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Kruskal's Algorithm

Prim's Algorithm

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Backtracking Algorithm

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DSA Tutorials

  • What is an Algorithm?

A greedy algorithm is an approach for solving a problem by selecting the best option available at the moment. It doesn't worry whether the current best result will bring the overall optimal result.

The algorithm never reverses the earlier decision even if the choice is wrong. It works in a top-down approach.

This algorithm may not produce the best result for all the problems. It's because it always goes for the local best choice to produce the global best result.

However, we can determine if the algorithm can be used with any problem if the problem has the following properties:

1. Greedy Choice Property

If an optimal solution to the problem can be found by choosing the best choice at each step without reconsidering the previous steps once chosen, the problem can be solved using a greedy approach. This property is called greedy choice property.

2. Optimal Substructure

If the optimal overall solution to the problem corresponds to the optimal solution to its subproblems, then the problem can be solved using a greedy approach. This property is called optimal substructure.

  • Advantages of Greedy Approach
  • The algorithm is easier to describe .
  • This algorithm can perform better than other algorithms (but, not in all cases).
  • Drawback of Greedy Approach

As mentioned earlier, the greedy algorithm doesn't always produce the optimal solution. This is the major disadvantage of the algorithm

For example, suppose we want to find the longest path in the graph below from root to leaf. Let's use the greedy algorithm here.

Apply greedy approach to this tree to find the longest route

Greedy Approach

1. Let's start with the root node 20 . The weight of the right child is 3 and the weight of the left child is 2 .

2. Our problem is to find the largest path. And, the optimal solution at the moment is 3 . So, the greedy algorithm will choose 3 .

3. Finally the weight of an only child of 3 is 1 . This gives us our final result 20 + 3 + 1 = 24 .

However, it is not the optimal solution. There is another path that carries more weight ( 20 + 2 + 10 = 32 ) as shown in the image below.

Longest path

Therefore, greedy algorithms do not always give an optimal/feasible solution.

  • To begin with, the solution set (containing answers) is empty.
  • At each step, an item is added to the solution set until a solution is reached.
  • If the solution set is feasible, the current item is kept.
  • Else, the item is rejected and never considered again.

Let's now use this algorithm to solve a problem.

  • Example - Greedy Approach
  • Create an empty solution-set = { } . Available coins are {5, 2, 1} .
  • We are supposed to find the sum = 18 . Let's start with sum = 0 .
  • Always select the coin with the largest value (i.e. 5) until the sum > 18 . (When we select the largest value at each step, we hope to reach the destination faster. This concept is called greedy choice property .)
  • In the first iteration, solution-set = {5} and sum = 5 .
  • In the second iteration, solution-set = {5, 5} and sum = 10 .
  • In the third iteration, solution-set = {5, 5, 5} and sum = 15 .
  • In the fourth iteration, solution-set = {5, 5, 5, 2} and sum = 17 . (We cannot select 5 here because if we do so, sum = 20 which is greater than 18. So, we select the 2nd largest item which is 2.)
  • Similarly, in the fifth iteration, select 1. Now sum = 18 and solution-set = {5, 5, 5, 2, 1} .
  • Different Types of Greedy Algorithm
  • Knapsack Problem
  • Minimum Spanning Tree
  • Single-Source Shortest Path Problem
  • Job Scheduling Problem
  • Prim's Minimal Spanning Tree Algorithm
  • Kruskal's Minimal Spanning Tree Algorithm
  • Dijkstra's Minimal Spanning Tree Algorithm

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Greedy Algorithms

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A greedy algorithm is a simple, intuitive algorithm that is used in optimization problems. The algorithm makes the optimal choice at each step as it attempts to find the overall optimal way to solve the entire problem. Greedy algorithms are quite successful in some problems, such as Huffman encoding which is used to compress data, or Dijkstra's algorithm , which is used to find the shortest path through a graph.

However, in many problems, a greedy strategy does not produce an optimal solution. For example, in the animation below, the greedy algorithm seeks to find the path with the largest sum. It does this by selecting the largest available number at each step. The greedy algorithm fails to find the largest sum, however, because it makes decisions based only on the information it has at any one step, without regard to the overall problem.

