IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. X, NO. X, JANUARY XXXX 1
Distributed Placement of Autonomic
Internet Services
Panagiotis Pantazopoulos, Merkouris Karaliopoulos, Member, IEEE,
and Ioannis Stavrakakis, Fellow, IEEE
Abstract—The optimal placement of service facilities largely determines the capability of a data network to efficiently support its
users’ service demands. As centralized solutions over large-scale distributed environments are extremely expensive, inefficient or
even infeasible, distributed approaches that rely on partial topology and demand information are the only credible approaches to
the service placement problem, even at the expense of non-guaranteed optimality. In this paper, we propose a distributed service
migration heuristic that iteratively solves instances of the 1-median problem pushing progressively the service to more cost-
effective locations. Key to our algorithm is a traffic-aware centrality metric, called weighted conditional betweenness centrality
(wCBC), that captures the ability of a node to act as service demand concentrator and is employed in both selecting the nodes and
setting their weights for the 1-median problem instance. The assessment of our heuristic proceeds in two steps. First, assuming
(ideal) knowledge of the invoked wCBC metric, we carry out a proof-of-concept study that demonstrates the effectiveness of
the heuristic over synthetic and real-world topologies as well as its advantages against comparable local-search-like migration
schemes. Next, we devise practical protocol implementations that approximate the heuristic using local measurements of transit
traffic and preserve the excellent accuracy and fast convergence properties of the algorithm for different routing policies. Our
solution applies to a broad range of networking scenarios, and is very relevant to the emerging trends for in-network storage and
involvement of the end-user in the creation and distribution of lightweight (autonomic) service facilities.
✦
1 INTRODUCTION
O
NE of the most significant changes in networked com-
munications over the last few years concerns the role
of the en d-user. Traditionally the end-user has been almost
exclusively the consumer of content and services generated
by explicit entities referred to as content and service
providers, respectively. Nowadays, Web2.0 technologies
have enabled a paradigm shift towards more user-centric
approa c hes to content generation and pr ovision. This shift
is strongly evidenced in the abundance of User-Generated
Content (UGC) in soc ia l networking sites, blog s, wikis,
or video distribution sites such as You Tube, which have
motivated even the rethinking of the Inte rnet architectur e
fundamentals [1], [2]. The generalization of the U GC
concept towards services is inc reasingly viewed as one of
the major trends in user-oriented networking [3].
The user-oriented service creation concept aims at engag-
ing end-users in the ge neration and distribution of service
components, more generally service facilities [4]. There
already exist onlin e applications that enable end-users to
compose their own customized combination of he te roge-
neous web sources through easy-to-use graphical interfaces.
Google App Engine [5] and Yahoo! Pipes [6] are typical
examples of what is often referred to as web-based mashup
tools. At the same time, efforts are under way to develop
platforms that will engage end users in the creation of
services with telecom-based features ( i.e., messaging, voice
• The authors are with the Dept. of Informatics, University of Athens.
E-mail: {ppantaz, mkaralio, ioannis}@di.uoa.gr
This work has been partially supported by the EC IST-FET RECOGNI-
TION project (FP7-IST-257756) and the EC Network of Excellence in
Internet Science project EINS (FP7-ICT-288021).
calls etc) integrated over the Next Generation Networks [7].
In parallel with the proliferation of the so-called User-
Generated Service (U GS) paradigm, significant research
effort is bein g carried out on th e design and deploy-
ment of energy-efficient data storage architecture s. Nano-
datacenters have been proposed with the a im to offload part
of the data management ope rations from the conventional
power-hung ry data-centers [8 ]. Numer ous ISP-owned home
gateways can be instrumented through virtualization tech-
nologies to host those lightweight peer servers and create
a distributed Internet service platform that leverages end-
user p roximity. This shift towards more distributed data
storage paradigms is further evidenced in a) the emerging
Information-centric networking (ICN) parad igm [9] and
its “in-n etwork storage” argument; b) the realiza tion of
distributed-fashion social networks. In ICN, the network is
equippe d with func tionality that allows it to contribute ac-
tively and reliably to the distribution of information objects.
Likewise, Diaspora presents an instanc e of a social network
implemented over interconnected nodes (i.e., pods) that are
hosted by numerous individual users on dedicated local
storage. Each node operates an instance of the Diaspora
software that turns it into a personal server [10].
