Beyond the Stars: Revisiting Virtual Cluster Embeddings


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1 Beyond the Stars: Revisiting Virtual Cluster Embeddings Matthias Rost Technische Universität Berlin September 7th, 2015, TélécomParisTech Joint work with Carlo Fuerst, Stefan Schmid Published in ACM SIGCOMM CCR Volume 45 Issue 3, July 2015 Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
2 Introduction 1 Introduction Problem Space: Running Applications in the Cloud Application Abstractions: The VC Abstraction 2 The classical VC Embedding Problem Overview Definition: VC Embedding Problem VCACE Algorithm 3 HoseBased Virtual Cluster Embeddings Motivation Definition: HoseBased VC Embedding Problem Computational Complexity of HVC Embeddings Computing (Fractional) HVC Embeddings 4 Computational Evaluation Experimental Setup Results Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
3 Introduction Problem Space: Running Applications in the Cloud Data Center Applications Virtualization has changed our view on complex computations Virtual machines (VMs) can be spawned within minutes and anywhere on the world (Microsoft Azure, Amazon EC2,... ) Complex tasks can quite easily be distributed to tens or hundreds of servers using frameworks like MapReduce Insight: Application performance does depend on bandwidth availability Facebook traces show that network transfers account for 33% of the execution time [4] Data centers exhibit oversubscription factors of upto 1:240 [6] Customer s application execution time can hardly be estimated! Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
4 Introduction Problem Space: Running Applications in the Cloud Data Center Applications Insight Performance does depend on bandwidth availability How to achieve resource isolation? New topologies limiting the oversubscription factor: Fat trees, MDCubes,... Figure: Fat tree topology [1] Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
5 Introduction Problem Space: Running Applications in the Cloud Data Center Applications Insight Performance does depend on bandwidth availability How to achieve resource isolation? New topologies limiting the oversubscription factor: Fat trees, MDCubes,... Figure: Fat tree topology [1] Novel descriptions of applications embedding algorithms Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
6 Introduction Problem Space: Running Applications in the Cloud Application Specification : Case Study Amazon AWS Customers can select from a huge number of instances Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
7 Introduction Problem Space: Running Applications in the Cloud Application Specification : Case Study Amazon AWS Customers can select from a huge number of instances Customers can select enhanced networking or cluster networking : Instances launched into a common cluster placement group are placed into a logical cluster that provides highbandwidth, lowlatency networking between all instances in the cluster. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
8 Introduction Problem Space: Running Applications in the Cloud Application Specification : Case Study Amazon AWS Customers can select from a huge number of instances Customers can select enhanced networking or cluster networking : Instances launched into a common cluster placement group are placed into a logical cluster that provides highbandwidth, lowlatency networking between all instances in the cluster. Resource Isolation? Besteffort! Level of Specification? Grouping of instances! Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
9 Application Abstractions: The VC Abstraction
10 Introduction Application Abstractions: The VC Abstraction The Right Level of Abstraction Early 2000s: Virtual Network Embedding Problem Requests are specified as graphs Nodes represent VMs Edges represent intervm links  CPU  RAM  Disk  latency  bandwidth Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
