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Mapping filter services on heterogeneous platforms To appear in IPDPS 2009 Anne Benoit,Fanny Dufoss e,Yves Robert January 9, 2009 Anne Benoit,Fanny Dufoss e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009

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SLIDE 1

Mapping filter services on heterogeneous platforms

To appear in IPDPS 2009 Anne Benoit,Fanny Dufoss´ e,Yves Robert January 9, 2009

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 1 / 42

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SLIDE 2

Introduction

The problem: treatment of a data flow filter services with selectivity σ and cost c precedence constraints between services servers with speed s

  • ne-to-one mappings

The objective: minimize the period minimize the latency

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 2 / 42

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Motivation

For services of selectivity less than one grep web services Select-Project-Join query optimization ... Related problems: component testing unsupervised systems

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 3 / 42

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SLIDE 4

1

Framework

2

Period General structure of optimal solutions Case of homogeneous servers NP-completeness of MinPeriod-Het Integer linear program

3

Latency General structure of optimal solutions Polynomial algorithm on homogeneous platforms NP-completeness of problem MinLatency-NoPrec-Het Integer linear program

4

Bi-criteria problem

5

Heuristics

6

Experiments

7

Conclusion

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 4 / 42

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SLIDE 5

The instances

The problems depend on: the criteria: MinPeriod, MinLatency or BiCriteria the platform: Hom or Het the dependence constraints: NoPrec or Prec

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 5 / 42

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SLIDE 6

The instances

The problems depend on: the criteria: MinPeriod, MinLatency or BiCriteria the platform: Hom or Het the dependence constraints: NoPrec or Prec The instances: A = (F, G, S) with: The services: F = {C1, C2, . . . , Cn} The precedence constraints: G ⊂ F × F The servers: S = {S1, S2, . . . , Sp}

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 5 / 42

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SLIDE 7

The problem

Example for 3 independent services: The plan?

C1 C2 C3 C1 C3 C2 C1 C2 C3 C2 C3 C1

The mapping? (C1, S2), (C2, S1), (C3, S3) (C1, S3), (C2, S2), (C3, S1)

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 6 / 42

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SLIDE 8

Example

C1 C2 C3

Figure: Chaining services.

C1 C3 C2

Figure: Combining selectivities

P = max

  • c1

s1 , σ1c2 s2 , σ1σ2c3 s3

  • P = max
  • c1

s1 , c2 s2 , σ1σ2c3 s3

  • L = c1

s1 + σ1c2 s2 + σ1σ2c3 s3

L = max

  • c1

s1 , c2 s2

  • + σ1σ2c3

s3

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 7 / 42

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SLIDE 9

Example

c1 = 1, c2 = 4, c3 = 10 σ1 = 1

2, σ2 = σ3 = 1 3

s1 = 1, s2 = 2 and s3 = 3

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 8 / 42

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Example

c1 = 1, c2 = 4, c3 = 10 σ1 = 1

2, σ2 = σ3 = 1 3

s1 = 1, s2 = 2 and s3 = 3

C1 C2 C3

Figure: Optimal plan for period.

C1 C3 C2

Figure: Optimal plan for latency

P = 1 L = 13

6

L = 5

2

P = 4

3

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 8 / 42

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SLIDE 11

1

Framework

2

Period General structure of optimal solutions Case of homogeneous servers NP-completeness of MinPeriod-Het Integer linear program

3

Latency General structure of optimal solutions Polynomial algorithm on homogeneous platforms NP-completeness of problem MinLatency-NoPrec-Het Integer linear program

4

Bi-criteria problem

5

Heuristics

6

Experiments

7

Conclusion

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 9 / 42

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General structure of optimal solutions

The instance : C1, ..., Cn, S1, ..., Sn with σ1, ..., σp ≤ 1 σp+1, ..., σn ≥ 1

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 10 / 42

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General structure of optimal solutions

The instance : C1, ..., Cn, S1, ..., Sn with σ1, ..., σp ≤ 1 σp+1, ..., σn ≥ 1

Cp+1 Cλ(3) Cλ(1) Cn Cλ(2) Cλ(p)

Figure: General structure

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 10 / 42

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SLIDE 14

Homogeneous case without precedence constraints

The instance : C1, ..., Cn with c1 ≤ c2 ≤ ... ≤ cp σ1, ..., σp < 1 σp+1, ..., σn ≥ 1 The matching: C1 → C2 → ... → Cp

