236601 - Coding and Algorithms for Memories Lecture 7

1 / 16

236601 - Coding and Algorithms for Memories Lecture 7 - PowerPoint PPT Presentation

236601 - Coding and Algorithms for Memories Lecture 7. Class Overview. What have we studied so far? Background on memories Flash memories: properties, structure and constraints Rewriting codes – WOM codes Other Rewriting Codes What’s next? Rank modulation codes ECC and constrained codes

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

PowerPoint Slideshow about '236601 - Coding and Algorithms for Memories Lecture 7' - lexine

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

236601 - Coding and Algorithms for MemoriesLecture 7

Class Overview
• What have we studied so far?
• Background on memories
• Flash memories: properties, structure and constraints
• Rewriting codes – WOM codes
• Other Rewriting Codes
• What’s next?
• Rank modulation codes
• ECC and constrained codes
• Wear leveling & memory management
• Coding for Storage
• HW 2 – due May 1st
Flash Memory Cell

3

2

1

0

The Overshooting Problem

3

2

1

0

Need to erase the whole block

Relative Vs. Absolute Values

Less errors

More retention

0

1

Jiang, Mateescu, Schwartz, Bruck, “Rank modulation for Flash Memories”, 2008

Absolute values  Relative values

Single cell  Multiple cells

Physical cell  Logical cell

1

2

3

4

Rank Modulation

Ordered set of n cells

Assume discrete levels

Relative levels define a permutation

Basic operation: push-to-the-top

1

4

3

2

Overshoot is not a concern

Writing is much faster

Increased reliability (data retention)

permutation in

lexicographical order

decimal

[Lehmer 1906, Laisant 1888]

0 1 2

0 2 1

1 0 2

1 2 0

2 0 1

2 1 0

0 0

0 1

1 0

1 1

2 0

2 1

0

1

2

3

4

5

New Number Representation System

an-1…a3a2a1 = an-1·(n-1)! + … +a3·3!+ a2·2!+a1·1! 0 ≤ai ≤i

1

2

3

1

2

3

3

1

2

3

1

2

2

3

1

2

3

1

2

1

3

2

1

3

3

2

1

3

2

1

1

3

2

1

3

2

1

2

3

Gray Codes for Rank Modulation

The problem: Is it possible to transition between all permutations?

Find cycle through n! states

by push-to-the-top transitions

n=3

3 cycles

1

2

3

Transition graph, n=3

1

2

3

3

1

2

2

3

1

2

1

3

3

2

1

1

3

2

~ (n-1)!

Gray Codes for Arbitrary n
• Recursive construction:
• Keep bottom cell fixed
• (n-1)! transitions with others

1 3 2 3 1 2

2 1 3 2 3 1

3 2 1 1 2 3

444444

4 1 2 1 4 2

2 4 1 2 1 4

1 2 4 4 2 1

333333

3 4 2 4 3 2

2 3 4 2 4 3

4 2 3 3 2 4

1 1 1 1 1 1

1

2

3

4

1

2

3

4

Gray Codes for Arbitrary n

1 3 2 3 1 2

2 1 3 2 3 1

3 2 1 1 2 3

444444

4 1 2 1 4 2

2 4 1 2 1 4

1 2 4 4 2 1

333333

3 4 2 4 3 2

2 3 4 2 4 3

4 2 3 3 2 4

111111

4 1 3 4 3 1

1 4 1 3 4 3

3 3 4 1 1 4

222222

Rewriting with Rank Modulation
• If we represent n! symbols then in the worst case we apply n-1 push-to-the-top operations to transfer from one permutation to another
• Problem: Is it possible to use less push-to-the-top operations in case less than n! symbols are represented?
• Rank Modulation Rewriting code (RMRC) (n,M) consists of
• Update function: E: Sn×[M] -> Sn
• Decoding function D: Sn-> [M]
1

2

3

1

2

3

3

1

2

3

1

2

2

3

1

2

3

1

2

1

3

2

1

3

3

2

1

3

2

1

1

3

2

1

3

2

Rewriting with Rank Modulation
• Definition: The cost of changing s1 into s2,α(s1->s2), is the min number of push-to-the-top operations needed to change s1 to s2
• Ex: α([123]->[213]) = 1, α([123]->[321]) = 2
• The rewriting cost of a RMRC is the maximum update cost
• The transition graph Gn=(Vn,En)
• Vn = Sn, En ={(s1,s2) : α(s1->s2)=1}
• The ballor radius r: Br(s)={ s’ : α(s->s’) ≤ r }
• The sphereor radius r: Sr(s)={ s’ : α(s->s’) = r }
• The balls and the sphere sizes do not depend on rBr,Sr
Rewriting with Rank Modulation
• For n,M, define r(n,M) to be the smallest integer such that Br(n,M) ≥ M
• Lemma (Lower Bound): For any RMRC (n,M), its rewriting cost is at least r(n,M)
• Upper bound on the rewriting cost is given by a construction