In the intersection of the aforementioned trends emerges
a rich ecosy stem of highly autonomic service facilities that
will be generated in pretty much every network location
and considered in dependent of other software entities; many
of these services will have strongly local scope, and will
require, in principle , access to storage resources in various
network locations. The technical challenge then is how to
optimally place these services to minimiz e th e ir access cost.
However, more than ever befor e, the search is for scala ble
distributed service placement approaches that can be car-
IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. X, NO. X, JANUARY XXXX 2
ried out b y typical network devices, withou t sp e cialized
processing capacity.
Motivated by these challenges, our work proposes a
scalable decentralized heuristic algorithm that iteratively
moves services from their generation location to the net-
work location that minimizes their access cost. We follow
earlier work on service placement in viewing the problem as
an in stance of the facility location problem [11]; precisely,
we employ the 1-median formulation that is d e emed more
suitable for the user-centric service paradigm. Contr a ry to
centralized approaches, where a single super-entity with
global information about network top ology and service
demand solves the problem in a single iteration, we let
it migrate towards its optimal location over a few hops. In
each hop, a small-scale and simpler 1-median problem is
solved so that the computational load is spread amongst the
nodes along the migration path.
The service migration p ath is derived by invoking a
node centrality metric we have d evised ea rlier in [12] and
call weighted Conditional Betweenness Cen trality (wCBC).
This metric assesses the capacity of a network node to
route service demand load from th e rest of the network
towards the current service location. Therefore, the metric
can effectively identify directions (fig. 1.b) o f high d e mand
attraction, i.e., network areas presenting high demand for
the service. The m etric help us competely determine the
subgrap h wher ein the small-scale 1-median problem will b e
solved ( 1-median subgraph): first, by selecting the nodes of
the subgraph and, then, by modulating the demand w e ights
with which each one participates in the 1-me dian problem
formu lation. We detail the metric and the way it is used by
our algorithm which we call cDSMA, in Sections 3 and 4.
Our contributions are b oth on the theoretical and practical
front. From theo retical point of view, we pr opose a novel
heuristic algorithm for the well-studied 1-median pro blem,
which comes under the broader family of local-search
techniques. The algorithm’s convergence and appr oxima-
tion properties are discussed in Section 4. The negative
result in this respect is that cDSMA is no t a constant-
ratio approximation algorithm, since synthetic examples
can be constructe d, where its deviation from the optimal
cannot be bo unded. On a practical note, we provid e a
systematic spec ifica tion and evaluation of our algorithm,
from the initial concept and properties down to practical
implementation concerns as presented is Sections 6 and 7.
First, we carry out a proof-of-concept analysis (Sec-
tion 6) over synthetic network topologies and under the
ideal assumption that nodes can obtain accurate topological
and d emand information for the whole network. Essentially,
this analysis tests the effectiveness of our metric as a
guide of the service migration and exposes main properties
and advantages of c D SM A. Next, maintaining these ideal
conditions, the algorithm is shown to achieve rema rkably
high accuracy and fast convergence over real- world ISP
topologies of hundreds of nodes, even when the 1-median
problem iterations are solved with no more than 6% of
the total network nodes. Hence, in realistic settings, and
contrary to the theoretical worst-case prescriptions, cDSMA
shows excellent potential to approximate the optimal so-
lution. Moreover, it demonstrates remarkable scalability
and robustness prop erties to service demand estimation
inaccuracies across the network. Finally, it needs much
fewer migration hops to y ie ld placements of given accuracy
than pure local-sear ch policies, which seek for the next
service m igration hop within the local neighborhood of its
current lo cation (fig. 1.a).
Later in Section 7, we relax the assumption of ideal
global information and propose a real-world distributed im-
plementation for cDSMA, ca tering for all challenges related
to distributed operation: how the node ea ch time hosting
the service collects topological an d demand information
and how it uses it to reconstruct the inputs ne e ded by the
algorithm . The implementation leverages the straightfor-
ward interpretation of the wCBC metric so that each node
can locally obtain estima te values of its own wCBC and
communicate them via de dicated messages to the current
service host. This information can then be processed by the
service host to extract partial topological informa tion about
the 1-median subgraph and determine the next service host
on its migration path. The im plementations can exercise
further flexibility regarding how many nodes will measure
and report their local estimates to the service ho st node. As
shown in Section 8, this way they effectively tradeoff the
algorithm a ccuracy with the generated message overhead.
