Greedy algorithm

Specifics

In general, greedy algorithms have five components:

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 will 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]

Cases of failure

Examples on how a greedy algorithm may fail to achieve the optimal solution.

Starting from A, a greedy algorithm that tries to find the maximum by following the greatest slope will find the local maximum at "m", oblivious to the global maximum at "M".

With a goal of reaching the largest sum, at each step, the greedy algorithm will choose what appears to be the optimal immediate choice, so it will choose 12 instead of 3 at the second step, and will not reach the best solution, which contains 99.

For many other problems, greedy algorithms fail to produce the optimal solution, and may even produce the unique worst possible solution. One example is the traveling salesman problem 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] For other possible examples, see horizon effect.

Types

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Greedy algorithms can be characterized as being 'short sighted', and also as 'non-recoverable'. They are ideal only for problems which have 'optimal substructure'. Despite this, for many simple problems, the best-suited algorithms are greedy algorithms. 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

Theory

Greedy algorithms have a long history of study in combinatorial optimization and theoretical computer science. Greedy heuristics are known to produce suboptimal results on many problems,^[4] and so natural questions are:

• For which problems do greedy algorithms perform optimally?
• For which problems do greedy algorithms guarantee an approximately optimal solution?
• For which problems is the greedy algorithm guaranteed not to produce an optimal solution?

A large body of literature exists answering these questions for general classes of problems, such as matroids, as well as for specific problems, such as set cover.

Matroids

A matroid is a mathematical structure that generalizes the notion of linear independence from vector spaces to arbitrary sets. If an optimization problem has the structure of a matroid, then the appropriate greedy algorithm will solve it optimally.^[5]

Submodular functions

A function ${\displaystyle f}$ defined on subsets of a set ${\displaystyle \Omega }$ is called submodular if for every ${\displaystyle S,T\subseteq \Omega }$ we have that ${\displaystyle f(S)+f(T)\geq f(S\cup T)+f(S\cap T)}$.

Suppose one wants to find a set ${\displaystyle S}$ which maximizes ${\displaystyle f}$. The greedy algorithm, which builds up a set ${\displaystyle S}$ by incrementally adding the element which increases ${\displaystyle f}$ the most at each step, produces as output a set that is at least ${\displaystyle (1-1/e)\max _{X\subseteq \Omega }f(X)}$.^[6] That is, greedy performs within a constant factor of ${\displaystyle (1-1/e)\approx 0.63}$ as good as the optimal solution.

Similar guarantees are provable when additional constraints, such as cardinality constraints,^[7] are imposed on the output, though often slight variations on the greedy algorithm are required. See ^[8] for an overview.

Other problems with guarantees

Other problems for which the greedy algorithm gives a strong guarantee, but not an optimal solution, include

• Set cover
• The Steiner tree problem
• Load balancing^[9]
• Independent set

Many of these problems have matching lower bounds; i.e., the greedy algorithm does not perform better, in the worst case, than the guarantee.

Applications

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Greedy algorithms typically (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 coloring 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 optimization methods like dynamic programming. Examples of such greedy algorithms are Kruskal's algorithm and Prim's algorithm for finding minimum spanning trees, and the algorithm for finding optimum Huffman trees.

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.

Examples

• The activity selection problem is characteristic to this class of problems, where the goal is to pick the maximum number of activities that do not clash with each other.
• 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. The artificial intelligence does not account for obstacles, so the demo mode often ends quickly.
• The matching pursuit is an example of greedy algorithm applied on signal approximation.
• A greedy algorithm finds the optimal solution to Malfatti's problem of finding three disjoint circles within a given triangle that maximize the total area of the circles; it is conjectured that the same greedy algorithm is optimal for any number of circles.
• A greedy algorithm is used to construct a Huffman tree during Huffman coding where it finds an optimal solution.
• In decision tree learning, greedy algorithms are commonly used, however they are not guaranteed to find the optimal solution.
  • One popular such algorithm is the ID3 algorithm for decision tree construction.
• Dijkstra's algorithm and the related A* search algorithm are verifiably optimal greedy algorithms for graph search and shortest path finding.
  • A* search is conditionally optimal, requiring an "admissible heuristic" that will not overestimate path costs.
• Kruskal's algorithm and Prim's algorithm are greedy algorithms for constructing minimum spanning trees of a given connected graph. They always find an optimal solution, which may not be unique in general.

See also

References

Sources

External links

• "Greedy algorithm", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Gift, Noah. "Python greedy coin example".