Lecture 8
Why do we need residual
networks?
• Residual networks allow one to reverse
flows if necessary.
If we have taken a bad path then residual
networks allow one to detect the condition
and reverse the flow.
A bad path is one which overlaps with too
many other paths.
Example
a
Source
1
1
Destination
1
1
b
1
Paths source, a, destinations and source, b destination
gives a flow of 2 units.
Path source, a, b, destination overlaps with both
the optimal paths.
If we initially choose source, a, b, destination as our path, then
no greedy strategy will be able to augment the network flow any
further (unless we use residual edges which allows recovery)
Verify how we recover in spite of the initial bad choice, if we use
the residual network to augment flows.
Defects of Augmenting Path
Algorithm
Sending flow along a path requires O(V) complexity.
1
1
1 2
1
1




.
Since there are n separate paths, we need to
1
. 1
push flow through n paths separately.
.
Every time complexity is O(V)
n
How about if we push flow through edges
separately without caring about flow conversation?
If there is excess flow at a node, then the excess
is sent to downstream nodes.
Finally, all excess flow is routed back to the
source, so that flow conservation is maintained. 1




1
1 2
1
1
.
1
.
.
n
1
Generic Pre-flow Push Algorithm
Augmenting path algorithms maintain feasibility at every
stage, and gradually strive towards optimality
Pre-flow push algorithm initially pushes the
maximum possible amount of flow, and strives
towards feasibility.
Initially, flow in every edge originating from the
source is set to the link capacity.
Distance label of every node is initially their
distance to the destination.
Set distance label of source node s, d(s)=V
A node (which is not source nor destination), with excess
flow is called an active node.
If a node has excess flow, send the flow along an
admissible edge (end node of an admissible edge
has less distance than the source node of the edge,
and hence is closer to the destination).
Let the excess flow at a node u be e(u).
Let the capacity of admissible edge (u, v) in
residual graph be ruv
Then flow routed along (u, v) is min(e(u), ruv )
If a node u with excess flow does not have an
admissible edge in the residual network, then it is
relabelled: d(u) is upgraded to d(v) + 1, where v is
the node with minimum distance label amongst all
nodes which have edges from u in the residual
network.
The algorithm terminates when there are no
active nodes.
Intuition
Edges are like water pipes
Nodes are joints
Distance is like height from the ground.
Destination is at the ground.
Initially source is at the heighest level and sends
water to all adjacent nodes.
Whenever a node has accumulated water, it sends water
to nodes at lower label.
So water moves towards the destination.
Sometimes water gets locally trapped as all neighboring
nodes are at a greater height (no admissible arc).
Then, the node label is raised (relabelling).
Push 4 units along 1-3 and 2 units
along 1-2
source
1
2
E=2
source
3 E=4
4
1
2
12
Push 4 unit along 3-4, and
then 1 unit along 1-4
1
2
3
5
2
1
0
E = 11
4
3
4
3
5
0
destination
1
2
1
4
destination
Original Network/Residual Network with 0
flow
source
source
1
2
2
12
E=1
1
3
4
3
1
1
4
1
2
Relabel node
2 to 2
0
destination
4
2
22
E=1
3
4
3
1
1
4
1
Route 1 unit
through 2-3
0
destination
4
source
1 4
3 E=1
2
2
22
source
1 4
1
2
1
1
4
1
Route 1 unit
through 3-4
2
2
40
destination
Now no node has any excess
Total flow out of source is 6
2
1
3
1
5
4
destination
Proof for Optimality
Distance labels always maintain requisite properties.
Initially, d(s) = V. Any difficulty in maintaining
requisite properties?
At termination d(s) is V or greater. So there
is no path from the source to destination in
the residual network (no augmenting path).
At termination, no node except the source and
destination has excess flow.
No node ever has any deficit flow.
So, flow allocation is feasible at termination
Since the residual network does not have an
augmenting path, the flow is a maxflow.
Complexity is O(V2E)
Specific Implementations of
Generic Preflow-Push algorithm
Order of selection of active nodes has not been
nailed down in the generic version
(FIFO)This algorithm pushes flow out of the active
nodes in first in first out order.
The algorithm initially orders the active nodes in
some manner.
It pushes flow from the first active node in the
list, till the active node is no longer active or till
it needs to be relabelled
While pushing flow from this active node, the newly
created active nodes are added to the back of the list
In the latter case, the node is relabelled and
added to the back of the list and picks the second
node from the list.
The algorithm terminates when the list is empty
Clearly, FIFO is a specific order for selecting
the active nodes, and hence the proof for
correctness is the same as that for generic
preflow push algorithm
O(V3) complexity
Problem with FIFO
Source
1
1


