![Fig. 11. Load (left) and stretch (right) performances of Bonsai arborescences [14], before and after improvement, when improvement targets load (top) or stretch (bottom), facing random failures. Each point represents the median metric value over 100 indep. trials. Ribbon delimits the 10 (resp. 90) quantile values.](/figures/fig-11-load-left-and-stretch-right-performances-of-bonsai-33cgsosk.webp)
![Fig. 10. Effect on Bonsai [14]: (left) Number of SRLG links in the last two arborescences before/after SRLG optimization, for varying SRLG sizes. Dashed line represents the ideal case where all SRLG links end up in the last two arborescences. (right) Distribution of the number of independent pairs before and after optimization, established over 3600 independent random graphs.](/figures/fig-10-effect-on-bonsai-14-left-number-of-srlg-links-in-the-29f63hxx.webp)
![Fig. 1. Example network from [14] with two different t-rooted arc-disjoint spanning arborescence decompositions, T1 left and T2 right. In both of them one arborescence is drawn with dotted red arrows, while the second arborescence is depicted with dashed blue arrows. Note that the mean path length of the arborescences of T1 is 2.5, while it is less than 2 in T2.](/figures/fig-1-example-network-from-14-with-two-different-t-rooted-a5o2g1fy.webp)
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Improved Fast Rerouting Using Postprocessing
Klaus-Tycho Foerster, Andrzej Kamisinski, Yvonne-Anne Pignolet, Stefan
Schmid, Gilles Trédan
To cite this version:
Klaus-Tycho Foerster, Andrzej Kamisinski, Yvonne-Anne Pignolet, Stefan Schmid, Gilles Tré-
dan. Improved Fast Rerouting Using Postprocessing. IEEE Transactions on Dependable and Se-
cure Computing, Institute of Electrical and Electronics Engineers, 2022, 19 (1), pp.537 - 550.
�10.1109/TDSC.2020.2998019�. �hal-03048830�
IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 1, NO. 1, MAY 2020 1
Improved Fast Rerouting Using Postprocessing
Klaus-Tycho Foerster , Andrzej Kamisi
´
nski , Yvonne-Anne Pignolet ,
Stefan Schmid , and Gilles Tredan
Abstract—To provide fast traffic recovery upon failures, most modern networks support static Fast Rerouting (FRR) mechanisms for
mission critical services. However, configuring FRR mechanisms to tolerate multiple failures poses challenging algorithmic problems.
While state-of-the-art solutions leveraging arc-disjoint arborescence-based network decompositions ensure that failover routes always
reach their destinations eventually, even under multiple concurrent failures, these routes may be long and introduce unnecessary loads;
moreover, they are tailored to worst-case failure scenarios.
This paper presents an algorithmic framework for improving a given FRR network decomposition, using postprocessing. In particular, our
framework is based on iterative arc swapping strategies and supports a number of use cases, from strengthening the resilience (e.g., in
the presence of shared risk link groups) to improving the quality of the resulting routes (e.g., reducing route lengths and induced loads).
Our simulations show that postprocessing is indeed beneficial in various scenarios, and can therefore enhance today’s approaches.
Index Terms—Resilience, fault-tolerance, computer networks, failover.
F
1 INTRODUCTION
Communication networks have become a critical infrastruc-
ture of our digital society: enterprises which outsource their
IT infrastructure to the cloud, as well as many applications
related to health monitoring, power grid management, or dis-
aster response [1], depend on the uninterrupted availability
of such networks. To meet their dependability requirements,
most modern networks provide static Fast Rerouting (FRR)
mechanisms [2], [3], [4], [5]. Since FRR mechanisms pre-
configure conditional failover behaviors, they enable a very
fast traffic recovery upon failures, which only involves the
data plane but not the (typically much slower [6]) control
plane.
However, while allowing to pre-configure conditional
failover behavior is the key benefit of FRR, enabling the
fast response to failures, it is also the key challenge when it
comes to designing algorithms for such mechanisms: as the
conditional failover behavior needs to be configured before
the failures are known, the algorithmic problem of how to
optimally configure the failover rules at the different routers,
for all possible failures, seems inherently combinatorial. The
problem is particularly challenging in scenarios where packet
headers cannot be used to carry meta-information about en-
countered failures: such header rewriting is often undesired
and introduces overhead (related to header rewriting itself,
but also in terms of additional rules required at the routers
to process such information).
