On Topological Design of Service Overlay Networks A. Sen∗, L. Zhou, B. Hao and B. H. Shen S. Ganguly Dept. of Computer Science & Engineering Dept. of Broadband & Mobile Networks Arizona State University, Tempe, AZ 85287 NEC Laboratories, America {asen,ling.zhou,binhao,bao}@asu.edu
[email protected]
Abstract The notion of service overlay network (SON) was proposed recently to alleviate difficulties encountered in providing end-to-end quality of service guarantees in the current Internet architecture. The main problem is that end-to-end traffic often traverses through multiple autonomous systems (AS) that do not have any multilateral service level agreement (SLA) for traffic from other domains. The SONs are able to provide QoS guarantees by purchasing bandwidth from individual network domains and building a logical end-to-end data delivery infrastructure on top of existing Internet. In this paper, we consider a generalized framework for SON categorized based on three different characteristic features: a) single-homed versus multi-homed b)usage-based versus leased cost model and c) capacitated versus uncapacitated network. We focus on the algorithmic analysis of the topology design problem for the above generalized SON. We prove that for certain cases polynomial time optimal algorithm exists while for other cases, the topology design problem is NP-complete. For the NP-complete cases, we provide approximate solutions. Experimental evaluation demonstrates that our approximate solutions produce reasonable quality solution in most of the instances in a fraction of time needed to find the optimal solution.
1
Introduction
The Internet today is comprised of multiple independently operated networks (autonomous systems or domains) joined at the peering points. The independently operated networks (often Internet Service Providers, ISPs) may have an interest in providing QoS guarantees within their own network but they in general do not have any incentive to provide service guarantees to customers of other remote ISPs. The notion of service overlay network (SON) was proposed in [6] to overcome this problem, so that end-to-end guarantees can be provided to the customers of different ISPs. Service ∗
Corresponding author: Arunabha Sen, Telephone: 480-965-6153, Fax: 480-965-2751, e-mail:
[email protected]
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overlay network is an outcomes of the recent studies on overlay networks such as Detour [13], resilient overlay network [2] and internet indirection infrastructure [14]. These studies shows that end-to-end performance can be significantly improved by utilizing alternate path created over a overlay network. Further studies in [15, 16, 17] shows that when compared with the ideal routing, network-level routing often yields suboptimal user performance due to routing hierarchy and BGP policy routing. Further empirical studies reported in [18] shows that 50% of the bottlenecks are at the intra-ISP links. The SONs are able to provide end-to-end QoS guarantees by building a logical delivery infrastructure on top of existing transport network by purchasing bandwidth from individual network domains. The SONs provide various flexibilities in deploying and supporting new services by allowing the creation of service-specific overlay network without incorporating changes in the underlying network infrastructure. This mechanism can be utilized to support applications for fault-tolerance [2], multi-cast communication [3], security [4], file sharing [5] and QoS [10]. We consider the SON model described in [11], where the SON is constructed on top of an infrastructure of ISPs and is capable of providing QoS guarantees to a set of customers. Because of this capability, the network is referred to as a QoS Provider Network or Provider Network. The provider network comprises of a collection of provider nodes and a set of customers, referred to as the end-systems or enterprises. The provider nodes and the end-systems gain access to the Internet through the ISPs. An illustration of the SON is given in Figure 1. The provider nodes are connected to each other through ISPs and the end-systems are also connected to the provider nodes through ISPs. Two provider nodes are said to be connected to each other, if they are connected to the same ISP. Similarly, an end-system is said to be connected to a provider node if they are connected to the same ISP. Figure 2 illustrates the relationships between provider nodes, ISPs and
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end-systems. The provider node buys services (guaranteed bandwidth) from the ISPs and sells them to the end-systems with end-to-end service guarantees. Currently there exists at least one commercial service overlay network (Internap [9]) that closely resembles the model used in this paper as well as in [11]. The topology design problem of a SON can be described as follows: Given a set of end-systems, provider nodes, access cost of traffic from an end-system to a provider node, transport cost of traffic among provider nodes, traffic demand for every pair of end-systems, find the least cost design that satisfies the traffic bandwidth demand between every pair of end-systems. Our work is inspired by the recent study done by Vieira et.al. [11] on topology design problem for a specific SON model. In this paper, we introduce a generalized framework for SON that provides a comprehensive view of the overall topology design space. We categorize the generalized SON model based on the following cases: • Single-homed vs multi-homed: The term multihoming is generally used to indicate that an end-system is connected to multiple ISPs [1]. Advantages of multihoming with overlay network is studied in [19, 20]. In the context of SON, we extend this notion where multihoming refers to the scenario where one end-system can be connected to multiple provider nodes instead of multiple ISPs. In a multi-homed environment, an end-system has more flexibility in connecting to a set of provider nodes. This flexibility enables the designer to find a lower cost solution. The figures 4 and 5 show an example of single-homed versus multi-homed scenario, where the cost of the single-homed design is 26 and that of multi-homed is 18. However, multihoming might require cooperation at the end-system in having specialized NAT based system [1] to connect to multiple provide nodes. • Usage-based vs leased(fixed) cost model: In the usage-based cost model, the cost of 3
the link is proportional to the volume of data sent through the link. In a leased or fixed cost model, we assume that each link has an associated cost that is independent of the traffic sent through it. Such fixed cost scenario is often applicable to enterprises who buy leased lines from the ISPs at a flat rate. In applying the cost model, we do not differentiate among the links connecting the end-systems and provider nodes. • Capacitated vs uncapacitated network: In case of a capacitated network, we assume that any link in the SON has a capacity bound that cannot be exceeded. While in an uncapacitated case, there exist no such constraints. We consider the topology design problem for the generalized SON scenarios. It is to be noted that authors in [11] provide solutions for the specific case of single-homed, uncapacitated network with usage-based cost model. In this paper, we provide a comprehensive study of the SON design problem.
