Coding and Algorithms for Memories Lecture 2 1

Overview Lecturer: Eitan Yaakobi Taub 638 Lectures hours: Thur Taub 8 Course website: Office hours: Thur 14:30-15:30 and/or other times (please contact by before) Final grade: – Class participation (10%) – Homeworks (50%) – Take home exam/final Homework + project (40%) 2

What is this class about? Coding and Algorithms to Memories Memories – HDDs, flash memories, and other non-volatile memories Coding and algorithms – how to manage the memory and handle the interface between the physical level and the operating system Both from the theoretical and practical points of view Q: What is the difference between theory and practice? 3

You do not really understand something unless you can explain it to your grandmother 4

One of the focuses during this class: How to ask the right questions, both as a theorist and as a practical engineer 5

Memory Storage Computer data storage (from Wikipedia): Computer components, devices, and recording media that retain digital data used for computing for some interval of time. What kind of data? – Pictures, word files, movies, other computer files etc. What kind of memories? – Many kinds… 6

Memories Volatile Memories – need power to maintain the information – Ex: RAM memories, DRAM, SRAM Non-Volatile Memories – do NOT need power to maintain the information – Ex: HDD, optical disc (CD, DVD), flash memories Q: Examples of old non-volatile memories? 7

Some of the main goals in designing a computer storage: Price Capacity (size) Endurance Speed Power Consumption 8

Optical Storage First generation – CD (Compact Disc), 700MB Second generation – DVD (Digital Versatile Disc), 4.7GB, 1995 Third generation – BD (Blu-Ray Disc) – Blue ray laser (shorter wavelength) – A single layer can store 25GB, dual layer – 50GB – Supported by Sony, Apple, Dell, Panasonic, LG, Pioneer 9

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The Magnetic Hard Disk Drive “1” “0” 12

Flash Memories

Gartner & Phison 14

SLC, MLC and TLC Flash High Voltage Low Voltage 1 Bit Per Cell 2 States SLC Flash High Voltage Low Voltage 2 Bits Per Cell 4 States MLC Flash High Voltage Low Voltage 3 Bits Per Cell 8 States TLC Flash 15

Flash Memory Structure A group of cells constitute a page A group of pages constitute a block – In SLC flash, a typical block layout is as follows page 0page 1 page 2page 3 page 4page page 62page 63 16

In MLC flash the two bits within a cell DO NOT belong to the same page – MSB page and LSB page Given a group of cells, all the MSB’s constitute one page and all the LSB’s constitute another page Row index MSB of first 2 14 cells LSB of first 2 14 cells MSB of last 2 14 cells LSB of last 2 14 cells 0page 0page 4page 1page 5 1page 2page 8page 3page 9 2 page 6page 12page 7page 13 3 page 10page 16page 11page 17 ⋮⋮⋮⋮⋮ 30page 118page 124page 119page page 122page 126page 123page MSB/LSB Flash Memory Structure 17

MLC Write Process voltage PV1 PV2 PV3 MSB=1 LSB=1 MSB=1 LSB=0 MSB=0 LSB=0 MSB=0 LSB=1 MSB=1MSB=0 Vread MSB=1MSB=0 18

Row index MSB of first 2 16 cells CSB of first 2 16 cells LSB of first 2 16 cells MSB of last 2 16 cells CSB of last 2 16 cells LSB of last 2 16 cells 0page 0page 1 1page 2page 6page 12page 3page 7page 13 2 page 4page 10page 18page 5page 11page 19 3 page 8page 16page 24page 9page 17page 25 4page 14page 22page 30page 15page 23page 31 ⋮⋮⋮⋮⋮ 62page 362page 370page 378page 363page 371page page 368page 376page 369page page 374page 382page 375page page 380page 381 MSB Page CSB Page LSB Page Flash Memory Structure 19

20 Flash Memories Programming Array of cells made from floating gate transistors Typical size can be 32 × 2 15 The cells are programmed by pulsing electrons via hot-electron injection

21 Flash Memories Programming Each cell can have q levels, represented by different amounts of electrons In order to reduce a cell level, thee cell and its containing block must be reset to level 0 before rewriting – A VERY EXPENSIVE OPERATION Array of cells made from floating gate transistors Typical size can be 32×2 15 The cells are programmed by pulsing electrons via hot-electron injection

