Follo lowin ing g De Derek ek's 's foots tstep eps s Feodor - - PowerPoint PPT Presentation

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

Co-comp Tree-width LBFS LDFS Interval Path cover

• Dom. pair
• Dom. path

Linear Efficient NP-hard Partial k-tree Tree spanner Tree power Clique-width Search minors decomposition Perfect graphs cographs domination Cliques AT-free

Derek’s Parallel Universe

 Not a complete picture  I didn’t want to block the Sun

Tal alk k ou

• utl

tlin ine

4

 collaborating with Derek

• fast estimation of diameters
• representing approximately graph distances with few tree

distances

 following Derek's footsteps

• tree- and path-decompositions and new graph parameters
• Approximating tree t-spanner problem using tree-breadth
• Approximating bandwidth using path-length
• Approximating line-distortion using path-length

Tal alk k ou

• utl

tlin ine

5

 collaborating with Derek

• fast estimation of diameters

x y

20 ) , ( ) (   y x d G diam

The e Di Diam ameter er Pr Prob

• blem

lem

6

• The eccentricity ecc 𝑤 = 𝑒𝑗𝑏𝑛 𝐻
• f a vertex 𝑤 is the maximum distance

from 𝑤 to a vertex in 𝐻

• The diameter 𝑒𝑗𝑏𝑛 𝐻 is the

maximum eccentricity of a vertex of 𝐻

• 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 eccentricity)

Ou Our App pproach

• ach

7

 Examine the naïve algorithm of

• choosing a vertex
• performing some version of BFS from this vertex and then
• showing a nontrivial bound on the eccentricity of the last vertex visited in this search.

 This approach has already received considerable attention

• (classical result [Handler’73]) for trees this method produces a vertex of maximum

eccentricity

• [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

• [Corneil et al’99] if LexBFS is used on AT-free graphs, then ecc(v) ≥ 𝑒𝑗𝑏𝑛 𝐻 − 1
• [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

• [Corneil et al’01] considered multi sweep LexBFSs …

Variants riants of BF BFS us used ed

8

u ) 6 , 7 (

8

7 6

) 6 , 7 ( ) 6 ( ) 7 ( ) (

Can be implemented to run in linear time

) (

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

asteroidal triple a,b,c b c a

9

Arbitrar itrary y k-Chor

• rdal

dal gr graphs phs

10

 a graph is 𝑙-chordal if it has no induced cycles of length greater than 𝑙.

• if 𝑀𝑀 is used for 𝑙-chordal graphs (𝑙 > 3), then ecc(v) ≥ 𝑒𝑗𝑏𝑛 𝐻 − 𝑙/2
• 𝑙 = 4𝑚
• 𝑒𝑗𝑏𝑛 𝐻 = 4𝑚 = 𝑙 = 𝑒(𝑏, 𝑐)
• 𝑓𝑑𝑑 𝑤 = 2𝑚 + 1= 4𝑚−2𝑚+1

= diam(G) − k/2+1

• Full power of LBFS is not needed
• Good bounds hold for other graph families

 Conclusion:

Hyperb perbolic

• lic gr

graphs phs

11

• if 𝑀𝑀 is used for 𝜀-hyperbolic graphs, then ecc(v) ≥ 𝑒𝑗𝑏𝑛 𝐻 − 2𝜀
• Chordal graphs: ℎ𝑐(𝐻) ≤ 1 [ Brinkmann, Koolen, Moulton: (2001) ]
• k-Chordal graphs (k>3): ℎ𝑐(𝐻) ≤ 𝑙 4

[ Wu, Zhang: (2011) ]

• ℎ𝑐 𝐻 = 0 iff 𝐻 is a block graph (metrically a tree)

ℎ𝑐 𝑇4 = 1

1 2

ℎ𝑐 𝐿𝑜 = 0 (is a tree metrically)

1 2 1 2 1 2

Autonomous Systems

Rea eal-Lif ife e datas atasets ts

12

Tal alk k ou

• utl

tlin ine

13

 collaborating with Derek

• fast estimation of diameters
• representing approximately graph distances with few tree

distances

G multiplicative tree 4- and

additive tree 3- spanner of G

Tree ee t t -Spanne anner Pr Prob

• blem

em

• Given unweighted undirected graph G=(V

,E) and integers t, s.