1.1
Contribution and summary of results
The key contribution of this paper are as follows: • We prove that the SON topology design problem with (a) multi-homed enterprise, (b) usagebased cost model and (c) uncapacitated network can be solved in polynomial time. • We prove that the SON topology design problem with (a) single-homed enterprise, (b) usagebased cost model and (c) uncapacitated network is NP-Complete. • We prove that the SON topology design problem with (a) single-homed/multi-homed enterprise and (b) fixed cost model is NP-Complete in both capacitated and uncapacitated network scenarios. We present approximation algorithms for the solution of uncapacitated versions of 4
Cost Model Usage-based cost Fixed cost
Uncapacitated Network single-homed Multi-homed NPC Poly. Solution NPC NPC
Capacitated Network single-homed Multi-homed NPC NPC (MCMCF) NPC NPC
Table 1: Complexity results for different versions of SON design problem End system 1
Provider Node 1
Provider Node 2
Provider Node 3
Provider Node 4
Provider Node 5
ISP 1
ISP 2
ISP 3
ISP 4
ISP 5
End System 1
End System 2
End System 3
End System 4
End System 5
End system 2
PN 2 ISP 1
End system 3 ISP 2
PN 3
PN 1 ISP 3 ISP 5
ISP 4
PN 4
PN 5
Provider Node End system 5
End system 4
End system
Figure 2:
Relationship between provider
Figure 1: Service Overlay Network Model nodes, end-systems and ISPs these problems. Our experimental evaluation demonstrates that our approximate algorithms produce reasonable quality solution in most of the instances in a fraction of time needed to find the optimal solution. • We show that all four problems in the capacitated versions of the SON design problem are NP-Complete. A summary of the complexities involved in the topology design problem for the various cases in shown in table 1.
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Table 2: Basic Notations ESi P Nj M N αij α lij L bij B ωij Ω
2
end-system i Provider node i Number of end-systems Number of provider nodes Access cost (per unit of reserved bandwidth) for traffic from ES i to P Nj Access cost matrix for traffic from all ES i to all P Nj Transport cost (per unit of reserved bandwidth) for traffic on the transport link from P N i to P Nj Transport cost matrix for traffic on the transport link from all P N i to all P Nj Cost of least-cost route (per unit of reserved bandwidth) for traffic between P N i to P Nj Cost of least-cost route matrix for traffic between all P N i to all P Nj Reserved bandwidth for traffic from ES i to ESj Reserved bandwidth matrix for traffic from all ES i to all ESj
Problem Formulation
As described in the previous section, the optimal topology design problem of a SON is as follows: Given a set of end-systems, provider nodes, access cost of traffic from an end-system to a provider node, transport cost of traffic from one provider node to another, demand for bandwidth between every pair of end-systems, find the least cost design that satisfies the traffic bandwidth demand between every pair of end-systems. In this paper we consider several different versions of this basic problem corresponding to three different scenarios: (i) the end-systems may be single-homed or multi-homed, (ii) the cost for network link may be usage-based or fixed [12], (iii) the capacity of the links may be infinite (very large) or finite (limited or capacitated). It may be noted that the results presented in [11] consider only the case where (i) the end-systems are single-homed, (ii) the cost for network link is usagebased and (iii) the capacity of the links is infinite. The notations used in this paper are essentially the same as in [11], and are shown in Table 2. The access cost αij of a link connecting an end-system ES i to a provider node P Nj refers to the 6
cost encountered to transfer one unit of data using that link in usage-based cost model or transfer any (greater than zero) amount of data using that link in fixed cost model. In case ES i can be connected to P Nj through more than one ISP, αij represents the least expensive way of connecting ESi to P Nj . If ESi cannot reach P Nj through any ISP, then the access cost α ij = ∞. If two provider nodes P Ni and P Nj are connected to the same ISP, then they are said to be connected by a transport link. The transport cost l ij of a link connecting P Ni to P Nj refers to the cost encountered to transfer one unit of data using that link in usage-based cost model or transfer any (greater than zero) amount of data through that link in fixed cost model. In case P N i can be connected to P Nj through more than one ISP, lij represents the least expensive way of connecting P Ni to P Nj . If P Ni cannot reach P Nj through any ISP, then the transport cost l ij = ∞. With these set of inputs we can construct