Online Algorithms and Competitive Analysis
Paging Algorithms
• Data brought from slower memory into cache
RAM
CPU
Paging Algorithms
• Data brought from slow memory into small fast memory (cache) of size k
• Sequence of requests: equal size pages
• Hit: page in cache, • Fault: page not in cache
Minimizing Paging Faults
• On a fault evict a page from cache
• Paging algorithm ≡ Eviction policy
• Goal: minimize the number of page faults
Worst case
• In the worst case page, the number of page faults on n requests is n.
E.g. cache of size 4, request sequence
p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12
Difficult sequences
Cache of size 4, request sequence p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12
Sequence is difficult, for one it never repeats pages so it is impossible to have a page hit
Compare to optimal
p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13
… is hard for everyone (i.e. 13 faults)
p1 p2 p3 p4 p5 p1 p2 p3 p4 p5 p1 p2 p3 p4 … 8 faults
Optimal algorithm knows the future
Offline optimum
Optimal algorithm knows the future, i.e. offline OPT.
Compare online paging strategy to offline paging strategy
Defined as :
cost of online algorithm on I cost of offline optimum on I
Competitive Ratio
supI
Paging• One of the earliest problems to be
studied under the online model• Competitive ratio defined by Sleator
and Tarjan
Competitive Ratio
• First appeared in the 1960’s in search context, e.g. – On the linear search problem (Beck, 1964)– More on the linear search problem (1965)– Yet more on the linear search problem (1970)– The return of the linear search problem (1972)– Son of the linear search problem (Beck & Beck 1984)– The linear search problem rides again (1986)– The revenge of the linear search problem (1992)
• Shmuel Gal – Search Games, Academic Press, 1980– The Theory of Search Games and Rendezvous (with Steve
Alpern), Kluwer Academic Press, 2002
Paging Algorithms • Least-Recently-Used (LRU)• First-In-First-Out (FIFO)• Flush-When-Full (FWF)
Classes of Paging algorithms• Lazy Algorithms: LRU and FIFO• Marking Algorithms: LRU and FWF
Paging
• Competitive analysis not always satisfactory, e.g. LRU “as good as” FWF
• Real life inputs well understood and characterized (temporal + spatial locality)
Goal: derive from first principles a new measure that better reflects observations in practice
Theory
1. Commonly studied under competitive ratio framework
2. Worst case analysis3. Marking algorithms
optimal4. In practice LRU is best5. LFD is offline optimal6. Competitive ratio is k 7. User is malicious
adversary8. No benefit from
lookahead
Systems
1. Commonly studied using fault rate measure
2. Typical case analysis3. LRU and friends is best4. LRU is impractical5. Huh?6. Competitive ratio is 2
or 37. User is your friend8. Lookahead helps
Fix the Theory-Practice disconnect
1. Make both columns matchHow?
• Fix reality
or
• Fix the model
A more realistic theoretical model is likely to lead to practical insights
Previous work1. Disconnect has been noted before.
2. Intense study of alternatives, viz.
1. Borodin and Ben David 2. Karlin et al. 3. Koutsoupias and Papadimitriou4. Denning5. Young 6. Albers et al.7. Boyar et al.8. Sleator and Tarjan + many others
Theory: competitive ratio of paging algorithms
• k for LRU, FIFO, and FWF• Thus LRU, FIFO, and FWF equally good• Lookahead does not helpPractice:
• LRU never encounters sequences with competitive ratio bigger than 4
• LRU better than FIFO and much better than FWF
• Lookahead helps
Previous work1. Partial separation :
• Access graph model. Borodin, Raghavan, Irani, Schieber [STOC ‘91]
• Diffuse adversary. Koutsopias and Papadimitrou [FOCS ‘94]
• Diffuse adversary. Young [SODA ‘98]• Accommodating ratio. Boyar, Larsen,
Nielsen [WADS ‘99] • Relative worst order ratio. Boyar,
Favrholdt, and Larsen [SODA ‘05]
Previous work• Concave Analysis. Albers, Favrholdt,
Gielet [STOC ‘02]
LRU ≠ certain marking algorithms
• Adequate Analysis. Pangiotou, Souza [STOC 06]
+ many others. See survey L-O, Dorrigiv [SIGACT News ’05]
None provide full separation between LRU and FIFO and/or FWF
Online motion planning
1. Commonly studied under competitive ratio framework
2. Worst case analysis3. Continuous curved motions4. Perfect scans 5. Flawless detection6. No error in motion7. Architects are your enemy
Robotics
1. Commonly studied using distance & scan cost
2. Typical case analysis3. Piecewise linear 4. Scanning error5. High detection error 6. Forward & rotational lag 7. Architects are your friend
“Architects are your friend”
Most of the time, anyhow.