11 Introduction Application Abstractions: The VC Abstraction The Right Level of Abstraction Early 2000s: Virtual Network Embedding Problem Requests are specified as graphs Nodes represent VMs Edges represent intervm links  CPU  RAM  Disk  latency  bandwidth Pro Concise specification Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
12 Introduction Application Abstractions: The VC Abstraction The Right Level of Abstraction Early 2000s: Virtual Network Embedding Problem Requests are specified as graphs Nodes represent VMs Edges represent intervm links  CPU  RAM  Disk  latency  bandwidth Pro Concise specification Contra Do customers know their requirements? Generally: Challenging NPhard problem! Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
13 Introduction Application Abstractions: The VC Abstraction The Right Level of Abstraction 2011: Virtual Cluster (VC) Abstraction [3] Allows only for star shaped graphs VMs are connected to logical switch logical switch VMs Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
14 Introduction Application Abstractions: The VC Abstraction The Right Level of Abstraction 2011: Virtual Cluster (VC) Abstraction [3] Allows only for star shaped graphs VMs are connected to logical switch logical switch VMs Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
15 Introduction Application Abstractions: The VC Abstraction The Right Level of Abstraction 2011: Virtual Cluster (VC) Abstraction [3] Allows only for star shaped graphs VMs are connected to logical switch Requests are specified by three parameters: N N number of virtual machines C N size of virtual machines B R + bandwidth towards logical switch C 1 B B 2 B N C C... C Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
16 Introduction Application Abstractions: The VC Abstraction The Right Level of Abstraction 2011: Virtual Cluster (VC) Abstraction [3] Allows only for star shaped graphs VMs are connected to logical switch Requests are specified by three parameters: N N number of virtual machines C N size of virtual machines B R + bandwidth towards logical switch C 1 B B 2 B N C C... C Pro Simple specification! Wellperforming heuristics for datacenter topologies [3, 8] Contra The VM size and the amount of bandwidth are dictated by the maximum wasteful Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
17 Introduction Application Abstractions: The VC Abstraction On Traffic Matrices Graph Abstraction VC Abstraction B {1,2} 1 B {1,3} 2 3 B {2,3} 1 B B 2 B 3 Allows for any traffic matrix M, where the bandwidth for edge {i, j} is less than B {i,j}. Allows for any traffic matrix M, where for any VM the sum of outgoing and incoming traffic is less than B Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
18 The classical VC Embedding Problem Overview Outlook Previous works... only considered (fat) trees only considered heuristics Ballani et al.: Oktopus [3] allocating virtual cluster requests on graphs with bandwidthconstrained edges is NPhard Xie et al.: Proteus [8] [Our algorithm] picks the first fitting lowestlevel subtree out of all such lowestlevel subtrees. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
19 The classical VC Embedding Problem Overview Outlook Previous works... only considered (fat) trees only considered heuristics Ballani et al.: Oktopus [3] allocating virtual cluster requests on graphs with bandwidthconstrained edges is NPhard Xie et al.: Proteus [8] [Our algorithm] picks the first fitting lowestlevel subtree out of all such lowestlevel subtrees. Main Questions Is the VC embedding problem really NPhard to solve? Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
20 Formal Definition of the VC Embedding Problem
21 The classical VC Embedding Problem Definition: VC Embedding Problem VC Embedding Problem Defintion VC request: N, B, C VC = (V VC, E VC ), V VC = {1, 2,..., N, center} E VC = {{i, center} 1 i N } Physical Network (Substrate) S = (V S, E S, cap, cost), cap : V S E S N cost : V S E S R 0 Task: Find a mapping of... VMs onto substrate nodes map V : V VC V S, and VC edges onto paths in the substrate map E : E VC P(E S ) Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
