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### More Hashing

Hashing Part Two

Better Collision Resolution

Small parts of this material stolen from

"File Organization and Access" by Austing and Cassel

Recap of Last Class

- Hash function converts key to file address
- Collision is when two or more keys hash to the same address
- Collision Avoidance
- Good Hash Function spreads out the keys evenly along the whole address space
- Non-Dense File decreases chance of collisions and decreases probes after a collision

Recap: Linear Probing

- Very simple collision resolution
- if H(key) = A, and A is already used, try A+1, then A+2, etc
- Advantages
- easy to implement
- guaranteed to use all addresses

- Disadvantages
- clustering / clumping

Clumping

- Given the following hashes and linear probing:
- adams = 20
- bates = 22
- cole = 20
- dean = 21
- evans = 23

- Result of either
- poor hash function
- dense file

Random Probing

- Instead of adding 1, spread out by random amount
- True random would not work. Instead use pseudo-random.
While A is in use

A = (A + R) mod T

A = address

R = prime

T = Table Size

But what if 25 and 30 already had keys directly hashed to those locations? Cole would be at 35 -- 4 probes away.

- adams = 20
- bates = 22
- cole = 20
- dean = 21
- evans = 23

Chaining those locations? Cole would be at 35 -- 4 probes away.

- Assuming a better hash function and less dense file are not options...
- And assuming linear and random probing lead to coalesced lists...
- Chaining : maintain a linked list of collisions, one head per address
- Example, after addition of Adams and Cole, and R=5:
19 : null

20 : 35 -> null

21 : null

- Example, after addition of Adams and Cole, and R=5:
- Advantage: Faster at resolving collisions
- Disadvantage : Space

Re-Cap from weeks ago those locations? Cole would be at 35 -- 4 probes away.

- File Read Time = seek time + latency + data read time
- Smallest Readable Portion = 1 cluster = 4KB (usually)
- To access portion of a file, most of the time is in seek time and latency, not read time
- so, number of file reads is more important than size of reads, until size gets really big

- SO... reading a few records from a file takes no more time than reading just one record

Buckets those locations? Cole would be at 35 -- 4 probes away.

- Given, collisions will occur...
- Why not just read 2, or 3, or 4 records instead of just 1 on each read operation?
- "Bucket" - a group of records at the same address
- "Hash File of Buckets" - hashed keys collide to small arrays of records in the data file

Bucket Size? those locations? Cole would be at 35 -- 4 probes away.

- use avg collisions and stddev?
- if 1000 records and 200 addresses
- then avg is 5.0
- but stddev might be 1.0

- start by determining how many records can fit in one or more disk clusters
- then design a good hash function to match that address space

Advantages and those locations? Cole would be at 35 -- 4 probes away.Disadvantages of Buckets

- Advantages:
- Can achieve relatively fast access
- Remember, the hash function tells us where the record is located, so only 1 read operation. And even with collisions, the list of possible records is read into memory, which searches fast.
- Search Time = time to read bucket + time to search the array

- Disadvantages:
- What do we do when the bucket is full?
- solutions are similar to collision resolution
- we end up reading multiple sets of records

Predicting Collision Rates those locations? Cole would be at 35 -- 4 probes away.

- Collisions will happen!
- Poisson Function:
- p(x) gives the probability that a given address will have had x records assigned to it.
(r/N)x e-(r/N)

p(x) = ---------------

x!

N = number of available addresses

r = number of records to be stored

x = number of records assigned to a given address

- p(x) gives the probability that a given address will have had x records assigned to it.

Analysis continued those locations? Cole would be at 35 -- 4 probes away.

- Given
- N = 1000
- r = 1000

- Probability that a given address will have exactly one, two, or three keys hashed to it:
p(1) = 0.368

p(2) = 0.184

p(3) = 0.061

Analysis Continued those locations? Cole would be at 35 -- 4 probes away.

- Given
- N = 10,000
- R = 10,000

- How many addresses should have one, two, or three keys hashed to them?
10,000 x p(1) = 10000x0.3679 = 3679

10,000 x p(2) = 10000x0.1839 = 1839

10,000 x p(3) = 10000x0.0613 = 613

- So, 1839 keys will collide once and 613 will collide at least twice.
- Many of those collisions will disrupt probing.

Impact of Packing Density those locations? Cole would be at 35 -- 4 probes away.

Records that never collide = 303

Records that cannot go at their home = 107

Records at their home, but cause collisions = 90

Total = 500

- Given
- r = 500
- N = 1000
- one record per address

- Addresses with exact one record?
N x p(1) = 1000 x 0.303 = 303

- How many overflow records?
1 x N x p(2) + 2 x N x p(3) + 3 x N x p(4) + ...

= N x [1 x p(2) + 2 x p(3) + 3 x p(4)]

= 1000 x [ 1 x 0.076 + 2 x 0.013 + 3 x 0.002]

= 107

- Percentage of Records NOT stored at home address
107 / 500 = 21.4%

Impact of Packing Density those locations? Cole would be at 35 -- 4 probes away.

Real Life those locations? Cole would be at 35 -- 4 probes away.

- We must balance many factors:
- file size
- e.g., wasted space in hashed files
- e.g., extra space for index files

- disk access times
- available memory
- frequency of additions and deletions compared to searches

- file size
- Best Solution of All?
- probably a combination of indexed files, hashing, and buckets

Next Classes… those locations? Cole would be at 35 -- 4 probes away.

- Thursday April 14
- No Class

- Tuesday April 19
- B-Trees

- Thursday April 21
- Review

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