a sharp threshold for minimum bound-depth/diameter spanning and steiner trees in random networks
DESCRIPTION
A sharp threshold for minimum bound-depth/diameter spanning and Steiner trees in random networks. arXiv:0810.4908. Omer Angel Abraham Flaxman David B. Wilson. U British Columbia U Washington Microsoft Research. Minimum spanning tree (MST). - PowerPoint PPT PresentationTRANSCRIPT
A sharp threshold forminimum bound-depth/diameter
spanning and Steiner treesin random networks
Omer Angel Abraham Flaxman David B. Wilson
U British Columbia U Washington Microsoft Research
arXiv:0810.4908
3
24
1
7
6
5
Minimum spanning tree (MST)• Graph with nonnegative edge weights• Connected acyclic subgraph,
minimizes sum of edge weights (costs)• Classical optimization problem
electric network, communication network, etc.• Efficiently computable:
Prim’s algorithm (explore tree from start vertex) Kruskal’s algorithm (add edges in order by weight)
MST on graph with random weightsWeight distribution irrelevant to MST
4 trees like 12 trees like
1 2
3
1 2
3
4
Clique K4
MST on graph with random weights
• Weight distribution irrelevant to MST• Not same as uniform spanning tree (UST)
(e.g. non-uniform on K4)• Diameter of MST on Kn is (n1/3)
[Addario-Berry, Broutin, Reed]• Diameter of UST on Kn is (n1/2) [Rényi, Szekeres]• Weight of MST with Exp(1) weights on Kn tends to
(3) a.a.s. [Frieze]• PDF of edge weights 1 at 0 weight (3) [Steele]
Minimum bounded-depth/diameterspanning tree
• Data in communication network, delay for each link, put a limit on number of links.
• Also known as “MST with hop constraints”• Tree with depth k from specified root has diameter
2k. Tree with diameter 2k has “center” from which depth is k
• NP-hard for any diameter bound between 4 and n-2, poly-time solvable for 2,3, & n-1 [Garey & Johnson]
• Inapproximable within factor of O(log n) unless P=NP [Bar-Ilan, Kortsarz, Peleg]
Minimum Steiner tree• In addition to graph, set of terminals is
specified. Tree must connect terminals, may or may not connect other vertices.
• Another classical optimization problem.• NP-hard to solve.
3
2
4
1
7
6
5
Steiner trees on Kn
When there are m terminals and Exp(1) weights, the Steiner tree weight tends to
when 2 m o(n)[Bollobás, Gamarnik, Riordin, Sudakov]
When m=(n), weight is unknown constant
Minimum bounded-depth/diameterSteiner tree
• Generalizes two different NP-hard problems, is NP-hard
• Solvable by integer programming [Achuthan-Caccetta, Gruber-Raidl]
• Fast approximation algorithms [Bar-Ilan- Kortsarz-Peleg, Althus-Funke-Har-Peled- Könemann-Ramos-Skutella]
• Heuristics [Abdalla-Deo-Franceschini, Dahl-Gouveia-Requejo, Voß, Gouveia, Costa-Cordeauc-Laporte, Raidl-Julstrom, Gruber-Raidl, Gruber-Van-Hemert-Raidl, Kopinitsch, Putz, Zaubzer, Bayati-Borgs-Braunstein-Chayes-Ramezanpour-Zecchina, …]
Minimum bounded depth/diameter spanning subgraph
• If depth-constrained, best subgraph is a tree, we give minimum weight
• If diameter-constrained, best subgraph is not a tree, possible to get smaller weight