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Survivable Network Design for Wireless Networks

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					Survivable Network Design for
      Wireless Networks
       Debmalya Panigrahi
             MIT
 (Classical) Survivable Network Design
• Input:
  – Graph G = (V, E) with edge/vertex cost
  – Connectivity requirement C
• Output:
  – min-cost subgraph H of G satisfying C
• Examples
  –   Shortest path
  –   Min spanning tree
  –   Steiner tree (Edge-weighted/Node-weighted)
  –   …

                  Survivable Network Design for Wireless Networks
Wireless Networks (Installation)

                                                           Cost


                                                                  Height




         Survivable Network Design for Wireless Networks
Wireless Networks (Power)




      Survivable Network Design for Wireless Networks
                General Framework
Input:
• Graph G = (V, E) [undirected in this paper]
• Domain D of possible parameter values
• Activation function of each edge:
   – fuv (d1, d2) = 1 if xu = d1, xv = d2 activates edge (u, v)
   – Given as truth tables
• Connectivity requirement C

Output:
• Choose xv Є D for each vertex v s.t.
   – activated edges satisfy C
   – Σv xv is minimized

                       Survivable Network Design for Wireless Networks
            Other Applications
• Node-weighted steiner tree [Klein-Ravi ’95]
   – Min-cost tree containing a set of terminals where
     vertices have weights
   – Edge activated if both end-points are chosen
   – fuv(d1, d2) = 1 iff d1 ≥ cu and d2 ≥ cv


• Min connected dominating set [Guha-Khuller ’98]


                 Survivable Network Design for Wireless Networks
                                    Results
                                   Generalized Steiner


Pairwise k-edge                                                              k-edge-connectivity
  connectivity                                                               Bicriteria approx for
 kDS-hardness                                                              important special cases
                                2-edge-connectivity
                                  O(log n)-approx


                                                                            Biconnectivity
                         Steiner Tree                                       O(log n)-approx
                      Ω (log n)-hardness
                       O(log n)-approx


                                                                     Spanning Tree
                   Shortest path
                                                                   Ω (log n)-hardness
                  Exact algorithm
                                                                    O(log n)-approx
                         Survivable Network Design for Wireless Networks
The Basic Technique




    Survivable Network Design for Wireless
                  Networks
                        Set cover
• Universe U of n elements
• Sets S1, S2, …, Sm with costs c1, c2, …, cm
• Find min-cost collection of sets covering U

• Greedy Algorithm [Chvatal ’79]:
   – In each round, add set with min cost-benefit ratio
   – Benefit of a set = # uncovered elements in it


                 Survivable Network Design for Wireless Networks
                      Set cover
• Key Lemma: If at any stage, k elements are left
  uncovered, then the cost-benefit ratio of the
  set added is ≤ OPT/k




             Approx factor O(log n)


               Survivable Network Design for Wireless Networks
Spanning Tree Algorithm




      Survivable Network Design for Wireless
                    Networks
Spanning Tree (Overview)
                               • In each round, add a
                                 star to the activated
                                 subgraph
                                      – Increase parameter
                                        values at central and
                                        peripheral vertices to
                                        activate edges in the star




     Survivable Network Design for Wireless Networks
Spanning Tree (Overview)
                               • In each round, add a
                                 star to the activated
                                 subgraph
                                      – Increase parameter
                                        values at central and
                                        peripheral vertices to
                                        activate edges in the star




     Survivable Network Design for Wireless Networks
Spanning Tree (Overview)
                               • In each round, add a
                                 star to the activated
                                 subgraph
                                      – Increase parameter
                                        values at central and
                                        peripheral vertices to
                                        activate edges in the star




     Survivable Network Design for Wireless Networks
      Spanning Tree (Algorithm)
• A Round of the Algorithm:

  – Activate the star that minimizes the cost-benefit
    ratio

  – Benefit of activating a star: Decrease in # of
    components in activated subgraph



                Survivable Network Design for Wireless Networks
   Spanning Tree (Running Time)
• A round: choosing the optimal star
  – Fix central vertex v and value of xv
  – For each neighbor u of v, interesting values of Δxu:
     (1) 0
     (2) Δx*u = min increase in xu to activate (u, v)
  – Remove redundant edges (u, v)
     • u already connected to v in activated subgraph
     • u and u’ in the same component AND Δx*u’ < Δx*u
  – Order by increasing Δx*u and wlog consider only
    prefixes of the order

                  Survivable Network Design for Wireless Networks
    Spanning Tree (Approx factor)
• Lemma: If k components are present in the
  activated subgraph at any stage of the
  algorithm, there is a star with cost-benefit
  ratio 2 OPT/(k-1).




Approx factor of O(log n) (cf. set cover analysis)
                Survivable Network Design for Wireless Networks
   Spanning Tree (Approx factor)
• Spanning Tree Decomposition:




             Survivable Network Design for Wireless Networks
   Spanning Tree (Approx factor)
• Spanning Tree Decomposition:
                                                               xv (OPT)




                                                 xu (OPT)         …       …


                              Sum of costs of stars ≤ 2 OPT
                              Sum of benefits of stars ≥ k-1
                              Min cost-benefit ratio ≤ 2 OPT/(k-1)

             Survivable Network Design for Wireless Networks
Biconnectivity Algorithm




      Survivable Network Design for Wireless
                    Networks
        Biconnectivity (Outline)
• Algorithm:
  – Stage 1: Run the spanning tree algorithm
  – Stage 2: Run an O(log n)-approximate algorithm
    for increasing connectivity from 1 to 2


• A round of stage 2 of the algorithm:
  – Activate the star that minimizes the cost-benefit
    ratio

                Survivable Network Design for Wireless Networks
                       Notation
• In a connected graph,

  – Block: maximal biconnected component

  – Cut vertex: removal disconnects graph




               Survivable Network Design for Wireless Networks
        Notation
                            Block-cut vertex tree T:
                            • Vertices: blocks B and
                            cut vertices u
                            • Edges: (B, u) for u Є B




                                    Partition number
                                    p = Σu (degT(u) – 1)
                                    (u is a cut vertex)


Survivable Network Design for Wireless Networks
      Biconnectivity (Algorithm)
• Stage 2 of the Algorithm:
  – Benefit of a star:
    Decrease in partition number
    of the activated subgraph




               Survivable Network Design for Wireless Networks
   Biconnectivity (Running Time)
• (Similar to spanning tree)
  – Fix central vertex v and xv
  – Interesting values of xu
    for neighbor u of v:
    (1) min {xu: fuv(xu, xv) = 1}
    (2) current value


• Benefit is non-additive!

                 Survivable Network Design for Wireless Networks
    Biconnectivity (Approx factor)
• Lemma: If the partition number of the
  activated subgraph is k at any stage of the
  algorithm, there is a star with cost-benefit
  ratio 2 OPT/k.




Approx factor of O(log n) (cf. set cover analysis)
                Survivable Network Design for Wireless Networks
               Open Problems
• Directed graphs
  – Some of the problems are challenging even for
    traditional models (e.g. Directed Steiner tree)
  – Min-cost arborescence
  – Min-cost strongly connected subgraph


• Questions?


                Survivable Network Design for Wireless Networks

				
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posted:11/23/2012
language:English
pages:27