236601 - Coding and Algorithms for Memories Lecture 7

1. 236601 - Coding and Algorithms for MemoriesLecture 7

2. 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

3. Flash Memory Cell 3 2 1 Can only add water (charge) 0

4. The Leakage Problem 3 2 Error! 1 0

5. The Overshooting Problem 3 2 1 0 Need to erase the whole block

6. Possible Solution – Iterative Programming 3 2 Slow… 1 0

7. Relative Vs. Absolute Values Less errors More retention 0 1 Jiang, Mateescu, Schwartz, Bruck, “Rank modulation for Flash Memories”, 2008

8. The New Paradigm Rank Modulation Absolute values  Relative values Single cell  Multiple cells Physical cell  Logical cell

9. 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)

10. permutation in lexicographical order FACTORADIC 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

11. 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

12. 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

13. 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

14. 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]

15. 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

16. 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