1. Lecture 1: Bloom Filters
Chen Qian
Department of Computer Engineering

2. Outline Pre-Knowledge Bloom filters Counting Bloom filters
Applications 2

3. Hash Functions
A hash function is any function that can be used to map data of arbitrary size to data of a fixed size 00
Hash function
Map names to integers from 0 to 15
keys 01
John Smith 02
Lisa Smith 03
Sam Doe 04
Sandra Dee 05
collision ⋮ 3 15

4. Hash Functions Requirements for hash functions
Can be applied to any sized key 𝑥
Produces fixed-length output d
Easy to compute d=ℎ(𝑥)
One way property: given 𝑑 is infeasible to find ℎ 𝑥 =𝑑
Weak collision resistance: given x is infeasible to find y that makes ℎ 𝑦 =ℎ(𝑥)
Strong collision resistance: it is infeasible to find any x, y to make ℎ 𝑥 =ℎ(𝑦)
Uniformity: Every hash value in the output range should be generated with roughly the same probability 4
Red ones are required for cryptographic hashes

5. Examples of Hash functions
mod
CRC-16, -32
Jenkins, CityHash, FarmHash, MetroHash, etc.
MD5, SHA-1, MD6, SHA-2, SHA-256
CRCs are based on the theory of cyclic error-correcting codes 5

6. Outline Pre-Knowledge Bloom filters Counting Bloom filters
Applications 6

7. Bloom Filters Membership query
A Bloom filter is a simple space-efficient randomized data structure
Represent a set in order to support membership queries
Burton Bloom introduced Bloom filters in the 1970s C A E B Z ⋯
Element P is in the S? P
set S 7

8. Bloom Filters Standard Bloom filters:
Represent a set 𝑆={ 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 } by an array of m bits
Use k independent hash functions ℎ 1 , ℎ 2 ,…, ℎ 𝑘
assumption
Hash functions map each item to a random number uniformly over the range {1,2,…,𝑚}
How to construct the bloom filters 8

9. Bloom Filters m bits array Initially all set to 0 𝑥 1 assume k is 3
𝑥 1
assume k is 3
Insert 𝑥 1 , 𝑥 2
ℎ 1 ( 𝑥 1 )
ℎ 3 ( 𝑥 1 )
ℎ 2 ( 𝑥 1 ) 9

10. Bloom Filters m bits array Initially all set to 0 𝑥 1 assume k is 3
𝑥 1
assume k is 3
Insert 𝑥 1 , 𝑥 2
ℎ 1 ( 𝑥 1 )
ℎ 3 ( 𝑥 1 )
ℎ 2 ( 𝑥 1 ) 1 10

11. Bloom Filters m bits array Initially all set to 0 𝑥 1 𝑥 2
𝑥 1
𝑥 2
assume k is 3
ℎ 1 ( 𝑥 2 )
Insert 𝑥 1 , 𝑥 2
ℎ 2 ( 𝑥 2 )
ℎ 3 ( 𝑥 2 ) 1 11

12. Bloom Filters m bits array Initially all set to 0 𝑥 1 𝑥 2
𝑥 1
𝑥 2
assume k is 3
ℎ 1 ( 𝑥 2 )
Insert 𝑥 1 , 𝑥 2
ℎ 2 ( 𝑥 2 )
ℎ 3 ( 𝑥 2 ) 1 12

13. Bloom Filters m bits array Initially all set to 0 assume k is 3 𝑥 1
assume k is 3
𝑥 1
𝑥 2
Insert 𝑥 1 , 𝑥 2 1
𝑦 1
𝑥 2
Query 𝑥 2 , 𝑦 1
(alien element)
All ones, 𝑥 2 is a member 1
𝑦 1 is not a member 13

14. Bloom Filters
False positive
False negative 14

15. Bloom Filters Have false positive No false negative
There is a clear tradeoff between m and the probability of a false positive
𝑥 1 𝑦
𝑥 2
Alien element: y
y is a member 1 15

16. Bloom Filters n keys, m bits array, k hash functions
After inserting n keys, let p denotes the probability that a particular bit is 0
𝑝= (1− 1 𝑚 ) 𝑘𝑛 ≈ 𝑒 −𝑘∗ 𝑛 𝑚
Let f denotes the probability of false positive
𝑓= (1−𝑝) 𝑘 = (1− (1− 1 𝑚 ) 𝑘𝑛 ) 𝑘
When 𝑘=𝑙𝑛2 ∗ 𝑚 𝑛 , f achieves optimal value 16

