Capacity Assignment in Bluetooth Scatternets – Analytical
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Transcript Capacity Assignment in Bluetooth Scatternets – Analytical
Multipath Routing
Ph.D. Research Proposal
Ron Banner
Supervisor: Prof. Ariel Orda
March 2004
Agenda
Introduction & summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
What is Multipath Routing?
Multipath Routing is the method of establishing multiple paths
between given source-destination nodes within the network.
Advantages of Multipath Routing
Survivability
Provides redundancy.
Congestion avoidance
Improves network utilization.
Provides load balancing.
Management and control
Provides better performance in the presence of selfish/unregulated
behavior
Previous Research
Survivability
Mainly solutions that focus on the establishment of pairs of disjoint
paths (e.g., the 1+1 and 1:1 protection architectures).
Congestion avoidance
Mainly heuristics (e.g., ECMP).
Online: no previous work for multipath routing.
Management and control
No previous work on the degradation of network performance due to
selfish behavior of users that employ multipath routing.
Notations
G (V,E) – Directed Graph
V - Collection of nodes
E – Collection of links (edges)
P(s,t) -Collection of all paths from s to t
(s,t) –flow demand from s to t
de-delay of link e
ce-capacity of link e
pe-failure probability of link e
fe-flow rate on link e
D(p) – the end-to-end delay of path p i.e., D p de .
e p
ce .
C(p) – the capacity of path p i.e., C p min
e p
P(p) – the reliability of path p i.e., P p 1 pe .
eE
Summary of results:
Survivability
We provide a quantitative framework that specifies the
desired level of survivability against single failures.
c=30, p=0.05
S
c=30,
p=0.05
c=30, p=0
T
Summary of results:
Survivability
We developed optimal polynomial schemes for 1:1 and 1+1
protection that consider important tradeoffs
Survivability vs. bandwidth
Survivability vs. feasibility.
…
No need to establish connections that consist of more than two
paths.
Derived a new “hybrid” protection architecture that has several
advantages over both the 1:1 and 1+1 protection architecture.
Show that by just slightly alleviating the requirement of full
survivability a major improvement is obtained.
Summary of results:
Congestion minimization-offline
Goal: Minimize network congestion when all demands are known in
advance.
Cope with constraints
Delay jitter
End-to-end delay
Number of paths
Minimizing the congestion under end-to-end delay and/or delay jitter
NP-hard
Pseudo polynomial solution
eoptimal approximation scheme
Minimizing the congestion while restricting the number of routing paths
NP-hard
2-approximation scheme
Summary of results :
Congestion minimization-online
Goal: Minimizing the network congestion when
demands arrive one at a time.
Derived a multipath routing algorithm for congestion
minimization with an O(logN)-competitive ratio.
Derived a lower bound of Ω(logN) for any online
multipath routing algorithm for congestion
minimization
Our algorithm is best possible.
Summary of results:
Selfish multipath routing
Goal: Investigating the degradation in network performance
due to selfish behavior of users.
Given a load-dependent performance function qe(fe) for each
link we consider bottleneck network objectives i.e., MaxeE{qe(fe)}
and additive network objectives i.e., qe fe .
eE
Assume that users are selfish and their performance is
dictated by their worst (bottleneck) elements.
Network
objective
Routing
approach
Bottleneck
Additive
Multipath
Routing
Single-path
Routing
1
∞
M
∞
Agenda
Introduction & summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The tunable survivability concept
Current survivability schemes typically offer two degrees of
protection against single failures
Full (100%) protection.
No protection at all.
In practice, the requirement of full protection is often too
restrictive
In many cases it is infeasible (N. Taft-Plotkin, B. Bellur and R. Ogier).
In other cases it is very limiting (G. Maier, A. Pattavina, S. De Patre and M. Martinelli).
Tunable survivability enables to consider valuable tradeoffs.
Survivability vs. bandwidth
Survivability vs. feasibility
Survivability vs. end-to-end delay
…
Survivable connections
p-survivable connection: a collection of paths (p1,p2,…, pk)P(s,t)×P(s,t) ×…× P(s,t)
that, upon a link failure, has a probability of at least p that at least one
path out of (p1,p2,…, pk) remains operational.
The bandwidth of a survivable connection with respect to the 1+1 protection
architecture is the maximum B≥0 such that n·B≤ce for each link e that is
common to n paths from (p1,p2,…, pk).
The bandwidth of a survivable connection with respect to the 1:1 protection
architecture is the maximum B≥0 such that B≤ce for each e that belongs to
a path in (p1,p2,…, pk).
It is also
min
ep1 p2 pk
ce .