Programming of Flash Memory Cells Flash memory cells are programmed in parallel in order to increase the write speed – Cells can only increase their value – In order to decrease a cell level, its entire containing block ( ~10 6 cells) has to be erased first Flash memory cells do not behave identically – When charge is injected, only a fraction of it is trapped in the cell – Easy cells – most of the charge is trapped in the cell – Hard cells – a small fraction of the charge is trapped in the cell 22

Programming of Flash Memory Cells Flash memory cells are programmed in parallel in order to increase the write speed – Cells can only increase their value – In order to decrease a cell level, its entire containing block ( ~10 6 cells) has to be erased first Flash memory cells do not behave identically – When charge is injected, only a fraction of it is trapped in the cell – Easy cells – most of the charge is trapped in the cell – Hard cells – a small fraction of the charge is trapped in the cell Goals: – Programming is done cautiously to prevent over-shooting – Programming should work for both easy and hard cells – And still… fast enough 23

Incremental Step Pulse Programming (ISPP) Incremental Step Pulse Programming – Gradually increase the program voltage – First the easy cells reach their level – On subsequent steps, only cells which didn’t reach their level are programmed – Enable fast programming of both easy and hard cells 24

Array of cells, made of floating gate transistors ─ Each cell can store q different levels ─ Today, q typically ranges between 2 and 16 ─ The levels are represented by the number of electrons ─ The cell’s level is increased by pulsing electrons ─ To reduce a cell level, all cells in its containing block must first be reset to level 0 A VERY EXPENSIVE OPERATION Rewriting Codes 25

Rewriting Codes Problem: Cannot rewrite the memory without an erasure However… It is still possible to rewrite if only cells in low level are programmed 26

From Wikipedia: One limitation of flash memory is that, although it can be read or programmed a byte or a word at a time in a random access fashion, it can only be erased a "block" at a time. This generally sets all bits in the block to 1. Starting with a freshly erased block, any location within that block can be programmed. However, once a bit has been set to 0, only by erasing the entire block can it be changed back to 1. In other words, flash memory (specifically NOR flash) offers random- access read and programming operations, but does not offer arbitrary random-access rewrite or erase operations. A location can, however, be rewritten as long as the new value's 0 bits are a superset of the over-written values. For example, a nibble value may be erased to 1111, then written e.g. as Successive writes to that nibble can change it to 1010, then 0010, and finally Essentially, erasure sets all bits to 1, and programming can only clear bits to 0. File systems designed for flash devices can make use of this capability, for example to represent sector metadata. 27

Rewriting Codes Store 3 bits once Store 1 bit 8 times Store 4 bits once Store 1 bit 16 times Rewrite codes significantly reduce the number of block erasures 28

One of the most efficient schemes to decrease the number of block erasures Floating Codes Buffer Codes Trajectory Codes Rank Modulation Codes WOM Codes Rewriting Codes 29

Write-Once Memories (WOM) Introduced by Rivest and Shamir, “How to reuse a write-once memory”, 1982 The memory elements represent bits (2 levels) and are irreversibly programmed from ‘0’ to ‘1’ 1 st Write 2 nd Write 30

Write-Once Memories (WOM) Examples: 1 st Write 2 nd Write 31

WOM Implementation in SLC Flash A scheme for storing two bits twice using only three cells before erasing the cells The cells only increase their level How to implement? (in SLC block) – Each page stores 2KB/1.5 = 4/3KB per write – A page can be written twice before erasing – Pages are encoded using the WOM code – When the block has to be rewritten, mark its pages as invalid – Again write pages using the WOM code without erasing – Read before write at the second write data 1 st write 2 nd write ⋮ … 10 WOM ENCODER … … … … … … 010 I N V A L I D … … … … … … …

BER for the First and Second Write 33

Why/When to Use WOM Codes? Disadvantage: sacrifice a large amount of the capacity – Ex: Two write WOM codes The best sum-rate is log3≈1.58 Can write (at most) only 0.79n bits so there is a lost of (at least) 21% of the capacity Advantage: Can increase the lifetime of the memory and reduce the write amplification 34