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 11 / 42

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SLIDE 15

Homogeneous case with precedence constraints

Computing the optimal subgraph for C in the graph G: Let D = maxi{log σi}. We construct a network flow graph W with: a source s a node fi by service in G a sink node t an edge s− > fi with capacity +∞ if Ci is ancestor of C in G, D else an edge fi− > fj of capacity +∞ if Cj is an ancestor of Ci in G an edge fi− > t with capacity D + log σi The set of services on the side of s in a min-cut is the optimal subset of predecessors for latency.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 12 / 42

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Homogeneous case with precedence constraints

Computing the optimal subgraph for C in the graph G: Let D = maxi{log σi}. We construct a network flow graph W with: a source s a node fi by service in G a sink node t an edge s− > fi with capacity +∞ if Ci is ancestor of C in G, D else an edge fi− > fj of capacity +∞ if Cj is an ancestor of Ci in G an edge fi− > t with capacity D + log σi The set of services on the side of s in a min-cut is the optimal subset of predecessors for latency. Optimal algorithm: at each step place the available service with minimal possible period.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 12 / 42

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SLIDE 17

1

Framework

2

Period General structure of optimal solutions Case of homogeneous servers NP-completeness of MinPeriod-Het Integer linear program

3

Latency General structure of optimal solutions Polynomial algorithm on homogeneous platforms NP-completeness of problem MinLatency-NoPrec-Het Integer linear program

4

Bi-criteria problem

5

Heuristics

6

Experiments

7

Conclusion

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 13 / 42

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SLIDE 18

Proof: NP-completeness of MinPeriod-Het

Problem (RN3DM)

Given an integer vector A = (A[1], . . . , A[n]) of size n, does there exist two permutations λ1 and λ2 of {1, 2, . . . , n} such that ∀1 ≤ i ≤ n, λ1(i) + λ2(i) = A[i]

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 14 / 42

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SLIDE 19

Proof: NP-completeness of MinPeriod-Het

Problem (RN3DM)

Given an integer vector A = (A[1], . . . , A[n]) of size n, does there exist two permutations λ1 and λ2 of {1, 2, . . . , n} such that ∀1 ≤ i ≤ n, λ1(i) + λ2(i) = A[i] The associated instance : ci = 2A[i] σi = 1/2 si = 2i P = 2

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 14 / 42

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Proof: NP-completeness of MinPeriod-Het

Problem (RN3DM)

Given an integer vector A = (A[1], . . . , A[n]) of size n, does there exist two permutations λ1 and λ2 of {1, 2, . . . , n} such that ∀1 ≤ i ≤ n, λ1(i) + λ2(i) = A[i] The associated instance : ci = 2A[i] σi = 1/2 si = 2i P = 2 ∀1 ≤ i ≤ n, λ1(i) + λ2(i) ≥ A[i] ⇐ ⇒ ∀1 ≤ i ≤ n, 1

2

λ1(i)−1 × 2A[i]

2λ2(i) ≤ 2

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 14 / 42

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SLIDE 21

Inapproximability of MinPeriod-Het

Proposition

For any K > 0, there exists no K-approximation algorithm for MinPeriod-NoPrec-Het, unless P=NP.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 15 / 42

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Inapproximability of MinPeriod-Het

Proposition

For any K > 0, there exists no K-approximation algorithm for MinPeriod-NoPrec-Het, unless P=NP. Reduction from RN3DM: ci = K A[i]−1 σi = 1/K si = K i P = 1

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 15 / 42

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SLIDE 23

Integer linear program

The variables: ti,u = 1 if service Ci is assigned to server Su si,j = 1 if service Ci is an ancestor of Cj M is the logarithm of the optimal period

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 16 / 42

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Integer linear program

The variables: ti,u = 1 if service Ci is assigned to server Su si,j = 1 if service Ci is an ancestor of Cj M is the logarithm of the optimal period The constraints: ∀i,

  • u ti,u = 1

∀u,

  • i ti,u = 1

∀i, j, k, si,j + sj,k − 1 ≤ si,k ∀i, si,i = 0 ∀i, log ci −

u ti,u log su + k sk,i log σk ≤ M

The objective function: Minimize M

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 16 / 42

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SLIDE 25

1

Framework

2

Period General structure of optimal solutions Case of homogeneous servers NP-completeness of MinPeriod-Het Integer linear program

3

Latency General structure of optimal solutions Polynomial algorithm on homogeneous platforms NP-completeness of problem MinLatency-NoPrec-Het Integer linear program

4

Bi-criteria problem

5

Heuristics

6

Experiments

7

Conclusion

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 17 / 42

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SLIDE 26

Structure of the optimal plan

Proposition

Let C1, ..., Cn, S1, ..., Sn be an instance of MinLatency. Then, the optimal latency is obtained with a plan G such that, for any v1 = (Ci1, Su1), v2 = (Ci2, Su2),

1

If di1(G) = di2(G), they have the same predecessors and the same successors in G.