Alternatives• Turn ratio• Performance Ratio• Search Ratio• Home Ratio• Travel Cost• Average Ratio• Effective Ratio• Smooth Analysis• Concave Analysis• Bijective Analysis• Others
Defined as :
maxp { length of search from s to p }
maxq { length of path from s to q }
Performance Ratio
• general idea: focus on distant targets searches, those should be efficient
• allows high inefficiency for target’s near-by
Defined as :
length of search from s to target p shortest off-line search from s to p
Search Ratio
supp
• finer, fairer measure than competitive ratio
• many on-line “unsearchable” scenarios suddenly practicable, e.g. trees
Example:
Searching for a node at depth h in a complete binary tree with unit length edges
Competitive ratio = O(2h / h)
Search ratio = O(1)
Search Ratio
Defined as :
length of search from s to target p shortest path from s to p
Home Ratio
supp
where the position of p is known
• e.g. target has radioed for help, position known, shape of search area unknown
• Surprisingly, in some cases is as large as competitive ratio, e.g. street polygons
• [L-O, Schuierer, 1996]
Home Ratio
Defined as :
sup { length of search from s to target p }
Travel Cost
p
• unnormalized
• similar to standard algorithmic analysis
Defined as :
length of search from s to target p shortest path from s to p
Average Ratio
avg supP p
• Searching on the real line is 4.59 competitive on the average
• Doubling is 5.27 competitive
Defined as :
Ψ ratio + F(cost of computing solution)
where Ψ є { Competitive, Performance, Turn, Search, Home, Average, etc.}
• Function F reflects the difference in computational and navigational costs
• Sensible adjustments n, n2, i.e.
F(solt’n) = time / n2
Effective Ratio
• Robustness • navigational errors • detection errors (revisit area)
• Known probability densities of target location
• Time sensitive considerations
Other considerations
• Rescue in high seas (man overboard)• High detection error• Known probability density from ocean currents data• Time sensitive (person must be found within 4 hours in North Atlantic)
Example
• Current coast guard strategy• scan high probability area • when done . . . scan again• if several vessels available, rescan
Search and Rescue (SAR)
Bijective Analysis
Σn = {σ1,σ2,…,σ10}:
the set of all possible input sequences of length n
B(σ)
6743
107668
10
A(σ)
5739753757
cost
Bijective Analysis
A ≤ B
B ≤ A
A < B
B(σ)
6743
107668
10
A(σ)
5739753757
cost
Bijective Analysis
Competitive ratio of A: 9
Competitive ratio of B: 4
B(σ)
6743
107668
10
A(σ)
5739753757
OPT(σ)
3241342523
Strict separation is complex
Theorem If the performance measure does not distinguish between sequences of the same length, then no such separation exists
Σ*Σ1Σ2
Σ3
Input sequences
Proof
We prove strong equivalence of all lazy marking algorithms.
I.e. given two marking algorithms A1 and A2, there is a one-to-one correspondence b() between inputs such that the performance characteristics of A1(I) are identical to A2(b(I)).
Proof
A
B
σ1 σ3σ2 σ1 σ4
σ1 σ3σ2 σ1 σ4
map any σ2 in A’s sequence to σ3 in B’s sequence
σ4
σ4 σ3
σ2
Partitioning Input Space
• We need a natural model to partition space. How?
Σ*Σ1Σ2
Σ3
Input sequences
Not all inputs are equal
• Adversarial model is wrong model for paging.