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. [1]

Limitations of Greedy Algorithms

Applications.

Structure of a Greedy Algorithm Greedy algorithms take all of the data in a particular problem, and then set a rule for which elements to add to the solution at each step of the algorithm. In the animation above, the set of data is all of the numbers in the graph, and the rule was to select the largest number available at each level of the graph. The solution that the algorithm builds is the sum of all of those choices.

If both of the properties below are true, a greedy algorithm can be used to solve the problem.

  • Greedy choice property: A global (overall) optimal solution can be reached by choosing the optimal choice at each step.
  • Optimal substructure: A problem has an optimal substructure if an optimal solution to the entire problem contains the optimal solutions to the sub-problems.

In other words, greedy algorithms work on problems for which it is true that, at every step, there is a choice that is optimal for the problem up to that step, and after the last step, the algorithm produces the optimal solution of the complete problem.

To make a greedy algorithm, identify an optimal substructure or subproblem in the problem. Then, determine what the solution will include (for example, the largest sum, the shortest path, etc.). Create some sort of iterative way to go through all of the subproblems and build a solution.

If there is a greedy algorithm that will traverse a graph, selecting the largest node value at each point until it reaches a leaf of the graph, what path will the greedy algorithm follow in the graph below?

What is the length of the longest path through the graph below? Calculate the length by adding the values of the nodes.

Sometimes greedy algorithms fail to find the globally optimal solution because they do not consider all the data. The choice made by a greedy algorithm may depend on choices it has made so far, but it is not aware of future choices it could make.

In the graph below, a greedy algorithm is trying to find the longest path through the graph (the number inside each node contributes to a total length). To do this, it selects the largest number at each step of the algorithm. With a quick visual inspection of the graph, it is clear that this algorithm will not arrive at the correct solution. What is the correct solution? Why is a greedy algorithm ill-suited for this problem? An example of greedy algorithm, searching the largest path in a tree [2] The correct solution for the longest path through the graph is \(7, 3, 1, 99\). This is clear to us because we can see that no other combination of nodes will come close to a sum of \(99\), so whatever path we choose, we know it should have \(99\) in the path. There is only one option that includes \(99\): \(7, 3, 1, 99\). The greedy algorithm fails to solve this problem because it makes decisions purely based on what the best answer at the time is: at each step it did choose the largest number. However, since there could be some huge number that the algorithm hasn't seen yet, it could end up selecting a path that does not include the huge number. The solutions to the subproblems for finding the largest sum or longest path do not necessarily appear in the solution to the total problem. The optimal substructure and greedy choice properties don't hold in this type of problem. \(_\square\)
Here, we will look at one form of the knapsack problem . The knapsack problem involves deciding which subset of items you should take from a set of items if you want to optimize some value: perhaps the worth of the items, the size of the items, or the ratio of worth to size. In this problem, we will assume that we can either take an item or leave it (we cannot take a fractional part of an item). We will also assume that there is only one of each item. Our knapsack has a fixed size, and we want to optimize the worth of the items we take, so we must choose the items we take with care. [3] Our knapsack can hold at most 25 units of space. Here is the list of items and their worths. Item Size Price Laptop 22 12 PlayStation 10 9 Textbook 9 9 Basketball 7 6 Which items do we choose to optimize for price? There are two greedy algorithms we could propose to solve this. One has a rule that selects the item with the largest price at each step, and the other has a rule that selects the smallest sized item at each step. Largest-price Algorithm: At the first step, we take the laptop. We gain \(12\) units of worth, but can now only carry \(25-22 = 3\) units of additional space in the knapsack. Since no items that remain will fit into the bag, we can only take the laptop and have a total of \(12\) units of worth. Smallest-sized-item Algorithm: At the first step, we will take the smallest-sized item: the basketball. This gives us \(6\) units of worth, and leaves us with \(25-7 = 18\) units of space in our bag. Next, we select the next smallest item, the textbook. This gives us a total of \(6+9 =15\) units of worth, and leaves us with \(18-9 = 9\) units of space. Since no remaining items are \(9\) units of space or less, we can take no more items. The greedy algorithms yield solutions that give us \(12\) units of worth and \(15\) units of worth. But neither of these are the optimal solution. Inspect the table yourself and see if you can determine a better selection of items. Taking the textbook and the PlayStation yields \(9+9=18\) units of worth and takes up \(10+9=19\) units of space. This is the optimal answer, and we can see that a greedy algorithm will not solve the knapsack problem since the greedy choice and optimal substructure properties do not hold. \(_\square\)

In problems where greedy algorithms fail, dynamic programming might be a better approach.