The probability of a survivable connection to remain operational upon a
single failure is the probability that all the common links are operational
upon that failure i.e.,
1- pe .
ep1 p2 pk
Two Paths are Enough
Theorem Let (p1,p2,…, pk)P(s,t)×P(s,t) ×…×P(s,t) be a p-survivable
connection. There exists a p-survivable connection p1 , p 2 P s,t × P s,t
that has at least the bandwidth of (p1,p2,…, pk) with respect to
the 1:1 (alternatively 1+1) protection architecture.
Proof (sketch for the 1:1 protection)
We shall construct p1 , p2 only from the links that belong to paths in (p1,p2,…,
pk). Therefore, the bandwidth of p1 , p2 is at least that of (p1,p2,…, pk).
Formal proof
Critical points
Most Survivable Connections with a Bandwidth of at Least B
Since two paths are enough, we focus on survivable connection that
consist of two paths.
The most survivable connection with a bandwidth of at least B for the
1+1 protection architecture is established by a reduction to the min
cost flow problem.
Links in the transformed network
Discard the link
A link in the original network
ce,pe
B≤Ce<2∙B
ce=B, we=0
ce=B, we=0
ce=B, we=-ln(1-pe)
The flow demand is set to 2∙B flow units.
Most Survivable Connections with a Bandwidth of at Least B
Since the flow demand and capacities are B-integral the min
cost flow is B-integral.
The flow decomposition algorithm can be applied in order to
decompose the B-integral link flow (that transfers 2·B flow
units) into a flow over two paths: p1, p2 such that f(p1)=f(p2)=B.
Since the flow has a minimum cost,
has a minimum value.
f
eE
e
we
ep1 p1
B ln 1 pe
Therefore, (p1,p2 ) is a connection with a bandwidth of at least B
that maximizes ln 1 pe ln 1 pe ; hence, it maximizes
ep1 p2
1 pe .
ep1 p1
ep1 p1
Establishing Most and Widest p-survivable Connections
The most survivable connection is the connection that has the
maximum probability to remain operational upon a failure
It is also the most survivable connection with a bandwidth of at least B=0.
The widest p-survivable connection is the p-survivable connection
with the maximum bandwidth.
How to establish the widest p-survivable connection?
Idea: search for the largest B such that the most survivable
It is enough to perform a binary search over the set ce e E , k 1,2 .
connection with a bandwidth of at least B is a p-survivable
connection.
Why
k
The widest p-survivable connection is therefore established within
O(logN) executions of any min cost flow algorithm.
Why
Establishing Survivable Connections for 1:1 protection
The only difference in the reduction lies for the links that have
capacities in the range [B,2B].
For 1:1 protection only one of the paths carries B flow units.
Hence, all links that have a capacity in the range [B,2B] can
concurrently be employed by both paths.
Links in the transformed network
A link in the original network
Discard the link
ce,pe
ce=B, we=0
ce=B, we=-ln(1-pe)
Go to 1+1
reduction
The Hybrid protection architecture
The tunable survivability concept gives rise to a third protection
architecture.
S
T
Reduces the congestion of all links that are shared by both
paths w.r.t 1+1 protection.
Upon a link has a faster restoration w.r.t 1:1 protection.
Provides the fastest propagation of data.
However, requires additional nodal capabilities.
The Hybrid protection architecture
The hybrid architecture transfers through each link exactly one
duplicate of the original traffic.
Hence, the bandwidth of (p1,p2) with respect to hybrid protection
is min ce .
ep1 p2
Hence, by definition, all schemes for 1:1 protection apply for
hybrid protection.
Go to Def
Simulation results
We quantify how much we gain by employing tunable survivability
instead of full survivability.
Random networks
10,000 Waxman topologies
10,000 Power-law topologies.
Explain the construction
2.4
2.2
Bandwidth ratio (1:1)
2
1.8
1.6
1.4
1.2
1
0.8
95
96
97
98
level of survivability p
Power-Law
Waxman
99
100
Simulation results
3
2.8
2.6
Feasibility ratio
Bandwidth ratio (1+1)
1.6
1.4
1.2
1
0.8
95
96
97
98
99
100
2.4
2.2
2
1.8
1.6
1.4
1.2
1
95
level of survivability p
Power-Law
Waxman
96
97
98
degree of survivability p
Power-Law
Waxman
99
100
Agenda
Introduction & summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Problem formulation
Goals:
Minimize network congestion when all demands are known in
advance
Cope with constraints (delay-jitter, delay, number of paths)
Performance Objective: network congestion factor
Minimizing
f
max e .
eE
ce
RFC 2702 and others.
No link becomes over-utilized.
More room for future traffic growth by maximizing the
common scaling factor.
Requirements for practical deployment
Restricting the delay-jitter among all routing paths
RFC 2991.
Avoid the “fast retransmit” mode.
Reduce buffering requirements.
Limiting the number of paths per destination
S. Nelakuditi and Zhi-Li Zhang.
Reduce the tendency of packet reordering.
