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Dynamic programming and greedy algorithms - Coggle Diagram
Dynamic programming and greedy algorithms
Dynamic Programming:
Dynamic programming is a problem-solving technique that involves breaking down a complex problem into simpler overlapping subproblems and solving them in a bottom-up manner.
It uses memoization or tabulation to store and reuse the results of subproblems to avoid redundant computations.
Dynamic programming is typically used for optimization problems where the objective is to maximize or minimize a certain value.
The key steps in dynamic programming are:
Characterize the structure of an optimal solution.
Define the value of an optimal solution recursively.
Compute the value of an optimal solution in a bottom-up manner.
Construct an optimal solution from computed information.
Greedy Algorithms:
Greedy algorithms make locally optimal choices at each step in the hope of finding a global optimum.
They are typically used for optimization problems where making the best choice at each step leads to an optimal solution overall.
Unlike dynamic programming, greedy algorithms do not always guarantee an optimal solution for every problem.
Greedy algorithms often have lower time complexity compared to dynamic programming, but they may not always produce the globally optimal solution.
Examples of Dynamic Programming and Greedy Algorithms:
Dynamic Programming:
The 0/1 Knapsack Problem: Given a set of items with weights and values, determine the most valuable combination of items that can be fit into a knapsack with a limited weight capacity.
Longest Common Subsequence: Find the longest subsequence common to two sequences.
Matrix Chain Multiplication: Determine the optimal order of multiplying a series of matrices to minimize the number of scalar multiplications.
Greedy Algorithms:
Fractional Knapsack Problem: Similar to the 0/1 Knapsack Problem, but fractional weights are allowed, and items can be divided.
Dijkstra's Algorithm: Find the shortest path between two nodes in a graph with non-negative edge weights.
Prim's Algorithm: Find the minimum spanning tree of a weighted undirected graph.
Trade-offs between Dynamic Programming and Greedy Algorithms:
Dynamic programming guarantees optimal solutions, but it may have higher time and space complexity compared to greedy algorithms.
Greedy algorithms provide fast solutions, but they may not always produce the globally optimal solution.
The choice between dynamic programming and greedy algorithms depends on the specific problem and the trade-offs between optimality and efficiency.