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  Theory of Greedy algorithms, benefits of greedy algorithms and greedy problems

Greedy algorithms:

Greedy algorithms are the algorithms which pursue the problem solving meta-heuristic of forming the locally optimum selection at each phase with the hope of finding the global optimum. For illustration, applying the greedy strategy to traveling salesman problem outcomes the following algorithm: "At each phase visit the nearby unvisited city to the current city”.

The Greedy algorithms do not consistently find the globally optimal solution, since they generally do not operate exhaustively on all the data. They can make commitments to certain selections too early that prevent them from finding the best overall solution later. For illustration, all known greedy algorithms for the graph coloring trouble and all other NP-complete problems do not constantly find optimum solutions. Nonetheless, they are helpful since they are quick to think up and frequently give good approximations to the optimum.

When a greedy algorithm can be proven to outcome the global optimum for a given problem class, it usually becomes the method of selection. Illustrations of such greedy algorithms are Kruskal's and Prim's algorithm for finding minimum spanning trees and the algorithm for finding the optimum Huffman trees. The theory of matroids, and also the even more general theory of greedoids, provides whole classes of these algorithms.

•    In common, greedy algorithms encompass five pillars:
•    A candidate set, from which an answer is formed. 
•    A selection function, that selects the best candidate to be added to solution. 
•    A feasibility function which is used to find out if a candidate can be employed to contribute to a solution 
•    An objective function, that assigns a value to solution, or a partial solution, and 
•    A solution function that will point out whenever we have discovered a complete solution.

The fundamental idea behind greedy algorithms is to make big solutions up from smaller ones. Dissimilar other approaches, though, greedy algorithms keep only the best solution they find as they go all along. Therefore, for the sample problem, to form the answer for N = 5, they find the best solution for N = 4, and then modify it to get a solution for N = 5. No other solution for N = 4 is ever thought.

Greedy algorithms are fast, usually linear to quadratic and need little extra memory. Unluckily, they generally are not correct. However whenever they do work, they are frequently easy to implement and fast adequate to execute. 

Problems with Greedy algorithms:

There are two fundamental problems to greedy algorithms.

A) How to Build:

How does one make bigger solutions from the smaller ones? In common, this is a function of the problem. For sample problem, the most apparent manner to go from four to five boards is to pick a board and eliminate a section, therefore creating two boards from one. You must select to eliminate the biggest section from any board which covers only stalls that do not require covering (so as to reduce the total number of stalls covered).

To eliminate a section of covered stalls, take the board that spans such stalls, and make into two boards: one of which covers the stalls prior to the section, and one of which covers the stalls subsequent to the second.

B) Does it work?

The actual challenge for the programmer lies in the fact that, greedy solutions do not always work. Even when they seem to work for the sample input, random input, and all the situations you can think of, if there is a case where it won't work, at least one (if not more) of the judge’s test cases will be of that form.

For sample problem, to see that the greedy algorithm explained the above works, let’s consider the following:

Suppose that the answer does not include the large gap which the algorithm eliminated, however does have a gap that is smaller. By combining the two boards at the end of smaller gap and splitting the board across the bigger gap, an answer is received which employs as many boards as the original solution however which covers fewer stalls. This new answer is better, therefore the supposition is wrong and we must always select to eliminate the biggest gap.

If the answer does not have this particular gap however does contain another gap that is just as large, doing similar transformation outcomes an answer that employs as many boards and covers as many stalls as other answer. This new outcome is just as good as the original solution however no better, therefore we might select either.

Therefore, there exists an optimal answer that consists of the big gap, therefore at each step; there is always an optimal answer that is a superset of the present state. Therefore, the final answer is optimal.

When a greedy solution exists, utilize it. They are simple to code, easy to debug, run speedily, and utilize little memory, fundamentally defining a good algorithm in contest terms. The only missing element from that list is accuracy. When the greedy algorithm finds the accurate answer, go for it; however do not get suckered into thinking the greedy solution will work for all troubles.

Sorting a three-valued series:

You are provided a three-valued (1, 2 or 3) sequence of length up to 1000. Determine a minimum set of exchanges to place the series in sorted order.

The series has three parts: the part that will be 1 whenever in sorted order, 2 whenever in sorted order, and 3 whenever in sorted order. The greedy algorithm swaps as many as probable of the 1's in 2 parts with 2's in the 1 part, as many as probable 1's in the 3 part with 3's in the 1 part, and 2's in 3 parts with 3's in the 2 part.

Once none of such type’s remains, the remaining elements out of place require being rotated one way or the other in the sets of 3. You can optimally sort such by swapping all the 1's into place and then all 2's into place.

Analysis: Apparently, a swap can put at most two elements in place, therefore all the swaps of the first kind are optimal. As well, it is apparent that they use various types of elements; therefore there is no ‘interference’ between such types. This signifies the order doesn’t matter.

Once such swaps have been executed, the best you can do is two swaps for each and every three elements not in the correct position, which is what the second part will attain (for illustration, all the 1's are put in position but no others; then all that remains are 2's in the 3's place and vice-versa, and that can be swapped). 

Topological Sort:

Given a collection of objects, all along with some ordering constraints, like ‘A must be before B,’ determine an order of the objects in such a way that all the ordering constraints hold.

Algorithm: Make a directed graph over the objects, where there is an arc from A to B if ‘A must be before B." Construct a pass via the objects in random order. Each time you find an object with in-degree of zero, greedily put it on the end of the present ordering, delete all of its out-arcs and recurse on its (former) children, executing the same check. When this algorithm gets through all the objects devoid of putting each and every object in the ordering, there is no ordering that satisfies the constraints.

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