2 THE SERVICE PLACEMENT PROBLEM
The optima l placement of service facilities within network
structures has been typically tackled as an instance of the
facility location problem [11]. Input to the problem is the
topology of the network nodes th a t may host services, their
costs of installation and the distribution of service demand
across the network users. The objective is to place services
in a way that minimizes the joint cost of their installation
and access over all users. Installation costs however are
more relevant to settings of diverse capacity nodes where
the service set-up cost can amount to a n on-negligible part
of a node’s c omputational capacity. Such is the case of the
traditional server-client model rather than the considered
distributed environment of equally powerful nodes. Here,
the lightweigh t services are expected to impose the same
minimal set-up cost across all nodes. Hence, the optimal
placement of up to k replicas of service in stances is treated
as a k-median problem [11]. Otherwise, when storage is re-
stricted, a knapsack problem [13] formulation is emp loyed.
We focus on the 1-median formulation that seeks to
minimize the access cost of a single service replica since it
matches better the expecte d features of the User-Generated
Service paradigm. Recent evidence confirms the existence
of f ew highly popu la r service/content objects and many
others of interest to significantly fewer users [14]. UGS
will enable the generation of service facilities in various
network locations from a highly versatile set of ama-
teur user-service providers. The huge majo rity of these
lightweight ser vice instances will be requiring min imum
storage resources and addressing users in the “proximity”
of the user-service provider, either geographical or social
(friend s, colleagues, etc.), so that their replication across
IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. X, NO. X, JANUARY XXXX 3
a. b. c. d.
Fig. 1. a,b)1-median subgraph nodes under local-search heuristics(a) and cDSMA(b). c) With node 7 as current
service host, two nodes (8 and 11) in the highlighted subgraph
e
G
7
map demand from the rest of the network
(terms w
map
(8; 7), w
map
(11; 7)). d) By crediting the demand of the G \
e
G
Host
nodes only on the entry nodes C
and L of the highlighted
e
G
F
subgraph, cDSMA moves the service to the optimal location C.
the network would not be justified
1
.
We assume that the network topology is represented by
an undir ected connected graph G = (V, E) o f |V | nodes
and |E| links. A subset V
S
⊆ V o f the total network nodes
are enabled ( or even willing) to act as service host sites and
along with the set E
S
⊆ E of ed ges linking them, form
the, generally disconne cted, subgra ph
e
G = (V
S
, E
S
). Each
potential service host k ∈ V
S
may serve one or more users
attached to some network node n ∈ V and accessing the
service with different intensity, generating demand w(n)
for it. The g oal is to minimize over all network users the
access cost of a service facility, which is
Cost(k) =
X
n∈V
w(n) · d(k, n) (1)
when the service is located at node k ∈ V
S
. The distances
d(k, n) may have different context, depending on routing
policies and th e network dimensioning process. The ex-
position of the algorith m hereafter assumes that minimum
cost paths coincide with minimum hopcount path s but
its adaptation to more general shortest-path co ncepts is
straightfor ward .
2.1 Why a distributed heuristic algorithm for ser-
vice migration
Centralized solutions are inefficient, if at all feasible, for
our problem. From a pure algorithmic point o f view, the
1-median p roblem complexity
2
is bounded by O(|V |
3
).
Hence, while not prohibitive, it does not scale well and
improvements are ne c essary, especially for larger networks.
More significantly, centralized approaches assume the exis-
tence of a super-entity with global to pological and service
demand information that has the resources and the manda te
to determine and realize the placements. This implies an
implicit logical hierarchy in the role of network nodes,
which in many cases is not present. Moreover, given that
(minor) user demand shifts or network topology changes
1. For instance, a customized tour service generated by a mashup tool
user to provide urban points of a certain interest, would most likely be
accessed only by those who share that interest and reside in the same area.
2. In comparison, the k-median problem is NP-hard in general topolo-
gies so that much of the research effort around it has been devoted to the
design of efficient approximation algorithms [15].
may be frequent and alter the optima l service location, it
is neither practical nor affordable to each time centrally
compute a new problem solution.