an input graph G ESP N = (VESP N , EESP N ) as follows. The vertex set VESP N can be divided into two groups VP N and VES , representing the provider nodes and the end-systems respectively. Similarly, the edge set E ESP N can be divided into two groups EP N,P N and EES,P N . If vi , vj ∈ VP N , each edge connecting these two nodes has a weight l ij associated with it. The lij values for all pairs of provider nodes is denoted by matrix L. The length of the shortest path between any pair of nodes v i , vj ∈ VP N , is denoted by bij . The bij values for all pairs of provider nodes is denoted by matrix B. If v i ∈ VES and vj ∈ VP N , each edge connecting these two nodes has a weight α ij associated with it. The αij values for all end-system to provider node pairs are denoted by matrix α. If v i , vj ∈ VES , then there is a traffic demand ωij associated with this ordered pair of nodes. The trafic demand for all pairs of end-systems is denoted by matrix Ω. An illustration of such a graph is shown in Figure 3. In this example, ES 1 wants to communicate with ES3 and ES4 , and ES2 wants to communicate with ES3 . In the fixed cost scenario, we do not need to pay attention to the actual bandwidth request by the end-systems.
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The optimal solution for the single-homed and multi-homed versions of the SON design problem are shown in figures 4 and 5 respectively. It may be observed that in this example, the optimal cost of the single-homed version is 26, whereas that of the multi-homed version is 18. In a single-homed enterprise model, each end-system is restricted to be connected to exactly one provider node. In a multi-homed model, there is no such restriction. In the capacited version of the SON topology design problem, each edge e, e ∈ E ESP N has a finite capacity c(e) associated with it. In the uncapaciated version of the problem, we make the same assumption as in [11] and take c(e) to be infinity (or larger than the total demand). The main difference between usage-based model and fixed cost model is how access and transport cost are calculated, especially in the case when the same edge is on more than one path between endsystem pairs. If the path from end-system ES i to ESj is ESi → P Nk1 → P Nk2 → . . . → P Nkp → ESj , then in a usuage based cost model, the cost of this path is (α ik1 + a fixed cost model it is (αik1 +
Pp−1
i=1 lki ki+1
Pp−1
i=1 lki ki+1 +αkp j )ωij ,
and in
+ αkp j ). If an edge epq ∈ EES,P N is used for transferring
ωi,j amount of data from ESi to ESj and also used for transferring ωi,k amount of data from ESi to ESk , then the cost of using this link will be α pq (ωi,j + ωi,k ) in usuage-based cost model and only αpq in the fixed cost model. Similarly, If an edge e pq ∈ EP N,P N is used for transferring ωi,j amount of data from ESi to ESj and ωr,s amount of data from ESr to ESs , then the cost of using this link will be lpq (ωi,j + ωr,s ) in usuage-based cost model and only l pq in the fixed cost model.
3
SON Topology Design Algorithms and their Complexities
In this paper we consider eight different versions of the SON topology design problem. We show that only one of the four different versions corresponding to uncapaciated network model can be solved in polynomial time and the other three versions are NP-complete. Since uncapacitated 8
1 2
2
2 1
2
20
2
2 20
4
1 2
2
20
20
3
4
2
2 1
2
2 10
8
2
2
2 1
15
20 10
1
2
2
8
4
8 2
3
4
3
2 3
4
2 3
4
3
Provider Node
Provider Node
Provider Node
End system
End system
End system
Figure 3: Service Overlay Net- Figure 4: single-homed Soluwork Model
tion for SON Design
Figure 5: multi-homed Solution for SON Design
version of the problem is a special case of the capacitated version, and we have proved that the uncapacitated versions of the problem are NP-complete, we can claim that the capacitated version of the corresponding problems are also NP-complete. It may be noted that the capacitated version of the problem with multi-homed enterprise and usage-based cost (MHE/C/UBC) model is nothing but minimum cost multi-commodity flow (MCMCF) [7] problem and the vast body of literature available for the solution MCMCF can be utilized for the solution of this version of SON design problem. The complexity results of various versions are summarized in table 1.
3.1
SON Design Problem with Multi-homed Enterprise, Uncapacitated Network and Usage-based Cost Model (SONDP-MHE/UN/UBC)
Instance: Given a graph GESP N = (VESP N , EESP N ), with matrices, L, α, Ω and a positive integer K. Question: Is it possible to construct a SON topology with total cost less than or equal to K so that all traffic demand given in matrix Ω can be satisfied? Theorem 1. SON design problem with MHE/UN/UBC can be solved in polynomial time.
