Online Algorithms

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Online Algorithms

• Morteza Alamgir
• Advance Algorithms

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Motivation

• till now, our algorithms start with input, work
  with it
• suppose input arrives a little at a time, need
  instant response
• eg stock market, paging
• question what is a good algorithm?
• problem isnt to decide fast, rather what to
  decide.

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Applications

• Resource Allocation
• Scheduling
• Memory Management
• Routing
• Robot Motion Planning
• Exploring an unknown terrain
• Finding a destination
• Computational Finance

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Methods of Analysis

• Competitive Analysis (Worst Case)
• For any input, the cost of our online algorithm
  is never worse than c times the cost of the
  optimal offline algorithm.
• Pros can make very robust statements about the
  performance of a strategy.
• Cons results tend to be pessimistic.

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Methods of Analysis

• Probabilistic Analysis
• Assume a distribution generating the input.
• Find an algorithm which minimizes the expected
  cost of the algorithm.
• Pros can incorporate information predicting the
  future.
• Cons can be difficult to determine probability
  distributions accurately.

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Competitive Analysis

• if knew whole input in advance, easy to
  optimize
• compare to full knowledge optimum
• k-competitive if for all sequences
• sometimes, to ignore edge effects

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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.
• 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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Online Scheduling problem

• We are given a set of m machines. A sequence of
  jobs arrives online. Each job has a processing
  time that may or may not be known in advance.
  Each job must be scheduled immediately on one of
  m machines without knowledge of any future jobs.
• The goal is to optimize a given objective
  function
• The must basic scheduling problem was introduced
  in 1966
• m identical machine
• The processing time of a new job is known
• The goal is to minimize the completion time of
  the last job
• Graham showed that the List Scheduling Algorithm
  is (2-1/m) competitive

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Online Scheduling problem

• In 1992 bartal gave an 1.986-competitive
• Karger in 1996 generalized the algorithm and
  proved an upper bound of 1.945
• The best algorithm known so far achieved a 1.923
  competitive ration
• 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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Online load balancing

• We have again a set of m machines and a sequence
  of jobs
• Each job has a weight and a duration that may be
  or may not be known in advance
• The load of a machine is some of weights of jobs
  present on the machine at that time
• The goal is to minimize the maximum load that
  occur during the processing of the job sequence
• For settings on m identical machine Azar and
  Epstein showed that greedy algorithm is (2-1/m)
  competitive
• Load balancing become more complicated with
  restricted assignments when each jab can only be
  assigned to a subset of admissible machines

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Network routing problems

• These problems were studied starting from the
  virtual circuit routing problem
• A communication network with every link has a
  given max capacity
• The input instance is formed by a seq. of
  communication request.
• For each request we must establish a virtual
  circuit on a path connecting two nodes of the
  network at a given bandwidth for a given duration
• Aspnes gave an O(log n)-competitive algorithm for
  the problem of minimizing the maximum load on a
  link of a network of n nodes when connection
  requests have unlimited duration
• The v.c.r problem has also been studied in its
  throughput version, the so called call control
  problem

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Randomized Online Algorithms

• An online algorithm is a two player zero sum game
  between algorithm and adversary. Well known that
  optimal strategies require randomization
• A randomized online algorithm is a probability
  distribution over deterministic online algorithms
• Algorithm is k-competitive if for any

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Restricting an adversary

• Competitive analysis is a strong worst-case
  performance measure
• For some problems, such as paging, the
  competitive ratios of online algorithms are much
  higher than practice.
• A line of research is concerned with evaluating
  online algorithms on restricted classes of
  request sequences.
• In other words, the power of an adversary is
  limited
• Statistical adversary The input generated by the
  adversary must satisfy certain statistical
  assumptions
• Diffuse adversary model an adversary must
  generate an input according to a probability
  distribution D that belongs to a class of
  possible distributions known to the online
  algorithm.

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Online Algorithms in machine learning

• The areas of Online Algorithms and Machine
  Learning are both concerned with problems of
  making decisions about the present based only on
  knowledge of the past.

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Predicting from Expert Advice

• Goal predicting whether or not it will rain that
  day
• Input the advice of n experts Each day, each
  expert predicts yes or no
• the learning algorithm must use this information
  in order to make its own prediction
• Suppose we make no assumptions about the quality
  or independence of the experts

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Online Learning from examples

• There is an example space typically
• Learning proceeds as a sequence of trials.
• In each trial, an example is presented
  to the learning algorithm. The algorithm then
  predicts either 1 or 0
• finally the algorithm is told the true label
• The algorithm is penalized for each mistake made
• Our goal is to make as few mistakes as possible.

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Streaming Algorithms

• In solving the problem, we only have a limited
  amount of storage
• Streaming Algorithms for Line Simplification
• Line Simplification Problem we are getting a
  possibly infinite sequence of points p0, p1, p2,
  . . . defining a polygonal path, and as we
  receive the points we wish to maintain a
  simplification of the path seen so far.

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• Any Question?