While FRR technology has been used for many years al-
ready in modern communication networks, a major algorith-
mic result on how to configure FRR mechanisms is relatively
•
K.-T. Foerster and S. Schmid are with the Faculty of Computer Science at
the University of Vienna, Austria.
•
Andrzej Kamisi´nski is with the AGH University of Science and Technology,
Poland.
• Yvonne-Anne Pignolet is with DFINITY, Switzerland.
• Gilles Tredan is with LAAS-CNRS, France.
Manuscript submitted December 15, 2019, revised May 11, 2020.
recent: Chiesa et al. [7], [8] showed that by decomposing the
network into
k
arc-disjoint spanning arborescences [9], highly
resilient FRR configurations can be defined. Edmonds [10]
proved that
k
-connected graphs always allow for
k
such
arborescences, and they can be computed rapidly [9].
However, Chiesa et al.’s conjecture that for any
k
-
connected graph, there exists a failover routing resilient
to any
k − 1
failures, remains an open problem. What is
more, while this network decomposition approach ensures
connectivity, the failover routes may be far from optimal
regarding latency (i.e., route length) and congestion.
The goal of this paper is to improve the network decom-
position approach, in terms of resilience, performance, and
flexibility. In particular, we are motivated by the observa-
tion that in practice, additional information about failure
scenarios and failover objectives may be available, e.g., about
shared risk link groups [11], [12], [13] or about critical flows
for which it is important to be routed along short paths, even
after failures. Existing optimizations of arborescence-based
failover schemes are oblivious to such aspects.
Model.
In a nutshell, we consider the problem of pre-
defining (static) conditional failover rules at network’s nodes
(i.e., switches or routers), which define to which link to
forward an incoming packet. These forwarding rules can
only depend on the destination
t
, the in-port at which a
packet arrives at the current node, as well as the status of the
links directly incident to the node. At the same time, they
should not depend on non-local failures or the packet source.
In particular, we do not allow for packet tagging (i.e., header
rewriting) or carrying failure information in the header.
More specifically, we consider FRR mechanisms leverag-
ing arc-disjoint arborescence network decompositions [7], [8]:
for each destination, a set of arborescences are defined which
are rooted at the destination and span the entire network
without two arborescences sharing an arc. As long as no
failure is encountered, a packet travels along an arbitrary
arborescence towards the root, being the destination. When
encountering a failure, a packet is rerouted onto the next
IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 1, NO. 1, MAY 2020 2
arborescence according to some pre-defined order. The logic
of the latter is defined by the arborescence routing strategy.
Contribution.
This paper presents an algorithmic framework
for postprocessing state-of-the-art FRR mechanisms based on
network decompositions, to improve resilience, performance,
and flexibility, of fast rerouting. The framework relies on
an iterative swapping of arcs, hence changing the network
decompositions towards a certain objective. More specifically,
such swapping operations can be used to account for specific
failure scenarios (e.g., given by shared risk link groups), to
improve traffic engineering properties of failover paths (such
as load and stretch), or to flexibly adjust the failover routes
to the specific requirements or priorities of flows (and their
applications).
We show that we do not limit ourselves by focusing
on arc-disjoint arborescence network decompositions by
proving that arborescence-based decompositions are as good
as any deterministic local failover method. Furthermore, we
demonstrate the potential of our arc-swapping framework
in four different use cases: two related to routing (i.e.,
improving stretch and load), and two related to properties
of the decomposition (namely, depth and independence of
paths). We report on extensive simulations using synthetic
network topologies, which illustrate the benefits of our
approach. Moreover, we also provide a novel Integer Linear
Program (ILP) formulation, to directly create optimized
arborescences, instead of postprocessing them.
Organization.
The remainder of this paper is organized
as follows. Section 2 provides intuition on why focusing
on arborescences-based network decompositions is not a
limitation. Our postprocessing framework is described in
Section 3. We discuss and evaluate case studies in Section 4
and present our ILP in Section 5. After reviewing related
work in Section 6 we conclude in Section 7.
2 IMPOSSIBILITY OF BEATING ARBORESCENCES
We first motivate our focus on failover algorithms based
on arborescence network decompositions, showing that this
approach does not only provide a high resilience but also
competitive route qualities (in terms of lengths).
In general, while the static fast rerouting algorithms
considered in this paper have the advantage that they do not
require header rewriting nor control plane reconvergence,
the resulting failover routes may have a high additive stretch.