There are many applications of greedy algorithms. Below is a brief explanation of the greedy nature of a famous graph search algorithm, Dijkstra's algorithm.

Dijkstra's Algorithm

Dijkstra's algorithm is used to find the shortest path between nodes in a graph. The algorithm maintains a set of unvisited nodes and calculates a tentative distance from a given node to another. If the algorithm finds a shorter way to get to a given node, the path is updated to reflect the shorter distance. This problem has satisfactory optimization substructure since if \(A\) is connected to \(B,\) \(B\) is connected to \(C\), and the path must go through \(A\) and \(B\) to get to the destination \(C\), then the shortest path from \(A\) to \(B\) and the shortest path from \(B\) to \(C\) must be a part of the shortest path from \(A\) to \(C\). So the optimal answers from the subproblems do contribute to the optimal answer for the total problem. This is because the algorithm keeps track of the shortest path possible to any given node.

a and b . It picks the unvisited vertex with the lowest distance, calculates the distance through it to each unvisited neighbor, and updates the neighbor's distance if smaller. Mark visited (set to red) when done with neighbors." /> Dijkstra's algorithm to find the shortest path between a and b . It picks the unvisited vertex with the lowest distance, calculates the distance through it to each unvisited neighbor, and updates the neighbor's distance if smaller. Mark visited (set to red) when done with neighbors. [4]

Huffman Coding

Huffman encoding is another example of an algorithm where a greedy approach is successful. The Huffman algorithm analyzes a message and depending on the frequencies of the characters used in the message, it assigns a variable-length encoding for each symbol. A more commonly used symbol will have a shorter encoding while a rare symbol will have a longer encoding.

The Huffman coding algorithm takes in information about the frequencies or probabilities of a particular symbol occurring. It begins to build the prefix tree from the bottom up, starting with the two least probable symbols in the list. It takes those symbols and forms a subtree containing them, and then removes the individual symbols from the list. The algorithm sums the probabilities of elements in a subtree and adds the subtree and its probability to the list. Next, the algorithm searches the list and selects the two symbols or subtrees with the smallest probabilities. It uses those to make a new subtree, removes the original subtrees/symbols from the list, and then adds the new subtree and its combined probability to the list. This repeats until there is one tree and all elements have been added. At each subtree, the optimal encoding for each symbol is created and together composes the overall optimal encoding.

For many more applications of greedy algorithms, see the See Also section.

  • Dijkstra's Shortest Path Algorithm
  • The Travelling Salesman Problem
  • Huffman Encoding
  • Dynamic Programming
  • , S. Greedy-search-path-example . Retrieved May 2, 2016, from https://en.wikipedia.org/wiki/File:Greedy-search-path-example.gif
  • , S. Greedy-search-path . Retrieved May 4, 2016, from https://commons.wikimedia.org/wiki/File:Greedy-search-path.gif
  • Okie , . Greedy Algorithms . Retrieved May 2, 2016, from http://www.radford.edu/~nokie/classes/360/greedy.html
  • , I. Dijkstra Animation . Retrieved May 2, 2016, from https://commons.wikimedia.org/wiki/File:Dijkstra_Animation.gif

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Greedy Algorithms

Article info.

  • Knapsack Problem
  • Huffman Coding
  • Minimum Spanning Tree
  • Selection Sort
  • Dynamic Programming
  • Algorithmic Complexity

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  • 3 2023-02-17 09:20:16 3914,3909 3,3914 By arvindpdmn Adding two images.
  • 2 2023-02-15 06:07:59 3909,2128 2,3909 By parrotGPT Initial version with minor edits from ChatGPT-gen content.
  • 1 2020-07-18 11:48:29 1,2128 By arvindpdmn Questions identified. Work in progress.
  • Submitting ... You are editing an existing chat message. All Versions 2023-02-17 09:20:16 by arvindpdmn 2023-02-15 06:07:59 by parrotGPT 2020-07-18 11:48:29 by arvindpdmn All Sections Summary Discussion Sample Code References Milestones Tags See Also Further Reading
  • 2023-02-15 06:08:00 - By arvindpdmn [Draft Approval Comment] Warnings to be addressed.

