CS547: Wireless Networking

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CS547 Wireless Networking

• Lecture 3 Ad Hoc Networks Connectivities And
  Power Assignments

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Outline

• Introduction to ad hoc networks
• Approximation algorithms for k-connectivity
• Approximation algorithm for 3-connectivity
• Approximation algorithm for 4-connectivity
• Open problems

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Wireless ad hoc networks
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Ad Hoc Devices
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Smart, Networked Sensors
Riding on Moores law, smart sensors get .
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Sample Sensor Hardware Berkeley motes
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Main Features of Ad Hoc/Sensor Nets

• Limited resouces
• processing and storage
• Communication bandwidth/capacity
• transmitter power/range
• energy
• Self organization
• Distributed algorithms
• Collaborative processing
• Inaccurate/local/partial view
• Low cost
• Flexible, rapid deployment

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Sensor Networks in Military

• Dynamic hostile environment
• battle field
• Large amount of wireless nodes
• sensors and other devices
• Achieve some global goals
• tracking enemy
• Detecting mine, gas,

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Habitat Monitoring http//www.greatduckisland.net
/
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Ecosystem Monitoring CENS, UCLA
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Ubiquitous Sensing will Change the Way People
Live, Work, and Play
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Hybrid Networks
internet
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Network Model

• A weighted graph G (V, E c)
• c(uv) the min. power required by u and v to
  communicated with each other

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Energy conservation via adjustable transmission
range/power
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Power Assignment Induced by H ? G

• Transmission power of u
• Power of H
• Weight of H

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Min-Power Assignment for Property P

• Input A weighted k-connected graph G.
• Output A spanning subgraph H of G with property
  P
• Measure

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k-connectivity

• Independent paths
• k-connected ? any pair of nodes are connected
  by ?k indep. paths
• ? every separator contains ?k nodes

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k-edge-connectivity

• Disjoint paths
• k-edge-connected ? any pair of nodes are
  connected by ?k disjoint paths
• ? every cut contains ?k edges

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Summary on best-known approximation ratios
k-connectivity
k-edge-connectivity 2k
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Definitions And Notations

• bidirected version of G
• bidirected version of H?G
• undirected version of

c(uv)
c(uv)
c(uv)
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Definitions And Notations

• Power Assignment Induced by Transmission power
  of u Power of D
• Weight of D
• Maximum outgoing degree ??(D).

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Weight vs. Weight, Power vs. Power

• For any H?G,
• For any D? ,

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Power vs. Weight

• For any D? , p(D) c(D).
• D is in-arborescence ? p(D) c(D).

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Power vs. Weight

• For any H?G, p(H) 2c(H).
• If H is a forest, then p(H) gt c(H).

T
A
c(T) c(A) p(A) lt p(T).
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k-restricted tree-decompostion

• A tree-decompostion of a tree T is an
  edge-partition of T into subtrees ?T1,, Tq.
• The power cost of ? p(?) p(T1) p(Tq)
• ? is k-restricted if every Tj contains at most k
  nodes.
• 2-restricted tree-decomp. is the set of edges and
  has power c(T)

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k-restricted power ratio

• ?k the supreme, over all trees T, of the ratio
  of the minimum power-cost of k-restricted
  tree-decompositions of T to p(T).
• If k2rs 2 with 0?slt2r, then any tree T has a
  k-restricted tree-decomposition of ? with
• Hence,

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1-connectivity

• MST a minimum spanning tree of G.
• OPT strongly connected spanning subdigraph
• r an arbitrary node
• OPT contains in-arborescence A rooted at r
• c(MST) c(A) p(A) lt opt
• p(MST) 2c(MST) lt 2 ? opt

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Tightness of the approximation ratio 2
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Better approximation algorithms

• Adapted from approximation algorithms for Minimum
  Steiner Tree
• Greedy Fork Contraction (?2 ?3)/211/6
• Stack-based algorithm ?2/2 ?3/6 ?4/316/9
• k-Restricted Relative Greedy Algorithm 1ln?2?
  1ln2?

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k-edge-inconnectivity

• A (di)graph is said to be k-inconnected to a node
  r if it contains k disjoint paths to r from any
  other node.
• Min-weight k-edge-inconnected spanning subgraph
  of a weighted digraph
• solvable in poly. time LawlerLa75,
  EdmondsEd79, GabowGa91

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k-edge-inconnectivity vs. k-edge-connectivity

• k-connectivity ?k-inconnectivity to every node
• D k-edge-inconnected to r ?
  k-edge-connected

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k-Edge-Connectivity Appr. Alg.

