The cooperative ratio of on-line algorithms

Abstract

On-line algorithms are usually analyzed using competitive analy-sis, in which the performance of an on-line algorithm on a sequence is normalized by the performance of the optimal off-line algorithm on that sequence. In this paper we introduce cooperative analysis as an alternative general framework for the analysis of on-line algorithms. The idea is to normalize the performance of an on-line algorithm by a measure other than the performance of the off-line optimal algo-rithm OPT. We show that in many instances the perform of OPT on a sequence is a coarse approximation of the difficulty or complexity of a given input. Using a finer, more natural measure we can separate paging and list update algorithms which were otherwise indistinguish-able under the classical model. This creates a performance hierarchy of algorithms which better reflects the intuitive relative strengths be-tween them. Lastly, we show that, surprisingly, certain randomized algorithms which are superior to MTF in the classical model are not so in the cooperative case, which matches experimental results. This confirms that the ability of the on-line cooperative algorithm to ignore pathological worst cases can lead to algorithms that are more efficient in practice.

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The Cooperative Ratio of On-line Algorithms
Reza Dorrigiv∗
Alejandro L´ opez-Ortiz∗
Technical Report CS-2007-39
Cheriton School of Computer Science
University of Waterloo
October 2007
Abstract
On-line algorithms are usually analyzed using competitive analy-
sis, in which the performance of an on-line algorithm on a sequence
is normalized by the performance of the optimal off-line algorithm on
that sequence. In this paper we introduce cooperative analysis as an
alternative general framework for the analysis of on-line algorithms.
The idea is to normalize the performance of an on-line algorithm by
a measure other than the performance of the off-line optimal algo-
rithm OPT. We show that in many instances the perform of OPT on
a sequence is a coarse approximation of the difficulty or complexity of
a given input. Using a finer, more natural measure we can separate
paging and list update algorithms which were otherwise indistinguish-
able under the classical model. This creates a performance hierarchy
of algorithms which better reflects the intuitive relative strengths be-
tween them. Lastly, we show that, surprisingly, certain randomized
algorithms which are superior to MTF in the classical model are not
so in the cooperative case, which matches experimental results. This
confirms that the ability of the on-line cooperative algorithm to ignore
pathological worst cases can lead to algorithms that are more efficient
in practice.
∗Cheriton School of Computer Science, University of Waterloo, Waterloo, Ont., N2L
3G1, Canada. {rdorrigiv, alopez-o}@uwaterloo.ca.
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1Introduction
There has been extensive research on the analysis of on-line algorithms. The
competitive ratio, first introduced formally by Sleator and Tarjan [ST85],
has served as a practical measure for the study and classification of on-
line algorithms. An algorithm (assuming a min problem) is said to be α-
competitive if the cost of serving any specific request sequence never exceeds
α times the optimal cost of an optimal off-line algorithm which knows the
entire sequence. The competitive ratio is a relatively simple measure to
apply yet powerful enough to quantify, to a large extent, the performance
of many an on-line algorithm. Notwithstanding the wide applicability of
competitive analysis, it has been observed by numerous researchers [BDB94,
BIRS95b, KP00, You94, BF03, PS06] that in certain settings the competitive
ratio produces results that are too pessimistic or otherwise found wanting.
A well known example for the shortcomings of competitive analysis is
the paging problem. A paging algorithm mediates between a slower and
a faster memory. Assuming a cache of size k, it decides which k memory
pages to keep in the cache without the benefit of knowing in advance the
sequence of upcoming page requests. After receiving the ithpage request the
on-line algorithm must decide which page to evict, in the event the request
results in a fault and the cache is full. The objective is to design efficient
on-line algorithms in the sense that on a given request sequence the total
cost, namely the total number of faults, is kept low. Three well known pag-
ing algorithms are Least-Recently-Used (LRU), First-In-First-Out (FIFO),
and Flush-When-Full (FWF). On a fault, if the cache is full, LRU evicts
the page that is least recently requested, FIFO evicts the page that is first
brought to the cache, and FWF empties the cache. All these paging algo-
rithms have competitive ratio k, which is the best among all deterministic
on-line paging algorithms [BEY98]. On the other hand, experimental stud-
ies show that LRU has a performance ratio at most four times the optimal
off-line [You94, SA88], as opposed to the competitive ratio k. Furthermore,
it has been empirically well established that LRU (and/or variants thereof)
are, in practice, preferable paging strategies to all other known paging al-
gorithms [SGG02]. This is in contrast to competitive analysis in which the
competitive ratio of LRU is the same as FWF and worse than some ran-
domized algorithms.
