Multirate Anypath Routing in Wireless Mesh Networks
Rafael Laufer
Computer Science Department
University of California at Los Angeles
Henri Dubois-Ferri
`
ere
Riverbed Technology, Inc.
Lausanne, Switzerland
Leonard Kleinrock
Computer Science Department
University of California at Los Angeles
Abstract—In this paper, we present a new routing paradigm
that generalizes opportunistic routing in wireless mesh networks.
In multirate anypath routing, each node uses both a set of next
hops and a selected transmission rate to reach a destination.
Using this rate, a packet is broadcast to the nodes in the set
and one of them forwards the packet on to the destination. To
date, there is no theory capable of jointly optimizing both the
set of next hops and the transmission rate used by each node.
We bridge this gap by introducing a polynomial-time algorithm
to this problem and provide the proof of its optimality. The
proposed algorithm runs in the same running time as regular
shortest-path algorithms and is therefore suitable for deployment
in link-state routing protocols. We conducted experiments in a
802.11b testbed network, and our results show that multirate
anypath routing performs on average 80% and up to 6.4 times
better than anypath routing with a fixed rate of 11 Mbps. If the
rate is fixed at 1 Mbps instead, performance improves by up to
one order of magnitude.
I. INTRODUCTION
The high loss rate and dynamic quality of links make
routing in wireless mesh networks extremely challenging [1].
Anypath routing
1
has been recently proposed as a way to
circumvent these shortcomings by using multiple next hops
for each destination [3]–[6]. Each packet is broadcast to a
forwarding set composed of several neighbors, and the packet
must be retransmitted only if none of the neighbors in the
set receive it. Therefore, while the link to a given neighbor
is down or performing poorly, another nearby neighbor may
receive the packet and forward it on. This is in contrast to
single-path routing where only one neighbor is assigned as
the next hop for each destination. In this case, if the link to
this neighbor is not performing well, a packet may be lost
even though other neighbors may have overheard it.
Existing work on anypath routing has focused on wireless
networks that use a single transmission rate. This approach,
albeit straightforward, presents two major drawbacks. First,
using a single rate over the entire network underutilizes
available bandwidth resources. Some links may perform well
at a higher rate, while others may only work at a lower rate.
Secondly and most importantly, the network may become
disconnected at a higher bit rate. We provide experimental
measurements from a 802.11b testbed which show that this
phenomenon is not uncommon in practice. The key problem is
that higher transmission rates have a shorter radio range, which
reduces network density and connectivity. As the bit rate in-
creases, links becomes lossier and the network eventually gets
disconnected. Therefore, in order to guarantee connectivity,
single-rate anypath routing must be limited to low rates.
In multirate anypath routing, these problems do not exist;
however, we face different challenges. First, we must find
1
We use the term anypath rather than opportunistic routing, since oppor-
tunistic routing is an overloaded term also used for opportunistic contacts [2].
not only the forwarding set, but also the transmission rate
at each hop that jointly minimizes its cost to a destination.
Secondly, loss probabilities usually increase with higher trans-
mission rates, so a higher bit rate does not always improve
throughput. Finally, higher rates have a shorter radio range
and therefore we have a different connectivity graph for each
rate. Lower rates have more neighbors available for inclusion
in the forwarding set (i.e., more spatial diversity) and less
hops between nodes. Higher rates have less neighbors available
for the forwarding set (i.e., less spatial diversity) and longer
routes. Finding the optimal operation point in this tradeoff is
the focus of this paper.
We thus address the problem of finding both a forwarding
set and a transmission rate for every node, such that the overall
cost of every node to a particular destination is minimized.
We call this the shortest multirate anypath problem. To our
knowledge, this is still an open problem [3], [4], [7] and we
believe our algorithm is the first practical solution for it.
We introduce a polynomial-time algorithm to the shortest
multirate anypath problem and present a proof of its optimality.
Our solution generalizes Dijkstra’s algorithm for the multirate
anypath case and is applicable to link-state routing protocols.
One would expect that the running time of such an algorithm
to be exponential. However, we show that it has the same
polynomial time as the corresponding shortest-path algorithm,
being suitable for implementation at current wireless routers.
We also generalize the expected transmission time (ETT)
routing metric [8] for multirate anypath routing.