• Construct and pick an arbitrary node r.
• Find a min-weight spanning subdigraph D of
  which is k-edge-inconnected to r.
• Output

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k-Edge-Connectivity Appr. Ratio

• OPT strongly k-edge-connected spanning
  subdigraph
• OPT contains k disjoint in-arborescences rooted
  at r A1,, Ak (by Edmonds Theorem)
• A1??Ak k-edge-inconnected to r
• c(D) c(A1) c(Ak) p(A1) c(Ak) lt kopt

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Min-Weight k-Connected Spanning Subgraph

• Input a weighted k-connected graph G.
• Output a k-connected spanning subgraph H of G
• Measure

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Min-Weight k-Connected Augmentation

• Input a weighted k-connected graph G, and H ? G
• Output H ? G such that H ? H is a k-connected
  spanning subgraph of G
• Measure c(H)
• ? MWkCSS G (V, E c)

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k-Connectivity Approximation Algorithm

• Construct the (k-1)-nearest-neighbor graph Gk-1
• Apply an algorithm A for Min-Weight k-CSS to find
  a minimal k-connected augmentation F to Gk-1
• Output Gk-1?F

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k-Connectivity Performance Analysis

• OPT an min-power k-CSS of G, opt p(OPT), ?
  appr. ratio of A.
• ? p(Gk-1) ? k opt, and p(F) ? 2? opt
• ? p(Gk-1?F) ? (k 2?) opt

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p(Gk-1) kopt

• Let D be the (k-1)-nearest-neighbor digraph of
• ? Gk-1 and p(D) opt
• ? p(Gk-1) kp(D) kopt

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Minimal k-Connected Augmentation

• Lemma H a spanning subgraph of G with min.
  degree k-1. F any minimal k-connected
  augmentation to H. ? F is a forest.
• Proof By Mader's theorem on cycle of critical
  edges every cycle of critical edges contains a
  node of degree k.

k-1
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p(F) lt 2? opt

• F?OPT min-weight k-connectivity augmentation
  to Gk-1
• ? F is a forest
• ? c(F) lt p(F) p(OPT) opt
• ? c(F) ? c(F) lt ? opt
• ? p(F) 2c(F) lt 2? opt

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k-inconnectivity

• A (di)graph is said to be k-inconnected to a node
  r if it contains k independent paths to r from
  any other node.
• Min-weight k-inconnected spanning subgraph of a
  weighted digraph
• solvable in poly. time Frank and Tardos FT89,
  GabowGa93
• if in addition the in-degree of the root is
  exactly k, still solvable in poly. time ADNP99

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k-inconnectivity vs. k-connectivity

• k-connectivity ?k-inconnectivity to every node
• H k-inconnected to r ? (k-?degH(r)/2?1)-connecte
  d
• degH(r) k ? (?k/2?1)-connected
• k 2 or 3 ? k-connected

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Independent rooted spanning trees

• Two or more rooted STs of G with the same root r
  are independent if for each v ? r, the (unique)
  paths between v and r along these STs are
  independent.
• Franks Conjecture any k-connected graph
  contains k indep. rooted STs with any common
  root.
• Proved for k 4

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2,3-Connectivity Appr. Alog.

• Construct
• Find a min-weight k-inconnected with
  the in-degree of the root equal to k
• Output

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2,3-Connectivity Appr. Ratio

• OPT has a node r of degree k in OPT (Halins
  theorem Hal69)
• ? OPT contains k indep. STs rooted at r Ti,, Tk
  
• ? Ai in-arborescence obtained by orienting Ti
  toward r
• A1??Ak k-inconnected to r, in-degree of r k
• c(D) c(A1) c(Ak)p(A1) c(Ak) ltkopt

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4-Connected Augmentation to G4

• Find a 4-connected subset S of 4 nodes in G4
• Construct G (V?r, E?rss?S c)
• 4 new edges and all edges of G4 have 0 weight
• Find a min-weight (4,r)-inconnected
• Output

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4-Connectivity Appr. Ratio

• F?OPT min-weight 4-connectivity augmentation
  to G4
• H G4 ? F ?rss?S 4-connected SS of G
• (4,r)-connected
• ?

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Open Problem

• Min-Weight k-Degree Spanning Subgraph
• Solvable in polynomial time
• Min-Power k-Degree Spanning Subgraph
• NP-hard KLL03
• k-NNG is a (k1)-approximation
• Better approximation??
• Geometric properties help?
• Distributed approximation algorithms?