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Optimal Placement of Replicas in Trees with Read, Write, and Storage Costs

Optimal Placement of Replicas in Trees with Read, Write, and Storage Costs. Outline. Introduction Preliminaries Finding optimal resident sets DP demo  Considering node capacities and load Conclusion. Introduction. Topologies General NP-hard

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Optimal Placement of Replicas in Trees with Read, Write, and Storage Costs

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  1. Optimal Placement of Replicas in Trees with Read, Write, and Storage Costs

  2. Outline • Introduction • Preliminaries • Finding optimal resident sets • DP demo  • Considering node capacities and load • Conclusion

  3. Introduction • Topologies • General • NP-hard • O(1)-time algorithm with edges of unit length • Ring • O(n^5)-time algorithm • Tree • O(n)-time • Without considering storage costs.

  4. Introduction • Model • Tree topology • Replicas • Cost： read、write、store • Goal • Minimize the total cost

  5. Introduction • The set of nodes that have a copy of the object is the residence set of the object • Consider the problem of placing copies of objects in a tree network in order to minimize the cost of read, write, storage • Finding an optimal residence set of size p for an object in a tree with n nodes

  6. Preliminaries • Read cost of S • Write cost of S • Storage cost of S • Total cost of S

  7. Preliminaries • Partial cost • Let

  8. Preliminaries • Define R(u, k, v, i) V Vi Tvi

  9. Finding optimal resident sets • Lemma1. • C(S1∪S2, T1∪T2∪{u}) = C(S1, T1∪{u}) + C(S2, T2∪{u}) - s(u)

  10. Lemma1 proof • It is sufficient to show that mst(S1∪S2) = mst(S1) + mst(S2)

  11. Lemma2 • u covers v1 but it doesn’t cover v2 • C(S1∪S2, T1∪T2∪{u}) = C(S1, T1∪{u}) + C(S2, T2) + Wtotal*δ(u, S2)

  12. Lemma2 proof • It is sufficient to show that mst(S1∪S2) = mst(S1) + mst(S2) + δ(u, S2)

  13. Lemma3 • Let S be an optimal modified u-restricted k-residence set for T(i)v • S1=(T(i-1)v∩S)∪{u} • S2=(S-S1)∪{u} • S1:an optimal modified u-restricted k1-residence set for T(i-1)v • S2:an optimal modified u-restricted (k-k1+1)-residence set for Tvi

  14. Lemma3 proof • Show that S1 is an optimal modified u-restricted k1-residence set for T(i-1)v • Hence, the total modified cost for S1’∪S2 is less than that of S1∪S2 = S. (contradiction) • The proof that S2 is an optimal modified u-restricted (k-k1+1)-residence set for Tvi is similar.

  15. Lemma4 • Let S be an optimal modified u-restricted k-residence set for T(i)v • S1=(T(i-1)v∩S)∪{u} • S2=(S-S1) • S1:an optimal modified u-restricted k2-residence set for T(i-1)v • S2:an optimal modified u-restricted (k-k2)-residence set for Tvi

  16. Lemma4 proof • Show that S1 is an optimal modified u-restricted k2-residence set for T(i-1)v .Hence, the total modified cost for S1’∪S2 is less than that of S1∪S2 = S. (contradiction) • Consider the set S2

  17. Lemma4 proof

  18. Theorem1

  19. Theorem1 • the cost of an optimal k-residence set for Tv is given by • the cost of an optimal k-resident set for T is given by • the cost of an optimal resident set for T is given by

  20. 測試 • fasdasd • Fsdfasdfa • fsdafasd

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