Reduce overhead.
Simplify the schemes that distribute traffic.
Bounding the end-to-end delay of each path.
Computational Intractability
Minimizing the network congestion factor under the
end-to-end delay restriction is NP- hard.
Proof
Minimizing the network congestion factor under the
delay jitter restriction is NP- hard.
Proof
.
.
Minimizing the network congestion factor under the
restriction on the number of paths is NP-hard.
Proof
.
Minimizing congestion while restricting the number of paths
Observation: The optimal network congestion factor of a /Kintegral path flow is larger by a factor of at most 2 than the
optimal network congestion factor of a path flow that admits at
most K paths.
Proof:
Let f* be a path flow that has the
Given a network
G(V,E) and a sourcedestination pair.
Round down the flow f(p) over each
path to a multiple of /K. Let fR be the
resulting path flow.
Since f transfer 2 flow units over at
most K paths fR transfers at least flow
units from S to T
smallest network congestion factor α*
among all path flows that transfers flow
units from S to T over at most K paths.
f=2∙f* is a path flow with a network
congestion factor 2∙α* that transfers
2 flow units from S to T over at most
K paths.
fR is a /K - integral path flow that
transfers at least flow units from S to
T and has a network congestion factor
of at most 2∙ α*.
Minimizing the congestion under integrality restrictions
A /K-integral path flow admits at most K paths.
Corollary: minimizing the congestion while restricting
the flow to be integral in /K is a 2-approximation
scheme.
The network congestion factor of all /K-integral path
flows belong to i e E, i 0, K .
K ce
The flow over each link is integral in /K and is at most .
Hence, for each eE it holds that
In particular,
fe
i
i 0, K .
ce K ce
f
max e i
e E , i 0, K .
eE
ce K ce
Minimizing the congestion under integrality restrictions
Goal: Find a /K-integral path flow that has the minimum network
congestion factor in i
Solution
e E , i 0, K .
K ce
Find a path flow with the smallest
e E , i 0, K
K ce
i
the following procedure succeeds.
multiply all link capacities by a factor of α.
Round down the capacity of each link to a multiply of /K.
such that
Since the flow must be /K-integral, such a rounding has no affect.
Apply a maximum flow algorithm that returns a /K-integral link flow
when all capacities are integral in /K.
If the link flow transfers flow units from S to T return Success
Else, return Fail
Minimizing the congestion under end-to-end
delay restrictions - linear program
Minimize
It is straight forward to
extend the linear program to
the multi-commodity case.
The path flow is constructed
using a variant of the flow
decomposition algorithm.
The complexity incurred by
solving the linear program is
polynomial in D
s.t.
fe
fe
fe0
eO ( v )
eO ( s )
eO ( s )
D
f
0
e
f
eI ( v )
de
e
f
eI ( s )
ce
de
e
0
0
v V \ s,t, 0, D
1, D
eE
fe 0
e E, 0, D de
fe 0
e E, 0, D
0
The number of variables is O(M·D).
The number of constraints is O(M·D).
Approximation Scheme
Goal: reduce the value of the end-to-end delay restriction D.
Delete from the network all the links with a delay de>D.
Delay scaling:
e D
de
D
d , D'= , where
.
N
'
e
Apply the linear program for the new instance.
As the new instance relax the original instance the congestion is not
worse then the optimum.
Convert each non-simple path into a simple path.
Total error for a path: N·.
New end-to-end delay: D+ N·=D∙(1+є).
Minimizing the congestion under delay-jitter restrictions
Idea: restrict the minimum end-to-end delay L and the maximum
end-to-end delay U of the routing paths.
It is sufficient to add the linear program a minimum end-to-end
delay restriction L.
New Linear Program
.
Given a delay-jitter restriction J and an end-to-end delay D
For each L[0,D-J] solve the new linear program with a minimum and
a maximum end-to-end delay restrictions L, L+J, respectively.
Scaling down the end-to-end delay restriction D produces an єoptimal approximation scheme for the case where dmax=O(J).
Details
.
Agenda
Introduction & summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
Selfish Routing
Network users are selfish.
Do not care about social welfare.
Want to optimize their performance.
A central Question: how much does the network performance
suffer from the lack of global regulation?
A flow is at Nash Equilibrium if no user can improve its
performance.
May not exist.
May not be unique.
The price of anarchy: The worst case ratio between the
performance of a Nash equilibrium and the optimal performance.
Previous Work
[Koutsoupias/Papadimitriou]
First paper to propose quantifying the cost of lack of
regulation.
Concentrated on two node networks.
[Roughgarden]
General networks.
Infinite number of users.
users route traffic along the minimum latency path.
The price of anarchy is unbounded.
Model
A set of users U.
For each user, a positive flow demand u and a sourcedestination pair (su,tu).