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Proof: One can make a few key observations regarding the characteristics of this version of the problem that enables one to develop a polynomial time algorithm for the solution. Observation 1: Multi-homed environment enables one end-system ES i to send data to two other end-systems ESp and ESq using different provider nodes P Nr and P Ns . Observation 2: Since the network is uncapacitated, we can construct the data path from ES i and ESj independently from the construction of the data path from ES p and ESq without concern about exceeding any link capacity. Observation 3: Since usage-based cost model is used, a path established to transfer w ij amount of data from ESi to ESj will incur some cost. This cost is independent of the cost encountered in establishing paths between other pairs of end-systems. From these three observations, we claim that path establishment between various pairs of endsystems can be carried out independently of one another, i.e., the problem of path establishment between a collection of source end-system nodes to a collection of destination end-system nodes can be de-coupled. We can establish the shortest path for each pair of end-systems separately and yet obtain a globally optimal solution. The claim can be proved by noting the fact that if a nonoptimal(shortest) path is used in connecting ES i to ESj , it can be replaced by the optimal path from ESi to ESj , thereby lowering the total cost of the SON topology design. The computational complexity of this algorithm is O(k(|V ESP N | log |VESP N | + |EESP N |) where k = number of end-systems pairs that needs to transfer data between them.
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3.2
SON Design Problem with single-homed Enterprise, Uncapacitated Network and Usage-based Cost Model (SONDP-SHE/UN/UBC)
From the transport cost matrix L (in Table2), we compute the least-cost route matrix B. The problem instance is stated as follows. Instance: Given a graph GESP N = (VESP N , EESP N ), with matrices, B, α, Ω and positive integer K. Question: Is it possible to construct a SON topology with total cost less than or equal to K so that all traffic demand given in matrix Ω can be satisfied and each end-system is connected to one provider node? Theorem 2. SON design problem with SHE/UN/UBC is NP-complete. Proof: We can restate the question more formally in the following way: Question:Is there a function g : {1, 2, ..., M } → {1, 2, ..., N }, namely an assignment of end-systems to provider nodes where each end-system is assigned to exactly one provider node, such that M X M X
wij (αig(i) + bg(i)g(j) + αjg(j) ) ≤ K?
(1)
i=1 j=1
Clearly SONDP-SHE/UN/UBC is in NP, as given a function g, one can easily verify whether the inequality is satisfied or not. In the rest, we will prove that the problem is also NP-Complete. We give a polynomial time reduction from Matrix Cover[6]to SONDP-SHE/UN/UBC. Since Matrix Cover is NP-Complete, this will prove the theorem. The reduction maps a Matrix Cover 0
instance (an n × n matrix A = {aij }, K) to a SONDP-SHE/UN/UBC instance (Ω, B, α, K ) so that there is a function f : {1, 2, ..., n} → {−1, +1} such that
Pn
only if there is a function g : {1, 2, ..., M } → {1, 2, ..., N } such that 0
αjg(j) ) ≤ K . 11
i=1
Pn
j=1 aij f (i)f (j)
PM PM i=1
≤ K if and
j=1 wij (αig(i) +bg(i)g(j) +
Given any instance of Matrix Cover: An n×n matrix A = {a ij } with nonnegative integer entries, and an integer K. We construct the instance for SONDP-SHE/UN/UBC problem as follows: 1. Let M = n, for the M × M bandwidth reservation matrix Ω = {w ij }, ∀1 ≤ i ≤ M, 1 ≤ j ≤ M, wij = 1. 2. Let N = 2M , for the N × N transport matrix B = {b ij }, ∀1 ≤ k ≤ M, 1 ≤ l ≤ M
akl + alk akl + alk + max; b2k−1,2l−1 = + max; 2 2 (2) akl + alk akl + alk + max; b2k−1,2l = − + max; b2k,2l−1 = − 2 2 where max is the maximum element of matrix A in the instance of Matrix Cover. The reason b2k,2l =
why max is added is that we want to make sure that b ij is nonnegative. 3. For the M × N Access Cost matrix α = {α ij }, ∀1 ≤ i ≤ M 0 if j = 2i − 1, or 2i αij = ∞ otherwise
(3)
4. Let K 0 = K + M 2 · max = K + n2 · max. It’s not hard to see that the construction can be done in polynomial time. To complete the proof, we show that this transformation is indeed a reduction. Suppose for the instance of matrix cover problem, there is a function f : {1, 2, .., n} → {−1, +1} such that
Pn Pn i=1
j=1 aij f (i)f (j)
≤ K, then we can build the corresponding g : {1, 2, .., M } →
