1. 236601 - Coding and Algorithms for Memories Lecture 8 1

2. Relative Vs. Absolute Values 01 Less errors More retention Jiang, Mateescu, Schwartz, Bruck, “Rank modulation for Flash Memories”, 2008 2

3. The New Paradigm Rank Modulation Absolute values  Relative values Single cell  Multiple cells Physical cell  Logical cell 3

4. Rank Modulation 3 2 1 4 1234 Ordered set of n cells Assume discrete levels Relative levels define a permutation Basic operation: push-to-the-top Overshoot is not a concern Writing is much faster Increased reliability (data retention) 4

5. New Number Representation System permutation in lexicographical order [Lehmer 1906, Laisant 1888] FACTORADIC decimal a n-1 …a 3 a 2 a 1 = a n-1 ·(n-1)! + … + a 3 ·3! + a 2 ·2! + a 1 ·1! 0 ≤ a i ≤ i 0 1 2 0 2 1 1 0 2 1 2 0 2 0 1 2 1 0 0 0 1 1 0 1 2 0 2 1 012345012345 5

6. 231231 312312 123123 132132 213213 321321 Gray Codes for Rank Modulation Find cycle through n! states by push-to-the-top transitions 231231 312312 123123 132132 213213 321321 Transition graph, n=3 n=3 3 cycles 2 3 1 123 The problem: Is it possible to transition between all permutations? 6

7. Gray Codes for Arbitrary n Recursive construction: – Keep bottom cell fixed – (n-1)! transitions with others 1234 4 ~ (n-1)! 2 3 1 231231 312312 123123 132132 213213 321321 7 1 3 2 3 1 2 2 1 3 2 3 1 3 2 1 1 2 3 4 4 4 4 1 2 1 4 2 2 4 1 2 1 4 1 2 4 4 2 1 3 3 3 3 4 2 4 3 2 2 3 4 2 4 3 4 2 3 3 2 4 1 1 1

8. 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: S n ×[M] -> S n – Decoding function D: S n -> [M] 8

9. Rewriting with Rank Modulation Definition: The cost of changing s 1 into s 2, α(s 1 ->s 2 ), is the min number of push-to-the-top operations needed to change s 1 to s 2 – Ex: α([123]->[213]) = 1, α([123]->[321]) = 2 The rewriting cost of a RMRC is the maximum update cost The transition graph G n =(V n,E n ) – V n = S n, E n ={(s 1,s 2 ) : α(s 1 ->s 2 )=1} The ball or radius r: B r (s)={ s’ : α(s->s’) ≤ r } The sphere or radius r: S r (s)={ s’ : α(s->s’) = r } The balls and the sphere sizes do not depend on r B r,S r 231231 312312 123123 132132 213213 321321 231231 312312 123123 132132 213213 321321 9

10. Rewriting with Rank Modulation Lemma: B r =n!/(n-r)! For n,M, define r(n,M) to be the smallest integer such that B r(n,M) ≥ M Lemma (Lower Bound): For any RMRC (n,M), its rewriting cost is at least r(n,M) A tight upper bound on the rewriting cost is given by a construction Theorem: There exists a RMRC with parameters (n,M≤B r ) and cost r – Ex. (n,n) with cost 1 – Ex. (n,n(n-1)) with cost 2 – Ex. (n,n!) with cost n-1 – Ex. (n,n!/2) with cost n-2 10

11. 3 2 1 4 2 3 1 4 3 2 4 1 11

12.  2134  2143 Kendall’s Tau Distance For a permutation  an adjacent transposition is the local exchange of two adjacent elements For ,π ∊ S m, d τ ( ,π) is the Kendall’s tau distance between  and π = Number of adjacent transpositions to change  to be π  =2413 and π=2314 2413 d τ ( ,π) = 3 It is called also the bubble-sort distance Lemma: Kendall’s tau distance induces a metric on S n The Kendall’s tau distance is the number of pairs that do not agree in their order  2143  2134  2314 12

13. Kendall’s Tau Distance Lemma: Kendall’s tau distance induces a metric on S n The Kendall’s tau distance is the number of pairs that do not agree in their order For a permutation , W τ (  ) = {(i,j) | i  -1 (i) } Lemma: d τ ( ,π)= |W τ (  )  W τ (π)| = |W τ (  )\W τ (π)| + |W τ (π)\W τ (  )| d τ ( ,id) = |W τ (  )| The maximum Kendall’s tau distance is n(n-1)/2 The inversion vector of  is x  =(x  (2),…,x  (n)) x  (i) = # of elements to the right of i and are less then i d τ ( ,id) = |W τ (  )| = Σ 2≤i≤n x  (i) 13

14. Kendall’s Tau Distance The Kendall’s tau ball: B r (  ) = {π|d τ ( ,π) ≤ r} The Kendall’s tau sphere: S r (  ) = {π|d τ ( ,π) = r} They do not depend on the center  |B 1 (  )| = n, |S 1 (  )| = n-1 14

15. How to Construct ECCs for the Kendall’s Tau Distance? Goal: Construct codes with some prescribed min Kendall’s tau dist d 15