For each link e, a performance function qe(∙).
qe(∙) is continuous and increasing for all links.
Users behavior
Users are selfish.
They optimize bottleneck objectives
bu ( f )
Network
qe fe .
Bottleneck objective B f Max
eE
Additive objective C f qe f e .
eE
Max qe f e .
eE f eu 0
Non-uniqueness of Nash Equilibrium
One user wants to transfer 1 unit from s to t.
Assume that qe(fe)=fe for each eE.
e1
p1
e3
s
e2
t
p2
(fp1=1, fp2=0) & (fp1=0, fp2=1) are Nash flows with respect to
unsplittable flow vectors.
(fp1=0.5, fp2=0.5) & (fp1=0.25, fp2=0.75) are Nash flows with
respect to splittable flow vectors.
We identified two different Nash flow for each routing approach.
Existence of Nash Equilibrium
Definition: N integral flow vector is a feasible flow
u
f
vector where p is integral in N for each user u U
and pP.
Theorem: Considering N1 integral flow vector there
exists a Nash equilibrium for each N+.
1
u
The existence of NEP for Single-path Routing corresponds to
the case where N=1.
The existence of NEP for Multipath Routing corresponds to
the case where N→∞.
However, still needs to prove for the case where “N=∞”.
The proof of the theorem
.
No price of anarchy for bottleneck network objectives
The price of anarchy is usually more than 1 and it is often
unbounded.
Roughgarden: the price of anarchy is unbounded.
log M
Papadimitriou: the price of anarchy is
.
log
log
log
M
Theorem: Given an instance [G(V,E), U,qe(·)]. If multipath routing
is allowed then the price of anarchy is 1.
Proof
.
Braess paradox: the addition of links to noncooperative networks
can negatively impact performance of all users.
However, cannot occur for multipath routing (when qe(0)=0).
Price of anarchy is at most M with additive objectives
Theorem: Given an instance [G(V,E), U,qe(·)]. If multipath routing
is allowed than the price of anarchy with respect to
additive network objectives is M.
Proof:
Let f and f* denote a Nash and an optimal flow, correspondingly.
Therefore, B(f)≤B(f*).
Therefore, maxeE {qe(f)} ≤maxeE {qe(f*)}.
Hence, ∑eE qe(f)≤ M∙maxE{qe(f)} ≤M∙maxeE {qe(f*)} ≤M∙∑eE qe(f*)
■
Corollary: Driving users to route traffic according to bottleneck
metrics bounds the price of anarchy of additive network
objectives to M.
Bad news for single-path-routing
The price of anarchy is unbounded for single path routing
Additive network objectives.
Bottleneck network objectives.
qe fe e
A=
2
f
3 e
S
T
B= 2∙
qe f e e
1
f
2 e
Optimal
flow
e
Bottleneck
Additive
e
2
3
Nash
flow
e
e
e
4
3
Price of
anarchy
4
3
e
e3
2
e
e
4
3
2
3
e2
e
Agenda
Introduction & summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Selfish multipath routing
Online multipath routing for congestion minimization
Future research
The Model
Requests arrive one at a time and there is no a priori knowledge
regarding future demands.
Each request specifies:
the source sr and destination tr.
the requested flow demand r.
the maximum number of routing paths kr that can carry the demand.
Goal: Route all demands while minimizing the network congestion
factor .
For the case were demands are limited to single an O(logN)competitive strategy was derived by Aspnes, Azar, Fiat, Plotkin,
Waarts.
Evaluating the Quality of Online Algorithms
A solution is offline if it is based on the entire input
sequence.
The competitive ratio is the worst case ratio between the
performance of the online algorithm and the performance of
the optimal offline algorithm.
In our case the performance is the network congestion
factor.
The entire requests sequence is denoted by R.
Minimizing the congestion under integrality restrictions
A path flow is /K-integral if the flow of each request rR over
each path is integral in r/Kr.
Theorem: The optimal network congestion factor of a /K-integral
path flow is larger by a factor of at most 2 than the optimal
network congestion factor of a path flow that admits at most Kr
paths for each request rR .
Proof
.
A /K-integral path flow employs at most Kr paths for each rR.
Corollary: minimizing the congestion while restricting the
flow to be integral in /K is a 2-approximation scheme.
Online solution
Upon the arrival of the nth request:
Split the request to Kn successive requests to transfer n/Kn flow
units.
n/Kn
n/Kn
sn
tn
n/Kn
Employ the online strategy of plotkin at el to route the demands over
single paths.
Plotkin’s online strategy produces a competitive ratio of O(logN).
Therefore, we establish an online strategy with a competitive
ratio of O(logN) for /K-integral path flows.
Therefore, we establish an online strategy for our original problem
with a competitive ratio of 2·O(logN)=O(logN).