A greedy algorithm is a type of algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage with the hope of finding a global optimum. While it may not find the global optimum, greedy algorithms are often simpler and faster while being not too far from the global optimum.

Greedy algorithms are being used in many areas of computer science. They're also used in operations research, engineering, finance, machine learning, bioinformatics, game theory, etc. They've become a well-established technique for solving optimization problems.

The key to developing an effective greedy algorithm is to identify the problem's specific structure and design a heuristic that makes the optimal local choice at each step. While they do have some limitations, there's ongoing research to address these limitations.

Suppose you are given an array of positive integers and you want to find the subset of those integers whose sum is as large as possible, but does not exceed a given value. For instance, if you were given the array [3, 4, 5, 6, 8] and the limit 13, the optimal subset would be [8, 5] , with a sum of 13.

  • Sort the array in decreasing order: [8, 6, 5, 4, 3]
  • Initialize an empty subset and a variable to track the current sum: subset = [] , sum = 0
  • Start the iterations: (i) subset = [8] , sum = 8; (ii) subset = [8, 5] , sum = 13.
  • Return the subset: [8, 5]
  • Simplicity : Greedy algorithms are often simple to implement and understand, making them a good choice for problems with large inputs or when efficiency is a concern.
  • Speed : Greedy algorithms are often very fast, especially when compared to more complex algorithms. This makes them a good choice for problems with large inputs.
  • Approximation : Even though the greedy algorithm does not always guarantee the optimal solution, it can often give a very good approximation of the optimal solution. In many cases, the difference between the optimal solution and the solution found by the greedy algorithm is not significant.
  • Starting Point : Greedy algorithms can be a good starting point for more complex algorithms. By using a greedy algorithm to quickly find a good solution, you can then refine the solution using other techniques.

Greedy algorithm applied to the coin changing problem. Source: Matthews 2016, slide 10.

  • Minimum Spanning Tree ( MST ) : MST is useful in network design and transportation planning. Kruskal's and Prim's algorithms are greedy approaches to this problem.
  • Huffman Coding : This is applied in data compression and transmission. It assigns shorter codes to more frequently occurring characters.
  • Knapsack Problem : This is considered a classic optimization problem. It asks what's the best way to fill a bag of fixed capacity with items of different sizes and values. A greedy algorithm selects items based on their value-to-size ratio.
  • Activity Selection : In scheduling problems, there's a need to select the maximum number of non-overlapping activities. A simple greedy algorithm solves this by selecting activities based on their finish time.
  • Shortest Path Algorithms : Dijkstra's algorithm is an example. It selects the shortest path from a given vertex to all other vertices in a graph.
  • Coin Changing Problem : This asks what's the minimum number of coins needed to make change for a given amount of money. This is solved by selecting the largest coin possible at each step.

Greedy algorithms and dynamic programming are two popular techniques for solving optimization problems, but they differ in several key ways.

Greedy algorithms make the locally optimal choice at each step in the hope of finding a global optimal solution. Dynamic programming breaks down a problem into smaller subproblems and then solves them in a bottom-up fashion. It stores the solutions to the subproblems and reuses them to solve the larger problem.

Greedy algorithms may give a sub-optimal solution , whereas dynamic programming always finds the optimal solution. Greedy algorithms typically make choices based only on the current state of the problem, while dynamic programming considers all possible subproblems and their solutions.

Greedy algorithms typically require less memory because they don't need to store the solutions to all possible subproblems.

Greedy algorithms are typically faster due to fewer calculations. However, the time complexity of greedy algorithms can be higher in certain cases.

Greedy algorithms are useful when there is a unique optimal solution . Dynamic programming can solve a wider range of problems, including problems with multiple optimal solutions or overlapping subproblems.

Greedy algorithm (c) is suboptimal and misses the optimal solution (b). Source: Simmons et al. 2019, fig. 1.