17. Bloom Filters
No deletion 1
𝑥 1
𝑥 2
delete 𝑥 1 17

18. Bloom Filters
No deletion 1
𝑥 2
𝑥 2 is not a member
delete 𝑥 1 18

19. Bloom Filters Features Low storage requirement
Fast membership checking
No false negative
Low false positive probability
No deletion is allowed 19

20. Outline Pre-Knowledge Bloom filters Counting Bloom filters
Applications 20

21. Counting Bloom Filters
Allow deletion
Each entry in the counting Bloom filters is a small counter
l bits per counter
𝑥 1
insert 𝑥 1 , 𝑥 2 1 21

22. Counting Bloom Filters
Each entry in the Bloom filters is a small counter
l bits per counter
𝑥 1
𝑥 2
insert 𝑥 1 , 𝑥 2 1 22

23. Counting Bloom Filters
Each entry in the Bloom filters is a small counter
l bits per counter 1 2
𝑥 1
𝑥 2
insert 𝑥 1 , 𝑥 2 23

24. Counting Bloom Filters
Each entry in the Bloom filters is a small counter
l bits per counter 1 2
𝑥 1
𝑥 2
insert 𝑥 1 , 𝑥 2
𝑥 2
delete 𝑥 1 1 24

25. Counting Bloom Filters
The size of the bits for the counter
𝑝 𝑐 𝑖 =𝑗 = 𝑛𝑘 𝑗 ( 1 𝑚 ) 𝑗 (1− 1 𝑚 ) 𝑛𝑘−𝑗
𝑝(𝑐(𝑖)≥𝑗)≤ 𝑛𝑘 𝑗 1 𝑚 𝑗 ≤ ( 𝑒𝑛𝑘 𝑗𝑚 ) 𝑗
The probability that the any counter is at least 𝑗 is bounded by 𝑚𝑝(𝑐(𝑖)≥𝑗)
𝑝(max⁡(𝑐(𝑖))≥𝑗)≤𝑚𝑝(𝑐(𝑖)≥𝑗)
𝑝(max⁡(𝑐(𝑖))≥𝑗)≤𝑚 𝑛𝑘 𝑗 1 𝑚 𝑗 ≤𝑚 ( 𝑒𝑛𝑘 𝑗𝑚 ) 𝑗
The count associated with the i-th counter
loose bound 25

26. Counting Bloom Filters
𝑝 max 𝑐 𝑖 ≥𝑗 ≤𝑚 𝑛𝑘 𝑗 1 𝑚 𝑗 ≤𝑚 𝑒𝑛𝑘 𝑗𝑚 𝑗
Let 𝑘≤ 𝑙𝑛2 𝑚 𝑛
𝑝(max⁡(𝑐(𝑖))≥𝑗)≤𝑚 ( 𝑒𝑙𝑛2 𝑗 ) 𝑗
𝑝 max 𝑐 𝑖 ≥16 ≤1.37∗ 10 −15 ∗𝑚
If we allow 4 bits per counter, the probability of overflow for practical values of m during the initial insertion in the CBF is minuscule. 26

27. Outline Pre-Knowledge Bloom filters Counting Bloom filters
Applications 27

28. Summary Cache: A Scalable Wide-Area Web Cache Sharing Protocol
Background and Problems
Internet cache protocol (ICP)
Summary cache
Evaluation 28

29. Background and Problems
Caching has been recognized as one of the most important techniques to reduce bandwidth consumption with the development of World Wide Web.
Users
Proxy Caches
Bottleneck
⋯ Rest of the Internet ⋯ 29

30. Background and Problems
Proxy caches should cooperate and serve each other’s misses.
If the object is missed in current proxy cache, we can get the object from other proxy caches
How do we know which proxy caches hold the requested objects that miss in current proxy cache?
This process is called “Web Cache Sharing”
Internet cache protocol (ICP) is one of the protocol
for web cache sharing 30

31. Overview Background and Problems Internet cache protocol (ICP)
Summary cache
Evaluation 31

32. Internet Cache Protocol (ICP)
ICP supports discovery and retrieval of documents from neighboring caches.
Proxy caches connect hierarchically.
Proxy Cache
2 Miss, ICP query
Parent
Web server
2 Miss, ICP query
Proxy Cache
Proxy Cache
2 Miss, ICP query
Proxy Cache
1. web page
request
Sibling
2 Hit, return result 32
Sibling
Client