A Lower Bound of Ω(logN) for Multipath Routing
Tlog N ,1
V1
V2
T3,1
V3
N
T2,1
T3,2
S
M
Tlog N ,2
M
T1,1
T3,3
T2,2
VN-1
T3,4
VN
The K-th request wishes to transfer a flow demand
of N flow units from S to some target in layer K.
2K
O
T
log N ,
N
2
A Lower Bound of Ω(logN) for Multipath Routing (cont.)
After logN requests the network congestion factor is at least
½∙logN.
The optimal offline algorithm can achieve a network congestion
factor of 1.
V1
V2
T3,1
V3
N
T2,1
T3,2
S
M
T1,1
T3,3
T2,2
VN-1
VN
T3,4
O
A Lower Bound of Ω(logN) for Multipath Routing (cont.)
There exists a lower bound of ½∙logN for networks with at
most N’=N∙logN+N≤2N∙logN nodes.
We have to show that ½∙logN=Ω(logN’).
Indeed, there exists C>0 and N>N0 such that
logN’=logN+log(2·logN)=logN+log2+loglogN ≤ C∙ ½∙logN.
There exists a lower bound of Ω(logN) for the
best possible competitive ratio.
Our online algorithm is best possible.
Agenda
Introduction & summary of results
Multipath routing schemes for survivable networks
Multipath routing schemes for congestion minimization
Online multipath routing for congestion minimization
Selfish multipath routing
Future research
Future research
Deepening the current work
Selfishness in multipath routing
Online multipath routing for finite holding time connections
Other congestion criteria
Multipath routing and security
Recovery schemes for multipath routing
Multipath routing and wireless networks
Fairness in multipath routing
Time dependent flow demands in multipath routing
Deepening the Current Work
Consider for the proposed schemes
Distributed implementation
Heuristic schemes with low complexity
Multi-commodity extensions (congestion minimization)
Already considered in the scheme that restricts the end-to-end
delay.
Establish a unifying scheme that bounds the number of
paths, the end to end delay of each path, and the
delay-jitter among all paths.
Online computation
Offline computation
Selfishness in Multipath Routing
In networks that have many users, the price of anarchy with
respect to additive metrics may be very large.
If all users route their traffic with respect to bottleneck
objectives, the price of anarchy with respect to additive network
objectives is at most M.
Driving users to route traffic according to bottleneck metrics
bounds the price of anarchy to M.
Advertising only the condition of the worst links may cause users
to route traffic according to bottleneck metrics.
In that case, what can be said on the price of anarchy when the
network manager advertises the condition of the K-worst links?
Online Multipath Routing for
finite holding time connections
We have established an online strategy for permanent connections
(i.e., connections with infinite holding times).
In practice, the holding times are usually finite.
There are online routing schemes with provable performance
guarantees for the finite holding time case.
The holding time may be specified upon arrival
Only the distribution on the holding time is known.
No information on the holding time.
Investigate multipath routing for the finite holding time model.
Investigate the lower bound.
Establish corresponding multipath routing schemes.
Other Congestion Criteria
Thus far, we measured congestion according to the most utilized
links in the network.
Although these links are the most severely affected by
congestion, other links are affected as well.
Moreover, there are cases where congestion is better modeled
through non-linear optimization functions.
Consider other optimization functions for congestion.
More general link congestion functions.
Already considered in the work on selfish routing.
Congestion functions that consider all the links in the network.
Multipath Routing and Security
Only the target sees the whole data stream when it is
split among several node-disjoint paths.
Reconstructing the data stream is possible only at the
target node.
It is essential to
Identify several node disjoint paths.
Assign a limited portion of the traffic over each path.
Develop multipath routing schemes that engage this
inherent advantage
The solution must consider the requirements of multipath
routing.
Recovery Schemes for Multipath Routing
Multipath Routing has the advantage of fast restoration upon a
failure.
Upon a path failure, the data stream that traveled over the failed
path may be split along the remaining paths.
Avoid additional path computation and resource reservation.
Requires that the sum of the spare capacities of the remaining
paths is not smaller than the flow on the failed path.
Establish multipath routing schemes that enable fast recovery
while considering the requirements of multipath routing.
Multipath Routing and Wireless networks
Energy Efficient Routing
In wireless networks nodes have a limited power resources
(batteries).
Energy consumption is proportional to node transmission rates.
Therefore, splitting the traffic among several paths can prolong the
time until the first battery is exhausted.
Establish schemes that maximizes the network’s lifetime while
considering the requirements of multipath routing.
Survivability in wireless networks
Standard survivability schemes establish pairs of disjoint paths.
If two links that belong to different paths are too near, noise can
affect both links.
Establish schemes that consider the minimum physical distance
between two links that belong to different paths.
Fairness in Multipath Routing
A commodity may attempt to establish too many paths
In order to maximize its bandwidth.