Because greedy algorithms make the locally optimal choice at each step, this may not lead to the global optimal solution. In some cases, making a suboptimal choice at an early stage can lead to a better global solution. The figure shows an example in which the objective is to maximize the sum of the nodes on a top-to-bottom path. Greedy algorithm leads to a sub-optimal solution.

Where the objective function has multiple conflicting objectives, or changes non-monotonically over time, greedy algorithms will not work well. The same can be said of problems with complex constraints or a large search space. For these cases, greedy algorithms would incur a larger time complexity as well.

Many optimization problems are NP-hard, which means that finding the optimal solution is computationally intractable. Greedy algorithms are not suitable for solving them, as they can't guarantee the optimal solution in a reasonable amount of time.

Adaptive greedy algorithms adjust their choices dynamically based on the problem's structure and input data. They have outperformed traditional greedy algorithms in many real-world optimization problems.

Hybrid algorithms combine greedy techniques with other optimization techniques, such as dynamic programming or local search. These hybrid algorithms can often provide better results than either technique used alone.

There are greedy algorithms capable of multi-objective optimization . They can be used to find a Pareto-optimal set of solutions.

In many real-world applications, input data is received incrementally over time. Online optimization algorithms must make decisions in real-time, without having access to all the input data in advance. Greedy algorithms have been shown to be effective in this setting because they can make quick decisions based on the available data.

Researchers have also been working on developing new methods for analysing the performance of greedy algorithms. There are new theoretical frameworks for understanding the behaviour of greedy algorithms in different types of optimization problems.

Indian mathematician Aryabhata uses a greedy algorithm to solve the problem of finding the largest number that can be written as the sum of two squares.

The modern concept of greedy algorithms was first introduced in the 1930s by mathematicians and computer scientists working on the design of algorithms for optimization problems.

The term greedy algorithm is coined by mathematician and computer scientist David A. Huffman in reference to the Huffman coding algorithm. The term "greedy" refers to the algorithm's strategy of always choosing the option that seems best at the time, without considering the possible consequences of that choice on future steps.

Computer scientist Joseph Kruskal develops a greedy algorithm for finding the minimum spanning tree of a graph. In 1957, Robert Prim develop's his own greedy algorithm for the solving the same problem. Also in 1956, Edsger Dijkstra develops a greedy algorithm for finding the shortest path in a graph. The same year L. R. Ford and D. R. Fulkerson develop a greedy algorithm for solving the maximum flow problem in a network.

In the 1980s, researchers develop a variety of approximation algorithms based on greedy techniques, which provide near-optimal solutions to NP-hard problems. This research continues into the 1990s.

  • Matthews, P. 2016. "Lecture 3: Greedy Method." Slides, on SlidePlayer. Accessed 2023-02-17.
  • Simmons, Benno I., Christoph Hoeppke, and William J. Sutherland. 2019. "Beware greedy algorithms." J. Animal Ecology, British Ecological Society, March 15. doi: 10.1111/1365-2656.12963. Accessed 2023-02-17.

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The Definitive Guide to Understanding Greedy Algorithm

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Do you know, during Huffman data compression, greedy algorithm is utilized to build a Huffman tree that compresses a given image, spreadsheet, or video into a lossless compressed file? Also, this greedy coding paradigm encapsulates algorithmic problems where selecting the most obvious answer for the current subproblem leads to the solution of a more complex problem. So, in this article, we will discover a greedy algorithmic paradigm in detail.

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What Is Greedy Algorithm?

A Greedy algorithm is an approach to solving a problem that selects the most appropriate option based on the current situation. This algorithm ignores the fact that the current best result may not bring about the overall optimal result. Even if the initial decision was incorrect, the algorithm never reverses it.

This simple, intuitive algorithm can be applied to solve any optimization problem which requires the maximum or minimum optimum result. The best thing about this algorithm is that it is easy to understand and implement.

The runtime complexity associated with a greedy solution is pretty reasonable. However, you can implement a greedy solution only if the problem statement follows two properties mentioned below: 

  • Greedy Choice Property: Choosing the best option at each phase can lead to a global (overall) optimal solution.
  • Optimal Substructure: If an optimal solution to the complete problem contains the optimal solutions to the subproblems, the problem has an optimal substructure.

