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

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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

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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]

Greedy Tree

Depth 2 Greedy Tree

Depth 3 Greedy Tree

Weight of tree vs. depth bound

Weight of tree vs. depth bound

Sharp threshold for depth bound

Sliced and spliced tree

Lower bound ingredients

Concentration of level weights

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.

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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, …]

Same threshold for Steiner trees(with linear number of terminals)

Everything works for Steiner trees(with linear number of terminals)

Steiner trees withsub-linear number of terminals

Don’t know asymptotic weight when depth bound is

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

Optimization problemswith side-constraints

Side-constraint (depth or diameter bound) has almost no effect on optimization (up to a point)

http://arXiv.org/abs/0810.4908