SLIDE 1
Caches are Everywhere
1
Web browser Web Cache
Server in Europe
# requests not served from cache total # requests
Goal: minimize cache miss ratio
Web App DB Cache
Slow database
with Variable Object Sizes Daniel S. Berger Nathan Beckmann Mor - - PowerPoint PPT Presentation
with Variable Object Sizes Daniel S. Berger Nathan Beckmann Mor - - PowerPoint PPT Presentation
Practical Bounds on Optimal Caching with Variable Object Sizes Daniel S. Berger Nathan Beckmann Mor Harchol-Balter Carnegie Mellon University ACM Sigmetrics, Irvine, June 19, 2018. Caches are Everywhere 10x Web browser Web Cache Server in
SLIDE 2
SLIDE 3
Everyone Has Worked in Caching
2
Key Question: how much further can miss ratios be improved?
lower is better
OPT
?
Results on 2016 CDN trace. Cache size: 4GB.
Def: OPT = lowest miss ratio possible on a given trace
Are we there yet?
SLIDE 4
Defining OPT
Constraints:
- 1. Limited cache size
- 2. Gets to see full request trace
- 3. No prefetching
⇒ admit an object only when it is requested
3
Def: OPT = lowest miss ratio possible on a given trace
a b c b d a c d a b c d
= offline optimal miss ratio on a given trace
SLIDE 5
Finding OPT
4
What is OPT? Belady? No! Belady assumes equal sizes! So, can we find OPT? Unfortunately, NP-hard In fact, strongly NP complete Can we approximate OPT? 9 orders of magnitude variability!
SLIDE 6
OPT Approximation Algorithms
5
Technique Time Complexity Approximation Guarantee
OFMA [STOC’97] O(N2) O(log cache size) LP rounding [SODA’99] O(N5) O(log (maxsize/minsize)) LocalRatio [JACM’01] O(N3) 4
OPT
State-of-the-art 4-approximation not practical
SLIDE 7
OPT Approximation Algorithms
6
Technique Time Complexity Approximation Guarantee
Best prior approximation O(N3)
4
Independent Reference Model (IRM)
Traces are not adversarial in practice
Large systems: #objects, cache size
⇒ Probabilistic assumptions
SLIDE 8
Our Main Result
7
Technique Time Complexity Approximation Guarantee
Best prior approximation O(N3) 4 Flow-offline optimum (FOO) O(N2log2N) 1
On trace with strong correlations: error < 0.14%
Independent Reference Model (IRM)
Traces are not adversarial in practice
Large systems: #objects, cache size
⇒ Probabilistic assumptions
SLIDE 9
8
Detailed OPT ILP Interval ILP FOO Min Cost Flow graph
How does FOO attain OPT for large systems?
How to get OPT fast:
Interval LP
relaxation
NP-hard NP-hard Ω(N3.5) O(N2log2N)
a b c b a x(a,1) x(a,2) x(a,3) x(a,4) x(a,5)
Trace: DVs for
- bject a:
SLIDE 10
9
Detailed OPT ILP Interval ILP FOO Min Cost Flow graph
How does FOO attain OPT for large systems?
How to get OPT fast:
Interval LP
relaxation
NP-hard NP-hard Ω(N3.5) O(N2log2N)
Non- integer decision vars (DVs) Precedence relation, which forces integer DVs
always exists
Coupon collector problem
under IRM
Integer DVs almost surely
large systems
How to prove FOO’s correctness:
SLIDE 11
Our Main Result
10
Technique Time Complexity Approximation Guarantee
Best prior approximation O(N3) 4 Flow-offline optimum (FOO) O(N2log2N) 1
Computable with up to 107 requests
SLIDE 12
lower is better
Empirical Results
11
30% gap Key Question: how much further can miss ratios be improved?
Could be
- ptimal…
Are we there yet?
- r not
SLIDE 13
30% gap 45% gap 61% gap 51% gap 15% gap 41% gap
Results for Other Configurations
12
CDN Small Cache Large Cache WebApp Storage
Key Question: how much further can miss ratios be improved?
SLIDE 14
Conclusion
13
Technique Time Complexity Approximation Guarantee
Best prior approximation O(N3) 4 Flow-offline optimum (FOO) O(N2log2N) 1
Actually can do: O(N log N)
Implication: large potential for new caching policies ⇒ e.g., 60% improvement possible for WebApps
SLIDE 15
Practical Bounds on Optimal Caching with Variable Object Sizes
Daniel S. Berger Nathan Beckmann Mor Harchol-Balter
ACM Sigmetrics, Irvine, June 19, 2018.
Carnegie Mellon University
Source code and data: available at
/dasebe/optimalwebcaching