Moving forward, we will learn how to create a greedy solution for a problem that adheres to the principles listed above.

Steps for Creating a Greedy Algorithm

By following the steps given below, you will be able to formulate a greedy solution for the given problem statement:

  • Step 1: In a given problem, find the best substructure or subproblem.
  • Step 2: Determine what the solution will include (e.g., largest sum, shortest path).
  • Step 3: Create an iterative process for going over all subproblems and creating an optimum solution.

Let’s take up a real-world problem and formulate a greedy solution for it. 

Problem: Alex is a very busy person. He has set aside time T to accomplish some interesting tasks. He wants to do as many tasks as possible in this allotted time T. For that, he has created an array A of timestamps to complete a list of items on his itinerary. 

Now, here we need to figure out how many things Alex can complete in the T time he has.

Approach to Build a Solution: This given problem is a straightforward greedy problem. In each iteration, we will have to pick the items from array A that will take the least amount of time to accomplish a task while keeping two variables in mind: current_Time and number_Of_Things. To generate a solution, we will have to carry out the following steps.

  • Sort the array A in ascending order.
  • Select one timestamp at a time.
  • After picking up the timestamp, add the timestamp value to current_Time.
  • Increase number_Of_Things by one.
  • Repeat steps 2 to 4 until the current_Time value reaches T.

Example of Greedy Algorithm

Problem Statement:  Find the best route to reach the destination city from the given starting point using a greedy method.

Thumbnail_Greedy_Problem

Greedy Solution: In order to tackle this problem, we need to maintain a graph structure. And for that graph structure, we'll have to create a tree structure, which will serve as the answer to this problem. The steps to generate this solution are given below:

  • Start from the source vertex.
  • Pick one vertex at a time with a minimum edge weight (distance) from the source vertex.
  • Add the selected vertex to a tree structure if the connecting edge does not form a cycle.
  • Keep adding adjacent fringe vertices to the tree until you reach the destination vertex.

The animation given below explains how paths will be picked up in order to reach the destination city.

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Limitations of Greedy Algorithm

Factors listed below are the limitations of a greedy algorithm:

  • The greedy algorithm makes judgments based on the information at each iteration without considering the broader problem; hence it does not produce the best answer for every problem.
  • The problematic part for a greedy algorithm is analyzing its accuracy. Even with the proper solution, it is difficult to demonstrate why it is accurate. 
  • Optimization problems (Dijkstra’s Algorithm) with negative graph edges cannot be solved using a greedy algorithm.

Moving forward, let’s look at some applications of a greedy algorithm.

Applications of Greedy Algorithm

Following are few applications of the greedy algorithm:

  • Used for Constructing Minimum Spanning Trees: Prim’s and Kruskal’s Algorithms used to construct minimum spanning trees are greedy algorithms.
  • Used to Implement Huffman Encoding: A greedy algorithm is utilized to build a Huffman tree that compresses a given image, spreadsheet, or video into a lossless compressed file.
  • Used to Solve Optimization Problems: Graph - Map Coloring, Graph - Vertex Cover, Knapsack Problem, Job Scheduling Problem, and activity selection problem are classic optimization problems solved using a greedy algorithmic paradigm.

Characteristics of a Greedy Method

The greedy method is a simple and straightforward way to solve optimization problems. It involves making the locally optimal choice at each stage with the hope of finding the global optimum. The main advantage of the greedy method is that it is easy to implement and understand. However, it is not always guaranteed to find the best solution and can be quite slow.

The greedy method works by making the locally optimal choice at each stage in the hope of finding the global optimum. This can be done by either minimizing or maximizing the objective function at each step. The main advantage of the greedy method is that it is relatively easy to implement and understand. However, there are some disadvantages to using this method. First, the greedy method is not guaranteed to find the best solution. Second, it can be quite slow. Finally, it is often difficult to prove that the greedy method will indeed find the global optimum.