{1, 2, ..., N } for the instance of SONDP-SHE/UN/UBC in the following way: ∀1 ≤ i ≤ M (M = n, N = 2M ) 2i if f (i) = +1 g(i) = 2i − 1 if f (i) = −1 12
(4)
Due to the construction of the instance of the SONDP-SHE/UN/UBC problem, we get the following table which indicates the relationship between the two objective functions of the two problems respectively. f (i) +1 −1 +1 −1
f (j) +1 −1 −1 +1
aij f (i)f (j) aij aij −aij −aij
aji f (j)f (i) aji aji −aji −aji
g(i) 2i 2i − 1 2i 2i − 1
g(j) 2j 2j − 1 2j − 1 2j
bg(i)g(j) aij +aji + max 2 aij +aji + max 2 aij +aji − 2 + max a +a − ij 2 ji + max
bg(j)g(i) aij +aji + max 2 aij +aji + max 2 aij +aji − 2 + max a +a − ij 2 ji + max
Table 3: relationship between two objective functions
Given the g function we have build, it is true that M M X X
wij (αig(i) + bg(i)g(j) + αjg(j) ) =
i=1 j=1
=
M M X X
i=1 j=1 n X n X
bg(i)g(j)
(wij = 1, αig(i) = αjg(j) = 0)
aij f (i)f (j) +
i=1 j=1
n X n X
max (5)
i=1 j=1
≤ K + n2 · max =K
0
Conversely, suppose that for the instance we have built for the SONDP-SHE/UN/UBC problem, there is a function g : {1, 2, .., M } → {1, 2, ..., N } such that
PM PM i=1
j=1 wij (αig(i)
+ bg(i)g(j) +
0
αjg(j) ) ≤ K . Then ∀1 ≤ i ≤ M , g(i) must be equal to 2i or 2i − 1, since otherwise α ig(i) will be equal to ∞, which cannot make the inequality satisfied. Then we can build the corresponding f : {1, 2, .., n} → {−1, +1} for the instance of Matrix Cover in the following way: ∀1 ≤ i ≤ n(n = M, N = 2M ); +1 if g(i) = 2i f (i) = −1 if g(i) = 2i − 1
(6)
Similarly, due to the construction of the instance for the SONDP-SHE/UN/UBC problem and 13
the relationship shown in Table 3.2, it is true that n X n X
aij f (i)f (j) =
bg(i)g(j) −
M M X X
=
wij (αig(i) + bg(i)g(j) + αjg(j) ) −
max
i=1 j=1
i=1 j=1
i=1 j=1
n X n X
M X M X
n n X X
max
i=1 j=1
i=1 j=1
(7)
0
≤ K − n2 · max =K This proves the theorem.
3.3
SON Design Problem with Multi-homed/single-homed Enterprise, Uncapacitated Network and Fixed Cost Model (SONDP-MHE/UN/FC)
In this section we consider both the multi-homed and single-homed versions of the uncapacitated network with fixed cost model. The multi-homed problem is as follows: Instance: Given a graph GESP N = (VESP N , EESP N ), with matrices, L, α, Ω and positive integer K. Question: Is it possible to construct a SON topology with total cost (under fixed cost model) less than or equal to K so that all traffic demand given in matrix Ω can be satisfied? The single-homed problem is as follows: Instance: Given a graph GESP N = (VESP N , EESP N ), with matrices, L, α, Ω and positive integer K. Question: Is it possible to construct a SON topology with total cost (under fixed cost model) less than or equal to K so that all traffic demand given in matrix Ω can be satisfied and each end-system is connected to one provider node? 14
Theorem 3. SON design problems with MHE/UN/FC and SHE/UN/FC are NP-complete. Proof. Clearly, the SONDP-SHE/UN/FC problem belongs to NP. We prove the NP-Completeness of SONDP-SHE/UN/FC by reduction from Steiner Tree Problem. Given any instance of Steiner Tree Problem: an undirected graph G(V, E), weight function c : E(G) → R + , the set of Steiner nodes S ⊆ V (G), and a positive integer K. We construct the instance for SONDP-SHE/UN/FC problem as follows: Take G as the set of Provider Nodes and transport links,i.e. V P N = V (G), ∀vi , vj ∈ VP N , lij = cij . For each Steiner node vi ∈ S, add a new end-system node ti , and connect ti and vi by a new edge with access cost αij = 0. We also require that any pair of end-systems t i , tj ∈ VES has bandwidth reservation ωij = 1. The positive integer K keeps unchanged. It’s no hard to see that the construction can be done in polynomial time. To complete the proof, we show that this transformation is indeed a reduction. Suppose for the instance of Steiner Tree problem, there is a Steiner tree T for S in G with total cost c(E(T )) less than or equal to K, then by adding all the edges (t i , vi ), ∀vi ∈ S, we get a solution for the SONDP-SHE/UN/FC problem: each pair of end-systems has been connected, each end-system is connected to one provider node, and the total cost of chosen edges is no greater than K. Similarly, given the solution for the instance we have constructed for SONDP-SHE/UN/FC problem, by removing all the edges (ti , vi ), ∀vi ∈ S,we get the solution for the instance of Steiner Tree problem, since all the Steiner nodes have been connected, and total cost of the steiner tree is no greater than K. Therefore, the transformation is a reduction and the SONDP-SHE/UN/FC problem is NP-Complete. It should be noted that the instance of SONDP-SHE/UN/FC problem we constructed can also be treated as a special instance for SONDP-MHE/UN/FC problem, since in the problem description, there is no constraint of how many provider nodes each end-system is connected to. In 15