Our approach is to re place the one -shot placement of
service with its few-step migration towards the optimal
location. This way we end up solving a few 1-median prob-
lems of dram atically smaller scale and complexity instead
of coping with the global 1-median optimization problem.
Central to the algorithm is a metric inspired from Co mplex
Network Analysis [16] that we call Weighted Conditional
Betweenness Centrality (wCBC) [12]. For every transit lo-
cation o f the service in the network, the wCBC is a measure
of the demand e a ch node routes towards the current service
host node and is used for two tasks. First, it identifies
nodes in
e
G with the highest wCBC values as candidates
for hosting the service in th e next iteration. These nodes
form the service-host-node-depe ndent 1-median subgraph,
wherein the optimization for the next-best service location
is solved. Secondly, the metric simplifies the map ping of the
service demand from the rest of the network nodes on this
subgrap h. This task is deemed mandatory to approp riately
weigh the service demand gradients across the network.
We detail the metr ic and its practical interpretation in
Section 3.
3 WEIGHTED CONDITIONAL BETWEEN-
NESS CENTRALITY
Central to our distributed approach is the Weighted Con-
ditional Betweenne ss Centrality (wCBC) metric. It orig-
inates from the well-known betweenness centrality metric
and captures both topolo gical and service demand informa-
tion for each node.
3.1 Capturing network topology: from BC to CBC
Betweenness centrality (BC) reflects to what extent a node
lies on the shortest paths linking other nodes. Let σ
st
denote
the n umber of shortest paths between any two nodes s and
t in a connected graph G = (V, E). If σ
st
(u) is the number
of shortest paths passing through the node u ∈V, then the
betweenness ce ntrality index of node u is given by:
BC(u) =
X
s,t∈V,s6=t6=u
σ
st
(u)
σ
st
(2)
IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. X, NO. X, JANUARY XXXX 4
BC(u) cap tures the ability of a no de u to control or assist
the establishment of paths between pairs of nodes. It is an
average value estimated over all network pairs.
In [17] we proposed the Conditional BC (CBC), as a way
to capture th e topological centrality of a random network
node with respect to a specific node t. It is defin ed as
CBC(u; t) =
X
s∈V,u6=t
σ
st
(u)
σ
st
(3)
with σ
st
(s) = 0. Note that the summation is over all
|V − 1| node pairs involving node t rather than all possible
|V |(|V |−1) node pairs, as in (2). Effectively, CBC assesses
to what extent a node u acts as a shortest path aggregator
towards the current service location t, by en umerating th e
shortest paths to t involving u from all other network nodes.
In the supp le mental material we compute the CBC values
and their distributions over simple network topologies.
3.2 Capturing service demand: from CBC to
wCBC
A high nu mber of shortest paths through the node u does
not necessarily mean that equally high demand load stems
from the sources of those paths. Weighted con ditional be-
tweenness centrality (wCBC) enhances the pure topology-
aware CBC metric in a way that takes into accou nt the
service demand that ca n be routed through the shortest
paths towards the service location [12]. The shortest path
ratios of σ
st
(u) to σ
st
in Eq. (3) are now weighted by the
demand loads generated by each node s as follows:
wCBC(u; t) =
X
s∈V,u6=t
w(s) ·
σ
st
(u)
σ
st
. (4)
Note that σ
ut
(u) = σ
ut
so that the wCBC(u; t) value
of node u is lower bounded by its own demand w(u).
Therefore, wCBC assesses to what extent a no de can serve
as demand load concentrator towards a given service loca-
tion. Clearly, when the demand for a service is uniformly
distributed across the ne twork nodes, the w CBC metric
degenerates to the CBC one, within a scale constant.
3.2.1 Approximating the metric with measurements
The wCBC(u; t) metric practically represents the service
demand that node u routes toward no de t, including its own
demand w(u) and the transit demand w
trans
(u; t) flowing
from other network nodes through u towards t. Therefore,
individual nodes may, in principle, estimate their own
metric values wC
ˆ
BC through passive measurements [18]
of the serv ic e demand they route towards the current service
host node. In other words, what is actually computed
theoretically in (4) for node u demandin g global infor-
mation about the network topology and service deman d,
can be locally approxima te d by u providing the basis for
the practical implementation of a distributed so lution. The
approximation lies in the fact th a t what is measu red, even
with perfect accuracy, is not always equal to the nominal
wCBC value, as specified in (4). For example, when there
are, say m, shortest paths between a given node pair (s, t)
but the routing protocol uses only one of those, a node will
measure the full demand of s, whereas, theoretically, the
contribution to the nominal wCBC value is the (1/m)
th
of the measured one. We will see later in section 7 that
what matters is the estimate of the ac tual demand routed
through the nod e rather than a wCBC approximation.