One of the most famous examples of the greedy method is the knapsack problem. In this problem, we are given a set of items, each with a weight and a value. We want to find the subset of items that maximizes the value while minimizing the weight. The greedy method would simply take the item with the highest value at each step. However, this might not be the best solution. For example, consider the following set of items:

Item 1: Weight = 2, Value = 6

Item 2: Weight = 2, Value = 3

Item 3: Weight = 4, Value = 5

The greedy method would take Item 1 and Item 3, for a total value of 11. However, the optimal solution would be to take Item 2 and Item 3, for a total value of 8. Thus, the greedy method does not always find the best solution.

There are many other examples of the greedy method. The most famous one is probably the Huffman coding algorithm, which is used to compress data. In this algorithm, we are given a set of symbols, each with a weight. We want to find the subset of symbols that minimizes the average length of the code. The greedy method would simply take the symbol with the lowest weight at each step. However, this might not be the best solution. For example, consider the following set of symbols:

Symbol 1: Weight = 2, Code = 00

Symbol 2: Weight = 3, Code = 010

Symbol 3: Weight = 4, Code =011

The greedy method would take Symbol 1 and Symbol 3, for a total weight of 6. However, the optimal solution would be to take Symbol 2 and Symbol 3, for a total weight of 7. Thus, the greedy method does not always find the best solution.

The greedy method is a simple and straightforward way to solve optimization problems. However, it is not always guaranteed to find the best solution and can be quite slow. When using the greedy method, it is important to keep these disadvantages in mind.

Components of a Greedy Algorithm

There are four key components to any greedy algorithm:

  • A set of candidate solutions (typically represented as a graph)
  • A way of ranking the candidates according to some criteria
  • A selection function that picks the best candidate from the set, according to the ranking
  • A way of "pruning" the set of candidates, so that it doesn't contain any solutions that are worse than the one already chosen.

The first two components are straightforward - the candidate solutions can be anything, and the ranking criteria can be anything as well. The selection function is usually just a matter of picking the candidate with the highest ranking.

The pruning step is important, because it ensures that the algorithm doesn't waste time considering candidates that are already known to be worse than the best one found so far. Without this step, the algorithm would essentially be doing a brute-force search of the entire solution space, which would be very inefficient.

Pseudo Code of Greedy Algorithms

One example of pseudo code for a greedy algorithm is given below:

function GreedyAlgorithm(problem) {

// currentState = initial state of problem while (currentState != goalState) { nextState = chooseNextState(currentState); currentState = nextState; } return currentState; }

The above pseudo code shows the basic structure of a greedy algorithm. The first step is to set the current state to the initial state of the problem. Next, we keep looping until the current state is equal to the goal state. Inside the loop, we choose the next state that we want to move to. This is done by using a function called chooseNextState(). Finally, we set the current state to be equal to the next state that we have just chosen and return to the goal state.

The pseudo code for the chooseNextState() function is given below:

function chooseNextState(currentState) { // find all possible next states nextStates = findAllPossibleNextStates(currentState); // choose the next state that is closest to the goal state bestState = null; for (state in nextStates) { if (isCloserToGoal(state, bestState)) { bestState = state; } } return bestState; }

The pseudo code for the chooseNextState() function shows how we can choose the next state that is closest to the goal state. First, we find all of the possible next states that we could move to from the current state. Next, we loop through all of the possible next states and compare each one to see if it is closer to the goal state than the best state that we have found so far. If it is, then we set the best state to be equal to the new state. Finally, we return the best state that we have found.

The above pseudo code shows how a greedy algorithm works in general. However, it is important to note that not all problems can be solved using a greedy algorithm. In some cases, it may be necessary to use a different type of algorithm altogether.

Disadvantages of Using Greedy Algorithms

The main disadvantage of using a greedy algorithm is that it may not find the optimal solution to a problem. In other words, it may not produce the best possible outcome. Additionally, greedy algorithms can be very sensitive to changes in input data — even a small change can cause the algorithm to produce a completely different result. Finally, greedy algorithms can be difficult to implement and understand.

In this greedy algorithm article, you learned what a greedy programming paradigm is and discovered properties and steps to build a greedy solution. The article also discusses applications and mentions the limitations of greedy algorithm. You also saw an example of this algorithm which will help grasp the concept.  If you want a quick and easy way to understand the greedy algorithmic paradigm, we advise you to watch this video: https://youtu.be/ilYwrsP7zzk .