22 The classical VC Embedding Problem Definition: VC Embedding Problem VC Embedding Problem Defintion VC request: N, B, C VC = (V VC, E VC ), V VC = {1, 2,..., N, center} E VC = {{i, center} 1 i N } Physical Network (Substrate) S = (V S, E S, cap, cost), cap : V S E S N cost : V S E S R 0 Task: Find a mapping of... VMs onto substrate nodes map V : V VC V S, and VC edges onto paths in the substrate map E : E VC P(E S ), such that 1 map E ({u, v}) connects map V (u) and map V (v) for {u, v} E VC Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
23 The classical VC Embedding Problem Definition: VC Embedding Problem VC Embedding Problem Defintion VC request: N, B, C VC = (V VC, E VC ), V VC = {1, 2,..., N, center} E VC = {{i, center} 1 i N } Physical Network (Substrate) S = (V S, E S, cap, cost), cap : V S E S N cost : V S E S R 0 Task: Find a mapping of... VMs onto substrate nodes map V : V VC V S, and VC edges onto paths in the substrate map E : E VC P(E S ), such that 1 map E ({i, j}) connects map V (i) and map V (j) for {i.j} E VC 2 v V VC \ {center} v = map V (v ) C cap(v) and e E VC e map E (e ) B cap(e) for v V S, e E S Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
24 The classical VC Embedding Problem Definition: VC Embedding Problem VC Embedding Problem Defintion Task: Find a mapping of... VMs onto substrate nodes map V : V VC V S, and VC edges onto paths in the substrate map E : E VC P(E S ), such that 1 map E ({u, v}) connects map V (u) and map V (v) for {u, v} E VC 2 v V VC \ {center} v = map V (v ) C cap(v) and 3 minimizing the cost C e E VC e map E (e ) v V VC \{center} B cap(e) for v V S, e E S cost(map V (v)) + B e E VC e map E (e ) cost(e). Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
25 VCACE Algorithm
26 The classical VC Embedding Problem VCACE Algorithm Key Insights Lemma We can assume B = C = 1. Proof idea. If B 1, C 1, we transform the substrate by scaling capacities and costs: cap S (u) = cap(u)/c for u V S cap S (e) = cap(e)/b for e E S cost S (u) = cost(u) C for u V S cost S (e) = cost(e) B for e E S Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
27 The classical VC Embedding Problem VCACE Algorithm Key Insights Lemma We can solve the edge embedding problem if all nodes are placed. Proof. 1 Construct extended graph with additional node s + and (parallel) edges: {(s +, map V (i)) i {1,..., N }} of capacity 1 and cost 0 2 Compute a minimum cost flow of value N from s + to map V (center). 3 Perform a pathdecomposition to obtain mapping for edges. 2 a b a b 1 source s + 3 d c d target c Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
28 The classical VC Embedding Problem VCACE Algorithm Key Insights Lemma We can solve the embedding problem if the logical switch is placed. Proof. 1 Construct extended graph with additional edges {(s +, u) u V S }, cap(s +, u) = cap(u) and cost(s +, u) = cost(u) for u V S 2 Compute a minimum cost flow of value N from s + to map V (center). 3 Perform pathdecomposition to obtain mapping for nodes and edges. 2? cap(a) = 3 cap(b) = 2 a b a b 1? source s + 3? d c cap(d) = 1 cap(c) = 1 d target c Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
29 The classical VC Embedding Problem VCACE Algorithm Key Insights Lemma We can solve the embedding problem if the logical switch is placed. Proof. 1 Construct extended graph with additional edges {(s +, u) u V S }, cap(s +, u) = cap(u) and cost(s +, u) = cost(u) for u V S 2 Compute a minimum cost flow of value N from s + to map V (center). 3 Perform pathdecomposition to obtain mapping for nodes and edges. flow = 2 a b 2 a b source s + flow = 2 1 flow = 1 d target c 3 d c Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
30 The classical VC Embedding Problem VCACE Algorithm VCACE Algorithm Idea Simply iterate over possible locations for the center. Algorithm 1: The VCACE Algorithm Input: Substrate S = (V S, E S ), request (N, B, C) Output: Optimal VC mapping map V, map E if feasible (ˆf, ˆv) (null, null) for v V S do V S = V S {s + } and E S = E S {(s +, u) u V S } cap { S (e) = cap(e)/b, if e E S cap(u)/c, if e = (s +, u) E { S cost(e) B, if e E S cost S (e) = cost(u) C, if e = (s +, u) E S f MinCostFlow(s +, v, N, V S, E S, cap S, cost S ) if f is feasible and cost(f ) < cost(ˆf ) then (ˆf, ˆv) (f, v) if ˆf = null then return null return DecomposeFlowIntoMapping(ˆf, ˆv) Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