Minimize
M X N X
αi,j yi,j +
i=1 j=1
Subject to
N X
N −1 X
l=1
xj,l i,k
+
i,k qk,j
−
N X
k,l qi,j >= 1,
i,k xl,j i,k − qi,j = 0,
l=1
M X M X
k,l qi,j ≤ M 2 × yi,j ,
k=1 l=1
N X N ³ X
xj,l i,k
i=1 k=1
+
xl,j i,k
´
li,j zi,j
(8)
i=1 j=i+1
j=1
N X
N X
≤ 2N 2 × zj,l ,
for 1 ≤ i ≤ M ; wk,l > 0
(9)
for 1 ≤ j ≤ N ; wi,k > 0
(10)
for 1 ≤ i ≤ M ; 1≤j≤N
(11)
for 1 ≤ j < l ≤ N
(12)
for 1 ≤ i ≤ M ; 1≤j≤N for 1 ≤ j < l ≤ N for 1 ≤ i, k, l ≤ M ; 1≤j≤N for 1 ≤ i, k ≤ M ; 1 ≤ j, l ≤ N
yi,j = 0/1, zj,l = 0/1, k,l qi,j = 0/1,
xj,l i,k = 0/1,
(13) (14) (15) (16)
Figure 6: ILP for SONDP-MHE/UN/FC Problem other word, the input to the SONDP-SHE/UN/FC problem and SONDP-MHE/UN/FC problem can be same, the difference of the two problems only exists in the requirement of output. Therefore we can use a similar proof to show that SONDP-MHE/UN/FC problem is also NP-Complete.
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3.4
Optimal Solution for SON Topology Design using Integer Linear Programming
In this section, we provide the 0-1 integer linear programming formulations of both multi-homing and single-homing problems. The formulation for multi-homing problem is shown in figure 6. For single-homing problem, we only need to add one more set of constraints for the ILP in figure 6 to ensure that exactly one access link is used for each end-system, e.g.,
PN
j=1 yi,j
= 1 for 1 ≤ i ≤ M.
The variable yi,j indicates that bandwidth is reserved on the access link from end-system i to provider node j if yi,j = 1. The variable zj,l indicates that bandwidth is reserved on the transport k,l link between provider node j and provider node l if z j,l = 1. The variable qi,j indicates that
the traffic from end-system k to end-system l is using the access link between end-system i and k,l provider node j for if qi,j = 1, wehre i is equal to k or l. The variable x j,l i,k indicates that traffic
from end-system i to end-system k using transport link between provider node j and provider node l if xj,l i,k = 1. The objective function in figure 6 is the sum of the costs of access links and transport links. Constraint (9) ensures that at least one access link is used to connect an end system to the overlay network. Constraint (10) is for flow conservation at each provider node. No traffic is initiated or terminated at a provider node. Constraints (11) and (12) determines the access links and the transport links used by the solution.
4
Approximate Algorithms for SON Topology Design
In this section we present approximate algorithms for the solution of the SHE/UN/FC and MHE/UN/FC versions of the topology design problem. It may be noted that we have shown that the optimal
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solution of the MHE/UN/UBC version of the problem can be obtained in polynomial time and approximate solution for the SHE/UN/UBC version of the problem is presented in [11]. Since in the fixed cost model the cost of each link is independent of the amount of data being sent through that link, we can mostly ignore the the amount of reserve bandwidth ω ij between the end-systems ESi and ESj . If ωij = 0, then in the resulting topology ES i and ESj need not be connected, otherwise, they should be connected. Therefore, from the reserve bandwidth matrix Ω we can construct a connectivity requirement set R of tuples such that (i, j) ∈ R if and only if Ω ij > 0. The set R will be a collection of end-system pairs {(s 1 , t1 ), (s2 , t2 ), (s3 , t3 ), . . . , (sk , tk ),}, where each si , ti represents an ordered pair of end-systems. We provide three different heuristics for the solution of SHE/UN/FC problem: (i) Randomized Heuristic, (ii) Gain-based Heuristic and (iii) Spanning Tree based heuristic. The heuristics are described below. In these algorithms, shortest path between any two nodes (end-systems or provider nodes) implies the shortest path that only use provider nodes as intermediate nodes. Heuristic 1: Randomized Approach Step 1: Intialize CRH = ∞ and DRH = 0 Step 2: Repeat steps 3 to 13 W of times. (The parameter W is set by the user and is used to determine the number of times the random process is repeated.) Step 3: Set R = {(s1 , t1 ), (s2 , t2 ), (s3 , t3 ), . . . , (sk , tk ), } Step 4: Randomly pickup a source-destination pair (si , ti ) from the set R. Remove this pair from the set R Step 5: Compute the shortest path from si to ti . Note the provider node Pj that si is connected to and Pk that ti is connected to in the shortest path. Call these provider nodes gateways for s i and ti respectively and denote them by G(si ) and G(ti ) respectively. Step 6: Set DRH = DRH + {weight of the shortest path computed in the previous step.} Step 7: Set the weights of all the links appearing on the shortest path to be equal to zero. Step 8: Repeat steps 9-12 till R is empty. Step 9: Randomly pickup a source-destination pair (si , ti ) from the set R. Remove this pair from the set R