33. Internet Cache Protocol (ICP)
ICP supports discovery and retrieval of documents from neighboring caches.
Proxy caches connect hierarchically.
Proxy Cache
2.1 ICP miss
Parent
Web server
2.1 ICP miss
Proxy Cache
Proxy Cache
2.1 ICP miss
Proxy Cache
1. web page
request
Sibling 33
Sibling
Client

34. Internet Cache Protocol (ICP)
ICP supports discovery and retrieval of documents from neighboring caches.
Proxy caches connect hierarchically.
Proxy Cache
3. web page request
Parent
Web server
Proxy Cache
Proxy Cache
Proxy Cache
Sibling
return result 34
Sibling
Client

35. Internet Cache Protocol (ICP)
ICP supports discovery and retrieval of documents from neighboring caches.
Proxy caches connect hierarchically.
Proxy Cache
2.2 ICP hit
Parent
Web server
2.2 ICP hit
Proxy Cache
Proxy Cache
2.2 ICP miss
Proxy Cache
1. web page
request
Sibling 35
Sibling
Client

36. Internet Cache Protocol (ICP)
ICP supports discovery and retrieval of documents from neighboring caches.
Proxy caches connect hierarchically.
Proxy Cache
Parent
Web server
3. Web page request
Proxy Cache
Proxy Cache
Proxy Cache
Sibling
return result 36
Sibling
Client

37. Overhead of ICP ICP is not a scalable protocol
ICP relies on query messages to find cache hits in other proxies
Each proxy cache needs to handle 𝑁−1 ∗ 1−𝐻 ∗𝑅 inquiries from neighboring caches
N: Number of proxy caches
H: Hit rate
R: Average requests
As N increases, the overhead quickly becomes prohibitive 37

38. Overhead of ICP 4 proxy caches
Exp1
Hit Ratio
Client Latency
User CPU
System CPU
UDP Msgs
TCP Msgs
Total Packets
no ICP
25%
2.75 (5%)
94.42 (5%)
(6%)
615 (28%)
334K (8%)
355K (7%)
ICP
3.07 (0.7%)
12%
(5%)
24%
(5%)
10%
54774 (0%)
9000%
328K (4%)
-2%
402K (3%)
13%
Exp2
45%
2.21 (1%)
80.83 (2%)
(2%)
540 (3%)
272K (3%)
290K (3%)
2.39 (1%) 8%
97.36 (1%)
20%
(1%) 7%
39968 (0%)
7300%
257K (2%)
-1%
314K (1%)
ICP incurs considerable overhead even when the number of proxies is as low as four. 38

39. Benefits of Web Cache Sharing
Use simulation results to illustrate.
Compare four different cooperation schemes
No cache sharing
Proxies do not collaborate
Simple cache sharing (ICP-style)
Do not coordinate cache replacement (each LRU)
Client
cache D locally
request D A B C … A B C D …
return D D E B …
request D
Proxy Cache 1
Proxy Cache 2
return D 39
miss

40. Benefits of Web Cache Sharing
Compare four different cooperation schemes
Single-copy cache sharing
Mark D as most-recently-accessed
&
Increase its caching priority
Client
request D A B C … A B C
return D D E B …
request D
Proxy Cache 1
Proxy Cache 2
return D
miss
Compared to simple cache sharing,
this scheme eliminates the storage of duplicate copies,
increases the utilization of available cache space. 40

41. Benefits of Web Cache Sharing
Compare four different cooperation schemes
Global cache
Client
share cache contents
request D
return D A B C D E … A B C … D E B A C … D E B …
Proxy Cache 1
Proxy Cache 2
Coordinate replacement 41
C is replaced by F from the cache

42. Benefits of Web Cache Sharing
Compare four different cooperation schemes
Global cache
Client
share cache contents
request D
return D A B F D E … D E B A F …
Proxy Cache 1
Proxy Cache 2
Coordinate replacement
We can regard global cache as one unified cache with global LRU replacement to the users 42

43. Benefits of Web Cache Sharing
Use simulation results to illustrate
Compare four different cooperation schemes
Collect five sets of traces of HTTP requests
DEC, UCB, Upisa, Questnet, NLANR 43

44. Benefits of Web Cache Sharing
Simulation result
Hit ratio under single-copy cache sharing and simple cache sharing are generally the same as or even higher than the hit ratio under global cache 44

45. Benefits of Web Cache Sharing
Simulation Result
All cache sharing schemes significantly improve the hit ratio over no cache sharing
Hit ratio under single-copy cache sharing and simple cache sharing are generally the same
A waste of memory has only a small effect
A smaller effective cache does not make a significant difference in the hit ratio 45

