Greedy algorithms.

A greedy algorithm is an algorithm that follows the problem solving heuristic of making the locally optimal choice at each stage, with the hope of finding a global optimum. In many problems, a greedy strategy does not in general produce an optimal solution, but nonetheless a greedy heuristic may yield locally optimal solutions that approximate a global optimal solution in a reasonable time.

For example, a greedy strategy for the traveling salesman problem (which is of a high computational complexity) is the following heuristic: "At each stage visit an unvisited city nearest to the current city". This heuristic need not find a best solution but terminates in a reasonable number of steps; finding an optimal solution typically requires unreasonably many steps. In mathematical optimization, greedy algorithms solve combinatorial problems having the properties of matroids.

In general, greedy algorithms have five components:

·         A candidate set, from which a solution is created

·         A selection function, which chooses the best candidate to be added to the solution

·         A feasibility function, that is used to determine if a candidate can be used to contribute to a solution

·         An objective function, which assigns a value to a solution, or a partial solution, and

·         A solution function, which will indicate when we have discovered a complete solution

Greedy algorithms produce good solutions on some mathematical problems, but not on others. Most problems for which they work, will have two properties:

Greedy choice property

We can make whatever choice seems best at the moment and then solve the sub problems that arise later. The choice made by a greedy algorithm may depend on choices made so far but not on future choices or all the solutions to the sub problem. It iteratively makes one greedy choice after another, reducing each given problem into a smaller one. In other words, a greedy algorithm never reconsiders its choices. This is the main difference from dynamic programming, which is exhaustive and is guaranteed to find the solution. After every stage, dynamic programming makes decisions based on all the decisions made in the previous stage, and may reconsider the previous stage's algorithmic path to solution.

Optimal substructure

    "A problem exhibits optimal substructure if an optimal solution to the problem contains optimal solutions to the sub-problems."[2]

Types

Greedy algorithms can be characterized as being 'short sighted', and as 'non-recoverable'. They are ideal only for problems which have 'optimal substructure'. Despite this, greedy algorithms are best suited for simple problems (e.g. giving change). It is important, however, to note that the greedy algorithm can be used as a selection algorithm to prioritize options within a search, or branch and bound algorithm. There are a few variations to the greedy algorithm:

    Pure greedy algorithms

    Orthogonal greedy algorithms

   Relaxed greedy algorithms

Applications

Greedy algorithms mostly (but not always) fail to find the globally optimal solution, because they usually do not operate exhaustively on all the data. They can make commitments to certain choices too early which prevent them from finding the best overall solution later.

If a greedy algorithm can be proven to yield the global optimum for a given problem class, it typically becomes the method of choice because it is faster than other optimization methods like dynamic programming.

Greedy algorithms appear in network routing as well. Using greedy routing, a message is forwarded to the neighboring node which is "closest" to the destination. The notion of a node's location (and hence "closeness") may be determined by its physical location, as in geographic routing used by ad hoc networks. Location may also be an entirely artificial construct as in small world routing and distributed hash table.

Example of failure.

The 0-1 Knapsack problem is posed as follows. A thief robbing a store finds n items ; the ith item is worth vi dollars and weights wi pounds, where vi and wi are integers. He wants to take as valuable a load as possible, but he can carry at most W pounds in his knapsack for some integer w.

According to Greedy strategy consider the problem instance ,as there are 3 items and the knapsack can hold 50 pounds.

Item 1 weight 10 pounds and is worth 60 dollars.

Item 2 weights 20 pounds and is worth 100 dollars.

Item 3 weight 30 pounds and is worth 120 dollars.

Thus , the value per pound of item 1 is 6 dollars per pound, which is greater than the value per pound of either item 2 (5 dollars per pound) or item 3 which is 4 dollars per pound.

Greedy strategy would take item 1 first, leaving behind item 2 and item 3.

However the optimal solution consider item 2 and item 3 , leaving behind item 1.

Hence greedy strategy fails.

Thank you Wiki and Algorithms book.