For the performance evaluation, we conducted experiments
in an 18-node 802.11b wireless testbed of embedded Linux
devices. Our results reveal that the network becomes partially
disconnected when we fix the transmission rate of every
node at 2, 5.5, and 11 Mpbs. A single-rate routing scheme
therefore performs poorly in this case, since 1 Mbps is the only
rate at which the network is fully connected. We show that
multirate anypath routing improves the end-to-end expected
transmission time by 80% on average and by up to 6.4 times
compared to single-rate anypath routing at 11 Mbps, while
still maintaining full network connectivity. The performance
is even higher over the single-rate case at 1 Mbps, with an
average gain of a factor of 5.4 and a maximum gain of a
factor of 11.3.
The remainder of the paper is organized as follows. Sec-
tion II reviews the basic theory of anypath routing, our
network model and assumptions. In Section III, we introduce
multirate anypath routing and the proposed routing metric.
Section IV presents the multirate anypath algorithm and proves
its optimality. Section V reveals our experimental results,
showing the benefits of multirate over single-rate anypath
routing. Section VI presents the related work in anypath
routing. Finally, conclusions are presented in Section VII.
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II. ANYPATH ROUTING
In this section we review the anypath routing theory in-
troduced by Zhong et al. [5] and Dubois-Ferri
`
ere et al. [6].
The main contributions of the paper are presented later in
Sections III and IV.
A. Overview
In classic wireless network routing, each node forwards a
packet to a single next hop. As a result, if the transmission to
that next hop fails, the node needs to retransmit the packet even
though other neighbors may have overheard it. In contrast, in
anypath routing, each node broadcasts a packet to multiple
next hops simultaneously. Therefore, if the transmission to one
neighbor fails, an alternative neighbor who received the packet
can forward it on. We define this set of multiple next hops
as the forwarding set and we usually use J to represent it
throughout the paper. A different forwarding set is used to
reach each destination, in the same way a distinct next hop is
used for each destination in classic routing.
When a packet is broadcast to the forwarding set, more than
one node may receive the same packet. To avoid unnecessary
duplicate forwarding, only one of these nodes should forward
the packet on. For this purpose, each node in the set has a
priority in relaying the received packet. A node only forwards
a packet if all higher priority nodes in the set failed to
do so. Higher priorities are assigned to nodes with shorter
distances to the destination. As a result, if the node with the
shortest distance in the forwarding set successfully received
the packet, it forwards the packet to the destination while
others suppress their transmission. Otherwise, the node with
the second shortest distance forwards the packet, and so
on. A reliable anycast scheme [9] is necessary to enforce
this relay priority. We talk more about this in Section II-B.
The source keeps rebroadcasting the packet until someone
in the forwarding set receives it or a threshold is reached.
Once a neighbor in the set receives the packet, this neighbor
repeats the same procedure until the packet is delivered to the
destination.
Since we now use a set of next hops to forward packets,
every two nodes are connected through a mesh composed of
the union of multiple paths. Figure 1 depicts this scenario
where each node uses a set of neighbors to forward packets.
The forwarding sets are defined by the multiple bold arrows
leaving each node. We define this union of paths between
two nodes as an anypath. In the figure, the anypath shown in
bold is composed by the union of 11 different paths between
a source s and a destination d. Depending on the choice
of each forwarding set, different paths are included in or
excluded from the anypath. At every hop, only a single node
of the set forwards the packet on. Consequently, every packet
from s traverses only one of the available paths to reach d. We
show a path possibly taken by a packet using a dashed line.
Succeeding packets, however, may take completely different
paths; hence the name anypath. The path taken is determined
on-the-fly, depending on which nodes of the forwarding set
successfully receive the packet at each hop.
s
d
Figure 1. An anypath connecting nodes s and d is shown in bold arrows.
The anypath is composed of the union of 11 paths between the two nodes.
Every packet sent from s traverses one of these paths to reach d, such as the
path shown with a dashed line. Different packets may traverse different paths,
depending on who receives the packet at each hop; hence the name anypath.
B. System Model and Assumptions
In order to support the point-to-multipoint links used in
anypath routing, we model the wireless mesh network as a
hypergraph. A hypergraph G = (V, E) is composed of a set V
of vertices or nodes and a set E of hyperedges or hyperlinks.
A hyperlink is an ordered pair (i, J), where i ∈ V is a node
and J is a nonempty subset of V composed of neighbors
of i. For each hyperlink (i, J) ∈ E, we have a delivery
probability p
iJ
and a distance d
iJ
. If the set J has a single
element j, then we just use j instead of J in our notation. In
this case, p
ij
and d
ij
denote the link delivery probability and
distance, respectively.
