Designing Floating Codes for Expected Performance

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.

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.

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 allp, 0 < p < 1. • Worst case optimal: approx 3q/2. D(0,0) = 00 D(1,3) = 11 R((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

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 allp, 0 < p < 1.

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.

Designing Floating Codes for Expected Performance