More formally, the (additive) stretch of a failover route from
v
to
t
is defined as the difference between the number of
hops taken
1)
along the failover route from
v
to
t
and the
hops
2)
along the shortest route from
v
to
t
. The additive
stretch of the routing scheme is then the maximum stretch
along all failover routes, i.e., from all v to t.
We will see that this is a feature inherent to all local fast
failover algorithms, though as the later evaluation sections
show, it is more of a rarely occurring worst-case scenario.
We start with some definitions for arborescence-based
re-routing. Let
(u, v)
denote a directed arc from node
u
to
v
.
A directed subgraph
T
is an
r
-rooted spanning arborescence
of
G
if (i)
r ∈ V (G)
, (ii)
V (T ) = V (G)
, (iii)
r
is the only
node without outgoing arcs and (iv), for each
v ∈ V \ {r}
,
there exists a single directed path from
v
to
r
. When it is
clear from the context, we use the term “arborescence” to
t
x u
w v
T
1
t
x u
w v
T
2
Fig. 1. Example network from [14] with two different
t
-rooted arc-disjoint
spanning arborescence decompositions,
T
1
left and
T
2
right. In both of
them one arborescence is drawn with dotted red arrows, while the second
arborescence is depicted with dashed blue arrows. Note that the mean
path length of the arborescences of
T
1
is 2.5, while it is less than 2 in
T
2
.
refer to a
t
-rooted spanning arborescence, where
t
is the
destination node. A set of arborescences
T = {T
1
, . . . T
k
}
is
arc-disjoint if no pair of arborescences in
T
shares common
arcs, i.e., if
(u, v) ∈ E(T
i
)
then
(u, v) /∈ E(T
j
)
for all
i 6= j
. A set of
t
-rooted arc-disjoint spanning arborescences
is a valid arborescence-based decomposition. See Fig. 1
for two examples of such arborescence decompositions. In
arborescence-based routing, packets follow an arborescence
towards its root. In case of encountering failures on its path
to the root, the packet switches to another arborescence. Let
the blue dashed arborescence be failover route for
x
if its
direct link to
t
fails. In this case the additive stretch is 3 for
the decomposition
T
1
, while it is 1 for
T
2
. This illustrates that
the choice of the decomposition has an impact on the quality
of service in case of failures.
In the following, we show that the arborescence-based
routing scheme depicted in Fig. 2 may lead to a detour
of length
Ω(n)
, even though a constant-length detour is
available. In our example, the arborescences (depicted with
different colours and line patters in Fig. 2) to be used have
been constructed such that a certain set of failures leads to a
long detour for packets emitted by node 22, even though 22
is very close to the destination
t
. The only link out of node
22 belongs to an arborescence that takes this long detour if
no other links fail as the packet will stay on this arborescence
until it reaches t.
In general though, no failover algorithm can obtain a
better stretch than
Ω(n)
for three failures: an adversary could
fail the links
(22, t), (21, 11), (23, 13)
, in which case even
algorithms with global information would take a detour of
length Ω(n).
However, what happens when we strengthen the defini-
tion of additive stretch to a competitive [15] point of view?
Recall that we so far defined the stretch in comparison
to the shortest path in the network without failures. In a
competitive setting, we compare the stretch under some
failure set, to the shortest path in the network with failures.
To give some intuition, in the simple network setting of a
cycle, a local failover strategy is to switch between clockwise
and counterclockwise routing. This strategy induces an
additive stretch according to the size of the cycle, for
the nodes neighboring the destination, when their direct
connection to the destination fails. However, for those nodes,
there is no shorter path to the destination after such a failure,
and hence from a competitive point of view, their failover
route is optimal. We next consider short failover routes.
In the failure example of Fig. 2, an algorithm with global
information could simply take a tour of length
5
from node
22
IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 1, NO. 1, MAY 2020 3
`3
`2
`1
C
`
13
t
11
C
1
23
22
21
C
2
33
32
31
C
3
. . .. . .
Fig. 2. Example of a
(4, `)
-clique-torus (see Definition 1), with
4 t
-rooted
arc-disjoint arborescences, in blue (dotted), red (dash-dotted), green
(dashed), and olive (loosely dashed). Three links (striked out) incident
to
22
have failed in this scenario, forcing a circular scheme to use the
olive (loosely dashed) arborescence at
22
, which takes a tour of length
at least ` − 1, even though a short 5-hop alternative is available.
to
t
, as already pointed out above. Is it possible to find better
deterministic local failover algorithms that can outperform
arborescence-based routing in this example?