2

If di1(G) > di2(G) and σi2 ≤ 1, then ci1/su1 < ci2/su2.

3

All nodes with a service of selectivity σi > 1 are leaves (di(G) = 0).

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 18 / 42

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SLIDE 27

1

Framework

2

Period General structure of optimal solutions Case of homogeneous servers NP-completeness of MinPeriod-Het Integer linear program

3

Latency General structure of optimal solutions Polynomial algorithm on homogeneous platforms NP-completeness of problem MinLatency-NoPrec-Het Integer linear program

4

Bi-criteria problem

5

Heuristics

6

Experiments

7

Conclusion

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 19 / 42

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Algorithm without dependence constraint

Data: n services of cost c1 ≤ · · · ≤ cn and of selectivities σ1, ..., σn ≤ 1 Result: a plan G optimizing the latency G is the graph reduced to node C1; for i = 2 to n do for j = 0 to i − 1 do Compute the completion time tj of Ci in G with predecessors C1, ..., Cj; end Choose j such that tj = mink{tk}; Add the node Ci and the edges C1 → Ci, . . . , Cj → Ci to G; end Algorithm 1: Optimal algorithm for MinLatency-NoPrec-Hom.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 20 / 42

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SLIDE 29

G is the graph reduced to the node C of minimal cost with no predecessor in G; for i = 2 to n do Let S be the set of services not yet in G and such that their set of predecessors in G is included in G; for C ∈ S do for C ′ ∈ G do Compute the set S′ minimizing the product of selectivities among services of latency less than LG(C ′), and including all predecessors of C in G; end Let SC be the set that minimizes the latency of C in G and LC be this latency; end Choose a service C such that LC = min{LC ′, C ′ ∈ S}; Add to G the node C, and ∀C ′ ∈ SC, the edge C ′ → C ; end Algorithm 2: Optimal algorithm for MinLatency-Prec-Hom.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 21 / 42

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SLIDE 30

Example

C1

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 22 / 42

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SLIDE 31

Example

C1

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 22 / 42

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SLIDE 32

Example

C1 C2

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 22 / 42

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SLIDE 33

Example

C1 C2

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 22 / 42

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SLIDE 34

Example

C1 C2 C3

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 22 / 42

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SLIDE 35

Example

C1 C2 C3

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 22 / 42

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SLIDE 36

Example

C1 C2 C3 C4

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 22 / 42

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SLIDE 37

Example

C1 C2 C3 C4

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 22 / 42

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SLIDE 38

Example

C1 C2 C3 C4 C5

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 22 / 42

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SLIDE 39

1

Framework

2

Period General structure of optimal solutions Case of homogeneous servers NP-completeness of MinPeriod-Het Integer linear program

3

Latency General structure of optimal solutions Polynomial algorithm on homogeneous platforms NP-completeness of problem MinLatency-NoPrec-Het Integer linear program

4

Bi-criteria problem

5

Heuristics

6

Experiments

7

Conclusion

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 23 / 42

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SLIDE 40

Proof:NP-completeness of MinLatency-Het

Lemma

Let C1, ..., Cn, S1, ..., Sn be an instance such that ∀i, ci and si are integer power of 2 and σi ≤ 1

  • 2. Then the optimal latency is obtained with a plan

G such that

1 Proposition 2 is verified; 2 for all nodes (Ci1, Su1) and (Ci2, Su2) with di1(G) = di2(G), we have

ci1 su1 = ci2 su2 .

