = 2𝑚 + 1 = 4 𝑚 − 2𝑚 +1 = diam(G) − k/2+1 Conclusion: o Full power of LBFS is not needed o Good bounds hold for other graph families 10
Hyperb perbolic olic gr graphs phs 1 2 1 2 1 2 1 2 ℎ𝑐 𝐿 𝑜 = 0 (is a tree metrically) ℎ𝑐 𝑇 4 = 1 • ℎ𝑐 𝐻 = 0 iff 𝐻 is a block graph (metrically a tree) Chordal graphs: ℎ𝑐(𝐻) ≤ 1 [ Brinkmann, Koolen, Moulton: (2001) ] • k-Chordal graphs (k>3): ℎ𝑐(𝐻) ≤ 𝑙 4 [ Wu, Zhang: (2011) ] • o if 𝑀𝑀 is used for 𝜀 -hyperbolic graphs, then ecc(v) ≥ 𝑒𝑗𝑏𝑛 𝐻 − 2𝜀 11
Autonomous Systems Rea eal-Lif ife e datas atasets ts 12
Tal alk k ou outl tlin ine collaborating with Derek o fast estimation of diameters o representing approximately graph distances with few tree distances 13
Defined this object Tree ee t t -Spanne anner Pr Prob oblem em Given unweighted undirected graph G=(V ,E) and integers t, s. Does G admit a spanning tree T =(V ,E’) such that (a multiplicative tree t-spanner of G) u , v V , dist ( v , u ) t dist ( v , u ) T G or (an additive tree s-spanner of G)? u v V dist u v dist u v s , , ( , ) ( , ) T G G multiplicative tree 4- and additive tree 3- spanner of G 14
Some me kn known wn res esult ults s for th the e tr tree ee sp spanner nner pr proble oblem (mostly multiplicative case) general graphs [CC’ 95] t 4 is NP-complete. ( t= 3 is still open, t 2 is P) approximation algorithm for general graphs [EP’ 04] O( log n) approximation algorithm chordal graphs [BDLL’ 02] t 4 is NP-complete. ( t= 3 is still open.) planar graphs [FK’ 01] t 4 is NP-complete. ( t =3 is polynomial time solvable.) AT-free graphs and their subclasses additive tree 3- spanner [Pr’ 99 , PKLMW’ 03] a permutation graph admits a multiplicative tree 3- spanner [MVP’ 96] an interval graph admits an additive tree 2-spanner 15
m Coll llectiv ective Addi ditiv tive e Tree ee r r -Spanner panners s Pr Proble blem ,E) and integers , r. Given unweighted undirected graph G=(V Does G admit a system of collective additive tree r-spanners { T 1 , T 2 …, T } such that u , v V and 0 i , dist ( v , u ) dist ( v , u ) r T i G (a system of collective additive tree r-spanners of G )? surplus , collective multiplicative tree t-spanners can be defined similarly 2 collective additive tree 2- spanners 16
m Coll llectiv ective Addi ditiv tive e Tree ee r r -Spanner panners s Pr Proble blem ,E) and integers , r. Given unweighted undirected graph G=(V Does G admit a system of collective additive tree r-spanners { T 1 , T 2 …, T } such that u , v V and 0 i , dist ( v , u ) dist ( v , u ) r T i G (a system of collective additive tree r-spanners of G )? , 2 collective additive tree 2- spanners
m Coll llectiv ective Addi ditiv tive e Tree ee r r -Spanner panners s Pr Proble blem ,E) and integers , r. Given unweighted undirected graph G=(V Does G admit a system of collective additive tree r-spanners { T 1 , T 2 …, T } such that u , v V and 0 i , dist ( v , u ) dist ( v , u ) r T i G (a system of collective additive tree r-spanners of G )? , 2 collective additive tree 2- spanners
m Coll llectiv ective Addi ditiv tive e Tree ee r r -Spanner panners s Pr Proble blem ,E) and integers , r. Given unweighted undirected graph G=(V Does G admit a system of collective additive tree r-spanners { T 1 , T 2 …, T } such that u , v V and 0 i , dist ( v , u ) dist ( v , u ) r T i G (a system of collective additive tree r-spanners of G )? , , 2 collective additive tree 0- spanners 2 collective additive tree 2- spanners
Ap Applic icati ations ons of of Col ollect lectiv ive e Tree ee Span anne ners message routing in networks Efficient routing schemes are known for trees but not for general graphs. For any two nodes, we can route the message between them in one of the trees which approximates the distance between them. - ( log 2 n)-bit labels, - O( ) initiation, O(1) decision solution for sparse t- spanner problem If a graph admits a system of collective additive tree r - spanners, then the graph admits a sparse additive r- spanner with at most (n-1) edges, where n is the number of nodes. 2 collective tree 2- spanners for G 20
s Some me resul sults ts on co collectiv llective e tr tree e span anne ners chordal graphs, chordal bipartite graphs log n collective additive tree 2 -spanners in polynomial time Ώ (n 1/2 ) or Ώ (n) trees necessary to get +1 no constant number of trees guaranties +2 (+3) circular-arc graphs 2 collective additive tree 2 -spanners in polynomial time k -chordal graphs log n collective additive tree 2 k/2 -spanners in polynomial time interval graphs log n collective additive tree 1-spanners in polynomial time no constant number of trees guaranties +1 21
hs Resul sults ts for AT-free free grap aphs AT-free graphs include: interval, permutation, trapezoid, co-comparability 2 collective additive tree 2 -spanners in linear time an additive tree 3 -spanner in linear time (before) graphs with a dominating shortest path an additive tree 4 -spanner in polynomial time (before) 2 collective additive tree 3 -spanners in polynomial time 5 collective additive tree 2 -spanners in polynomial time graphs with asteroidal number an(G)=k k(k-1)/2 collective additive tree 4-spanners in polynomial time k(k-1) collective additive tree 3-spanners in polynomial time 22
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