• Does G admit a spanning tree T =(V

,E’) such that

) , ( ) , ( , , u v dist t u v dist V v u

G T

   

s v u dist v u dist V v u

G T

    ) , ( ) , ( , ,

(a multiplicative tree t-spanner of G)

• r

(an additive tree s-spanner of G)?

14

Defined this object

Some me kn known wn res esult ults s for th the e tr tree ee sp spanner nner pr proble

• blem

 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(logn) 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

(mostly multiplicative case)

15

Coll llectiv ective Addi ditiv tive e Tree ee r r -Spanner panners s Pr Proble blem m

• Given unweighted undirected graph G=(V

,E) and integers , r.

• Does G admit a system of  collective additive tree r-spanners {T1, T2…, T}

such that

) , ( ) , ( , , r u v dist u v dist i and V v u

G Ti

       

(a system of  collective additive tree r-spanners of G )? 2 collective additive tree 2-spanners collective multiplicative tree t-spanners can be defined similarly

,

surplus

16

Coll llectiv ective Addi ditiv tive e Tree ee r r -Spanner panners s Pr Proble blem m

• Given unweighted undirected graph G=(V

,E) and integers , r.

• Does G admit a system of  collective additive tree r-spanners {T1, T2…, T}

such that

) , ( ) , ( , , r u v dist u v dist i and V v u

G Ti

       

(a system of  collective additive tree r-spanners of G )? 2 collective additive tree 2-spanners

,

Coll llectiv ective Addi ditiv tive e Tree ee r r -Spanner panners s Pr Proble blem m

• Given unweighted undirected graph G=(V

,E) and integers , r.

• Does G admit a system of  collective additive tree r-spanners {T1, T2…, T}

such that

) , ( ) , ( , , r u v dist u v dist i and V v u

G Ti

       

(a system of  collective additive tree r-spanners of G )? 2 collective additive tree 2-spanners

,

Coll llectiv ective Addi ditiv tive e Tree ee r r -Spanner panners s Pr Proble blem m

• Given unweighted undirected graph G=(V

,E) and integers , r.

• Does G admit a system of  collective additive tree r-spanners {T1, T2…, T}

such that

) , ( ) , ( , , r u v dist u v dist i and V v u

G Ti

       

(a system of  collective additive tree r-spanners of G )? 2 collective additive tree 2-spanners

,

2 collective additive tree 0-spanners

,

Ap Applic icati ations

• ns of
• f Col
• llect

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.

• ( log2n)-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

chordal graphs, chordal bipartite graphs

 log n collective additive tree 2-spanners in polynomial time  Ώ(n1/2) 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

Some me resul sults ts on co collectiv llective e tr tree e span anne ners s

21

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

Resul sults ts for AT-free free grap aphs hs

22

Resul sults ts for AT-free free grap aphs hs

Any AT-free graph G admits an additive tree

3-spanner [PKLMW’03]

Thm: Any AT-free graph G admits a system

• f 2 collective additive tree 2-spanners

which can be constructed in linear time.

To get +2, one needs at least 2 spanning

trees

To get +1, one needs at least (n) spanning

trees

an AT-free graph with its backbone

23

2 collective additive tree 2-spanners of G

caterpillar-tree cactus-tree

Resul sults ts for AT-free free grap aphs hs

24

Talk lk out utline ine

25

 collaborating with Derek

• fast estimation of diameters
• representing approximately graph distances with few tree distances

Pa Paper pers s th that t infl fluenced uenced my (later) er) work rk

26

(among many others)

 Graph searches and their algorithmic use AT-free graphs

Pa Paper pers s th that t infl fluenced uenced my (later) er) work rk

27

(among many others)

 Tree spanners, tree powers  Graph decompositions and their parameters  first paper that I got from Derek (long time ago)

Pa Paper pers s th that t infl fluenced uenced my (later) er) work rk

28

(among many others)