The hyperlink delivery probability p
iJ
is defined as the
probability that a packet transmitted from i is successfully
received by at least one of the nodes in J. One would expect
that the receipt of a packet at each neighbor is correlated due
to noise and interference. However, we conducted experiments
which suggest that the loss of a packet at different receivers
occur independently in practice [10], which is consistent with
other studies [11]. With the assumption of independent losses,
the probability p
iJ
is
p
iJ
= 1 −
Y
j∈J
(1 − p
ij
) . (1)
Previously proposed MAC protocols have been designed to
guarantee the relay priority among the nodes in the forwarding
set [4], [9]. Such protocols can use different strategies for this
purpose, such as time-slotted access, prioritized contention
and frame overhearing. Reliable anycast is an active area
of research [9] and we assume that such mechanism is in
place to make sure that the relaying priority is respected.
The details of the MAC, however, are abstracted from the
routing layer. Practical routing protocols only incorporate
the delivery probabilities into the routing metric in order to
abstract from the MAC details [8], [12] and we take the
same approach. The only MAC aspect that is important is the
effectiveness of the relaying node selection. As long as the
relaying node is actually the one with the shortest distance to
the destination, there should be no significant impact on the
routing performance.
C. Anypath Cost
We are interested in calculating the anypath cost from a
node i to a given destination via a forwarding set J. The
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anypath cost D
i
is defined as D
i
= d
iJ
+ D
J
, which is
composed of the hyperlink cost d
iJ
from i to J and the
remaining-anypath cost D
J
from J to the destination.
The hyperlink cost d
iJ
depends on the routing metric used.
Most previous works on anypath routing have adopted the
expected number of anypath transmissions (EATX) as the
routing metric [3], [5], [6]. The EATX is a generalization
of the unidirectional ETX metric [12], which is defined as
d
ij
= 1/p
ij
. The distance d
ij
for ETX represents the expected
number of transmissions necessary for a packet sent by i to
be successfully received by j. For EATX, the distance d
iJ
is defined as d
iJ
= 1/p
iJ
, which is the average number of
transmissions necessary for at least one node in J to correctly
receive the transmitted packet.
The remaining-anypath cost D
J
is intuitively defined as
a weighted average of the distances of the nodes in the
forwarding set as
D
J
=
X
j∈J
w
j
D
j
, with
X
j∈J
w
j
= 1, (2)
where the weight w
j
in (2) is the probability of node j
being the relaying node. For example, let J = {1, 2, . . . , n}
with distances D
1
≤ D
2
≤ . . . ≤ D
n
. We refer to the
probability p
ij
simply by p
j
for convenience. Node j will
be the relaying node only when it receives the packet and
none of the nodes closer to the destination receives it, which
happens with probability p
j
(1 − p
j−1
)(1 − p
j−2
) . . . (1 − p
1
).
The weight w
j
is then defined as
w
j
=
p
j
j−1
Y
k=1
(1 − p
k
)
1 −
Y
j∈J
(1 − p
j
)
, (3)
with the denominator being the normalizing constant.
As an example, consider the network depicted in Figure 2.
The distance via J in Figure 2(a) is calculated as
D
i
= d
iJ
+ D
J
=
1
1 − (1 − 1/4)(1 − 1/5)
+
(1/4)3 + (3/4)(1/5)3
1 − (1 − 1/4)(1 − 1/5)
= 2.5 + 3.0 = 5.5. (4)
One would expect that adding an extra node to the forwarding
set is always beneficial because it increases the number of
possible paths a packet can take. However, this is not always
true, as shown in Figure 2(b). The anypath distance via
J
′
= J ∪ {j} is D
i
= d
iJ
′
+ D
J
′
= 1.8 + 4.6 = 6.4. On
one hand, using J
′
instead of J reduces the hyperlink cost,
that is, d
iJ
′
≤ d
iJ
. On the other hand, the extra node increases
the remaining anypath cost, that is, D
J
′
≥ D
J
. If the increase
D
J
′
− D
J
is higher than the decrease d
iJ
− d
iJ
′
, adding this
extra node is not worthy since the total cost to reach the
destination increases. The intuition here is that when node j
is the only one in J
′
that received the packet, it is cheaper to
retransmit the packet to one of the two nodes in J and take a
shorter path from there than to take the long path via node j.
i
5
4
4
3
9
3
i
j
J
(a)
i
5
4
4
3
9
3
i
j
J’
(b)
Figure 2. An anypath cost calculation example. The weight of each link is
the expected number of transmissions (ETX), which is the inverse of the link
delivery probability. The anypath cost in (a) is lower than the cost in (b).