4 THE CDSM ALGORITHM DESCRIPTION
Our centrality-driven Distributed Service Migration Algo-
rithm (cDSMA) progressively steers the service towards its
optimal location via a finite number of steps.
Step 1: Initia liza tion. The first algorithm iteration is
executed at node s in
e
G that initially generates the service
facility (pseudocode line 2). In subsequent iterations, the
new reference node is the one each time hosting the service.
Step 2: Metric comp utation and 1-median subgraph
derivation. Next, the wCBC(u; s) metric is computed
3
for
every node u in the network graph
e
G. Nodes in
e
G featu ring
the top α% wCBC values, together with the node cur rently
hosting the service (Host) form the 1 -median subgraph
e
G
Host
over which the 1-me dian problem will be solved
(lines 3−4 and 15−16). Clearly, its size and the algorithm
complexity are directly affected by the α parameter choice.
Step 3: Mapping the demand of the remaining nodes on
the subgraph. To account for the contribution of the “out-
side world” to the service provisioning cost, the demand
for service from nodes in G \
e
G
Host
(i.e., the non-shaded
nodes in fig. 1.c) is mapped on the
e
G
Host
ones. To do this
correctly and with no redund ancy, the algo rithm credits the
demand of some outside node z only to the first “entry”
e
G
Host
node enc ountered on each shortest path (over G),
from z towards the service host. Thus, the weights w(n) for
calculating the service access cost at node n in the
e
G
Host
subgrap h (see Section 2) are replaced by effective de mands:
w
eff
(n; Host) = w(n) + w
map
(n; Host), where (a ssum-
ing that Host is node t):
w
map
(n; t) =
X
z∈{G\
e
G
t
}
w(z)
σ
′
zt
(n)
σ
zt
(5)
σ
′
zt
(n) =
σ
zt
X
j=1
1I
{n∈SP
zt
(j)
T
n= argmin
u∈SP
zt
(j)
d(z,u)}
with SP
zt
(j) standing for the j
th
element of the shortest
path set fro m node z to node t. For example, in fig. 1.c the
original service demand of node, say, 16 is not mapped on
all the
e
G
7
nodes lying on the shortest path s from 16 to the
Host 7 (i.e., 11, 12 and 8), but only on 11.
The mapping step and its rationale can be better under-
stood in the following example. In the network of fig. 1.d
the service mig rates towards the lowest cost lo cation, which
under uniform demand is n ode C. Assume that the
e
G
Host
subgrap h size is 4 and at some migration step the service re-
sides at node F . The top wCBC nodes arou nd F are C, K
and L. The c D SMA assigns w
eff
values only to the entry-
nodes C and L, w
eff
(C; F ) = 6 and w
eff
(L; F ) = 3,
and while setting the w
map
values of nodes F and K to
3. For the actual wCBC computation, which involves solving the all-
pairs shortest path problem, we properly modified the scalable algorithm
in [19] for betweenness centrality computation, with runtime O(|V ||E|).
IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. X, NO. X, JANUARY XXXX 5
zero, effectively identifies the gr a dient direction towards
the node C. Thus, it better projects the demand attraction
forces on the selected nodes, C being the stronger. On the
contrary, if we map th e demand of node s in G \
e
G
Host
on all
e
G
Host
nodes, we en d up with w
eff
(F ; F ) = 8 and
w
eff
(K; F ) = 3. The serv ice then c a nnot identify node C
as th e next-best loc a tion and locks at node F .
