Symmetric Allocations for Distributed Storage Derek Leong1, Alexandros G. Dimakis2, Tracey Ho1 1California Institute of Technology, USA 2University of Southern California, USA GLOBECOM 2010 2010-12-09

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

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 ≈ 0.0329218

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 ≈ 0.00411523

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

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

A Motivating Example 1 010010101010010101000101010101000101010111010101001001010001010100 2 01101010001010101110101010010010100010101001 3 1010010101000101001110 4 1010010101000101001110 5

? 1 2 3 4 5 A Motivating Example (1/3)2 (2/3)3 ≈ 0.0329218 010010101010010101000101010101000101010111010101001001010001010100 2 01101010001010101110101010010010100010101001 3 1010010101000101001110 4 1010010101000101001110 5 ?

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

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

A Motivating Example 1 2 3 4 5

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 0.79012 B 7/6 7/6 0 0 0 0.88889 C C 2/3 2/3 1/3 1/3 1/3 0.90535

#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

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

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

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 .

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

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

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

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

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.

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.

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

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

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

Thank you!