![Example
• Consider the example: arr[] = {{100, 20}, {60, 10}, {120, 30}}, W = 50.
Sorting: Initially sort the array based on the profit/weight ratio. The sorted array will be {{60,
10}, {100, 20}, {120, 30}}.
Iteration:
• For i = 0, weight = 10 which is less than W. So add this element in the knapsack. profit =
60 and remaining W = 50 – 10 = 40.
• For i = 1, weight = 20 which is less than W. So add this element too. profit = 60 + 100 =
160 and remaining W = 40 – 20 = 20.
• For i = 2, weight = 30 is greater than W. So add 20/30 fraction = 2/3 fraction of the element.
Therefore profit = 2/3 * 120 + 160 = 80 + 160 = 240 and remaining W becomes 0.
• So the final profit becomes 240 for W = 50.](https://image.slidesharecdn.com/greedymethoddaa-240924134655-0e392e92/85/Greedy-Method-unit-2-Design-and-analysis-of-algorithms-pptx-12-320.jpg)
Greedy Method unit-2(Design and analysis of algorithms).pptx
- 1. Bachelor of Science in Artificial Intelligence and Machine Learning Course Name : Design and analysis of algorithms Subject Code : UGCA1979 Semester : B.S.c AI&ML-3rd by Associate Professor, DCA, CBSA
- 2. Course Objectives CO# Course Outcomes CO1 relate the characteristics of different algorithms and their analysis methods. CO2 compare and contrast the time and space trade-offs in algorithm analysis. CO3 apply various fundamental algorithmic strategies CO4 analyze the application techniques in solving specific problems. CO5 utilize different traversal and shortest path algorithms.
- 4. • A greedy algorithm always makes the choice that looks best at the moment. That is, it makes a locally optimal choice in the hope that this choice will lead to a globally optimal solution. • The greedy method is quite powerful and works well for a wide range of problems. • A greedy algorithm, as the name suggests, always makes the choice that seems to be the best at that moment. This means that it makes a locally-optimal choice in the hope that this choice will lead to a globally-optimal solution. Greedy Algorithms
- 5. Steps for Creating a Greedy Algorithm • The steps to define a greedy algorithm are: • Define the problem: Clearly state the problem to be solved and the objective to be optimized. • Identify the greedy choice: Determine the locally optimal choice at each step based on the current state. • Make the greedy choice: Select the greedy choice and update the current state. • Repeat: Continue making greedy choices until a solution is reached.
- 6. Greedy Algorithm Examples • Examples of greedy algorithms are the best way to understand the algorithm. Some greedy algorithm real-life examples are: • Fractional Knapsack: Optimizes the value of items that can be fractionally included in a knapsack with limited capacity. • Dijkstra’s algorithm: Finds the shortest path from a source vertex to all other vertices in a weighted graph. • Kruskal’s algorithm: Finds the minimum spanning tree of a weighted graph.
- 7. Applications of Greedy Algorithm • There are many applications of the greedy method in DAA. Some important greedy algorithm applications are: • Assigning tasks to resources to minimize waiting time or maximize efficiency. • Selecting the most valuable items to fit into a knapsack with limited capacity. • Dividing an image into regions with similar characteristics. • Reducing the size of data by removing redundant information.
- 8. Advantages of the Greedy Method • The greedy approach is easy to implement. • Typically have less time complexity. • Greedy algorithms can be used for optimization purposes or finding close to optimization in case of Hard problems. • Greedy algorithms can produce efficient solutions in many cases, especially when the problem has a substructure that exhibits the greedy choice property. • Greedy algorithms are often faster than other optimization algorithms, such as dynamic programming or branch and bound, because they require less computation and memory.
- 9. Disadvantages/Limitations of Using a Greedy Algorithm • Greedy algorithms may not always find the best possible solution. • The order in which the elements are considered can significantly impact the outcome. • Greedy algorithms focus on local optimizations and may miss better solutions that require considering a broader context. • Greedy algorithms are not applicable to problems where the greedy choice does not lead to an optimal solution.
- 10. Difference between Greedy Algorithm and Divide and Conquer Algorithm S.No Divide and conquer Greedy Algorithm 1 Divide and conquer is used to obtain a solution to the given problem, it does not aim for the optimal solution. The greedy method is used to obtain an optimal solution to the given problem. 2 In this technique, the problem is divided into small subproblems. These subproblems are solved independently. Finally, all the solutions to subproblems are collected together to get the solution to the given problem. In Greedy Method, a set of feasible solutions are generated and pick up one feasible solution is the optimal solution. 3 Divide and conquer is less efficient and slower because it is recursive in nature. A greedy method is comparatively efficient and faster as it is iterative in nature.
- 11. Fractional Knapsack Problem using Greedy algorithm • The basic idea of the greedy approach is to calculate the ratio profit/weight for each item and sort the item on the basis of this ratio. Then take the item with the highest ratio and add them as much as we can (can be the whole element or a fraction of it). • This will always give the maximum profit because, in each step it adds an element such that this is the maximum possible profit for that much weight.
- 12. Example • Consider the example: arr[] = {{100, 20}, {60, 10}, {120, 30}}, W = 50. Sorting: Initially sort the array based on the profit/weight ratio. The sorted array will be {{60, 10}, {100, 20}, {120, 30}}. Iteration: • For i = 0, weight = 10 which is less than W. So add this element in the knapsack. profit = 60 and remaining W = 50 – 10 = 40. • For i = 1, weight = 20 which is less than W. So add this element too. profit = 60 + 100 = 160 and remaining W = 40 – 20 = 20. • For i = 2, weight = 30 is greater than W. So add 20/30 fraction = 2/3 fraction of the element. Therefore profit = 2/3 * 120 + 160 = 80 + 160 = 240 and remaining W becomes 0. • So the final profit becomes 240 for W = 50.
- 13. Conclusions Learning outcomes 1. Students are able to understand vitualization concept. 2. Students are able to identify key technologies like hypervisors and containers, and their roles in cloud environments.