Step 4: 1-median problem solution an d service migration
to the new host node. Any cen tralized techniqu e ( e.g., [15])
may be used to solve this sma ll-scale optimization problem
and determine the next best lo c ation of the service in
e
G
Host
. Note that the pairwise physical distances between
e
G
Host
nodes in Eq. (1) are computed over the original
graph G (see fig. 2). We call s the current serv ice locatio n
(line 21) while the optimal one in
e
G
Host
is assigned
to Host (li ne 22). As long as node s (a) yields higher
cost than the candidate Host node (line 10); and (b) the
candidate Host has not been u sed as a service host before
(lines 11−13), the service is moved there and the algo rithm
iterates through steps 2-4. Progressively, cDSMA steers the
service to the (globally) lowest-cost loc a tion.
Fig. 2. Left: Selected 1-median subgraph
e
G
C
com-
prised solely of (highlighted) nodes that represent ser-
vice host candidates over a non-weighted grid topol-
ogy. Right: The corresponding overlay that reflects the
physical distances between all
e
G
C
node pairs. Aggre-
gated physical links are presented with thick lines.
Algorithm 1 cDSMA in
e
G(V
S
, E
S
)
1. choose randomly node s
2. place SERV ICE @ s
3. for all u ∈
e
G do compute wCBC(u; s), set f lag(u) = 0
4.
e
G
s
← {α% of
e
G with top wCBC values} ∪ {s}
5. for all u ∈
e
G
s
do
6. compute w
map
(u; s)
7. w
ef f
(u; s) ← w
map
(u; s) + w(u)
8. compute cost C(u) in
e
G
s
9. Host ← 1-median solution in
e
G
s
10. while C
Host
< C
s
do
11. if f lag (Host) == 1 then
12. abort
13. else
14. move SERV ICE to Host, flag(s) = 1
15. for all u ∈
e
G do compute wCBC(u; Host)
16.
e
G
Host
← {α% of
e
G with top wCBC values} ∪ {Host}
17. for all u ∈
e
G
Host
do
18. compute w
map
(u; Host)
19. w
ef f
(u; Host) ← w
map
(u; Host) + w(u)
20. compute cost C(u) in
e
G
Host
21. s ← Host
22. Host ← 1-median solution in
e
G
Host
23. end if
24. end while
Fig. 3. A ring topology of N = 2k nodes under a
non-uniform demand pattern results in symmetric (with
respect to B) demand mapping on the G
B
entry nodes
K and L. This blocks the service migration process and
yields a highly suboptimal solution.
Convergence and approxima tion properties: We com-
plete the descrip tion of cDSMA by elaborating on its con-
vergence properties and theoretic capability to approximate
the optimal solution. Clearly, a service facility following the
migration process of Alg orithm 1 will visit any
e
G network
node at most once (see condition in line 12). Thus, our
heuristic will take O(|V |) steps to terminate. Its theoretical
performance bounds are studied next.
Proposition 4.1: cDSMA provides no constant factor
approximation guarantee.
Proof: We sketch a counterexample tha t leads to
arbitrarily bad solution quality: Assume, witho ut loss of
generality, that a ring topology consists of N = 2k
(k ∈ Z
+
) potential service ho st nodes i.e.,
e
G ≡ G. Every
node but one aggregates a unit of demand load from the
users it ser ves (see fig. 3); the single heavy hitter node A
generates W units of demand load and the service facility is
generated at the anti-diametric ring node B. Under cDSMA
the current service host B will select αN nodes with α < 1,
that will form its G
B
subgrap h. Interestingly enough, the
demand that will be mapped on the G
B
-chain entry nodes
(i.e., K and L) is su ch that the in itial location becomes a
local minimum for every value of α < 1. Therefore, the
service re mains with node B without initiating at all the
migration process. The global access cost C
cDSMA
(B) is
C
cDSMA
(B) = 2
k−1
X
i=1
i + W k =
N
2
+ 2N(W − 1)
4
(6)
On the other hand, the optimal service location is at node
A, where the cost is:
C
OP T
= 2
i=k−1
X
i=1
i + k =
N
2
4
(7)
Therefore, their ratio equals:
C
cDSMA
(B)
C
OP T
= 1 + 2
W − 1
N
(8)
Eq. 8 shows that the resulting placement may bec ome
arbitrary bad as the demand of the heavy hitter rises.
Proposition 4.1 su ggests that there are combination s of
network topology and dema nd that may generate such sym-
metric 1-median subgra ph mappings that tra p the service in
a (local) minimum and prematurely terminate the migration
process. On the positive side of the particular unfavorable
example, the approxima tion ratio improves fast as the net-