 Tree spanners, tree powers  Graph decompositions and their parameters  first paper that I got from Derek (long time ago)

Follo lowing wing De Derek' ek's s foots tsteps eps

29

LBFS  Derek’s journey  How I envision it

Follo lowing wing De Derek' ek's s foots tsteps eps

30

Tal alk k ou

• utl

tlin ine

31

 collaborating with Derek

• fast estimation of diameters
• representing approximately graph distances with few tree

distances

 following Derek's footsteps

• tree- and path-decompositions and new graph parameters
• Approximating tree t-spanner problem using tree-breadth
• Approximating bandwidth using path-length
• Approximating line-distortion using path-length

Tal alk k ou

• utl

tlin ine

32

following Derek's footsteps

• tree- and path-decompositions and new graph parameters
• Approximating tree t-spanner problem using tree-breadth

 Graph decompositions and their parameters +  Tree spanners = =

Tree ree-Decom Decomposi position tion

 Tree-decomposition 𝑈(𝐻) of a graph 𝐻 = (𝑊, 𝐹) is a pair

𝑌𝑗: 𝑗 ∈ 𝐽 , 𝑈 = (𝐽, 𝐺) where 𝑌𝑗: 𝑗 ∈ 𝐽 is a collection of subset of V (bags) and 𝑈 is a tree whose nodes are the bags satisfying:

1)

𝑌𝑗 = 𝑊

𝑗∈𝐽

2)

∀ 𝑣𝑤 ∈ 𝐹, ∃ 𝑗 ∈ 𝐽 𝑡. 𝑢. 𝑣, 𝑤 ∈ 𝑌𝑗

3)

∀ 𝑤 ∈ 𝑊, 𝑢ℎ𝑓 𝑡𝑓𝑢 𝑝𝑔 𝑐𝑏𝑕𝑡 𝑗 ∈ 𝐽, 𝑤 ∈ 𝑌𝑗 𝑔𝑝𝑠𝑛 𝑏 𝑡𝑣𝑐𝑢𝑠𝑓𝑓 𝑈

𝑤 𝑝𝑔 𝑈

[ Robertson, Seymour ]

33

Tree ree-Decom Decomposi position tion and Gr Graph ph Param rameter ers

 Tree-width 𝒖𝒙(𝑯):  Width of 𝑈 𝐻 is max

𝑗∈𝐽 𝑌𝑗 − 1

 𝒖𝒙(𝑯): minimum width over all tree-decompositions  Tree-length 𝒖𝒎(𝑯):  Length of 𝑈 𝐻 is max

𝑗∈𝐽

max

𝑣,𝑤∈𝑌𝑗 𝑒𝐻(𝑣, 𝑤)

 𝒖𝒎(𝑯): minimum length over all tree-decompositions  Tree-breadth 𝒖𝒄(𝑯):  Breadth is minimum 𝑠 such that ∀𝑗 ∈ 𝐽, ∃𝑤𝑗 with 𝑌𝑗

⊆ 𝐸𝑠(𝑤𝑗, 𝐻)

 𝒖𝒄 𝑯 : minimum breadth over all tree-decompositions

Tree-length was introduced in [ Dourisboure, Gavoille: DM (2007) ] and [ Dragan,Lomonosov: DAM (2007) ] Tree-breadth was introduced in [ Dragan,Lomonosov: DAM (2007) ] and [ Dragan, Köhler: APPROX (2011) ] (R,D)-acyclic clustering 34

 Tree-width 𝒖𝒙(𝑯):  Width of 𝑈 𝐻 is max

𝑗∈𝐽 𝑌𝑗 − 1

 𝒖𝒙(𝑯): minimum width over all tree-decompositions  Tree-length 𝒖𝒎(𝑯):  Length of 𝑈 𝐻 is max

𝑗∈𝐽

max

𝑣,𝑤∈𝑌𝑗 𝑒𝐻(𝑣, 𝑤)

 𝒖𝒎(𝑯): minimum length over all tree-decompositions  Tree-breadth 𝒖𝒄(𝑯):  Breadth is minimum 𝑠 such that ∀𝑗 ∈ 𝐽, ∃𝑤𝑗 with 𝑌𝑗