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 24 / 42

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SLIDE 41

Proof:NP-completeness of MinLatency-Het

Lemma

Let C1, ..., Cn, S1, ..., Sn be an instance such that ∀i, ci and si are integer power of 2 and σi ≤ 1

  • 2. Then the optimal latency is obtained with a plan

G such that

1 Proposition 2 is verified; 2 for all nodes (Ci1, Su1) and (Ci2, Su2) with di1(G) = di2(G), we have

ci1 su1 = ci2 su2 .

ci = 2A[i]×n+(i−1) σi = 1

2

n si = 2n×(i+1) L = 2n − 1

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 24 / 42

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SLIDE 42

Inapproximability of MinLatency-Het

Proposition

For any K > 0, there exists no K-approximation algorithm for MinLatency-NoPrec-Het, unless P=NP.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 25 / 42

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SLIDE 43

Inapproximability of MinLatency-Het

Proposition

For any K > 0, there exists no K-approximation algorithm for MinLatency-NoPrec-Het, unless P=NP. Reduction from RN3DM ci = K A[i]×n+(i−1) σi = 1

K

n si = K n×(i+1) L = K n−1

K−1

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 25 / 42

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SLIDE 44

1

Framework

2

Period General structure of optimal solutions Case of homogeneous servers NP-completeness of MinPeriod-Het Integer linear program

3

Latency General structure of optimal solutions Polynomial algorithm on homogeneous platforms NP-completeness of problem MinLatency-NoPrec-Het Integer linear program

4

Bi-criteria problem

5

Heuristics

6

Experiments

7

Conclusion

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 26 / 42

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SLIDE 45

The variables: z(i, u, e) = 1 if the service Ci is associated to the server Su and its set of predecessors is e ⊂ C. t(i) is the completion time of Ci M is the optimal latency

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 27 / 42

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SLIDE 46

The variables: z(i, u, e) = 1 if the service Ci is associated to the server Su and its set of predecessors is e ⊂ C. t(i) is the completion time of Ci M is the optimal latency The constraints: ∀u ∈ S,

  • i∈C
  • e⊂C z(i, u, e) = 1

∀i ∈ C,

  • u∈S
  • e⊂C z(i, u, e) = 1

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 27 / 42

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SLIDE 47

The variables: z(i, u, e) = 1 if the service Ci is associated to the server Su and its set of predecessors is e ⊂ C. t(i) is the completion time of Ci M is the optimal latency The constraints: ∀u ∈ S,

  • i∈C
  • e⊂C z(i, u, e) = 1

∀i ∈ C,

  • u∈S
  • e⊂C z(i, u, e) = 1

∀i, i′ ∈ C, ∀u, u′ ∈ S, ∀e, e′ ⊂ C, e e′, i ∈ e′, z(i, u, e) + z(i′, u′, e′) ≤ 1 ∀u ∈ S, ∀e ⊂ C, ∀i ∈ e, z(i, u, e) = 0

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 27 / 42

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SLIDE 48

The variables: z(i, u, e) = 1 if the service Ci is associated to the server Su and its set of predecessors is e ⊂ C. t(i) is the completion time of Ci M is the optimal latency The constraints: ∀u ∈ S,

  • i∈C
  • e⊂C z(i, u, e) = 1

∀i ∈ C,

  • u∈S
  • e⊂C z(i, u, e) = 1

∀i, i′ ∈ C, ∀u, u′ ∈ S, ∀e, e′ ⊂ C, e e′, i ∈ e′, z(i, u, e) + z(i′, u′, e′) ≤ 1 ∀u ∈ S, ∀e ⊂ C, ∀i ∈ e, z(i, u, e) = 0 ∀i ∈ C, ∀e ⊂ C, ∀k ∈ e, t(i) ≥

  • u∈S z(i, u, e)
  • ci

su ∗ Cj∈e σj + t(k)

  • ∀i ∈ C,

t(i) ≥

u z(i, u, e) ci su ∗ Cj∈e σj

∀i ∈ C, t(i) ≤ M The objective function: Minimize M

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 27 / 42

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SLIDE 49

1

Framework

2

Period General structure of optimal solutions Case of homogeneous servers NP-completeness of MinPeriod-Het Integer linear program

3

Latency General structure of optimal solutions Polynomial algorithm on homogeneous platforms NP-completeness of problem MinLatency-NoPrec-Het Integer linear program

4

Bi-criteria problem

5

Heuristics

6

Experiments

7

Conclusion

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 28 / 42

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SLIDE 50

Data: n services of cost c1 ≤ · · · ≤ cn and of selectivities σ1, ..., σn ≤ 1 and a maximum throughput K Result: a plan G optimizing the latency with a throughput less than K G is the graph reduced to node C1; for i = 2 to n do for j = 0 to i − 1 do Compute the completion time tj of Ci in G with predecessors C1, ..., Cj; end Let S = {k|ci