46. Benefits of Web Cache Sharing
Simulation Result
Hit ratio under single-copy cache sharing and simple cache sharing are generally the same as or even higher than the hit ratio under global cache
Global LRU sometimes performs less well than group-wise LRU
In global cache setting, a burst of rapid successive requests from one user might disturb the working set of many other users
Conclusion
ICP-style simple cache sharing obtains most of the benefits of more elaborate cooperative caching. 46

47. Dilemma
Summary Cache
Benefits of ICP
High overhead of ICP 47

48. Overview Background and Problems Internet cache protocol (ICP)
Summary cache
Evaluation 48

49. Summary Cache Basic idea Two tolerate errors
Each proxy stores a summary of directory of cached documents in every other proxies
When a user request misses in the local cache, find proxies that store the requested document with summaries
Two tolerate errors
False misses: summary does not reflect that the document is cached at some other proxies
False hit: summary wrongly indicates that the document is cached at some other proxies 49

50. Summary Cache Challenges of scalability Naive summary representations
Network overhead: interproxy traffic
Memory to store summaries: summaries need to be stored in DRAM for performance reasons
Naive summary representations
Exact-directory approach: cache directory, each URL is represented by its 16-byte MD5 signature
Problem: Consume too much memory
E.g. 16 proxies of 8GB each and an average file size of 8KB, the exact-directory approach will consume 16−1 ∗16∗ 8𝐺𝐵 8𝐾𝐵 𝑏𝑦𝑡𝑒=240𝑀𝐵 50

51. Summary Cache Summary representations in summary cache Good news
Bloom filter
Memory-efficient and with low false hits
Good news
Summaries do not have to be up-to-date or accurate
Summaries do not have to be updated every time the cache directory is changed
The update can occur upon regular time intervals
Use an experiment to illustrate it
Delaying the update of summaries until percentage of cached documents that are new reaches a threshold 51

52. Bloom Filter as Summaries
Construction
A proxy builds a Bloom filter from the list of URL’s of cached documents
A proxy sends the Bloom filter plus the corresponding hash functions to other proxies
Update
Use counting Bloom filter to update the local filter
Specify which bits in the bit array are flipped or send the whole array to update Bloom filter in other proxies 52

53. Overview Background and Problems Internet cache protocol (ICP)
Summary cache
Evaluation 53

54. Evaluation
Three configurations (m denotes the size of Bloom filter array)
Bloom_filter_8/16/32: m is 8/16/32 times the average number of documents in the cache
Four hash functions
First calculate the MD5 signature of the URL (128 bits)
Divides 128 bits into four 32-bit word
Take the modulus of each 32-bit word by m 54

55. Evaluation Memory consumption Low memory consumption Approach DEC
NLANR
Exact_dir
2.8%
0.70%
Server_name
0.19%
0.08%
Bloom_filter_8
0.038%
Bloom_filter_16
0.38%
0.075%
Bloom_filter_32
0.75%
0.15%
Low memory consumption 55

56. Evaluation Exp1 Hit Ratio Client Latency User CPU System CPU UDP Msgs
TCP Msgs
Total Packets
no ICP
25%
2.75 (5%)
94.42 (5%)
(6%)
615 (28%)
334K (8%)
355K (7%)
ICP
3.07 (0.7%)
12%
(5%)
24%
(5%)
10%
54774 (0%)
9000%
328K (4%)
-2%
402K (3%)
13%
SC-ICP
2.85 (1%) 4%
95.07 (6%)
0.7%
(6%)
1079 (0%)
75%
330K (5%)
-1%
351K (5%)
Exp2
45%
2.21 (1%)
80.83 (2%)
(2%)
540 (3%)
272K (3%)
290K (3%)
2.39 (1%) 8%
97.36 (1%)
20%
(1%) 7%
39968 (0%)
7300%
257K (2%)
314K (1%)
2.25 (1%) 2%
82.03 (3%) 1%
(3%)
799 (5%)
48%
269K (5%)
287K (5%) 56

57. Conclusion Summary Cache enhanced ICP
Reduces the number of interproxy protocol message by factor of 25 to 60
Reduces the bandwidth consumption by over 50%
Almost no degradation in the cache hit ratios
Reduces CPU overhead between 30% to 95% 57

58. Chen Qian cqian12@ucsc.edu https://users.soe.ucsc.edu/~qian/
Thank You
Chen Qian