Once the cost of an anypath is defined, it is of interest to
find the anypath with the lowest cost to the destination, that
is, the shortest anypath. This is called the shortest-anypath
problem [6]. Interestingly enough, the shortest anypath will
always have an equal or lower cost than the shortest single
path. This is a direct consequence of the definition of an
anypath as a set of paths. Among all possible anypaths between
two nodes, we also have the anypath composed only of the
path with the shortest ETX. Therefore, if we are to choose
the shortest anypath among all these possibilities, we know
for sure that its cost can never be higher than the cost of the
shortest single path.
III. MULTIRATE ANYPATH ROUTING
Previous work on anypath routing focused on a single
bit rate [3]–[6]. Such an assumption, however, considerably
underutilizes available bandwidth resources. Some hyperlinks
may be able to sustain a higher transmission rate, while others
may only work at a lower rate. To date, the problem of
how to select the transmission rate for anypath routing is
still open [7]. We provide a solution to this problem and
incorporate the multirate capability inherent in IEEE 802.11
networks into anypath routing. In this case, besides selecting
a set of next hops to forward packets, a node must also select
one among multiple transmission rates. For each destination,
a node then keeps both a forwarding set and a transmission
rate used to reach this set. As a result, every two nodes
will be connected through a mesh composed of the union of
multiple paths, with each node transmitting at a selected rate.
Figure 3 depicts the scenario where nodes use a selected bit
rate to forward packets to a set of neighbors. We define this
union of paths between two nodes, with each node using a
potentially different bit rate as a multirate anypath. In the
figure, assume that a packet is sent from s to d over the
multirate anypath. Only one of the available paths is traversed
depending on which nodes successfully receive the packet at
each hop. We show a path possibly taken by the packet using
dashed lines. We use different dash lengths to represent the
different transmission rates used by each node. A shorter dash
represents a shorter time to send a packet, hence a higher
transmission rate. Succeeding packets may take completely
different paths with other transmission rates along its way.
In order to support multirate, we must extend the system
model in Section II-B. Let R be the set of available bit rates
that nodes can use to transmit their packets. For each hyperlink
(i, J) ∈ E, we now have a delivery probability p
(r)
iJ
and a
distance d
(r)
iJ
associated with each transmission rate r ∈ R.
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s
d
Figure 3. A multirate anypath connecting nodes s and d is shown in bold
arrows. Every packet sent from s traverses a path to reach d, such as the
path shown with dashed lines. Different dash lengths represent the different
bit rates used by each node, with a shorter dash for higher rates.
In real wireless networks, we usually have different delivery
probabilities and distances for each transmission rate, which
justifies this model extension.
The EATX metric described in Section II-C was originally
designed considering that nodes transmit at a single bit rate.
To account for multiple bit rates, we introduce the expected
anypath transmission time (EATT) metric. For EATT, the
hyperlink distance d
(r)
iJ
for each rate r ∈ R is defined as
d
(r)
iJ
=
1
p
(r)
iJ
×
s
r
, (5)
where p
(r)
iJ
is the hyperlink delivery probability defined in (1),
s is the maximum packet size, and r is the bit rate. The
distance d
(r)
iJ
is basically the time it takes to transmit a packet
of size s at a bit rate r over a lossy hyperlink with delivery
probability p
(r)
iJ
. The EATT metric is a direct generalization
of the expected transmission time (ETT) metric [8] commonly
used in single-path wireless routing. Note that for each bit rate
r ∈ R, we have a different delivery probability p
(r)
iJ
, which
usually decreases for higher rates. This behavior imposes a
tradeoff; a higher bit rate decreases the time of a single packet
transmission (i.e., s/r decreases), but it usually increases
the number of transmissions required for a packet to be
successfully received (i.e., 1/p
(r)
iJ
increases).
The remaining-anypath cost D
(r)
J
now also depends on the
transmission rate, since the delivery probabilities change for
each rate. Since both the hyperlink distance and the remaining
anypath cost depend on the bit rate, node i has a different
anypath cost D
(r)
i
= d
(r)
iJ
+ D
(r)
J
for each forwarding set J
and for each transmission rate r ∈ R.