Why/When to Use WOM Codes? Advantage: Can increase the lifetime of the memory and reduce the write amplification Example: – User has 3GB of flash with lifetime 100 P/E cycles – Each day the user writes 2GB of new data (no need to store the old data) – Without WOM, the memory lasts 3/2*100=150 days – With WOM (the Rivest Shamir scheme) every two days the memory is erased once the memory lasts 2*100=200 days 35

Write-Once Memories (WOM) Introduced by Rivest and Shamir, “How to reuse a write-once memory”, 1982 The memory elements represent bits (2 levels) and are irreversibly programmed from ‘0’ to ‘1’ Q: How many cells are required to write 100 bits twice? P1: Is it possible to do better…? P2: How many cells to write k bits twice? P3: How many cells to write k bits t times? P3’: What is the total number of bits that is possible to write in n cells in t writes? 1 st Write 2 nd Write 36

Binary WOM Codes k 1,…,k t :the number of bits on each write – n cells and t writes The sum-rate of the WOM code is R = (Σ 1 t k i )/n – Rivest Shamir: R = (2+2)/3 = 4/3 37

Definition: WOM Codes Definition: An [n,t;M 1,…,M t ] t-write WOM code is a coding scheme which consists of n cells and guarantees any t writes of alphabet size M 1,…,M t by programming cells from zero to one – A WOM code consists of t encoding and decoding maps E i, D i, 1 ≤i≤ t –E 1 : {1,…,M 1 }  {0,1} n – For 2 ≤i≤ t, E i : {1,…,M i }×{0,1} n  {0,1} n such that for all (m,c) ∊ {1,…,M i }×{0,1} n, E i (m,c) ≥ c – For 1 ≤i≤ t, D i : {0,1} n  {1,…,M i } such that for D i ( E i (m,c)) =m for all (m,c) ∊ {1,…,M i }×{0,1} n The sum-rate of the WOM code is R = (Σ 1 t logM i )/n Rivest Shamir: [3,2;4,4], R = (log4+log4)/3=4/3 38

Definition: WOM Codes There are two cases – The individual rates on each write must all be the same: fixed-rate – The individual rates are allowed to be different: unrestricted-rate We assume that the write number on each write is known. This knowledge does not affect the rate – Assume there exists a [n,t;M 1,…,M t ] t-write WOM code where the write number is known – It is possible to construct a [Nn+t,t;M 1 N,…,M t N ] t-write WOM code where the write number is not-known so asymptotically the sum-rate is the same 39

The Capacity of WOM Codes The Capacity Region for two writes C 2-WOM ={(R 1,R 2 )| ∃ p ∊ [0,0.5],R 1 ≤h(p), R 2 ≤1-p} h(p) – the binary entropy function h(p) = -plog(p)-(1-p)log(1-p) The maximum achievable sum-rate is max p ∊ [0,0.5] {h(p)+(1-p)} = log3 achieved for p=1/3: R 1 = h(1/3) = log(3)-2/3 R 2 = 1-1/3 = 2/3 Capacity region (Heegard ‘86, Fu and Han Vinck ‘99) C t-WOM ={(R 1,…,R t )| R 1 ≤ h(p 1 ), R 2 ≤ (1–p 1 )h(p 2 ),…, R t-1 ≤ (1–p 1 )  (1–p t–2 )h(p t–1 ) R t ≤ (1–p 1 )  (1–p t–2 )(1–p t–1 )} The maximum achievable sum-rate is log(t+1) 40

The Capacity for Fixed Rate The capacity region for two writes C 2-WOM ={(R 1,R 2 )| ∃ p ∊ [0,0.5],R 1 ≤h(p), R 2 ≤1-p} When forcing R 1 =R 2 we get h(p) = 1-p The (numerical) solution is p = , the sum-rate is 1.54 Multiple writes: A recursive formula to calculate the maximum achievable sum-rate R F (1)=1 R F (t+1) = (t+1)  root{h(zt/R F (t))-z} where root{f(z)} is the min positive value z s.t. f(z)=0 – For example: R F (2) = 2  root{h(z)-z} = 2  = R F (3) = 3  root{h(2z/1.54)-z}=3  =