31 The classical VC Embedding Problem VCACE Algorithm VCACE Algorithm Idea Simply iterate over possible locations for the center. Theorem Correctness follows from the lemma on the previous slide. Algorithm 2: The VCACE Algorithm Input: Substrate S = (V S, E S ), request (N, B, C) Output: Optimal VC mapping map V, map E if feasible (ˆf, ˆv) (null, null) for v V S do V S = V S {s + } and E S = E S {(s +, u) u V S } cap { S (e) = cap(e)/b, if e E S cap(u)/c, if e = (s +, u) E { S cost(e) B, if e E S cost S (e) = cost(u) C, if e = (s +, u) E S f MinCostFlow(s +, v, N, V S, E S, cap S, cost S ) if f is feasible and cost(f ) < cost(ˆf ) then (ˆf, ˆv) (f, v) if ˆf = null then return null return DecomposeFlowIntoMapping(ˆf, ˆv) Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
32 The classical VC Embedding Problem VCACE Algorithm VCACE Algorithm Theorem The runtime is O ( N (n 2 log n + n m) ) with n = V S and m = E S, when using the successiveshortest path for the flow computation. Corollary. The VC Embedding Problem is not NPhard. Algorithm 3: The VCACE Algorithm Input: Substrate S = (V S, E S ), request (N, B, C) Output: Optimal VC mapping map V, map E if feasible (ˆf, ˆv) (null, null) for v V S do V S = V S {s + } and E S = E S {(s +, u) u V S } cap { S (e) = cap(e)/b, if e E S cap(u)/c, if e = (s +, u) E { S cost(e) B, if e E S cost S (e) = cost(u) C, if e = (s +, u) E S f MinCostFlow(s +, v, N, V S, E S, cap S, cost S ) if f is feasible and cost(f ) < cost(ˆf ) then (ˆf, ˆv) (f, v) if ˆf = null then return null return DecomposeFlowIntoMapping(ˆf, ˆv) Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
33 We can compute optimal solutions in polynomialtime.
34 We can compute optimal solutions in polynomialtime. Can we do even better?
35 Can we do even better? Yes,...
36 HoseBased Virtual Cluster Embeddings
37 HoseBased Virtual Cluster Embeddings Motivation Starting from Scratch VC Abstraction 1 B B 2 B 3 Allows for any traffic matrix M, where for any VM the sum of outgoing and incoming traffic is less than B Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
38 HoseBased Virtual Cluster Embeddings Motivation Starting from Scratch VC Abstraction 1 B B 2 B 3 Question: What is the purpose of the switch? Allows for any traffic matrix M, where for any VM the sum of outgoing and incoming traffic is less than B Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
39 HoseBased Virtual Cluster Embeddings Motivation Starting from Scratch VC Abstraction 1 B B 2 B Allows for any traffic matrix M, where for any VM the sum of outgoing and incoming traffic is less than B 3 Question: What is the purpose of the switch? Ballani et al. Oktopus [3] Oktopus allocation algorithms assume that the traffic between a tenant s VMs is routed along a tree. Answer: To route the traffic along a tree. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
40 HoseBased Virtual Cluster Embeddings Motivation Starting from Scratch VC Abstraction 1 B B 2 B 3 Question: Can we do without the switch? Allows for any traffic matrix M, where for any VM the sum of outgoing and incoming traffic is less than B Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
41 HoseBased Virtual Cluster Embeddings Motivation Starting from Scratch VC Abstraction Question: Can we do without the switch? 1 B B 2 B 3 Ballani et al. Oktopus [3] Alternatively, the NM [Network Manager] can control datacenter routing to actively build routes between tenant VMs [..] Allows for any traffic matrix M, where for any VM the sum of outgoing and incoming traffic is less than B We defer a detailed study of the relative merits of these approaches to future work. Answer: Yes! Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
