Routing, Anycast, and Multicast for Mesh and Sensor Networks

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Routing, Anycast, and Multicastfor Mesh and
Sensor Networks
RAM
Roland Flury Roger Wattenhofer
Distributed
Computing Group
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(No Transcript)
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All-In-One Solution
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Routing in Mesh and Sensor Networks

• Unicast Routing (send message to a given node)
• Multicast Routing (send message to a given set of
  nodes)

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Routing in Mesh and Sensor Networks (2)

• Anycast Routing (send message to any node of a
  given set)

Unicast
Multicast
Anycast
Unicast Anycast Multicast
ALL IN ONE
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Modeling Wireless Networks

• Routing in limited, wireless networks
• Limited Storage
• Limited Power
• No central unit, fully distributed algorithms
• Description of network topology
• Undirected Graph G(V,E)
• Vertices V Set of network nodes
• Edges E present between any two connected nodes
• Wireless Networks
• Nodes tend to be connected to other nodes in
  proximity
• Connectivity graphs constant doubling

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Constant Doubling Metrics

• Ball Bu(r) v v 2 V and dist(u, v) r
• 8 u 2 V, r gt 0 9 S µ Bu(r) s.t.
• 8 x 2 Bu(r) 9 s 2 S dist(x, s) r/2
• S 2? O(1) doubling dimension ?

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Routing, Anycast, and Multicast

• Labeled routing scheme
• (1?)-approximation for Unicast Routing
• Constant approximations to Multicast and Anycast
• -approximate Distance Queries
• Label size O(log ?) (bits)
• ? is the diameter of the network
• Routing table size O(1/?)? (log ?) (O(?) log
  ?) (bits)
• ? is the doubling dimension of the graph (2..5)
• ? is the max degree of any node

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Some Related Work

• Gavoille, Gengler. Space Efficiency for Routing
  Schemes of Stretch Factor Three.
• No labeling and stretch lt 3 requires routing
  tables of size ?(n)

Routing Table (bits) Label (bits)
Talwar, 2004 O(1/(? ?))? log2? ? O(? log ?)
Chan et al. 2005 (? / ?)O(?) log ? log ? O(? log(1/?)) log ?
Slivkins, 2005 ?-O(?) log ? log ? O(? log(1/?)) log ?
Slivkins, 2005 ?-O(?) log ? log log ? log n 2O(?) log n log(?-1 log ?)
Abraham et al. 2006 ?-O(?) log n log(min(?,n)) d log n e
This work O(1/?)? log ? (O(?) log ?) 2O(?) log ?
n Number of nodes in network
? Diameter of network
? Doubling dimension of network
? Max. degree of any node
? Approximation factor for unicast routing
Not only Unicast Routing, but also Multicast,
Anycast, and distance queries distributed
construction
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Outline

• Introduction
• Related Work
• Construction of Labels
• Construction of Routing Tables
• Unicasting
• Multicasting
• Anycasting

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Node Labeling ?-net

• Given a graph G(V,E)
• U ½ V is a ?-net if
• a) 8 v 2 V 9 u 2 U d(u,v) ?
• b) 8 u1, u2 2 U d(u1, u2) gt ?

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Dominance Net Hierarchy

• Build ?-nets for ? 2 1, 2, 4, , 2d log ? e

Level 3
? 8
Level 2
? 4
Level 1
? 2
? 1
Level 0
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Naming Scheme

• Select parent from next higher level
• Parent enumerates all of its children
• At most 22? children
• 2? bits are sufficient for the enumeration
• Name of net-center obtained by concatenation of
  enum values
• Name at most 2? log ? bits long

Root
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Node Labeling

• Each net-center c of a ?-net advertises itself to
  Bc(2?)
• Any node n stores ID of all net-centers from
  which it receives advertisements
• Per level at most 22? net-centers to store
• If net-center c covers n, then also the parent of
  c covers n
• The set of net-centers to store form a tree

• Per level at most 22? 2? bits
• d log ? e levels
• ) Label size of O(log ?) for a constant ?

Level 3
Level 2
Level 1
Level 0
n
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Outline

• Introduction
• Related Work
• Construction of Labels
• Construction of Routing Tables
• Unicasting
• Multicasting
• Anycasting

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Routing Tables

• Routing tables to support (1?) stretch routing
• Recall Routing table size of O(1/?)? (log ?)
  (O(?) log ?) bits
• Every net-center c 2 ?-net of the dominance net
  advertises itself to Bc(? (8/? 6))
• Every node stores direction to reach all
  advertising net-centers

c
Bc(? (8/? 6))
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Routing Tables Analysis

• Each node needs to store direction for at most
  22?(8/? 6)? net-centers per level
• If a node needs to store a routing entry for
  net-center c, then it also needs to store a
  routing entry for the parent of c.
• The routing table can be stored as a tree
• For each net-center, we need to store its
  enumeration value, and the next-hop information,
  which takes at most 2? log ? bits
• Total storage cost is 22?(8/? 6)? log ? (2?
  log ?) bits.

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Unicast Routing

• Problem From a sender node s, send a message to
  a target node t, given the ID and label L(t).
• Algorithm From all net-centers listed in L(t),
  s picks the net-center c on the lowest level to
  which it has routing information and forwards the
  message towards c.
• Main idea Once we reach a first net-center of t,
  we are sure to find a closer net-center on a
  lower layer.
• The path to the first net-center causes only
  little overhead as the net-centers advertise
  themselves quite far.

c1
s
t
c2
c3
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Multicast Routing

• Problem From a sender node s, send a message to
  a set of target nodes U.
• Algorithm Build a Minimum Spanning Tree (MST)
  on s ? U and send the message on this tree.
• Main idea The MST is a 2-approximation to the
  Minimum Steiner Tree problem.
• Our distance estimates may be wrong up to a
  factor 6, and message routing on the tree has
  stretch (1?), which results in a
  12(1?)-approximation.

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Anycast Routing

• Problem From a sender node s, send a message to
  an arbitrary target chosen from a set U.
• Algorithm Determine u 2 U with minimal distance
  to s, and send the message to u.
• Main idea Our distance estimates may be wrong up
  to a factor 6, and message routing on the tree
  has stretch (1?), which results in a
  6(1?)-approximation.

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Summary
(1?) approximation
Unicast
Unicast Anycast Multicast
Multicast
12 (1?) approximation
Anycast
6 (1?) approximation
ALL IN ONE
Label size O(log ?) Routing table
size O(1/?)? (log ?) (O(?) log ?)
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Questions / Comments

• Thank you!
• Questions / Comments?

Roland Flury Roger Wattenhofer
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?-nets on ? Doubling Metrics

• Property 1 (Sparseness) Any ball Bv(2x ?)
  covers at most 2(1x) ? nodes from an arbitrary
  ?-net.

2-net
v
Bv(4?)
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?-nets on ? Doubling Metrics (2)

• Property 2 (Dominance) Given a ?-net, and each
  net-center u covers Bu(2?), then any network node
  is covered by at most 22? net-centers.

2-net
Bu(4)
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Distance Queries

• Problem Determine the distance between two nodes
  u and v, given their labels L(u) and L(v)
• Solution Determine the smallest level i for
  which the labels of u and v have at least one
  common net-center
• Theorem The distance query is a
  -approximation.
• Proof Lower bound dist(u, v) gt 2i -1, otherwise
  L(u) and L(v) have a common net-center on level
  i-1.
• Upper bound By distinction of cases and
  applying triangle inequality (see paper)

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Constant Doubling Metrics