# Topics. Hamming Algorithm

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1 Topics Hamming algorithm Magnetic disks RAID Hamming Algorithm In a Hamming code r parity bits added to m-bit word Forms codeword with length (m + r) bits Bit numbering Starts at with leftmost (high-order) bit All powers of 2 are parity bits Remaining bits are for data 2

2 Bit Numbering for Hamming Algorithm Given an 8-bit data word to encode Data bits (remaining bits) Parity bits (powers of 2) 3 Bit No Parity Bit Assignment 8 s Bit No. in Binary 4 s 2 s s Encoded by 2, 4, 4, 2 4, 2, 8, 8, 2 8, 2, 8, 4 Encodes 3, 5, 7, 9, 3, 6, 7,, 5, 6, 7, 2 9,,, 2 4 2

3 For parity bit Method for Parity Check b b 3 b 5 b 7 b 9 b For parity bit 2 b 2 b 3 b 6 b 7 b b For parity bit 4 Even parity Odd parity b 4 b 5 b 6 b 7 b 2 For parity bit 8 b 8 b 9 b b b 2 5 Boolean Exclusive Sum - XOR Function Logic operation A B F Output is true when either input is true but not both Corresponds to addition in base 2 F = A B Truth table for XOR function A B F 6 3

4 Class Exercise Compute the codeword for the 8-bit data: Class Exercise Find the bit that is incorrect in the following codeword. This 2 bit codeword encodes 8 data bits. Bit checks:, 3, 5, 7, 9, Bit 2 checks: 2, 3, 6, 7,, Bit 4 checks: 4, 5, 6, 7, 2 Bit 8 checks: 8, 9,,, 2 incorrect incorrect correct incorrect Bit 8 4

5 For Further Study Article on parity and non-parity main memory 9 Memory Hierarchy Down the hierarchy: Increased access time Increased storage capacity Increased number of bits per dollar 5

6 Storing Data on a Magnetic Disk Tracks are concentric circles on a platter Sectors are pie-shaped wedges on a track Cylinders are a set of tracks at a given radial distance Data Transfer Time Seek move to correct radial position (cylinder) Rotational latency desired sector located under read/write head 2 6

7 Formatting a Track 3 Photo of Disk Head, Arm, Actuator Spindle Arm Head Actuator Platters (2) Adapted from David Patterson s CS 252 lecture notes. Copyright 2 UCB. 4 7

8 (Not So) Floppy Disk Small removable medium to distribute S/W General characteristics the same as hard disk Except the floppy disk read/write heads touch the disk, causing extra wear and tear 5 Integrated Drive Electronics (IDE) Standard way to connect storage device to computer Hard drives, internal floppy drives, CD-ROM drives, and even some tape backup drives Extended IDE for Logical Block Addressing (LBA) Required to address larger drives (2 24 ) sector numbers are converted to head, sector, and cylinder addresses 6 8

9 Small Computer System Interface (SCSI) Benefits of SCSI: Speed up to 6 megabytes per second (MB/s) Reliability Can put multiple devices on one bus Compatible with most computer systems Potential problems when using SCSI: Requires configuration for a specific computer. Limited system BIOS support. Numerous variations (speeds, connectors) No common software interface 7 Small Computer System Interface (SCSI) 8 9

10 IDE/EIDE vs. SCSI IDE/EIDE Low-cost Intel-based systems Allow only one active device at a time SCSI High-cost Standard in UNIX workstations All devices run at once Lower data transfer rate Higher data transfer rate 9 Redundant Arrays of (Inexpensive) Disks Files are "striped" across multiple disks Redundancy yields high data availability Availability: service still provided to user, even if some components failed Disks will still fail Contents reconstructed from data redundantly stored in the array Capacity penalty to store redundant info Bandwidth penalty to update redundant info Adapted from David Patterson s CS 252 lecture notes. Copyright 2 UCB. 2

11 RAID Levels Through 5 Backup and parity drives are shaded 2 RAID Level Disk Mirroring Each disk is fully duplicated onto its mirror Most expensive solution % capacity overhead 22

12 Exercise RAID level 3 is able to correct single-bit errors using only one parity drive. What is the point of RAID level 2? After all, it also can only correct one error and takes more drives to do so. 23 Exercise Answer RAID level 2 can not only recover from crashed drives, but also from undetected transient errors. If one drive delivers a single bad bit, RAID level 2 will correct this with its Hamming code, but RAID level 3 will not. 24 2

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