In order to maximize its survivability.
This may come at the expense of other commodities.
E.g., a commodity may use too many entries in a (limited)
routing table.
Seek suitable fairness criteria for multipath routing
and establish schemes that incorporate the new
criteria.
Time Dependent Flow Demands in Multipath Routing
We have assumed that flow demands are constant in
time.
Often, flow demands are not constant.
Users send/receive data for short periods of time.
The TCP congestion control mechanism changes transmission
rates with time.
Extend our model to cases where → (t).
The End
Two Paths are Enough
Theorem Let (p1,p2,…, pk)P(s,t)×P(s,t) ×…×P(s,t) be a p-survivable
connection. There exists a p-survivable connection p1 , p 2 P s,t × P s,t
that has at least the bandwidth of (p1,p2,…, pk) with respect to
the 1:1 (alternatively 1+1) protection architecture.
Proof
Remove from the network all the links that are not used by the paths of
(p1,p2,…, pk). We have to show that there exists a pair of paths p1 , p 2 P s,t × P s,t
in the resulting network such that p1 p2 p .
k
i
k
Assign to each link e p i
i=1
one unit of capacity.
There exists a pair of paths
from
k
pi
i=1
two units of capacity, and assign to all other links
p , p P
1
s,t
2
× P s,t that intersect only on links
iff it is possible to define an integral link flow that transfers
i=1
two flow units from s to t.
Hence, it is sufficient to show that it is possible to define an integral link flow
that transfers two flow units from s to t.
Two Paths are Enough
Proof (cont)
However, since all capacities are integral, the maximum flow that can be
transferred from s to t is equal to the maximum flow that can be
transferred from s to t when the integrality restriction is omitted. Hence,
we left to show that it is possible to transfer two flow units from s to t.
Suppose by the way of contradiction that it is impossible to transfer two
flow units from s to t.
Hence, according to the max-flow min cut theorem there exists a cut (S,T)
with sS and tT such that C S,T
cxy < 2.
xS,yT
Therefore, since the capacity of all links is integral it follows that C(S,T)≤1.
Hence, since each link has at least one unit of capacity, it follows that at
most one link crosses (S,T).
Denote this link by e. Since C(S,T)≤1 it follows that ce≤1.
Obviously all paths from (p1,p2,…, pk) must traverse through e. Hence, e p i .
Therefore, by construction ce=2, which contradicts the fact that ce≤1. i=1
k
Establishing the widest p-survivable connection
Why is it enough to perform the search over the set ce e E , k 1,2 ?
k
If one path admits a link e then the bandwidth of the
connection is at most ce.
If both paths admit a link e then the bandwidth of the
connection is at most ce/2.
Hence, by definition, there exists at least one tight link eE
such that the bandwidth of the connection is either ce or ce/2.
Why O(logN) executions of a min cost flow algorithm ?
The set contains 2·M elements.
A binary search over the set enables to consider O(log2·M)=O(logN)
values.
The end-to-end delay restriction is intractable
A special case of our problem: Is there a path flow that transfers flow units
from s to t such that if path p transfers a positive amount of flow then
D(p)≤D?
The partition problem: Given an ordered set of elements a1, a2 ,…, a2n that
constitute a set A with a size s(a)+ for each a A, is there a subset
A’A such that A’ contains exactly one element of a2i-1, a2i for 1≤i≤n such that
∑aA’ s(a)=∑aA\A’ s(a)?
S(a1)
S(a3)
S(a5)
S(a2n-1)
S
T
S(a2)
S(a4)
S(a6)
S(a2n)
All link capacities are 1.
Claim: It is possible to transfer 2 flow units over paths whose end-to-end
delays are not larger than ½∑aA s(a) iff there is a subset A’A such that A’
contains exactly one element of a2i-1, a2i for 1≤i≤n and ∑aA’ s(a)=∑aA\A’ s(a).
The end-to-end delay restriction is intractable
<=
There is a a subset A’A such that A’ contains exactly one element of a2i-1, a2i for
1≤i≤n and ∑aA’ s(a)=∑aA\A’ s(a).
The selection of the links that correspond to the elements of A’ and the zero
delay links that connect these links constitutes a path p.
Path p is disjoint to the path that the complement subset A\A’ defines.
Since all capacities are equal to 1, we have two disjoint paths that can transfer
together 2 units of flow.
The end-to-end delay of each path is ½∑aA s(a).
=>
There is a path flow that transfers two flow units over paths that are not larger
than ½∑aA s(a).
Let p be a path that carries a positive flow; by construction, p contains exactly one
element of a2i-1, a2i for 1≤i≤n.
Since all the links have one unit of capacity p can transfer at most 1 flow unit.
