Online Algorithms
Amrinder Arora
Permalink:
http://standardwisdom.com/softwarejournal/presentations/
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Ski Rental Problem
• Classic toy problem of rent/buy nature
• Rental cost = 1$, Buying cost = $k
• Algorithm
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Rent for k-1 days
Buy skis on day #k
If you stop skiing during the first (k-1) days, it costs the same
If you stop skiing after day #k, your cost is $(2k-1) which is (21/k) times more than best possible ($k)
• This is the best algorithm. (But we will look for a better
one shortly.)
http://en.wikipedia.org/wiki/Ski_rental_problem
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Agenda
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What are online algorithms? (And why?)
How to analyze them?
Some example problems and algorithms
Categories of online algorithms
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What does “Online” mean?
• “Online” means - Input arrives a little at a
time, need instant response
– E.g., stock market, paging
• Question: what is a “good” algorithm?
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Applications
• Variety of Applications
– Scheduling
– Memory Management
– Routing
• Sample Problems
– Ski Rental
– Job Scheduling
– Paging
– Ambulances
– Graph Problems
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Analyzing Online Algorithms
• Competitive Analysis
• Probabilistic Analysis
• Other/newer methods
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Competitive Analysis
For any input, the cost of our online
algorithm is never worse than c times the
cost of the optimal offline algorithm.
Very robust
Pessimistic
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Probabilistic Analysis
• Assume a distribution generating the input
• Find an algorithm which minimizes the
expected cost of the algorithm.
Can incorporate
information
predicting the future
Can be difficult to
determine
accurately
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Competitive Analysis

• Input sequence:
• Full knowledge optimum (also called
offline optimum): C ( )
• k-competitive if for all input sequences :
MIN
• Sometimes, to ignore boundary conditions
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1966 – Online Era Begins!
• We are given a set of m identical machines.
• A sequence of jobs arrives online.
• Each job must be scheduled immediately on one
of m machines without knowledge of any future
jobs.
• The goal is to minimize the completion time of
the last job
• The must basic scheduling problem was
introduced in 1966
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List Scheduling
• Schedule the job on the least loaded
machine.
• Graham showed that the List Scheduling
Algorithm is (2-1/m) competitive
• This analysis is tight.
• This is the optimal algorithm for m = 2 and
m = 3.
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Fancier Results
• In 1992 Bartal gave an 1.986-competitive
"New Algorithms for an Ancient Scheduling Problem"
• Karger in 1996 generalized the algorithm
and proved an upper bound of 1.945
"A Better Algorithm for an Ancient Scheduling Problem"
• The best algorithm known so far achieved
a 1.923 competitive ration
• Lower Bounds
– Susanne Albers proved a bound of 1.852
– Current best lower bound of 1.88
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Extending Research
• Many variants of basic problem is studied:
– Jobs may be preempted
– Jobs may be rejected at a penalty
– Online algorithms may use randomization
– In addition they are results for different
machine types
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Paging Problem
• Maintain a two level memory system, consisting
of a small fast memory and a large slow memory
• The goal is to serve a sequence of requests to
memory pages so as to minimize number of
page faults.
• Hit: Page found
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Minimizing Paging Faults
• On a fault evict a page from cache
• Paging algorithm ≡ Eviction policy
• Goal: minimize the number of page faults
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Paging Algorithms
• LRU, FIFO, LIFO, Least freq use
• LIFO, LFU not competititive
• LRU, FIFO k-competitive.
• Theorem: No deterministic online algorithm can
be better than k-competitive
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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
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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 p 2 p3 p 4 p5 p1 p2 p 3 p4 p5 p1 p 2 p3 p4 …
8 faults
Optimal algorithm knows the future
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k-Server problem
• Ambulance location problem
• Metrical Task Systems
• Somewhat, ice cream vending problem
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Online Graph Theory
• Online Minimum Spanning Tree
• Online Graph Coloring
• Online Shortest Paths
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Randomization
• If ski rental costs $1, and buying costs
$10, then with probability=0.5, buy after 8
days, and with probability=0.5, buy after
10 days.
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Randomization
3 kinds of adversaries
• Oblivious (weak): Does not know the
randomized results
• Adaptive Online (medium): Knows the
decision/random output of algorithm, but
needs to make their decision first.
• Adaptive Offline (strong): Knows
everything – randomization does not help.
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Summary
Basic Concepts
• Online Problem – Problem is revealed, one step at a
time.
• Online Algorithm – Needs to respond to the request
sequence that has been shown, not knowing next items
in the sequence
• Competitive Ratio – The worst ratio of the online
algorithm to the optimal offline algorithm, considering all
possible request sequences.
Importance and Research Topics
• Online algorithms show up in many practical problems.
• Even if you are considering an offline problem, consider
what would be the online version of that problem.
• Research areas including improving algorithms,
improving analysis of existing algorithms, proving
tightness of analysis, considering problem variations,
etc.
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Extra Material
• Adversary Approach to prove that 3n/2 – 2
comparisons are always needed to find
both the minimum and the maximum given
an unsorted array of n numbers:
• Notes at: http://is.gd/csp1ab
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