A careful study of the competitive ratio reveals the nature of the short-
comings. Chiefly among them are its focus on worst case behaviour and
indirect comparison of online algorithms via an off-line optimal algorithm.
In the former case, this might lead, as observed above, to the competitive
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ratio declaring two wildly differing algorithms “equal” if they happen to err
in the same way in the worst possible input, even though in most other
inputs one is superior to the other (e.g. LRU versus FWF). In the latter
case, the indirect comparison to an off-line optimal can introduce spurious
artifacts due to the comparison of two objects of different types, namely an
online and an off-line algorithm.1
Such anomalies have lead to the introduction of many alternatives to
competitive analysis of on-line algorithms (see [DLO05] for a comprehensive
survey). Here we briefly describe some of these alternatives. Loose compet-
itiveness, which was first proposed by Young in [You94] and later refined
in [You02], considers an off-line adversary that is oblivious to the parameter
k (the cache size). The adversary must produce a sequence that is bad for
most values of k rather than for just a specific value. It also ignores the
sequences on which the on-line algorithm incurs a cost less than a certain
threshold. This results in a weaker adversary and hence in paging algorithms
with constant performance ratios. The diffuse adversary model by Koutsou-
pias and Papadimitriou [KP00] as well as Young [You98, You00] refines the
competitive ratio by restricting the set of legal request sequences to those
derived from a class (family) of probability distributions. This restriction fol-
lows from the observation that although a good performance measure could
in fact use the actual distribution over the request sequences, determining
the exact distribution of real-life phenomena is a difficult task (e.g. depend-
ing on the intended application different distributions might arise). The
Max/Max ratio, introduced by Borodin and Ben-David [BDB94] compares
on-line algorithms based on their amortized worst-case behaviour (here the
amortization arises by dividing the cost of the algorithm over the length of
the request sequence). The relative worst order ratio [BF03, BFL05, BM04]
combines some of the desirable properties of the Max/Max ratio and the ran-
dom order ratio (introduced in [Ken96] in the context of on-line bin packing).
As with the Max/Max ratio, it allows for direct comparison of two on-line
algorithms. Informally, for a given request sequence the measure consid-
ers the worst-case ordering (permutation) of the sequence, for each of the
two algorithms, and compares their behaviour on these orderings. It then
finds among all possible sequences the one that maximizes this worst-case
1To illustrate, consider a consumer wishing to purchase a mountain bike. There are
two choices which the user evaluates indirectly by comparing them to an “optimal” racing
bike. While in general good racing bikes and mountain bikes have common characteristics,
such a comparison would award no points for shock absorbers. Similarly, lightness, which
is essential in a racing bike is secondary to sturdiness in the case of the mountain bike,
and so on and so forth.
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performance.
There are several measures that model paging with locality of reference.