⊆ 𝐸𝑠(𝑤𝑗, 𝐻)

 𝒖𝒄 𝑯 : minimum breadth over all tree-decompositions

• ∀ 𝐻, 𝑢𝑐(𝐻) ≤ 𝑢𝑚(𝐻) ≤ 2𝑢𝑐(𝐻) as ∀ 𝑇𝑊(𝐻), 𝑠𝑏𝑒𝐻 𝑇 ≤ 𝑒𝑗𝑏𝑛𝐻 𝑇 ≤ 2𝑠𝑏𝑒𝐻 𝑇
• 𝑢𝑥 𝐻 and 𝑢𝑚(𝐻) are not comparable (check cycles and cliques)

𝑢𝑥(𝐷3𝑙) = 2, 𝑢𝑚(𝐷3𝑙) = 𝑙 𝑢𝑥(𝐿𝑜) = 𝑜 − 1, 𝑢𝑚(𝐿𝑜) = 1

Tree ree-Decom Decomposi position tion and Gr Graph ph Param rameter ers

35

Tree ee-str stretch tch vs tree s tree-brea eadth dth

36

• Given unweighted undirected graph G=(V

,E) and integer t.

• Does G admit a spanning tree T =(V

,E’) such that

) , ( ) , ( , , u v dist t u v dist V v u

G T

   

Tree t-spanner problem:

Tree ree sp span anner ners s in n boun unded ded tre ree-bread breadth th gr graph aphs

37

Ap Approximatin ximating g tree ree t-sp spann anner er probl blem em in n ge gene nera ral l un unweight eighted ed gr graphs aphs

38

Ou Our resu esults vs ts vs kno nown wn resu esults ts

39

Autonomous Systems

Rea eal-Lif ife e datas atasets ts

40

Tal alk k ou

• utl

tlin ine

41

following Derek's footsteps

• Approximating bandwidth using path-length
• Approximating line-distortion using path-length

 Graph decompositions and their parameters +  AT-free graphs = =

[ F. Dragan, E. Köhler, A. Leitert: Line-distortion, Bandwidth and Path-length of a graph, SWAT 2014 ]

Path-Decom Decompositi position

• n

 Path-decomposition 𝑄(𝐻) of a graph 𝐻 = (𝑊, 𝐹) is a pair

𝑌𝑗: 𝑗 ∈ 𝐽 , 𝑄 = (𝐽, 𝐺) where 𝑌𝑗: 𝑗 ∈ 𝐽 is a collection of subset of V (bags) and 𝑄 is a path whose nodes are the bags satisfying:

1)

𝑌𝑗 = 𝑊

𝑗∈𝐽

2)

∀ 𝑣𝑤 ∈ 𝐹, ∃ 𝑗 ∈ 𝐽 𝑡. 𝑢. 𝑣, 𝑤 ∈ 𝑌𝑗

3)

∀ 𝑤 ∈ 𝑊, 𝑢ℎ𝑓 𝑡𝑓𝑢 𝑝𝑔 𝑐𝑏𝑕𝑡 𝑗 ∈ 𝐽, 𝑤 ∈ 𝑌𝑗 𝑔𝑝𝑠𝑛 𝑏 𝑡𝑣𝑐𝑞𝑏𝑢ℎ 𝑝𝑔 𝑄

[ Robertson, Seymour ]

42

Path-Decom Decompositi position

• n and

d new Gr Graph ph Param rameter ers

 path-width 𝒒𝒙(𝑯):  Width of 𝑄 𝐻 is max

𝑗∈𝐽 𝑌𝑗 − 1

 𝒒𝒙(𝑯): minimum width over all path-decompositions  path-length 𝒒𝒎(𝑯):  Length of 𝑄 𝐻 is max

𝑗∈𝐽

max

𝑣,𝑤∈𝑌𝑗 𝑒𝐻(𝑣, 𝑤)

 𝒒𝒎(𝑯): minimum length over all path-decompositions  path-breadth 𝒒𝒄(𝑯):  Breadth is minimum 𝑠 such that ∀𝑗 ∈ 𝐽, ∃𝑤𝑗 with 𝑌𝑗 ⊆

