Ski Rental Problem
• Classic toy problem of rent/buy nature
• Rental cost = 1$, Buying cost = $k
– 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 $(2k-1) which is (2-
1/k) times more than best possible ($k)
• This is the best algorithm. (But we will look for a better
• What are online algorithms? (And why?)
• How to analyze them?
• Some example problems and algorithms
• Categories of online algorithms
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?
• Variety of Applications
– Memory Management
• Sample Problems
– Ski Rental
– Job Scheduling
– Graph Problems 6
Analyzing Online Algorithms
• Competitive Analysis
• Probabilistic Analysis
• Other/newer methods
For any input, the cost of our online
algorithm is never worse than c times the
cost of the optimal offline algorithm.
• Assume a distribution generating the input
• Find an algorithm which minimizes the
expected cost of the algorithm.
predicting the future
Can be difficult to
• Input sequence:
• Full knowledge optimum (also called
offline optimum): C ( )
• k-competitive if for all input sequences :
• Sometimes, to ignore boundary conditions
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
• The goal is to minimize the completion time of
the last job
• The must basic scheduling problem was
introduced in 1966
• Schedule the job on the least loaded
• 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.
• 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
• 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
• 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
• Hit: Page found
Minimizing Paging Faults
• On a fault evict a page from cache
• Paging algorithm ≡ Eviction policy
• Goal: minimize the number of page faults
• 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
• 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
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 p 3 p4 p5 p1 p 2 p3 p4 …
Optimal algorithm knows the future
• Ambulance location problem
• Metrical Task Systems
• Somewhat, ice cream vending problem
Online Graph Theory
• Online Minimum Spanning Tree
• Online Graph Coloring
• Online Shortest Paths
• 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
3 kinds of adversaries
• Oblivious (weak): Does not know the
• 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.
• Online Problem – Problem is revealed, one step at a
• 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,
• 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