Greedy algorithm

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The greedy algorithm determines the minimum number of US coins to give while making change. These are the steps a human would take to emulate a greedy algorithm. The coin of the highest value, less than the remaining change owed, is the local optimum. (Note that in general the change-making problem requires dynamic programming to find an optimal solution; US and other currencies are special cases where the greedy strategy works.)

A greedy algorithm is any algorithm that follows the problem solving metaheuristic of making the locally optimum choice at each stage^[1] with the hope of finding the global optimum.

For example, applying the greedy strategy to the traveling salesman problem yields the following algorithm: "At each stage visit the unvisited city nearest to the current city".

[edit] Specifics

In general, greedy algorithms have five pillars:

1. A candidate set, from which a solution is created
2. A selection function, which chooses the best candidate to be added to the solution
3. A feasibility function, that is used to determine if a candidate can be used to contribute to a solution
4. An objective function, which assigns a value to a solution, or a partial solution, and
5. 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 well have two properties:

Greedy choice property 

We can make whatever choice seems best at the moment and then solve the subproblems 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 subproblem. 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]

[edit] When greedy-type algorithms fail

For many other problems, greedy algorithms may produce the unique worst possible solutions. One example is the nearest neighbor algorithm mentioned above: for each number of cities there is an assignment of distances between the cities for which the nearest neighbor heuristic produces the unique worst possible tour. ^[3]

[edit] 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. For example, all known greedy algorithms for the graph coloring problem and all other NP-complete problems do not consistently find optimum solutions. Nevertheless, they are useful because they are quick to think up and often give good approximations to the optimum.

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 optimisation methods like dynamic programming. Examples of such greedy algorithms are Kruskal's algorithm and Prim's algorithm for finding minimum spanning trees, Dijkstra's algorithm for finding single-source shortest paths, and the algorithm for finding optimum Huffman trees.

The theory of matroids, and the more general theory of greedoids, provide whole classes of such algorithms.

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.

[edit] Examples

• In the Macintosh computer game Crystal Quest the objective is to collect crystals, in a fashion similar to the travelling salesman problem. The game has a demo mode, where the game uses a greedy algorithm to go to every crystal. Unfortunately, the artificial intelligence does not account for obstacles, so the demo mode often ends quickly.

[edit] Notes

1. ^ Paul E. Black, "greedy algorithm" in Dictionary of Algorithms and Data Structures [online], U.S. National Institute of Standards and Technology, February 2005, webpage: NIST-greedyalgo.
2. ^ Introduction to Algorithms (Cormen, Leiserson, Rivest, and Stein) 2001, Chapter 16 "Greedy Algorithms".
3. ^ (G. Gutin, A. Yeo and A. Zverovich, 2002)

[edit] References

• Introduction to Algorithms (Cormen, Leiserson, and Rivest) 1990, Chapter 17 "Greedy Algorithms" p. 329.
• Introduction to Algorithms (Cormen, Leiserson, Rivest, and Stein) 2001, Chapter 16 "Greedy Algorithms" .
• G. Gutin, A. Yeo and A. Zverovich, Traveling salesman should not be greedy: domination analysis of greedy-type heuristics for the TSP. Discrete Applied Mathematics 117 (2002), 81-86.
• J. Bang-Jensen, G. Gutin and A. Yeo, When the greedy algorithm fails. Discrete Optimization 1 (2004), 121-127.
• G. Bendall and F. Margot, Greedy Type Resistance of Combinatorial Problems, Discrete Optimization 3 (2006), 288-298.