an optimal solution to the problem

http://www.personal.kent.edu/~rmuhamma/Algorithms/MyAlgorithms/Greedy/greedyIntro.htm

Greedy Introduction

Greedy algorithms are simple and straightforward. They are shortsighted in their approach in the sense that they take decisions on the basis of information at hand without worrying about the effect these decisions may have in the future. They are easy to invent, easy to implement and most of the time quite efficient. Many problems cannot be solved correctly by greedy approach. Greedy algorithms are used to solve optimization problems

Greedy Approach

Greedy Algorithm works by making the decision that seems most promising at any moment; it never reconsiders this decision, whatever situation may arise later.

As an example consider the problem of "Making Change".

Coins available are:

• dollars (100 cents)
• quarters (25 cents)
• dimes (10 cents)
• nickels (5 cents)
• pennies (1 cent)

Problem    Make a change of a given amount using the smallest possible number of coins.

Informal Algorithm

• Start with nothing.
• at every stage without passing the given amount.
  • add the largest to the coins already chosen.

Formal Algorithm

Make change for n units using the least possible number of coins.

MAKE-CHANGE (n)

C ← {100, 25,
10, 5, 1}     // constant.

Sol ←
{};            
            // set that
will hold the solution set.

Sum ← 0 sum of
item in solution set

WHILE sum not = n

x =
largest item in set C such that sum + x ≤ n

IF
no such item THEN

RETURN    "No Solution"

S ← S {value of
x}

sum ← sum + x

RETURN S

Example
    Make a change for 2.89 (289 cents) here n =
2.89 and the solution contains 2 dollars, 3 quarters, 1 dime and 4
pennies. The algorithm is greedy because at every stage it chooses the
largest coin without worrying about the consequences. Moreover, it
never changes its mind in the sense that once a coin has been included
in the solution set, it remains there.

Characteristics
and Features of Problems solved by Greedy Algorithms

To construct the solution in an optimal way. Algorithm maintains
two sets. One contains chosen items and the other contains rejected
items.

The greedy algorithm consists of four (4) function.

1. A function that checks whether chosen set of items provide a solution.
2. A function that checks the feasibility of a set.
3. The selection function tells which of the candidates is the most promising.
4. An objective
   function, which does not appear explicitly, gives the value of a solution.

Structure
Greedy Algorithm

• Initially the set of chosen items is empty i.e.,
  solution set.
• At each step
  • item will be added in a solution set by using
    selection function.
  • IF the set would no longer be feasible
    • reject items under consideration (and is
      never consider again).
  • ELSE IF set is still feasible THEN
    • add the current item.

Definitions of
feasibility

A feasible set (of
candidates) is promising if it can be extended to produce not merely a
solution, but an optimal solution to the problem. In particular, the
empty set is always promising why? (because an optimal solution always
exists)

Unlike Dynamic Programming, which solves the
subproblems bottom-up, a greedy strategy usually progresses in a
top-down fashion, making one greedy choice after another, reducing each
problem to a smaller one.

Greedy-Choice
Property

The "greedy-choice property" and "optimal
substructure" are two ingredients in the problem that lend to a greedy
strategy.

Greedy-Choice
Property

It says that a globally optimal solution can be
arrived at by making a locally optimal choice.