We address the problem of finding both the forwarding
set and the transmission rate that minimize the overall cost
to reach a particular destination. We call this the shortest
multirate anypath problem, which generalizes the shortest-
anypath problem [6] for the multirate scenario. Interestingly,
the shortest multirate anypath will always have equal or lower
cost than the shortest single path. Among all possible multirate
anypaths between two nodes, we also have the single path with
the shortest ETT. As a result, the cost of the shortest multirate
anypath can never be higher than the cost of the shortest path.
Likewise, due to the same argument, the shortest multirate
anypath will also have equal or lower cost than any shortest
anypath using a single rate.
IV. FINDING THE SHORTEST MULTIRATE ANYPATH
In this section we introduce the proposed shortest-anypath
algorithms. In Section IV-A, we present the Shortest Anypath
First (SAF) algorithm used in a single-rate network with the
EATX metric. Our SAF algorithm, while derived indepen-
dently, is similar to the single-rate algorithm proposed by
Chachulski [3]. Our main contribution is to provide a proof
of its optimality. We also use the single-rate case as the basis
for the multirate generalization introduced in Section IV-B.
Surprisingly, the Shortest Multirate Anypath First (SMAF)
algorithm has the same running time as a shortest single-path
algorithm for multirate, being suitable for deployment in link-
state routing protocols. We only show the proof of optimality
of the SMAF algorithm, since by definition this also implies
the optimality of the SAF algorithm.
A. The Single-Rate Case
We now present the Shortest Anypath First algorithm used
in the simpler single-rate scenario. Given a graph G = (V, E),
the algorithm calculates the shortest anypaths from all nodes to
a destination d. For every node i ∈ V we keep an estimate D
i
,
which is an upper-bound on the distance of the shortest
anypath from i to d. In addition, we also keep a forwarding
set F
i
for every node, which stores the set of nodes used as
the next hops to reach d. Finally, we keep two data structures,
namely S and Q. The S set stores the set of nodes for which
we already have a shortest anypath defined. We store each
node i ∈ V − S for which we still do not have a shortest
anypath in a priority queue Q keyed by their D
i
values.
SHORTEST-ANYPATH-FIRST(G, d)
1 for each node i in V
2 do D
i
← ∞
3 F
i
← ∅
4 D
d
← 0
5 S ← ∅
6 Q ← V
7 while Q 6= ∅
8 do j ← EXTRACT-MIN(Q)
9 S ← S ∪ {j}
10 for each incoming edge (i, j) in E
11 do J ← F
i
∪ {j}
12 if D
i
> D
j
13 then D
i
← d
iJ
+ D
J
14 F
i
← J
As in the shortest-path algorithm, the Shortest Anypath First
algorithm is composed of |V | rounds, dictated by the number
of elements initially in Q. At each round, the EXTRACT-MIN
procedure extracts the node with the minimum distance to
the destination from Q. Let this node be j. At this point,
j is settled and inserted into S, since the shortest anypath
from j to the destination is now known. For each incoming
edge (i, j) ∈ E, we check if the distance D
i
is larger than the
distance D
j
. If that is the case, then node j is added to the
forwarding set F
i
and the distance D
i
is updated.
Figure 4 shows the execution of Shortest Anypath First
algorithm using the EATX metric. We see in Figure 4(a) the
graph right after the initialization. Figures 4(b)–4(h) show
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8
8
8
8
8
8
0
4
1
3
1
5
6
5
1
3
2
7
4
d
(a)
8
8
8
4
2
0
7
4
1
3
1
5
6
5
1
3
2
7
4
d
(b)
8
8
0
7
4
1
3
1
5
6
5
1
3
2
7
4
7
2.5
d
2
(c)
8
0
7
4
1
3
1
5
6
5
1
3
2
7
4
2.5
6.5
5.2
d
2
(d)
0
7
4
1
3
1
5
6
5
1
3
2
7
4
2.5
6.5
5.2
6.2
d
2
(e)
0
7
4
1
3
1
5
6
5
1
3
2
7
4
2.5
6.5
5.2
6.2
d
2
(f)
0
7
4
1
3
1
5
6
5
1
3
2
7
4
2.5
6.5
5.2
6.2
d
2
(g)
0
7
4
1
3
1
5
6
5
1
3
2
7
4
2.5
6.5
5.2
6.2
d
2
(h)
Figure 4. Execution of the Shortest Anypath First (SAF) algorithm from every node to d. The weight of each link is the expected number of transmissions
(ETX), which is the inverse of the link delivery probability. (a) The situation just after the initialization. (b)–(h) The situation after each successive iteration
of the algorithm. Part (h) shows the situation after the last node is settled.
each iteration of the algorithm. At each step, the value inside
a node i presents the distance D
i
from that node to the
destination d and the arrows in boldface present the shortest
anypath to d. Nodes with two circles are the settled nodes in S.