2
1 1

1


n -1
Destination
1
Highest Label Preflow push
Pushes flow from the active node with the
highest label
Terminates when there are no more active nodes
O(V2E0.5)
Multicommodity network flow
We have several source destination pairs.
Refer to each as a separate commodity.
Need to maximize a weighted sum of the different
flows
Flow allocation must be feasible:
Capacity Constraints
Total flow in any link is between 0 and the
link capacity
Flow Conservation
Input flow for any commodity at a node equals the output
flow of the commodity, for every node except the
source/destination for the commodity.
Solution techniques use optimization ideas
However, we may not have integral flows even if
link capacities are integers
In fact, unlike the single commodity flow problem,
multicommodity flow problem with the constraint
that flows in the links are integers is NP-hard
Example Application
Wireless nodes have low battery power and hence
limited lifetime.
Every transmission consumes energy.
Reception consumes less energy.
Lifetime of a node is related to the total energy spent in
transmission.
Transmission energy increases with increase in
distance to the receiving node.
Signal strength decreases as distance increases.
So if the node needs to transmit to a distant node,
it needs to use greater power.
Power increases as d4 where d is the distance
between the two nodes.
It would be interesting to ask the question
that given a particular transmission
requirement what is the maximum lifetime of
a network?
Lifetime of a network is the maximum time in which
all nodes are active.
Energy Conserving Routing in Wireless Ad-hoc
Networks
J.H. Chang and L. Tassiulas, INFOCOM 2000
Clearly, routing strategy will affect the network
lifetime.
For example, if all traffic is routed through
a single node, it will die in a short time,
and according to our definition network
dies as well.
Routing strategy will depend on traffic
requirements, and the node energies.
Problem Formulation
Node i has energy Ei
Transmission energy from node i to j is eij for each
transmission (depends on the distance between the
two nodes)
Let there be C sessions
Let Qic be the rate of packet generation in node i for
session c
Let qijc be the rate of information transmission in
link i, j for session c under the current routing
(routing determines the qijc )
Every node has a maximum power. So there is a set of
neighbors Si of node i.
Lifetime of a node i under flow q is Ti(q)= Ei/j  Si eijc  Cqijc
Lifetime of a system is minimum of all its nodes
Tsys (q) = mini Ti (q)
Let Dc be the set of destinations of session C
Session c information must reach some node in Dc
Lifetime Maximization Objective
Maximize Tsys (q) = mini Ei/j  Si eijc  Cqijc
Subject to:
qijc0
i  Sj qjic + Qic= j  Si qijc  session c, and all
nodes i Dc
Reformulation
Maximize Tsys (q)
Subject to:
j  Si eijc  CqijcTsys (q)  Ei  session c, and all nodes i
qijc0
i  Sj qjic + Qic= j  Si qijc  session c, and all
nodes i Dc
Let qijc = qijcTsys (q)
Reformulation
Maximize Tsys (q)
Subject to:
j  Si eijc  Cqijc  Ei  session c, and all nodes i
qijc0
i  Sj qjic + Tsys (q) Qic= j  Si qijc  session c, and all
nodes i Dc
Special Case
Consider the special case where every session has a single
source, i.e., Qic = 0 at all nodes except the source of
session i. Thus, Qc is the input rate for session c.
Also, eij = ei  neighbors j of i.
Also, pijc = qijcTsys (q)/ Qc
The problem can be reformulated as follows:
Reformulation
Maximize Tsys (q)
Subject to:
j  Si c  Cpijc  Ei /ei session c, and all nodes i
pijc0
i  Sj pjic = j  Si pijc session c, and all
nodes i except source and destination
of session c.
i  Sj pjic + Tsys (q) = j  Si pijc
• This is the multi-commodity flow problem with node
capacities, i.e., there is an additional constraint that sum of
the output flow through a node must not exceed the node
capacity.
– Problem with node capacity can be reduced to the problem without
node capacity, replacing each node by two nodes with a link
between them with capacity equal to the node capacity
– Note that there is no link capacity in this problem (equivalently all
link capacities are infinity).
• Object of this multi-commodity flow problem is to find
the minimum flow supported for each session.
In general, eij may not be the same for all neighbors of
node i, since all neighbors may not be equidistant from
node i.
The paper suggests several heuristics for the purpose.
Minimum Cut
Consider the augmenting path algorithm.
The set S reachable from the source node in the
residual network when the algorithm terminates and
the remaining nodes V-S constitute a minimum cut.
Why?
Thus we know how to find a minimum cut.
How about finding the edges of all minimum
cuts?
Why do we need the above?
Consider an edge (u, v) in the residual network
corresponding to the max-flow. Let such an edge
satisfy the following properties:
Edge (u, v) is filled to capacity.
Node v is not reachable from source s in the residual
network.
Node u can not reach destination t in the residual
network.
There is no path from u to v in the residual
network.
A non-zero capacity edge (u, v) belongs to some
minimum cut if and only if it satisfies all the above
properties in the residual network corresponding to a
max flow.
Prove the above.
Your grade for the second homework is the maximum of your
grade in this problem, and the second homework!
I need the answer by Nov 14th