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Step 10: If G(si ) and G(ti ) are known, compute the shortest path between G(si ) and G(ti ) else if G(si ) is known and G(ti ) is not known, compute the shortest path between G(si ) and ti else if G(si ) is not known and G(ti ) is known, compute the shortest path between si and G(ti ) else if neither G(si ) nor G(ti ) is known, compute the shortest path between si and ti Step 11: Set DRH = DRH + {weight of the shortest path computed in the previous step.} Step 12: Set weights of all the links appearing on the shortest path to be equal to zero. Step 13: Set CRH = min(CRH , DRH ) Step 14: Output CRH . This is the cost of the solution.
Computational complexity: The computational complexity of the Randomized Heuristic is O(kW (M + N )log(M + N )), where M is the number of endsytems, N is the number of provider nodes, k is the number of connections to be established and W is the number of times the random process is repeated. Heuristic 2: Gain Based Approach Step 1: Initailize CGBH = 0 Step 2: Set R = {(s1 , t1 ), (s2 , t2 ), (s3 , t3 ), . . . , (sk , tk ), } Step 3: Compute shortest paths between all pairs of end-systems in R. Step 4: Identify the source-destination pair (si , ti ) from the set R that has the longest path length. Remove this pair from the set R Note the provider node Pj that si is connected to and Pk that ti is connected to in the shortest path. Call these provider nodes gateways for si and ti respectively and denote them by G(si ) and G(ti ) respectively. Step 5: Set CGBH = CGBH + {weight of the path chosen in the previous step.} Step 6: Set the weights of all the links appearing on the path chosen in the previous step to be equal to zero. Step 7: Repeat steps 8-12 till R is empty. Step 8: Compute shortest paths between all pairs of end-systems in R. (If either G(si ) and G(ti ) is identified in one of the earlier iterations, in shortest path computation G(si ) and G(ti ) should replace si and ti respectively. Step 9: Note the gain, i.e., the change in path length of all the paths in the set R. Step 10: Identify the end-system pair (si , ti ) that has the largest gain. Remove this pair from the set R. Step 11: Set CGBH = CGBH + {weight of this path.} Step 12: Set the weights of all the links appearing on the path chosen in the step 10 to be equal to zero. Step 13: Output CGBH . This is the cost of the solution.
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Computational complexity: The computational complexity of the Gain-based Heuristic is O(k(M + N )3 ), where M is the number of endsytems, N is the number of provider nodes, k is the number of connections to be established. A different implementation can realize this in O(k 2 (M + N )2 ). The implementation should be chosen based on the values of M, N and k. Heuristic 3: Spanning-Tree Based Approach Step 1: Initialize CST H = 0 Step 2: Set R = {(s1 , t1 ), (s2 , t2 ), (s3 , t3 ), . . . , (sk , tk ), } Step 3: Compute the minimum spanning tree M STP N of the subgraph induced by the Provider Nodes. Set CST H = Cost of M STP N . Step 4: Connect the endsytems in the set R to the nearest Provider node in the spanning tree. Update CST H with the additional cost connecting the end-systems to the provider nodes. Step 5: Remove those Provider Nodes from M STP N that are not used to connect any end-systems. Step 6: Output CST H . This is the cost of the solution.
Computational complexity: The computational complexity of the Spanning Tree based Heuristic is O((M + N )log(M + N )), where M is the number of endsytems, N is the number of provider nodes. Approximate Solution for SON Topology Design - SHE/UN/FC version The multi-homed version of the problem is identical to the single-homed version, except that the end-systems are no longer required to be connected to at most one Provider Node.