Therefore, there exists a path p’ that is disjoint to p that transfers a positive
flow; by construction, p’=A\p
Hence, D(p) ≤½∑aA s(a) and D(p’) ≤½∑aA s(a).
Therefore, since D(p)+ D(p’)=∑aA s(a) it follows that ∑ap s(a)=∑ap’ s(a)=½∑aA
s(a).
The delay jitter restriction is intractable
A special case of our problem: Is there a path flow that
transfers flow units from s to t such that if path p1, p2
transfers a positive amount of flow then D(p1)-D(p2)≤J?
Reduction from the problem with end-to-end delay restriction.
S
A link with a
capacity ∑ce
and a zero
delay.
S
T
T
A
B
It is possible to transfer flow units in network A over
paths with end-to-end delay at most W iff it is possible to
transfer +∑ce flow units in network B over paths with
delay jitter restriction W.
The restriction on the number of paths is intractable
A special case of our problem: Is there a path flow that transfers flow units
from s to t over at most K paths?
The single source unsplittable flow problem: Given a network G with a source s,
targets t1, t2 ,…, tk and corresponding demands D1, D2 ,…, Dk , is there an assignment
of traffic to paths such that for each 1≤i≤k demand Di is routed over a single path
without violating the capacity constraints?
S
t1
t2
D2
D1
Dk
tk
T
Claim: There exists a path flow that transfers = D1+ D2 +…+ Dk flow units from S
to T over at most K paths iff it is possible to find an assignment of the demands
D1, D2 ,…, Dk to paths such that Di, 1≤i≤k is routed over a single path without
violating the capacity constraints
There is exactly one path from S to ti for each 1≤i≤k. Hence, there are exactly K paths from S to T
that carry a positive flows.
There is at least one path from S to ti for each 1≤i≤k. However, since there are at most K paths there is
exactly one path from S to ti for each 1≤i≤k.
Waxman and Power-law topologies
Waxman networks:
Source and destination are located at the diagonally opposite corner
of a square area of unit dimension.
198 nodes are uniformly spread over the square.
A link between two nodes u,v exists with a probability, which depends
on the distance between them δ(u,v):
where α=1.8, β=0.05.
u, v
p u, v exp
2
Power-law networks:
We assigned a number of out-degree credits to each node, using the
power-law distribution β∙x-α where α=0.75 and β=0.05.
Then, we connected the nodes so that every node obtained the
assigned out-degree.
Minimizing the congestion under delay-jitter restrictions
Minimize
s.t.
fe
fe
fe0
eO ( v )
eO ( s )
eO ( s )
eI ( v )
f
eI ( s )
f
L ,D eI ( t )
D
f
0
e
fe 0
0
de
e
ce
fe 0
fe
e
de
0
v V \ s,t, 0, D
de
0
1, D
fe
L ,D eO ( t )
eE
e E, 0, D de
e E, 0, D
Approximation scheme for the restriction on the delay jitter
We impose a restriction 1≤H≤N-1 on the hop count.
Important in order to cope with routing loops.
We present an approximation scheme for the case where
dmax=O(J).
The number of variables is in the order of M∙H∙min{ D,H∙dmax}≤
M∙H2·dmax.
The delay of each link is reduced to smaller integral value.
J e
de
D
d e , D= , where
.
2 N
Total error in the evaluation of the delay of each path is H∙Δ.
A pair of paths that originally have a delay jitter J may now have a
delay jitter J+H∙Δ.
Therefore, in order to relax the new instance the delay jitter
restriction is:
J
J= H
Approximation scheme for the restriction on the delay jitter
Assume that p1, p2 transfers a positive flow in the output. We will
show that D(p1)-D(p2)≤J(1+є).
de
de
D p1 D p2 de de
e p2
e p1
e p2
e p1
de
de
1 d°e 1 d°e
e p2
e p1
e p2
e p1
d° d° p J° p J° H
e p1
e
e p2
e
1
1
J
J
H N 1 1 H N 1
J e
J 1 e .
J N H J 2 N
2 N
Approximation scheme for the restriction on the delay jitter
Assume that p transfers a positive flow in the output. We will
show that D(p) ≤D(1+є).
de
de
D p ep de ep ep 1 ep d°e p
D
J e
°
D H N 1 D N D N
2 N
D e
D N
D 1 e .
2 N
Existence of Nash Equilibrium
The joint strategy space is finite.
Each user selects at most N out of |P(s,t)| possible paths.
There are at most |U| users.
By the way of contradiction assume that there is no Nash equilibrium.
Each profile in the joint strategy space has a player that can improve its
bottleneck.
Let <f1,f2,… > be a sequence of profiles such that for each two profiles fi,
fi+1<f1,f2,… > exactly one user in fi+1 reroutes its traffic and improves its
bottleneck with respect to fi.
After a finite number of transitions between successive profiles we must
encounter the same profile.