• The user is not out to get you (contrast this with crypto case).
• Compilers go to great lengths to preserve locality of reference.
• Designers go to great lengths to create cache friendly (conscious) algorithms.
Updated model
Competitive ratio
Friendly models:
• Fault model, Torng [FOCS ’95]
• Concave analysis, Albers et al. [STOC ’02]
ALG(I)
nice(I)
Cooperative
Cooperative ratio
• Agreement between user and algorithm about inputs which are:
– likely– common– good– important
Cooperative ratio
• Badly written code (not cache conscious)– (Rightly) considered the programmer’s fault – Paging strategy not responsible for
predicting non-standard paging behaviour
• Well written code (cache conscious)– Code can rely on well defined paging
behaviour to produce better code (e.g. I/O model, cache oblivious model)
Friendly models
• Torng fault model doesn’t fully separate • Albers et al. concave analysis doesn’t
either
• Bijective + concave analysis separates
Concave analysisf( ): an increasing concave function
Definition A request sequence is consistent with f if the number of distinct pages in any window of size n is at most f(n)
Intuition Not too many consecutive newly requested pages, i.e. locality of reference
Proposed by Albers et al. [AFG05], based on Denning’s working set model
Subpartitioning of Input Space
• Subsets compatible with f( )
Σ*
Input sequences
Σf3
Σf4 Σf
5
Bijective analysis
Theorem. LRU ≤f,b A
for all paging algorithms A
Proof. Construct a continuation of b() such that b(σ∙r)= b(σ)∙r’ and LRU(σ∙r) ≤ A(b(σ∙r))
Corollary. avg(LRU) ≤ avg(A)
Strict separation
Theorem For any given paging algorithm A there exists f such that
A >f,avg LRU
i.e. LRU is unique best [SODA ’07]
Strict separation
Theorem For any algorithm A there exists f such that
A >f,avg LRU
Proof. We want to show avg(A) > avg(LRU).
I.e. Σσ A(σ) > Σσ LRU(σ)
Use double counting technique: Compute the number of faults at time i across all sequences, then add for i = 1..n.
Strict separation
Σσ A(σ) > Σσ LRU(σ) [sum by rows]
Compute the number of faults at time i across all sequences, then add for i = 1..n. [sum by columns]
This shows A(i, σ) ≥ LRU(i, σ). Finally we exhibit one sequence σ for which A(i, σ) > LRU(i, σ).
Other results
• Lookahead: the next L items in the list are known in advance
• It does not improve competitive ratio of any of the standard strategies
Theorem LRU with lookahead L is strictly better than LRU with lookahead L’ for all L > L’
Applications to other problems
• Similar open problem for List Update: “An important open question is whether
there exist alternative ways to define competitiveness such that MTF and other good online algorithms for the list update problem would be competitive“ [Martinez and Roura].
• Bijective analysis separates MTF from rest
Applications to other problems
• Leads to better compression results for Burrows-Wheeler based compression
• BWT reorders text in a way that increases repetitions in text
• Reordering is compressed using MTF
Applications to other problems
Observation: BWT permutations have high locality of reference
• Use list update algorithms which are designed under locality of reference assumptions, instead of adversarial worst case inputs
• Leads to better compression
Cooperative ratio for motion planning
• Robot must search efficiently scenes which are “reasonable”
• Can perform somewhat worse in “unreasonable” scenes
• Leads to adaptive-style analysis. E.g. define niceness measure of office floor plan in terms of orthogonality of scene, number of rooms/corridors, size of smallest relevant feature, etc.
Other results
• It leads to deterministic algorithms that outperform randomized ones in the classical model.
single bad case
simple cases
Under competitive ratio algorithm must tend to single bad case, even if at the expense of the simple case
Randomized algorithm can toss a coin and sometimes do one, sometimes do the other
Bijective analysis naturally prefers the majority of good cases
Conclusions• Introduced refined measurement
technique for online analysis• Resolved long standing open problem:
LRU is unique optimum• Bridged theory-practice disconnect• Result applicable to
– List update (MTF is unique optimum)– BWT compression– Motion planning
• Leads to new cooperative analysis model