In this context, a deterministic algorithm makes all
decisions on which out-port is used by a packet entirely
depend on available information only, no randomness is
used, e.g., when switching to another arborescence. An
algorithm is local if its failover decisions do not take the state
of other routers into account, but only the locally available
information (inport, dst). In particular a router does not know
where in the network other failures have happened.
Succinctly stated, the answer is no—all deterministic local
fast failover routing schemes perform badly in such cases,
i.e., they do not outperform arborescence-based routing. To
this end, we will show that there are
k
-connected
k
-regular
graphs where every deterministic local algorithm has to take
large detours, even though short routes are available.
The intuition behind this statement lies in the fact that
even with the freedom of taking other decisions, there are
cases that lead to long detours and/or high load when
only local knowhow can be used (i.e., the router does not
know where else failures have happened). Thus the power
of algorithms that make deterministic decisions without
knowing anything about the state of other flows and routers
coincides with the power of arborescence based algorithms.
For our proof we define the following graph class: start
with a cycle of
`
nodes and replace each link with
k − 1
parallel links. Observe that these graphs are
k
-connected and
k
-regular, but have parallel links between neighboring nodes.
In order to obtain a simple graph without parallel links, we
expand each node into a clique of
k − 1
nodes, preserving
connectivity and regularity. For example for
k − 1 = 3
, this
results in a 3 × ` torus graph, as in Fig. 2.
Definition 1.
Let
k, ` ∈ N
with
k ≥ 3
,
` ≥ 3
. A
(k, `)
-clique-
torus is a graph with
(k−1)`
nodes and
(`(k−1))+(`
(k−1)(k−2)
2
)
links, constructed as follows: create
`
cliques
C
j
,
1 ≤ j ≤ `
, of
k − 1
nodes, i.e., so far every node has degree
k − 2
. Denote the
k − 1
nodes of each clique
C
j
as
v
j,1
, v
j,2
, . . . , v
j,k−1
. Next and
last, for each
1 ≤ j ≤ `
and each
1 ≤ i ≤ k − 1
, connect
v
j,i
with v
(j mod `)+1,i
.
We show that every deterministic local fast failover
algorithm sometimes has to take detours with a length in
the order of the diameter of the graph, even though a route
with a constant number of hops is available. We note that
our results here refer to deterministic algorithms.
Theorem 1.
For all
k ≥ 3
,
` ≥ 6
: for deterministic local fast
failover algorithms
ALG
resilient to
k − 1
failures on
k
-connected
k
-regular graphs, matching on in-port (from which link the packet
arrives) and destination, the competitive additive stretch of
ALG
vs. a globally optimal algorithm is ≥ ` − 6.
We utilize the following Lemma for the theorem proof:
Lemma 1.
Let
G = (V, E)
be a connected graph with a link
failure set
F ⊂ E
. Let
U
be a
(V
1
, V
2
)
-node separator of
G
s.t.
1) F ⊆ E(V
1
)
,
2)
all links in
F
are not of the type
(v
1
, v
2
), v
1
∈
V
1
, v
2
∈ V
2
,
3)
all nodes in
V
1
, which are adjacent to nodes in
V
2
(denoted as
N
V
2
(V
1
)
) have at most degree
|F | + 1
. Let
t ∈ V
2
s.t. there is no path to
t
in
E(V
1
) \ F
from any node in
N
V
2
(V
1
)
,
but in
E \ F
. Let
R
(V
1
,V
2
)
t,F
be the shortest path from any node in
N
V
2
(V
1
)
to
t
only using edges from
E \ F
and, over all nodes
v ∈ V
2
, v
0
∈ V
1
, with
F
0
= E(v
0
)
, let
R
(V
1
,V
2
)
t,F
0
be the maximum
length, of all shortest paths from nodes
v ∈ N
V
1
(V
2
)
to
t
only
using edges from
E \ E(v
0
)
. Then, the competitive additive stretch
of any
|F |
-resilient local fast failover algorithm
A
, matching on
in-port and destination, on
G
is at least, for all eligible
t, V
1
, V
2
, F
:
max
t,V
1
,V
2
,F
R
(V
1
,V
2
)
t,F
− R
(V
1
,V
2
)
t,F
0
− 1 . (1)
Proof: We start by considering fixed
t, V
1
, V
2
, F
fulfill-
ing the lemma requirements. Observe that
A
must provision
routes from all nodes in
N
V
2
(V
1
)
to
t
when the links
F
fail,
where the shortest of such routes has the length
R
(V
1
,V
2
)
t,F
.