Borodin, Raghavan, Irani, and Schieber [BIRS95a] proposed the access graph
model in which the universe of possible request sequences is reduced to re-
flect that the actual sequences that can arise depend heavily on the structure
of the program being executed. The space of request sequences can then be
modeled by a graph in which paths between vertices correspond to actual
sequences. In a generalization of the access graph model, Karlin, Phillips,
and Raghavan [KPR00] proposed a model in which the request sequences are
distributed according to a Markov chain process. Becchetti [Bec04] refined
the diffuse adversary model of Koutsoupias and Papadimitriou by consider-
ing only probabilistic distributions in which temporal locality of reference
is present.Albers, Favrholdt, and Giel [AFG05] introduced a model in
which input sequences are classified according to a measure of locality of
reference. The measure is based on Denning’s working set [Den68] which
is supported by extensive experimental results. The technique they used,
which we term concave analysis, reflects the fact that efficient algorithms
must perform competitively in each class of inputs of similar locality of refer-
ence, as opposed to the worst case alone. Panagiotou and Souza proposed a
model that explains the good performance of LRU in practice [PS06]. They
classify input sequences according to some parameters and prove an upper
bound on the competitive ratio of LRU as a function of these parameters.
Then they argue that sequences in practice have parameters that lead to
constant competitive ratio for LRU. Unfortunately, their model is rather
complicated and hence unlikely to be extended to paging algorithms other
than LRU. Recently, Angelopoulos et al. introduced Bijective Analysis and
Average Analysis [ADLO07] which combined with the locality model of Al-
bers et al. [AFG05], shows that LRU is the sole optimal paging algorithm
on sequences with locality of reference. This resolved an important dispar-
ity between theory and practice of on-line paging algorithms, namely the
superiority in practice of LRU. A remaining question however, is how to
characterize the full spectrum of performance of the various known paging
algorithms. We address this question in this paper.
In general most the alternative models for the competitive ratio build
upon each other either directly by extending a certain approach or implicitly
by applying the lessons learned from other approaches which did not seem
to succeed. For example, there is now substantial evidence that locality of
reference must be built into the model and most if not all of the proposals
now routinely incorporate this. Similarly the approaches of Young; Albers
et al.; Panagiotou and Souza; and Angelopoulos et al. seem to be converg-
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ing towards a common idea. Each successive paper has generally provided
stronger and finer separation between the various competing algorithms.
This long scientific journey reflects both the difficulty of the challenge and
its centrality to both on-line algorithm analysis and systems research.
In this paper we introduce a new model, cooperative analysis, by apply-
ing adaptive analysis ideas to on-line algorithms. The model incorporates
locality of reference assumptions while mediating between the competitive
ratio which compares an on-line algorithm versus an off-line algorithm and
direct on-line versus on-line algorithm comparisons of the various models
described above. We show that this new model produces the finest sepa-
ration yet of list-update algorithms while also being applicable to paging
and other online algorithms. Paging and list update are the best testbeds
for developing alternative measures, given our extensive understanding of
these problems. We know why competitive analysis fails, what are typical
sequences in practice and we can better evaluate whether a new technique
indeed overcomes known shortcomings. It is important to note that even
though well studied, most models for this problems are only partially suc-
cessful in resolving the issues posed by them and as such they are still
challenging case studies. Note that cooperative ratio is readily generaliz-
able to other settings, since the difficulty of an instance can be defined for
any problem. As such it improves over other list-update and paging specific
models, while building upon the lessons learned. Indeed we have obtained
results for geometric searching using this model (to appear elsewhere).
In the list update or the list accessing problem, we have an unsorted list
of m items. The input is a sequence of n requests that should be served
in an on-line manner. Let A be an arbitrary on-line list update algorithm.
To serve a request to an item x, A should linearly search the list until it
finds x. If x is ithitem in the list, A incurs cost i to access x. Immediately
after accessing x, A can move x to any position closer to the front of the
list at no extra cost. This is called a free exchange. Also A can exchange
any two consecutive items at a cost of 1. These are called paid exchanges.
The idea is to use free and paid exchanges to minimize the overall cost
of serving a sequence. Three well known deterministic on-line algorithms
are Move-To-Front (MTF), Transpose, and Frequency-Count (FC). MTF
moves the requested item to the front of the list and Transpose exchanges
the requested item with the item that immediately precedes it. FC main-
tains a frequency count for each item, updates this count after each access,
and makes necessary moves so that the list always contains items in non-
increasing order of frequency count. Sleator and Tarjan showed that MTF
is 2-competitive, while Transpose and FC do not have constant competitive
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