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Algorithms

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Title: Algorithms


1
Algorithms
  • L. Grewe

2
Algorithms and Programs
  • Algorithm a method or a process followed to
    solve a problem.
  • A recipe.
  • An algorithm takes the input to a problem
    (function) and transforms it to the output.
  • A mapping of input to output.
  • A problem can have many algorithms.

3
Algorithm Properties
  • An algorithm should (ideally) possess the
    following properties
  • It must be correct.
  • It must be composed of a series of concrete
    steps.
  • There can be no ambiguity as to which step will
    be performed next.
  • It must be composed of a finite number of steps.
  • It must terminate.
  • A computer program is an instance, or concrete
    representation, for an algorithm in some
    programming language.

4
Algorithm Efficiency
  • There are often many approaches (algorithms) to
    solve a problem. How do we choose between them?
  • At the heart of computer program design are two
    (sometimes conflicting) goals.
  • To design an algorithm that is easy to
    understand, code, debug.
  • To design an algorithm that makes efficient use
    of the computers resources.

5
Algorithms
  • Any problem can have a large number of algorithms
    that could be used to solve it.
  • but there are some algorithms that are
    effecitve in solving many problems.

6
Common Algorithms / Algorithm Methods
  • Greedy Algorithms
  • Divide and Conquer
  • Dynamic Programming
  • Backtracking
  • Branch and Bound

7
Other (common) Algorithms / Algorithm Methods
  • Linear Programming
  • Integer Programming
  • Neural Networks
  • Genetic Algorithms
  • Simulated Annealing
  • Typically these are covered in application
    specific courses that use them (e.g. Artificial
    Intelligence)

8
Algorithms specific to a data structure
  • Also algorithms can be specifically designed for
    common operations on a particular data structure.
  • Example - Graph Algorithms
  • Graph matching
  • Find shortest path(s)

9
Defining the Problem
  • Before even trying to design or reuse an existing
    algorithm, you must define your problem.
  • many problems can be defined as an optimization
    problem.

10
Optimization Problem
  • Problem Function to X Constraints
  • Function to X
  • This is where you describe the problem as a
    formula/function. Example, find the shortest
    path can be stated as Sum (distances) is
    minimum. Here the X is to minimize. The
    Function is the Sum(distances).
  • Sometimes these are called COST functions
  • Constraints
  • In our shortest path problem this might be to
    never visit the same node in the path twice.

11
One Solution to any Problem..The Brute Force
Algorithm
Try every solution!
  • Exponential Time, because exponentially many
  • This is WHY we discuss algorithms!!!!

12
Optimization Problem Elements
  • Instances The possible inputs to the problem.
  • Solutions for Instance Each instance has an
    exponentially large set of solutions.
  • Cost of Solution Each solution has an easy to
    compute cost or value.
  • Specification
  • ltpreCondgt The input is one instance.
  • ltpostCondgt An valid solution with optimal cost.
    (minimum or maximum)

13
Greedy Algorithms
  • Class of algorithms that solve Optimization
    Problems in a greedy way
  • Some greedy algorithms will generate an optimal
    solution, others only a good (enough) solution.
  • Greedy Method at each point in the algorithm a
    decision is make that is best at that point.
    Decisions made are not changed at a later point.
  • Greedy Criterion criterion used to make a
    greedy decision at each point.

14
Divide and Conquer
  • Problem Set of Several Independent (smaller)
    sub-problems.
  • Divide problem into several independent
    sub-problems and solve each sub-problem. Combine
    solutions to derive final solution.
  • Can work well on parallel computers.
  • Many times the sub-problems are the same problem
    and need to only develop one algorithm to solve
    them and then the algorithm to combine the
    results.

15
Divide and Conquer
  • Example
  • You are given a bag with 16 coins and told one is
    counterfeit and lighter than the others. Problem
    determine if bag contains the counterfeit coin.
    You have a machine that compares the weight of
    two sets of coins and tells you which is lighter
    or if they are the same.
  • Starting Thoughts
  • You could start with comparing coin 1 and 2. If
    one is lighter you are done. You can then
    compare coin 3 and 4 and so on. If you have N
    coins this can take N/2 times.
  • The Divide and Conquer Way
  • If you have N coins divide into two N/2 groups.
    Weigh them. If one is lighter then we are done
    and the bag does contain a counterfeit coin. If
    they are the same, there is no counterfeit coin.
    This takes ONLY 1 operation.
  • What happens if you want to find the counterfeit
    coin?

16
Dynamic Programming
  • dynamic programming is a method of solving
    complex problems by breaking them down into
    simpler steps.
  • Bottom-up dynamic programming simply means
    storing the results of certain calculations,
    which are then re-used later because the same
    calculation is a sub-problem in a larger
    calculation.
  • Top-down dynamic programming involves formulating
    a complex calculation as a recursive series of
    simpler calculations.

17
More on Dynamic Programming
  • Some programming languages can automatically
    memorize the result of a function call with a
    particular set of arguments
  • Some languages make it possible portably (e.g.
    Scheme, Common Lisp or Perl), some need special
    extensions (e.g. C, see 2). Some languages
    have automatic memoization built in. In any case,
    this is only possible for a referentially
    transparent function.

18
Backtracking
  • Backtracking is a general algorithm for finding
    all (or some) solutions to some computational
    problem, that incrementally builds candidates to
    the solutions, and abandons each partial
    candidate c ("backtracks") as soon as it
    determines that c cannot possibly be completed to
    a valid solution
  • problems which admit the concept of a "partial
    candidate solution
  • and a relatively quick test of whether it can
    possibly be completed to a valid solution.

19
Branch and Bound
  • Divides a problem to be solved into a number of
    subproblems, similar to the strategy
    backtracking.
  • Systematic enumeration of all candidate
    solutions, where large subsets of fruitless
    candidates are discarded en masse, by using upper
    and lower estimated bounds of the quantity being
    optimized
  • Efficiency of the method depends strongly on the
    node-splitting procedure and on the upper and
    lower bound estimators. All other things being
    equal, it is best to choose a splitting method
    that provides non-overlapping subsets.
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