Let
e
be the first link used by
R
(V
1
,V
2
)
t,F
, i.e.,
e
is from some
v
1
∈ V
1
to some
v
2
∈ V
2
. After traversing
e
, the route is
deterministically predefined, never encountering incident
links from
F
. We can furthermore enforce that
e
will be
traversed by
A
, by setting the failure set
F
0
as all links
incident to
v
1
except
e
, with
|F
0
| ≤ |F |
. Now, being at
node
v
2
, both failure sets
F, F
0
are indistinguishable to
A
,
i.e., the remaining route of
R
(V
1
,V
2
)
t,F
will be used. On the
other hand, the globally optimal route from
v
2
to
t
has at
most the the length
R
(V
1
,V
2
)
t,F
0
, with one additional hop from
v
1
. As such, we proved a competitive additive stretch of
(R
(V
1
,V
2
)
t,F
) − (R
(V
1
,V
2
)
t,F
0
+ 1)
for fixed eligible
t, V
1
, V
2
, F
, from
which the lemma statement follows directly.
We can now prove Theorem 1 in a succinct fashion, using
Lemma 1 as follows for
(k, `)
-clique-torus graphs: a local
algorithm cannot distinguish the situation where
1)
all links
between two cliques failed, forcing a long detour, and
2)
being enforced to take a hop on the long detour, by a dense
cluster of failures which leaves a short detour intact.
Proof of Theorem 1: We pick
t
from clique
1
and set
F
as the
k − 1
links between clique
1, 2
, with
V
1
being clique
2
and
V
2
being
V \ V
1
, where the picked
t, V
1
, V
2
, F
fulfill the
requirements of Theorem 1. The theorem statement follows
from
R
(V
1
,V
2
)
t,F
≥ ` − 1
and that
R
(V
1
,V
2
)
t,F
0
+ 1 ≤ 5
(
1
hop for
e
,
1 in C
3
, 2 to reach C
1
, at most 1 extra to reach t ∈ C
1
).
Combining the fact that no deterministic local algorithm
can have a better competitive additive stretch than
Ω(`)
with
the fact that a
(k, `)
-clique-torus graph has
(k − 1)`
nodes,
i.e., ` ∈ Ω(n/(k − 1)), yields the following:
Corollary 1.
For all
k ≥ 3
, deterministic local fast failover
algorithms resilient to
k − 1
failures, matching destination and
in-port, have competitive additive stretch of Ω(n/(k − 1)).
IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 1, NO. 1, MAY 2020 4
Algorithm 1: Basic Arc-Swap Operation
Input: valid arborescence-based decomposition
Output: modified valid arborescence-based
decomposition
1 given a node v and two outgoing arcs e, e
0
2 if arborescence conditions hold for e, e
0
then
3 swap arborescences
3 THE POSTPROCESSING FRAMEWORK
This section presents our algorithmic framework to post-
process arborescence-based network decompositions for im-
proved resilience and performance. In the following, we first
present the general framework, before we discuss concrete
use cases. In particular, we do not make any assumptions on
the given arborescence-based network decompositions nor
the (re)routing strategy used, which can be arbitrary (specific
examples will be considered in our simulations).
The framework can be used to optimize a large set of
objectives. We consider two classes of objectives in this paper
and present two examples each. In the first class, we aim to
improve traffic-engineering metrics of failover routes (like
load or stretch) or account for flow or application priorities,
given certain assumptions about a traffic scenario and failure
model, without sacrificing maximum resilience. As a shorthand,
we will refer to this first class as the
traffic scenario
. In the
second class, the concrete routing mechanism is ignored and
properties of the decomposition, e.g., depth and independence,
are improved, which can lead to shorter paths and higher
resilience respectively.
We will refer to this second class as the
decomposition
.
In the remainder of this section, we introduce our framework
without concrete instantiations of objective functions, which
we will cover in the next section.
At the heart of our postprocessing framework lies an
arc-swapping algorithm which can come in different flavors,
depending on the use case. All the different variants of the
arc-swapping algorithm have in common that they always
preserve connectivity: if a source-destination pair features
a certain property that influences the objective, then this
property can only be improved in each arc-swap operation.
