Designing Floating Codesfor Expected Performance

Hilary Finucane

Zhenming Liu

Michael Mitzenmacher

Floating Codes

• A new model for flash memory.• State is an n-ary sequence of q-ary numbers.

– Represents block of n cells; each cell holds an electric charge.

• State mapped to variable values.– Gives k-ary sequence of l-ary numbers.

• State changes by increasing one or more cell values, or reset entire block.– Adding charge is easy; removing charge requires

resetting everything.– Resets are expensive!!!!

Floating Codes: The Problem

• As variable values change, need state to track variables.

• How do we choose the mapping function from states to variables AND the transition function from variable changes to state changes to maximize the time between reset operations?

History

• Write-once memories (WOMs)– Rivest and Shamir, early 1980’s.– Punch cards, optimal disks.– Can turn 0’s to 1’s, but not back again.

• Many related models: WOMs, WAMs, WEMs, WUMs.

• Floating codes (Jiang, Bohossian, Bruck) use model for Flash Memory.– Designed to maximize worst-case time between

resets.

Contribution : Expected Time

• Argument: Worst-case time between resets is not right design criterion.– Many resets in a lifetime.– Mass-produced product.– Potential to model user behavior.

• Statistical performance guarantees more appropriate.– Expected time between resets.– Time with high probability.– Given a model.

Specific Contributions

• Problem definition / model

• Codes for simple cases

Formal Model

• General Codes

• We consider limited variation; one variable changes per step.

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Variable values : (d1,d2,...,dk )∈ {0,1..., l −1}k

Cell states : (c1,c2,...,cn )∈ {0,1...,q−1}n

Decoding function : D :{0,1...,q−1}n → {0,1..., l −1}k

Rewriting function : R :{0,1...,q−1}n ×{0,1..., l −1}k →{0,1...,q−1}n

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R :{0,1...,q−1}n ×{1,2...,k} ×{0,1..., l −1} →{0,1...,q−1}n

Formal Model Continued

• Above : when– Cost is 0 when R moves to cell state above previous, 1

otherwise.

• Assumption : variables changes given by Markov chain.– Example : ith bit changes with prob. pi

– Given D, R, gives Markov chain on cell states. – Let be equilibrium on cell states.

• Goal is to minimize average cost:

– Same as maximize average time between resets.

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(c1,c2,...,cn ) ≥ (b1,b2,...,bn )

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c i ≥ bi∀ i

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min π x Pr(x → y)y not above x

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Variations

• Many possible variations – Multiple variables change per step– More general random processes for values– Rules limiting transitions– General costs, optimizations

• Hardness results?– Conjecture some variations NP-hard or worse.

Specific Case

• Binary values : l = 2• 2 bits : k = 2• Markov model: only bit changes at each step.

First bit changes with probability p.• Result : Asympotically optimal code.

– Code handles n(q-1)-o(nq) value changes with high probability.

– Same code works for every value of p.

Code : n = 2, k = 2, l = 2

• 2 bit values.• 2 cells.• Code based on

striped Gray code.• Expected time/time with

high probability before reset = 2q - o(q)

• Asymptotically optimal for all p, 0 < p < 1.

• Worst case optimal: approx 3q/2.

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00 01 11 10

10 00 01 11

11 10 00 01

01 11 10 00

D(0,0) = 00D(1,3) = 11R((1,0),2,1) = (2,0)

Proof Sketch

• “Even cells”: down with probability p, right with probability 1-p.

• “Odd cells” : right with probability p, down with probability 1-p.

• Code hugs the diagonal.

• Right/down moves approximately balance for first 2q-o(q) steps.

Performance Results

Scheme 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2DWC,4 0.209 0.210 0.213 0.218 0.222 0.227 0.232 0.238 0.244

2DGC,4 0.212 0.215 0.217 0.218 0.218 0.218 0.216 0.216 0.212

2DGC+,4 0.176 0.183 0.187 0.190 0.191 0.190 0.187 0.183 0.176

2DWC,8 0.092 0.093 0.094 0.094 0.095 0.096 0.097 0.098 0.100

2DGC,8 0.080 0.081 0.082 0.083 0.083 0.083 0.082 0.081 0.080

2DGC+,8 0.075 0.077 0.078 0.079 0.079 0.079 0.078 0.077 0.075

Code : n = 3, k = 2, l = 2

• Layer Gray codes for n = 3.

• Expected time/time whp before reset = 3q - o(q)– Slightly hard argument.

• Asymptotically optimal for all p, 0 < p < 1.

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00 01 11 10

10 00 01 11

11 10 00 01

01 11 10 00

10 00 01 11

11 10 00 01

01 11 10 00

00 01 11 10

00 01 11 10

10 00 01 11

11 10 00 01

01 11 10 00

10 00 01 11

11 10 00 01

01 11 10 00

00 01 11 10

Codes for k = l = 2

• Glue together codes for larger n.

• Example : n = 4. Go 2q - o(q) in first two dimension, 2q - o(q) in next two, so 4q - o(q) overall.

• Some further results in paper.

Conclusions

• Introduce problem of maximizing expected time until a reset for floating codes.

• Simple schemes for k = 2, l = 2 case based on Gray codes.– Building block for larger parameters?

Open Questions• Lots and lots of open questions.

– Complexity of finding optimal designs for given parameters.– Asymptotically good codes for larger parameters.– Lower bounds.– Reasonable models for real systems.– Small “families” of codes good over ranges of different user

behaviors.– Multi-objective: tradeoffs between average/worst-case

performance.– Incorporating error-correction.– Extending to buffer codes, or other models.– And more.