42 HoseBased Virtual Cluster Embeddings Motivation Starting from Scratch VC Abstraction HoseBased VC Abstraction 1 B 2 B B 1 2 B B 3 B 3 Allows for any traffic matrix M, where for any VM the sum of outgoing and incoming traffic is less than B Allows for any traffic matrix M, where for any VM the sum of outgoing and incoming traffic is less than B Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
43 HoseBased Virtual Cluster Embeddings Motivation Motivating Example cap = 1 cap = N = 6, B = C = 1 ring substrate Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
44 HoseBased Virtual Cluster Embeddings Motivation Motivating Example cap = 1 cap = N = 6, B = C = 1 ring substrate There exists no solution in the classic VC embedding model. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
45 HoseBased Virtual Cluster Embeddings Motivation Motivating Example allocation = N = 6, B = C = 1 6 Embedding on the same substrate 5 Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
46 HoseBased Virtual Cluster Embeddings Motivation Motivating Example allocation = N = 6, B = C = 1 Embedding on the same substrate There exists a solution in the hosebased VC model! Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
47 HoseBased Virtual Cluster Embeddings Motivation Motivating Example allocation = 2 5 Embedding on the same substrate 4 Why allocations of 2 are sufficient: Consider edge e between VMs 6 and 5. The edge is used by routes R(e) = {(1, 5), (2, 5), (3, 6), (4, 6), (5, 6)}. Any valid traffic matrix M will respect: M 1,5 + M 2,5 1 M 3,6 + M 4,6 + M 5,6 1 Hence (i,j) R(e) M i,j 2 holds. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
48 HoseBased Virtual Cluster Embeddings Motivation Motivating Example II cap = 0 cap = 1 cap = 2 cap = cost = 6 5 N = 6, B = C = 1 extended ring topology Solution costs can be arbitrarily higher under the classic starinterpretation! Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
49 HoseBased Virtual Cluster Embedding Problem
50 HoseBased Virtual Cluster Embeddings Definition: HoseBased VC Embedding Problem HoseBased VC Embedding Problem (HVCEP) Definition (Clique Graph) V C = {1,..., N }, E C = {(i, j) i, j V C, i < j} Task: Find a mapping of... VC nodes onto substrate nodes map V : V C V S, and VC routes onto paths in the substrate map E : E C P(E S ), such that 1 route (i, j) E C connects map V (i) and map V (j), 2 the mapping of VMs must not violate node capacities (cf. slide 12), Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
51 HoseBased Virtual Cluster Embeddings Definition: HoseBased VC Embedding Problem HoseBased VC Embedding Problem (HVCEP) Definition (Clique Graph) V C = {1,..., N }, E C = {(i, j) i, j V C, i < j} Task: Find a mapping of... VC nodes onto substrate nodes map V : V C V S, and VC routes onto paths in the substrate map E : E C P(E S ), and integral bandwidth reservations l u,v cap(u, v) for {u, v} E S, s.t. 1 route (i, j) E C connects map V (i) and map V (j), 2 the mapping of VMs must not violate node capacities (cf. slide 12), 3 for all valid traffic matrices M ij i.e. (j,i) E C M ji + M ij B holds the bandwidth reservation is not exceeded on any edge {u, v} E S : {i,j} E C :{u,v} map E ({i,j}) M ij l u,v, Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
52 HoseBased Virtual Cluster Embeddings Definition: HoseBased VC Embedding Problem HoseBased VC Embedding Problem (HVCEP) Definition (Clique Graph) V C = {1,..., N }, E C = {(i, j) i, j V C, i < j} Task: Find a mapping of... VC nodes onto substrate nodes map V : V C V S, and VC routes onto paths in the substrate map E : E C P(E S ), and integral bandwidth reservations l u,v cap(u, v) for {u, v} E S, such that 1 route (i, j) E C connects map V (i) and map V (j), 2 the mapping of VMs must not violate node capacities (cf. slide 12), 3 for all valid traffic matrices M i.e. (j,i) E C M ji + M ij B holds the bandwidth reservation is not exceeded on any edge {u, v} E S : {i,j} E C :{u,v} map E ({i,j}) M ij l u,v, 4 minimizing C cost(map V (i)) + B l u,v cost(e). i V C e E S Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