In particular, these swaps must maintain the arborescence
character of the decompositions, i.e., they cannot introduce
cycles.
The general principle is quite simple (see Algorithm 1):
we only swap the arborescences of two outgoing arcs of the
same node
v
and ensure that no cycles are generated. For
simplicity we refer to the set of arcs that do not belong to
any arborescence as the
0
-arborescence, even though they do
not form an arborescence. This allows us to treat all arcs
in a uniform manner and simplifies the description of our
algorithm.
More formally, we revisit the approach use to gener-
ate arborescences as in e.g. [9], where arcs are added to
arborescences incrementally until no further arcs can be
added. When this situation is reached, arcs belonging to
different arborescences and possibly the 0-arborescences are
swapped to allow the process to continue. The (incomplete)
arborescence set is denoted by
{T
1
, . . . , T
k
}
. When growing
an arborescence
T
i
, the following minimal conditions must
t
v
1
v
2
Before swapping After swapping
t
v
1
v
2
e
0
e
e
0
e
Fig. 3. Introductory example with three nodes where growing arbores-
cences sequentially can end in a deadlock. On the left side, the dotted
blue arborescence uses both arcs to
t
, leaving no possibility for the
remaining dashed red arborescence to route to the destination. However,
after swapping
(v
2
, t)
with
(v
2
, v
1
)
, the dashed red arborescence may
use the link
(v
2
, t)
on the right side (and subsequentially, the link
(v
1
, v
2
)
to complete the construction).
hold when swapping
e = (u, v) ∈ 0 − arborescence
with
e
0
= (u, v
0
) ∈ E(T
j
)
to ensure that the resulting arbores-
cences are valid [14]:
1
(1) u has a neighbor v
0
∈ V (T
i
)
(2) e = (u, v)
does not belong to any arborescence yet, i.e.,
e /∈ ∪
ρ=1..k
E(T
ρ
)
(3) u /∈ V (T
i
)
(4) ∃j, s.t. e
0
= (u, v
0
) ∈ E(T
j
)
(5) v ∈ V (T
j
)
(6) v
0
is not on the path to from v to the root in T
j
.
Let us consider an example. Let us assume that arbores-
cences have different colors. In Fig. 3, when we swap the
dotted blue arc
(v
1
, t)
to the unused arc
(v
1
, v
2
)
, the dashed
red arborescence may now take over
(v
1
, t)
, removing the
current deadlock situation. In general, when we cannot add
an arc to
T
i
in the normal round-robin fashion as explained
in [14] and discussed further in Section 4.4, we can check for
candidate arc pairs
e = (u, v), e
0
= (u, v
0
)
leaving node
u
if
we could perform a swapping operation. Analogously to the
above conditions, we can formulate the criteria for swapping
two arcs belonging to arborescences T
i
and T
j
.
In contrast to the swapping checks necessary when
constructing arborescences, we do not have to test whether
each node is incident to an arborescence in this case: this is
guaranteed already by the existing decomposition (condition
(1,4,5)). Thus, in contrast to the swapping conditions during
the arborescence decomposition, there are two cases to
consider.
Case
(i)
:
e = (u, v) ∈ E(T
i
), e
0
= (u, v
0
) ∈ E(T
j
)
. From
the above correctness conditions (1,4,5) are always satisfied,
while (2,3) are irrelevant. In addition to (6), it must hold that
v
is not on the path to from
v
0
to the root in
T
i
. If these
conditions are satisfied, then
e
can be added to
T
j
and
e
0
can
be added to
T
i
. An example is provided in Figure 4, which
improves the depth of both arborescences.
Case
(ii)
,
e = (u, v) ∈ E(T
i
)
and
e
0
= (u, v
0
)
does not
belong to any (real) arborescence,
e
0
/∈ ∪
ρ=1..k
E(T
ρ
)
. In this
case, (1) is always satisfied and (2-6) are irrelevant. Instead,
to be able to remove
e
from
T
i
and replace it with
e
0
it must
hold that
v
does not belong to the path from
v
0
to the root in
T
i
. An example is provided in Figure 5, which gives a better
depth for the dashed red arborescence.
If the conditions are met, then the arborescence set after
the swap is still valid. The time complexity of picking all
1
[9] and similar approaches use additional criteria which are immaterial
to this discussion