Optimal Placement of Replicas
in Trees with Read, Write, and
Storage Costs
Outline
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Introduction
Preliminaries
Finding optimal resident sets
Considering node capacities and load
Conclusion
Introduction
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
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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
Preliminaries
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u cover v
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Read cost of S
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Write cost of S
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Storage cost of S
Preliminaries
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Let
Preliminaries
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Define R(u, k, v, i)
V
Vi
Tvi
Finding optimal resident sets
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Lemma1.
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C(S1∪S2, T1∪T2∪{u}) = C(S1, T1∪{u})
+ C(S2, T2∪{u})
- s(u)
Lemma1 proof
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It is sufficient to show that
mst(S1∪S2) = mst(S1) + mst(S2)
Lemma2
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u covers v1 but it doesn’t cover v2
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C(S1∪S2, T1∪T2∪{u}) = C(S1, T1∪{u})
+ C(S2, T2)
+ Wtotal*δ(u, S2)
Lemma2 proof
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It is sufficient to show that
mst(S1∪S2) = mst(S1) + mst(S2) + δ(u, S2)
Lemma3
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Let S be an optimal modified u-restricted
k-residence set for T(i)v
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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
Lemma3 proof
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Show that S1 is an optimal modified u-restricted k1-residence set
for T(i-1)v
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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.
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Lemma4
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Let S be an optimal modified u-restricted
k-residence set for T(i)v
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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
Lemma4 proof
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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
Lemma4 proof
Theorem1
Theorem1
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the cost of an optimal k-residence set for Tv is given
by
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he cost of an optimal k-resident set for T is given by
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the cost of an optimal resident set for T is given by
Simple example
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Just consider read cost and residence set k≦2
A
1
B
2
C
1
D
Node
No. Reads
A
100
B
1
C
2
D
200
R(B, 1, C, 0)=1*2=2 (C)
R(B, 1, D, 0)=1*2=2 (D)
R(B, 2, C, 0)=0 (B, C)
R(B, 2, D, 0)=0 (B, D)
R(u, 2, B, 2)
R(A, 2, B, 2)=R(A, 1, B, 1)+R(A, 2, D, 0)=7 (A, D)
=R(A, 2, B, 1)+R(A, 1, D, 0)=401 (A, C)
R(A, 1, B, 1)=1*1+2*(2+1)=7 (A)
R(A, 2, B, 1)=R(A, 1, B, 0)+R(A, 2, C, 0)=1 (A, C)
=R(A, 2, B, 0)+R(A, 1, C, 0)=2*(2+1)=6 (A)
R(A, 2, D, 0)=0 (A, D)
R(A, 1, D, 0)=200*(1+1) (A)
R(A, 1, B, 0)=1*1 (A)
R(A, 2, C, 0)=0 (A, C)
R(D, 2, B, 2)=R(D, 1, B, 1)+R(D, 2, A, 0)=7 (A, D)
= R(D, 2, B, 1)+R(D, 1, A, 0)=201 (C, D)
R(D, 1, B, 1)=1*1+2*(2+1)=7
R(D, 2, B, 1)=R(D, 1, B, 0)+R(D, 2, C, 0)=1 (C, D)
=R(D, 2, B, 0)+R(D, 1, C, 0)=6 (B, D)
R(D, 1, B, 0)=1*1 (D)
R(D, 1, C, 0)=2*(2+1)=6 (B, D)
R(D, 2, A, 0)=0 (A, D)
R(D, 2, B, 0)=0 (B, D)
R(D, 2, C, 0)=0 (C, D)
A
1
B
2
C
1
D
Node
No. Reads
A
100
B
1
C
2
D
200
Theorem2
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Let T be a tree with n vertices and p be an
integer 1≦p≦n
We can find an optimal p-resident set for T in
O(n3p2) time
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R(u, k, v, i) is defined for at most n2p entries
Each entry: O(np)
Considering node capacities and load
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Vertex v imposes an integer load λ(v)≧0 on
the vertex u
Each vertex u on V has an integer capacity
Λ(u)≧0
Theorem4
Conclusion
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Described a O(n3p2)-time algorithm for finding
minimum total read-write-storage cost normal
residence sets for trees
Some issues:
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Finding optimal or near-optimal residence sets for multiple
objects
Finding near-optimal residence sets, that consider read,
write, and storage costs as well as capacity constraints, in
a distributed manner
Finding optimal or near-optimal replica schemes for other
network topologies
Reference

K. Kalpakis, K. Dasgupta, and O. Wolfson,
“Optimal Placement of Replicas in Trees with
Read, Write, and Storage Costs,” IEEE Trans.
Parallel and Distributed Systems, June 2001