𝐸𝑠(𝑤𝑗, 𝐻)

 𝒒𝒄 𝑯 : minimum breadth over all path-decompositions

43

Line e dist stor

• rtion

tion and ba bandwidth ndwidth

 Line-distortion 𝒎𝒆 𝑯 :

: 𝒈: 𝑾 → 𝒎 with minimum k such that ∀𝑦, 𝑧 𝜗 𝑊

 Non-contractiveness: 𝑒𝐻 𝑦, 𝑧 ≤ |𝑔 𝑦 − 𝑔(𝑧)|  minimum distortion k : |𝑔 𝑦 − 𝑔(𝑧)| ≤ 𝑙 𝑒𝐻 𝑦, 𝑧  Bandwidth 𝒄𝒙 𝑯 :

𝒄: 𝑾 → 𝑶 with minimum k such that ∀𝑦𝑧 𝜗 𝐹

 minimum bandwidth k : 𝑐 𝑦 − 𝑐 𝑧

≤ 𝑙

a b c d e f g h a b c d e f g h 6 7

44

Line e dist stor

• rtion

tion and ba bandwidth ndwidth

 Line-distortion 𝒎𝒆 𝑯 :

: 𝒈: 𝑾 → 𝒎 with minimum k such that ∀𝑦, 𝑧 𝜗 𝑊

 Non-contractiveness: 𝑒𝐻 𝑦, 𝑧 ≤ |𝑔 𝑦 − 𝑔(𝑧)|  minimum distortion k : |𝑔 𝑦 − 𝑔(𝑧)| ≤ 𝑙 𝑒𝐻 𝑦, 𝑧  Bandwidth 𝒄𝒙 𝑯 :

𝒄: 𝑾 → 𝑶 with minimum k such that ∀𝑦𝑧 𝜗 𝐹

 minimum bandwidth k : 𝑐 𝑦 − 𝑐 𝑧

≤ 𝑙

a b c d e f g h a b c d e f g h 6 7

𝑐𝑥 𝐻 ≤ 𝑚𝑒 𝐻 𝑐𝑥 𝐷𝑙 = 2 𝑚𝑒 𝐷𝑙 = 𝑙 − 1

k=5

45

Line e dist stor

• rtion

tion and ba bandwidth ndwidth

 Line-distortion 𝒎𝒆 𝑯 :

: 𝒈: 𝑾 → 𝒎 with minimum k such that ∀𝑦, 𝑧 𝜗 𝑊

 Non-contractiveness: 𝑒𝐻 𝑦, 𝑧 ≤ |𝑔 𝑦 − 𝑔(𝑧)|  minimum distortion k : |𝑔 𝑦 − 𝑔(𝑧)| ≤ 𝑙 𝑒𝐻 𝑦, 𝑧  Bandwidth 𝒄𝒙 𝑯 :

𝒄: 𝑾 → 𝑶 with minimum k such that ∀𝑦𝑧 𝜗 𝐹

 minimum bandwidth k : 𝑐 𝑦 − 𝑐 𝑧

≤ 𝑙

a b c d e f g h a b c d e f g h 6 7

46

Hard to approximate within a constant factor Hard to approximate within a constant factor

47

Li Line ne-di dist stor

• rti

tion

• n vs pat

s path-length length

   Line-distortion is hard to approximate within a constant factor

Pat ath-length ength an and A d AT-free ee grap aphs hs

48

Li Line ne-di dist stor

• rti

tion

• n vs pat

s path-length length

     Line-distortion is hard to approximate within a constant factor

Ap Approximatin ximating g li line ne-dist distor

• rtion

ion

    k ≤ 𝑞𝑚 𝐻 ≤ 𝑚𝑒 𝐻

49 ([BDGRRRS: SODA’05])

hard to approximate within a constant factor in general graphs

Ban andwi dwidth dth ap approximat ximation ion

50

   hard to approximate within a constant factor in general graphs k ≤ 𝑞𝑚 𝐻 ≤ 𝑚𝑒 𝐻

AT AT-free free gr graph aphs

51

  

52