1. Symmetric Allocations for Distributed Storage
Derek Leong1, Alexandros G. Dimakis2, Tracey Ho1 1California Institute of Technology, USA 2University of Southern California, USA GLOBECOM

2. A Motivating Example
Suppose you have a distributed storage system comprising 5 storage devices (“nodes”)… 1 2 3 4 5

3. 2 4 1 2 3 4 5 A Motivating Example (1/3)2 (2/3)3 ≈ 0.0329218
Each node independently fails with probability 1/3, and survives with probability 2/3 … 2 4 1 2 3 4 5
(1/3)2 (2/3)3 ≈

4. 1 2 3 4 5 1 2 3 4 5 A Motivating Example (1/3)5 ≈ 0.00411523
Each node independently fails with probability 1/3, and survives with probability 2/3 … 1 2 3 4 5 1 2 3 4 5
(1/3)5 ≈

5. A Motivating Example
You are given a single data object of unit size, and a total storage budget of 7/3 … 1 2 3 4 5

6. A Motivating Example
You can use any coding scheme to store any amount of coded data in each node, as long as the total amount of storage used is at most the given budget 7/3 … 1 2 3 4 5

7. A Motivating Example 1 2 3 4 5

8. ? 1 2 3 4 5 A Motivating Example (1/3)2 (2/3)3 ≈ 0.0329218 2 3 4 5 ?

9. A Motivating Example
For maximum reliability, we need to find (1) an optimal allocation of the given budget over the nodes, and (2) an optimal coding scheme that jointly maximize the probability of successful recovery

10. A Motivating Example S 1 2 3 4 5 t1 t2
Using an appropriate code, successful recovery occurs whenever the data collector accesses at least a unit amount of data (= size of the original data object) S 1 2 3 4 5 t1 t2

11. A Motivating Example 1 2 3 4 5

12. A Motivating Example
Recovery Probability 1 2 3 4 5
for p = 2/3 A
7/15 7/15 7/15 7/15 7/15 B
7/6 7/ C C
2/3 2/ /3 1/3 1/3

13. #P-hard to compute for a given allocation and choice of p
Problem Formulation
#P-hard to compute for a given allocation and choice of p
Given n nodes, access probability p, and total storage budget T, find an optimal allocation (x1; …; xn) that maximizes the probability of successful recovery
recovery probability
The optimal allocation also tells us whether coding is beneficial for reliable storage
budget constraint
Trivial cases of minimum and maximum budgets:
when T = 1, the allocation (1, 0, …, 0) is optimal
when T = n, the allocation (1, 1, …, 1) is optimal

14. Related Work
Discussion between R. Karp, R. Kleinberg, C. Papadimitriou, E. Friedman, and others at UC Berkeley, 2005
S. Jain, M. Demmer, R. Patra, K. Fall, “Using redundancy to cope with failures in a delay tolerant network,” SIGCOMM 2005

15. Symmetric Allocations
We are particularly interested in symmetric allocations because they are easy to describe and implement
Successful recovery for the symmetric allocation occurs if and only if at least out of the m nonempty nodes are accessed
Therefore, the recovery probability of is

16. Asymptotic Optimality of Max Spreading
The symmetric allocation that spreads the budget maximally over all n nodes is asymptotically optimal when the budget T is sufficiently large
RESULT 1 The gap between the recovery probabilities for an optimal allocation and for the symmetric allocation is at most
If p and T are fixed such that , then this gap approaches zero as

17. Asymptotic Optimality of Max Spreading
Proof Idea: Bounding the optimal recovery probability…
By conditioning on the number of accessed nodes r, we can express the probability of successful recovery as where Sr is the number of successful r-subsets
We can in turn bound Sr by observing that we have Sr inequalities of the form , which can be summed up to produce
, where

18. Asymptotic Optimality of Max Spreading
Proof Idea: Bounding the optimal recovery probability…
We therefore have
Applying the bound to leads to the conclusion that the optimal recovery probability is at most

19. Asymptotic Optimality of Max Spreading
Proof Idea: Bounding the suboptimality gap for max spreading…
The recovery probability of the allocation is
The suboptimality gap for this allocation is therefore at most the difference between the upper bound for the optimal recovery probability and 1, which is
For , we can apply the Chernoff bound to obtain
As , this upper bound approaches zero

20. Optimal Symmetric Allocation
number of nonempty nodes in the symmetric allocation
The problem is nontrivial even when restricted to symmetric allocations…

21. Optimal Symmetric Allocation
Maximal spreading is optimal among symmetric allocations when the budget T is sufficiently large
RESULT 2
If , then either or is an optimal symmetric allocation.

22. Optimal Symmetric Allocation
Minimal spreading is optimal among symmetric allocations when the budget T is sufficiently small
Coding is unnecessary for such an allocation
RESULT 3
If , then is an optimal
symmetric allocation.

23. Optimal Symmetric Allocation
Proof Idea: Finding the optimal symmetric allocation…
Observe that we can find an optimal m* from among candidates:
For , where , the recovery probability is
RESULT 2 (max spreading optimal) is a sufficient condition on p and T for to be nondecreasing in k
To obtain RESULT 3 (min spreading optimal) , we first establish a sufficient condition on p and T for to be nonincreasing in k; we subsequently expand the condition to include other points for which remains optimal m …
For constant p and k, is a nondecreasing function of m
Recall that the recovery probability of the symmetric allocation is given by

24. Optimal Symmetric Allocation
maximal spreading is optimal among symmetric allocations
other symmetric allocations may be optimal in the gap
minimal spreading is optimal among symmetric allocations

25. Conclusion The optimal allocation is not necessarily symmetric
However, the symmetric allocation that spreads the budget maximally over all n nodes is asymptotically optimal when the budget is sufficiently large
Furthermore, we are able to specify the optimal symmetric allocation for a wide range of parameter values of p and T

26. Thank you!