Models and Techniques for
Communication in
Dynamic Networks
Christian Scheideler
Dept. of Computer Science
Johns Hopkins University
Dynamic networks
Peer-to-peer
Wireless
Faulty networks
networks
networks
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Overview
In this talk:
Basic problems
Basic assumptions
Faulty networks
Wireless networks
Adversarial networks
More can be found in the paper…
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Basic Problems - Connectivity
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Basic Problems – Routing
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Basic Problems – Admission
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Basic Assumptions
All packets are atomic
(unsplittable, uncompressable)
Time proceeds in synchronous steps
Only point-to-point connections (links)
Each link can forward one packet per step
in each direction
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Focus of Talk
Communication in dynamic networks
(connectivity, routing, admission control)
Online, distributed algorithms


online: no knowledge about future
distributed: runs locally at each node
Adversarial or random network changes

adversary: tries to harm algorithm
Adversarial packet injections
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Faulty Networks
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Faulty Networks
Basic models:
Adversarial or random node/edge faults
Central questions:
What repair rate needed to ensure


large connected component
similar routing properties
How to ensure uninterrupted service
How to achieve a high throughput
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Large Connected Components
Information dispersal [Rabin 89]:
no information lost
Assumption: random edge faults
Is there critical survival probability p* s.t.
p>(1+e)p* : constant fraction CC
p<(1-e)p* : no constant fraction CC
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Large connected components
p*<=[Ajtai,
1/(n-1)
[Erdos
and
Renyi
0.337
p*=1/d
< p*
0.436
p*=1/2
Komlos
[Karlin,
p*=1/d
[Kesten
and
Nelson
Szemeredi
80]
and 60]
Tamaki
82] 94]
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Large connected components
Robustness of graph of ave. degree d:
r = d . p*
complete graph: r = 1
d-dimensional hypercube: r = 1
random graph of dn/2 edges: r = 1
d-dimensional butterfly: 1.348 < r < 1.744
2-dimensional mesh: r = 2
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Structurally robust networks
When can faulty network still keep original
routing properties?
When can faulty network still simulate original
network with (amortized) constant slowdown?
Constant slowdown:
1 step in original - O(1) steps in faulty network
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Simulations with O(1) slowdown
1-eworst-case
upup
tonto
n1-e
worst-case
faults
Maggs
and
Sitaraman
98]
up
to
[Leighton,
Maggs
and
Sitaraman
97]
failure
with
any
const.faults?
p<1
OK[Leighton,
[Hastad,
Leighton
and
Newman
87]
n1-e
worst-case
faults
[Cole,
Maggs
and
Sitaraman
97]
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Open questions
Hypercube very robust!
What about constant degree networks??
How to exploit knowledge about
robustness of networks for the design of
efficient routing protocols?
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Fault-tolerant circuit switching
How can uninterrupted service be ensured?
Use more than one path.
Allocate more bandwidth than needed.
A
B
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Fault-tolerant circuit switching
k-edge disjoint paths problem (k-EDP):
Given a graph G and a set T of pairs of nodes, find
a maximum set of pairs s.t. each pair is connected
by k disjoint paths and all paths are disjoint.
k-disjoint flows problem (k-DFP):
Given a network G with edge capacities and a set T
of pairs of nodes with demands d i , find a set of
pairs of maximum total demand s.t. pair i in set is
connected by k disjoint paths with flow d i /k and all
capacity constraints are kept.
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Fault-tolerant circuit switching
Bounded greedy algorithm (BGA)
For each pair: if k disjoint paths available that
use at most L edges: take them
[Bagchi & Chaughari & Kolman & Sch. 02]
L= O(k 3D a -1 log n):
BGA is O(k D a log n) - competitive
3
-1
D: maximum degree, a: expansion, n: number of nodes
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Wireless ad hoc networks
How How
to send
How
to select
packets
to ensure
andalong
connectivity?
maintain
selected
routes?
routes?
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Connectivity
[Yao 82, Ruppert & Seidel 91,Lukovszki 99]:
partition space into sectors
connect to nearest neighbor in each sector
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Route selection
Strategy:
Always go to neighbor that lies in sector of
destination (if possible…)
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Wireless ad hoc networks
Problem with sector-based approach:
Nodes have to know location (GPS)
Solution when nodes move at random:
Just connect to k nearest neighbors
What if nodes have limited range??
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Adversarial networks
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Adversarial networks
Situation
Network and packet injections under
adversarial control
Problems
How to select paths?
How to schedule packet movements?
How to perform admission control?
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Adversarial networks
“Friendly” adversary
All injected packets can be delivered
Buffer size for this at most B
Only gathering scenario
All packets have the same destination
Is there online algorithm s.t.
number of packets in system finite?
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Adversarial gathering
Balancing algorithm:
For every time step t and every edge (v,w)
#packets(v) - #packets(w) > T: send packet
receive all incoming and injected packets
[Awerbuch et al 89, Aiello et al 98,…]
v
w
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Adversarial gathering
[Awerbuch & Berenbrink & Brinkmann & Sch. 01]
For T>2D, number of packets in node at most
(B+T)(n-1)+B
where D is maximum degree of node.
Bound tight up to constant factor.
How about more complex communication modes?
(unicasting, anycasting, multicasting,…)
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Communication modes
Anycasting
Unicasting
Multicasting
: one
optional
: multiple
destination
set destinations
of destinations
per packet
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Anycasting and multicasting
Approach:
Adversary controls topology, injections,
now unfriendly
Compare number of packet deliveries
(throughput) with optimal algorithm
Algorithm is (c,s)-competitive:
Reaches c-fraction of optimal throughput
with s times more buffer space
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Anycasting
Anycasting can be reduced to gathering!
Option set
Originial
model:
model:
each node
buffer:
virtual
has
node
buffer for each anycast set
each step
edge:
set of
adversary
virtual edges
offersbetween
edges corr. buffers
combine all virt. dest. nodes in single dest.
v
w
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Gathering in option set model
Balancing algorithm:
For every time step t and every option set S
select edge (v,w) with largest #pack(v)-#pack(w)
#pack(v) - #pack(w) > T: send packet
receive all incoming and injected packets
(if not possible: delete packet)
[Awerbuch, Brinkmann and Sch. 02]
balancing is (1-e,O(L/e)) – competitive
(L: ave. path length used by packets in OPT)
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Multicasting
Major problem:
Multicast packets can split
Approach:
Option set model for anycasting can be
generalized to multicasting
Balancing algorithm can be extended
accordingly
Balancing (1-e, O(f(e)))-competitive
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Adversarial networks
Major problem:
Solutions show feasibility but FAR from
being practical
Possible solutions:
Use clustering strategies
Combine balancing with other methods
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Conclusion
Routing in dynamic networks is interesting
(also systems people are searching in dark)
Techniques and models are available
Many open problems left:


Robust, fault-tolerant networks of constant
degree
Routing in fault-tolerant and wireless networks
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