The graph in Figure 4(h) shows the result of SAF algorithm
right after settling the last node.
The running time of the Shortest Anypath First algorithm
depends on how Q is implemented. Assuming that we have
a Fibonacci heap, the cost of each of the |V | EXTRACT-MIN
operations in line 8 takes O(log V ), with a total of O(V log V )
aggregated time. The running time to calculate both d
iJ
and
D
J
in line 13 depends on the size of J; however, if we store
additional state, it can be reduced to a constant time [10]. The
for loop of lines 10–13 takes O(E) aggregated time and as a
result the total complexity of the algorithm is O(V log V +E),
which is the same complexity of Dijkstra’s algorithm.
B. The Multirate Case
We now generalize the SAF algorithm to support multiple
transmission rates, introducing the Shortest Multirate Anypath
First (SMAF) algorithm. For each node i ∈ V , we now
keep a different distance estimate D
(r)
i
for every rate r ∈ R.
The estimate D
(r)
i
is an upper-bound on the distance of the
shortest anypath from i to d using transmission rate r. In
addition, we also keep its corresponding forwarding set F
(r)
i
,
which stores the set of next hops used for i to reach d
using r. We use D
i
and F
i
without the indicated rates to
store the minimum distance estimate among all rates and its
corresponding forwarding set, respectively. We also keep a
transmission rate T
i
for every node, which stores the rate used
to reach d.
The key idea of the SMAF algorithm is that each node
i ∈ V has an independent distance estimate D
(r)
i
for each rate
r ∈ R and we keep the minimum of these estimates as the
node distance D
i
. At each round of the while loop, the node
with the minimum distance from Q is settled. Let this node
be j. For each incoming edge (i, j) ∈ E, we check for every
rate r ∈ R if the distance D
(r)
i
is larger than the distance D
j
of
the node just settled. If that is the case, then node j is added to
the forwarding set F
(r)
i
of that specific rate and distance D
(r)
i
is updated accordingly. If the new distance D
(r)
i
is shorter than
the node distance D
i
, we update the node distance D
i
as well
SHORTEST-MULTIRATE-ANYPATH-FIRST(G, d)
1 for each node i in V
2 do D
i
← ∞
3 F
i
← ∅
4 T
i
← NIL
5 for each rate r in R
6 do D
(r)
i
← ∞
7 F
(r)
i
← ∅
8 D
d
← 0
9 S ← ∅
10 Q ← V
11 while Q 6= ∅
12 do j ← EXTRACT-MIN(Q)
13 S ← S ∪ {j}
14 for each incoming edge (i, j) in E
15 do for each rate r in R
16 do J ← F
(r)
i
∪ {j}
17 if D
(r)
i
> D
j
18 then D
(r)
i
← d
(r)
iJ
+ D
(r)
J
19 F
(r)
i
← J
20 if D
i
> D
(r)
i
21 then D
i
← D
(r)
i
22 F
i
← F
(r)
i
23 T
i
← r
as the forwarding set F
i
and transmission rate T
i
to reflect the
new minimum.
The running time of the Shortest Multirate Anypath First
algorithm also depends on the implementation of Q. The
initialization in lines 1–10 takes O(V R) time. Assuming that
we have a Fibonacci heap, the EXTRACT-MIN operations in
line 12 take a total of O(V log V ) aggregated time. We assume
that the distance calculation of d
(r)
iJ
and D
(r)
J
in line 18 is
optimized to take a constant time [10]. As a result, the for
loop in lines 15–23 takes O(ER) aggregated time. The total
running time is therefore O(V log V + (E + V )R), which is
O(V log V +ER) if all nodes are able to reach the destination.
This is the same running time of the shortest single-path
algorithm for multiple rates. Compared to the SAF algorithm,
the SMAF algorithm allows nodes to take advantage of their
multiple transmission rates at the cost of just a small increase
in the running time.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 2009 proceedings.
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