53 Computational Complexity of HVC Embeddings
54 HoseBased Virtual Cluster Embeddings Computational Complexity of HVC Embeddings Excursion: VPN Embeddings and the VPN Conjecture Definition (VPN Embedding Problem (VPNEP) [7]) Given: Substrate network G = (V, E) with edge costs cost : E R + 0 Set of terminals W V with demands b(i) R + for i W Task: Find paths P {i,j} for all pairs i, j W, i j, and bandwidth allocations x e R + 0 on edges e E, s.t. i,j W :i j,p {i,j} M {i,j} x e holds for traffic matrices M, minimizing the cost e E cost(e) x e. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
55 HoseBased Virtual Cluster Embeddings Computational Complexity of HVC Embeddings Excursion: VPN Embeddings and the VPN Conjecture Definition (VPN Embedding Problem (VPNEP) [7]) Given: Substrate network G = (V, E) with edge costs cost : E R + 0 Set of terminals W V with demands b(i) R + for i W Task: Find paths P {i,j} for all pairs i, j W, i j, and Theorem bandwidth allocations x e R + 0 on edges e E, s.t. i,j W :i j,p {i,j} M {i,j} x e holds for traffic matrices M, minimizing the cost e E cost(e) x e. Finding a feasible solution for the capacitated VPNEP is NPhard [7]. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
56 HoseBased Virtual Cluster Embeddings Computational Complexity of HVC Embeddings Excursion: VPN Embeddings and the VPN Conjecture Definition (VPN Embedding Problem (VPNEP) [7]) Given: Substrate network G = (V, E) with edge costs cost : E R + 0 Set of terminals W V with demands b(i) R + for i W Task: Find paths P {i,j} for all pairs i, j W, i j, and Theorem bandwidth allocations x e R + 0 on edges e E, s.t. i,j W :i j,p {i,j} M {i,j} x e holds for traffic matrices M, minimizing the cost e E cost(e) x e. Finding a feasible solution for the capacitated VPNEP is NPhard [7]. Theorem (By reduction from the VPNEP) Finding a feasible solution for the HVCEP is NPhard. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
57 HoseBased Virtual Cluster Embeddings Computational Complexity of HVC Embeddings Excursion: VPN Embeddings and the VPN Conjecture Definition (VPN Embedding Problem (VPNEP) [7]) Given: Substrate network G = (V, E) with edge costs cost : E R + 0 Set of terminals W V with demands b(i) R + for i W Task: Find paths P {i,j} for all pairs i, j W, i j, and bandwidth allocations x e R + 0 on edges e E, s.t. i,j W :i j,p {i,j} M {i,j} x e holds for traffic matrices M, minimizing the cost e E cost(e) x e. Theorem (VPN Conjecture) Tree routing and arbitrary routing solutions coincide for the VPNEP on uncapacitated graphs. [5]. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
58 HoseBased Virtual Cluster Embeddings Computational Complexity of HVC Embeddings Excursion: VPN Embeddings and the VPN Conjecture Definition (VPN Embedding Problem (VPNEP) [7]) Given: Substrate network G = (V, E) with edge costs cost : E R + 0 Set of terminals W V with demands b(i) R + for i W Task: Find paths P {i,j} for all pairs i, j W, i j, and bandwidth allocations x e R + 0 on edges e E, s.t. i,j W :i j,p {i,j} M {i,j} x e holds for traffic matrices M, minimizing the cost e E cost(e) x e. Theorem (VPN Conjecture) Tree routing and arbitrary routing solutions coincide for the VPNEP on uncapacitated graphs. [5]. Theorem (Via the VPN Conjecture) Algorithm VCACE solves the HVCEP when capacities are sufficiently large! Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
59 Computing (Fractional) HVC Embeddings
60 HoseBased Virtual Cluster Embeddings Computing (Fractional) HVC Embeddings A MixedInteger Programming Formulation for the HVCEP Variables x i u y i,j u.v l u.v mapping of VM i onto node u mapping of link (i, j) V C onto (directed) substrate edge (u, v) load on substrate edge {u, v} ω i uv dual variable for allocation of communications of VM i on edge {u, v} MixedInteger Program 1: HVCOSPE min cost u xu i + cost u,v l uv (1) i V C,u V S {u,v} E S xu= i 1 i V C. (2) u V S σ u (xu i xu i+1 ) 0 i V C \ {N }. (3) u V S C xu i cap u u V S. (4) i V C (u,v) δ + u y ij uv (v,u) δ u l uv cap uv {u, v} E S. (5) y ij vu = x i u x j u (i, j) E C, (6) u V S. i V C B ω i uv l uv {u, v} E S. (7) y ij uv + y ij vu ωi uv + ω j uv (i, j) E C, (8) {u, v} E S. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