Let u be a user that achieves the worst (not constant) bottleneck in all
profiles <f1,f2,…fn >.
Let fk be the profile where u achieves for the first time the worst bottleneck.
There exists in profile fk-1 exactly one user u’ that improves its bottleneck.
However, since u’ ships traffic through the bottleneck of u in fk, u’ is not
improving its bottleneck.
No price of anarchy for bottleneck network objectives
Theorem: Given an instance [G(V,E), U,qe(·)]. If multipath routing is
allowed than the price of anarchy is 1.
proof :
Notations
f- Nash flow.
G(f)- The collection of users that ship traffic through a network
bottleneck in f.
g- Path flow f without the users U\G(f) and their respective flows.
E’ – The collection of all network bottlenecks with respect to g.
P(e)- The collection of all paths that traverse through link e.
Lemma: g is a Nash flow that satisfies
B(f)=B(g)
bu(g)=B(g) for each user uG(f).
Proof
.
No price of anarchy for bottleneck
network objectives (cont.)
By contradiction assume the existence of a flow vector h, B(h)<B(g)
Since g is a Nash flow, every path pP(su,tu) where uG(f) must
traverse through at least one network bottleneck from E’.
s
Therefore, P
u ,t u
eE '
su ,tu
P e for each bottleneck uG(f).
P e.
Therefore,
Therefore, since the total traffic of every feasible flow vector that
P
uG f
eE '
traverses through the paths
P
uG f
traffic that traverse through
h.
eE '
su ,tu
equals to
uG f
P e equals to
u
, the total
u
uG f
both in g and in
No price of anarchy for bottleneck
network objectives (cont.)
Since B(h)<B(g) it follows that qe(he)<qe(ge) for each eE’.
Therefore, he< ge for each eE’.
Therefore, the traffic that traverses through P(e) is smaller in h
than in g for each eE’.
Therefore, the traffic that traverses through
P e is smaller in h
eE '
than in g.
However, this contradicts the fact that the total traffic of the
paths in
P e is the same in flow vector h and g.
eE '
Since B(g) is optimal and since B(f)=B(g), it follows that B(f) is
optimal (the bottleneck of f that also satisfy the demands of all the
users in G(f) can only be worse than the bottleneck of g)
Proof of the Lemma
Let E’’ be the collection of all bottlenecks with respect to f.
B(f)=B(g):
By definition, the traffic that is carried over E’’ belongs only to G(f).
u
u
Therefore, since f p g p for each uG(f) and pP, it holds that
f e ge for each eE’’.
Therefore, B(f)=B(g).
bu(g)=B(g) for each user uG(f).
Consider a user uG(f).
u must ship traffic through at least one link from E’’ in flow vector f.
u
u
Since f p g p for each uG(f) and pP, it follows that u must also
ship positive traffic through a link from E’’ in flow vector g.
Since qe(ge)=qe(fe)=B(f) for each e E’’, it follows that bu(g)=B(g).
g is at Nash equilibrium:
Since f is a Nash flow, every path pP(su,tu) where uG(f) must
traverse through at least one network bottleneck from E’’.
Proof of the Lemma
We have shown that all bottlenecks of f remain unchanged in g.
Therefore, every path p P(su,tu) where uG(f) traverses through one
network bottleneck with respect to g.
By contradiction, assume there exists a user uG(f) in g, that can
improve its bottleneck.
Let E(su,tu) be the collection of all network bottlenecks in g on paths
from P(su,tu).
Let P(e) be the collection of all paths that traverse through e.
u can improve its bottleneck only if it reduces the total traffic that
it carries over paths from P(e) for each employed link eE(su,tu).
Therefore, it must ship traffic to other paths from P(su,tu).
However, we have shown that all other paths already traverse
through at least one bottleneck from E(su,tu).
Minimizing congestion while restricting the number of paths
Theorem: The optimal network congestion factor of a /K-integral
path flow is larger by a factor of at most 2 than the optimal
network congestion factor of a path flow that admits at most Kr
paths for each request rR .
Proof:
Given a network
G(V,E) and a sourcedestination pair.
Let f* be a path flow that has the
smallest network congestion factor α*
among all path flows that transfers for
each rR, r flow units from Sr to Tr over
at most Kr paths.
For each rR, round down the flow f(p)
over each path pP(sr,tr) to a multiple of
r/Kr. Let fR be the resulting path flow.
f=2∙f* is a path flow with a network
congestion factor 2∙α* that transfers
2r flow units from Sr to Tr over at
most Kr paths for each rR.
For each rR, f transfers 2r flow units
over at most Kr paths. Therefore, fR
transfers at least r flow units from Sr
to Tr for each rR
fR is a /K - integral path flow that
transfers at least r flow units from Sr
to Tr for each rR and has a network
congestion factor of at most 2∙ α*.