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The matching pursuit is an example of a 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.
Once the greedy choice is made, the problem reduces to finding an optimal solution for the subproblem. If A is an optimal solution to the original problem S containing the greedy choice, then A ′ = A ∖ { 1 } {\displaystyle A^{\prime }=A\setminus \{1\}} is an optimal solution to the activity-selection problem S ′ = { i ∈ S : s i ≥ f 1 ...
Kruskal's algorithm [1] finds a minimum spanning forest of an undirected edge-weighted graph.If the graph is connected, it finds a minimum spanning tree.It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle. [2]
But in complex situations it can easily fail to find the optimal scheduling. HEFT is essentially a greedy algorithm and incapable of making short-term sacrifices for long term benefits. Some improved algorithms based on HEFT look ahead to better estimate the quality of a scheduling decision can be used to trade run-time for scheduling performance.
This result guarantees the optimality of many well-known algorithms. For example, a minimum spanning tree of a weighted graph may be obtained using Kruskal's algorithm, which is a greedy algorithm for the cycle matroid. Prim's algorithm can be explained by taking the line search greedoid instead.
Longest-processing-time-first (LPT) is a greedy algorithm for job scheduling. The input to the algorithm is a set of jobs, each of which has a specific processing-time. There is also a number m specifying the number of machines that can process the jobs. The LPT algorithm works as follows:
A later simplification by Garsia and Wachs, the Garsia–Wachs algorithm, performs the same comparisons in the same order. The algorithm works by using a greedy algorithm to build a tree that has the optimal height for each leaf, but is out of order, and then constructing another binary search tree with the same heights. [7]
The notion of matroid has been generalized to allow for other types of sets on which a greedy algorithm gives optimal solutions; see greedoid and matroid embedding for more information. Korte and Lovász would generalize these ideas to objects called greedoids , which allow even larger classes of problems to be solved by greedy algorithms.