  • 0≤k<i σk ≤ K};

Choose j such that tj = mink∈S{tk}; Add the node ci and the edges C1 → Ci, . . . , Cj → Ci to G; end Algorithm 3: Optimal algorithm for latency with a fixed throughput.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 29 / 42

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SLIDE 51

1

Framework

2

Period General structure of optimal solutions Case of homogeneous servers NP-completeness of MinPeriod-Het Integer linear program

3

Latency General structure of optimal solutions Polynomial algorithm on homogeneous platforms NP-completeness of problem MinLatency-NoPrec-Het Integer linear program

4

Bi-criteria problem

5

Heuristics

6

Experiments

7

Conclusion

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 30 / 42

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SLIDE 52

sigma-inc We place services on a chain in increasing order of σ. short service/fast server We associate the service with shortest cost to the server with fastest speed. long service/fast server We associate the service with largest cost with the server with fastest speed.

  • pt-homo We randomly associate services to servers.

greedy min This simple heuristic consists in running successively the four previous heuristics on the problem instance, and returning as a result the best of the four solutions. random This last heuristic is fully random: we randomly associate services and servers, and we randomly place these pairs on a linear chain.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 31 / 42

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SLIDE 53

1

Framework

2

Period General structure of optimal solutions Case of homogeneous servers NP-completeness of MinPeriod-Het Integer linear program

3

Latency General structure of optimal solutions Polynomial algorithm on homogeneous platforms NP-completeness of problem MinLatency-NoPrec-Het Integer linear program

4

Bi-criteria problem

5

Heuristics

6

Experiments

7

Conclusion

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 32 / 42

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SLIDE 54

The instances: independent services the cost of services: 1 ≤ c ≤ 100 the selectivities: 0, 01 ≤ σ ≤ 1 the speed of servers: 1 ≤ s ≤ 100

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 33 / 42

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SLIDE 55

0.5 1 1.5 2 2.5 3 3.5 4 4.5 2 4 6 8 10 12 14 16 period n sigma-inc short service/fast server long service/fast server

  • pt-homo

greedy min random

  • pt

Figure: Experiment 1: general experiment.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 34 / 42

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SLIDE 56

1 2 3 4 5 6 7 10 20 30 40 50 60 70 80 90 100 period n sigma-inc short service/fast server long service/fast server

  • pt-homo

greedy min random

Figure: Experiment 1: general experiment.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 35 / 42

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SLIDE 57

0.5 1 1.5 2 2.5 3 3.5 4 4.5 1 2 3 4 5 6 7 8 9 10 period n sigma-inc short service/fast server long service/fast server

  • pt-homo

greedy min random

  • pt

Figure: Experiment 2: with small selectivity.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 36 / 42

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SLIDE 58

1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 10 period n sigma-inc short service/fast server long service/fast server

  • pt-homo

greedy min random

  • pt

Figure: Experiment 3: with high selectivity.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 37 / 42

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SLIDE 59

5 10 15 20 25 30 35 1 2 3 4 5 6 7 8 9 10 period n sigma-inc short service/fast server long service/fast server

  • pt-homo

greedy min random

  • pt

Figure: Experiment 4: with low speed.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 38 / 42

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SLIDE 60

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 period n sigma-inc short service/fast server long service/fast server

  • pt-homo

greedy min random

  • pt

Figure: Experiment 5: with low heterogeneity.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 39 / 42

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SLIDE 61

1e-05 0.0001 0.001 0.01 0.1 1 10 100 2 4 6 8 10 12 14 16 computing time(s) n heuristics LP

Figure: Experiment 1: Computing times.

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 40 / 42

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SLIDE 62

References

  • M. P. Alessandro Agnetis, Paolo Detti and M. S. Sodhi.

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Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 41 / 42

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SLIDE 63

Conclusion

The results: MinLatency-Hom is polynomial MinPeriod-Het is NP-complete MinLatency-Het is NP-complete BiCriteria-Hom is polynomial The experiments on MinPeriod-NoPrec-Het: heuristics close to the optimal for small instances better performance than random Future work: model with communication costs

Anne Benoit,Fanny Dufoss´ e,Yves Robert () Mapping filter services on heterogeneous platforms January 9, 2009 42 / 42