61 HoseBased Virtual Cluster Embeddings Computing (Fractional) HVC Embeddings A MixedInteger Programming Formulation for the HVCEP Explanation (2)  (4) control the VM embedding (5)  (8) is adapted from Altin et al. [2] for computing the optimal hose allocations on edges MixedInteger Program 2: HVCOSPE min cost u xu i + cost u,v l uv (1) i V C,u V S {u,v} E S xu= i 1 i V C. (2) u V S σ u (xu i xu i+1 ) 0 i V C \ {N }. (3) u V S C xu i cap u u V S. (4) i V C (u,v) δ + u y ij uv (v,u) δ u l uv cap uv {u, v} E S. (5) y ij vu = x i u x j u (i, j) E C, (6) u V S. i V C B ω i uv l uv {u, v} E S. (7) y ij uv + y ij vu ωi uv + ω j uv (i, j) E C, (8) {u, v} E S. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
62 HoseBased Virtual Cluster Embeddings Computing (Fractional) HVC Embeddings A MixedInteger Programming Formulation for the HVCEP Explanation (2)  (4) control the VM embedding (5)  (8) is adapted from Altin et al. [2] for computing the optimal hose allocations on edges Observation There are O(N 2 E S ) binary variables for computing paths. MixedInteger Program 3: HVCOSPE min cost u xu i + cost u,v l uv (1) i V C,u V S {u,v} E S xu= i 1 i V C. (2) u V S σ u (xu i xu i+1 ) 0 i V C \ {N }. (3) u V S C xu i cap u u V S. (4) i V C (u,v) δ + u y ij uv (v,u) δ u l uv cap uv {u, v} E S. (5) y ij vu = x i u x j u (i, j) E C, (6) u V S. i V C B ω i uv l uv {u, v} E S. (7) y ij uv + y ij vu ωi uv + ω j uv (i, j) E C, (8) {u, v} E S. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
63 HoseBased Virtual Cluster Embeddings Computing (Fractional) HVC Embeddings A MixedInteger Programming Formulation for the HVCEP Observation There are O(N 2 E S ) binary variables for computing paths. Initial Computational Results Solving this formulation may take up to 1800 seconds for embedding a 10VM VC onto a 20 node substrate. MixedInteger Program 4: HVCOSPE min cost u xu i + cost u,v l uv (1) i V C,u V S {u,v} E S xu= i 1 i V C. (2) u V S σ u (xu i xu i+1 ) 0 i V C \ {N }. (3) u V S C xu i cap u u V S. (4) i V C (u,v) δ + u y ij uv (v,u) δ u l uv cap uv {u, v} E S. (5) y ij vu = x i u x j u (i, j) E C, (6) u V S. i V C B ω i uv l uv {u, v} E S. (7) y ij uv + y ij vu ωi uv + ω j uv (i, j) E C, (8) {u, v} E S. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
64 HoseBased Virtual Cluster Embeddings Computing (Fractional) HVC Embeddings A MixedInteger Programming Formulation for the HVCEP Further Observations The hardness result has shown that the problem is hard even if the VMs are fixed. The large number of variables necessary for computing each endtoend path between VMs renders solving even the linear relaxation i.e. dropping integrality constraints computationally hard. Assumptions for obtaining a solvable formulation Assume that the VMs are already mapped. Assume that the hosepaths are splittable, i.e. each VMs are connected by a set of (weighted) paths. Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
65 HoseBased Virtual Cluster Embeddings Computing (Fractional) HVC Embeddings Computing Splittable HVC Embeddings Assumptions Assume that the VMs are already mapped. Assume that the hosepaths are splittable. arbitrarily many paths. map V (i) map V (j) y ij e = 0.5 Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
66 HoseBased Virtual Cluster Embeddings Computing (Fractional) HVC Embeddings Computing Splittable HVC Embeddings (u,v) δ + u y ij uv (v,u) δ u l uv cap uv {u, v} E S. (5) y ij vu = x i u x j u (i, j) E C, (6) u V S. i V C B ω i uv l uv {u, v} E S. (7) y ij uv + y ij vu ωi uv + ω j uv (i, j) E C, (8) {u, v} E S. map V (i) map V (j) W e δ + (W ) y ij e = 1 This type of constraint is equivalent to (6). Matthias Rost (TU Berlin) Revisiting Virtual Cluster Embeddings Paris, September
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