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Experimental Results
In this section, we compare the performance of our three heuristics for the SONDP-SHE/UN/FC problem, against the optimal solution obtained by solving ILP. Simulation experiments are carried out on randomly generated input sets. We developed a random graph generator that takes as input the number of nodes and average degree, and generates connected undirected graphs. It also randomly generates the weights on the links from a uniform distribution over a specified range (we 20
used the ranges of 3 to 8, and 3 to 80 for our experiments). The graphs produced by the generator were used as the network for the Provider Nodes. The random weights on the edges were used as the fixed transport cost among Provider Nodes. Once the Provider Network is generated, a specified number of end-systems nodes are connected to the nodes in the provider network in the following way: Step 1: The degree of an end-system node is randomly generated from a uniform distribution over the interval of 1 to 10; Step 2: The provider node neighbors of an end-system node are randomly generated with uniform distribution. Step 3:
The access cost from the end-system node to the provider node is randomly generated
with a uniform distribution over the interval of 3 to 8 (small access cost variation) or 3 to 80 (large access cost variation); Step 4: Communication requests between end-systems is also randomly generated.
The results of our simulation experiments are presented in table 5, which includes the optimal cost of connection between a set of end-systems as well as the costs obtained by the heuristic approaches. Similarly, the time taken by the optimal solution as well as the heuristic solutions are also presented. We present results from nineteen problem instances. In the table M and N represent the number of end-systems and provider nodes respectively. With M end-systems, there could potentially be M(M-1/)/2 possible requests for data transfer. The term Req% used in the table represents the percentage of M(M-1/)/2 possible requests that were considered for that problem instance. It may be noted that this actually indicates the number of connection requests. The term Cost-variation ratio is defined to be the difference in cost between the heuristic
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solution(s) and the optimal solution, divided by the optimal solution. The values of cost-variation for three different heuristics are presented in the table. It may be observed from the table that ILP fails to find a solution within a reasonable amount of time when the problem instance increases in size. The heuristics however are able to produce solutions in these instances. As noted earlier, the link cost distribution was taken to be 3 to 8 for some of the instances and 3 to 80 over the rest. The goal of this was to see if the variation of the link weights have any significant impact of the quality of the solution. At the end of the experiments, we did not notice any such impact. The performance of the heuristics 1, 2 and 3 are comparable. Since among the three heuristics, the Spanning Tree based approach takes the least amount of time, we recommend that one should use that approach for the solution of this problem. This approach provides a reasonable quality solution in a fraction of time needed to find the optimal solution.
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Conclusion
In this paper, we present results of our comprehensive study on SON topology design problem. The results presented in this paper supplement the results presented in an earlier paper [11]. Given the importance of the topic, further study of the problem under capacitated network environment is warranted. We are currently studying such problems.
References [1] A. Akella, B. Maggs, S. Seshan, A. Shaikh, and R. Sitaraman, “ A Measurement-Based Analysis of Multihoming,” Proceedings of the 2003 ACM SIGCOMM, August 2003 [2] D. G. Anderson, H. Balakrishnan, M.F. Kaashoek and R. Morris, “Resilient Overlay Network”, Proc. 18th ACM SOSP 2001, Banff, October 2001. [3] Y. Chu, S. G. Rao and h. Zhang, “A Case for End System Multicast”, Proc. of ACM Sigmetrics, June 2000. [4] A. Keromytis, V. Mishra and D. Rubenstein, “Secure Overlay Networks”, Proc. of SIGCOMM’02, pp. 61-72, 2002.
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Table 4: Simulation Results for SONDP-SHE/UN/FC Problem Instance Cost # M N Req% Opt H1 H2 1* 10 10 44 124 134 134 2* 12 10 36 93 131 128 3* 15 10 29 103 141 165 4 20 15 53 94 130 124 5 25 15 42 102 150 140 6 30 25 34 143 186 191 7 35 25 29 129 195 185 8* 40 30 26 704 1083 1247 9 45 35 23 169 301 285 10* 50 40 20 904 1646 1475 11 55 45 19 237 393 364 12* 60 50 17 1285 2245 2208 13 65 55 16 242 408 392 14* 70 60 14 1464 2038 1962 15 75 65 14 295 496 482 16 85 75 12 313 566 552 17 95 85 11 N/A 608 592 18* 80 70 13 N/A 2721 2851 19 105 95 10 N/A 673 674 +: Result with best cost variation ratio *: Link cost is between 3 and 80; otherwise, it N/A: ILP failed to find an optimal solution.
H3 142 141 145 132 144 202 205 1073 309 1441 371 2004 445 2248 548 594 638 2726 729
Cost-var. Ratio (%) H1 H2 H3 + + 8.1 8.1 14.5 40.9 37.6+ 51.6 36.9+ 60.2 40.8 38.3 31.9+ 40.4 47.1 37.3+ 41.2 + 30.1 33.6 41.3 + 51.2 43.4 58.9 53.8 77.1 52.4 + 78.1 68.6+ 82.8 82.1 63.2 59.4 + + 65.8 53.6 56.5 74.7 71.8 56.0 + + 68.6 62.0 83.9 + 39.2 34.0 53.6 68.1 63.4+ 85.8 80.8 76.4+ 89.8 N/A N/A N/A N/A N/A N/A N/A N/A N/A
Running Time (s) Opt H1 H2 H3 1