Every day, new products, tools, and apps are being introduced and launched into the market. And there are hundreds of programming languages and frameworks being utilized every day in the realm of software development . Hence, it is crucial for you to go beyond data structure concepts and cover the foundations of interactive application development.

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Algorithms

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Basics of Greedy Algorithms

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  • Breadth First Search
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  • Minimum Spanning Tree
  • Shortest Path Algorithms
  • Flood-fill Algorithm
  • Articulation Points and Bridges
  • Biconnected Components
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  • Topological Sort
  • Hamiltonian Path
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When to Use Greedy Algorithms – And When to Avoid Them [With Example Problems]

Greedy algorithms try to find the optimal solution by taking the best available choice at every step.

For example, you can greedily approach your life. You can always take the path that maximizes your happiness today. But that doesn't mean you'll be happier tomorrow.

Similarly, there are problems for which greedy algorithms don't yield the best solution. Actually, they might yield the worst possible solution.

But there are other cases in which we can obtain a solution that is good enough by using a greedy strategy.

In this article, I'll write about greedy algorithms and the use of this strategy even when it doesn't guarantee that you'll find an optimal solution.

The first section is an introduction to greedy algorithms and well-known problems that are solvable using this strategy. Then I'll talk about problems in which the greedy strategy is a really bad option. And finally, I'll show you an example of a good approximation through a greedy algorithm.

Note : Most of the algorithms and problems I discuss in this article include graphs. It would be good if you are familiar with graphs to get the most out of this post.

How greedy algorithms work

Greedy algorithms always choose the best available option.

In general, they are computationally cheaper than other families of algorithms like dynamic programming, or brute force. This is because they don't explore the solution space too much. And, for the same reason, they don't find the best solution to a lot of problems.

But there are lots of problems that are solvable with a greedy strategy, and in those cases that strategy is precisely the best way to go.

One of the most popular greedy algorithms is Dijkstra's algorithm that finds the path with the minimum cost from one vertex to the others in a graph.

This algorithm finds such a path by always going to the nearest vertex. That's why we say it is a greedy algorithm.

This is pseudocode for the algorithm. I denote with G the graph and with s the source node.

After running this algorithm, we get a list of distances such that distances[u] is the minimum cost to go from node s to node u .

This algorithm is guaranteed to work only if the graph doesn't have edges with negative costs. A negative cost in an edge can make the greedy strategy choose a path that is not optimal.

Another example that is used to introduce the concepts of the greedy strategy is the Fractional Knapsack.

In this problem, we have a collection of items. Each item has a weight Wi greater than zero, and a profit Pi also greater than zero.

We have a knapsack with a capacity W and we want to fill it in such a way that we get the maximum profit. Of course, we cannot exceed the capacity of the knapsack.

In the fractional version of the knapsack problem, we can take either the entire object or only a fraction of it. When taking a fraction 0 <= X <= 1 of the i-th object, we obtain a profit equal to X*Pi and we need to add X*Wi to the bag.

We can solve this problem by using a greedy strategy. I won't discuss the solution here. If you don't know it, I recommend that you try to solve it by yourself and then look for the solution online.

The number of problems that we can solve by using greedy algorithms is huge. But the number of problems that we cannot solve this way is even bigger. The next section is about the latter problems - those we shouldn't solve this way.

When being greedy is the worst

In the previous section, we saw two examples of problems that are solvable using a greedy strategy. This is great because those are pretty fast algorithms.

But, as I said, Dijkstra's algorithm doesn't work in graphs with negative edges.

And the problem is even bigger. I can always build a graph with negative edges in a way that Dijkstra's solution would be as bad as I wanted! Consider the following example that was extracted from Stackoverflow

rmowk.png

Dijkstra's algorithm fails to find the distance between A and C . It finds d(A, C) = 0 when it should be -200. And if we decrease the value of the edge D -> B , we'll obtain a distance that we'll be even farther from the actual minimum distance.

Similarly, when we can't break objects in the knapsack problem (the 0-1 Knapsack Problem), the solution that we obtain when using a greedy strategy can be pretty bad, too. We can always build an input to the problem that makes the greedy algorithm fail badly.

Another example is the Travelling Salesman Problem (TSP). Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?

We can greedily approach the problem by always going to the nearest possible city. We select any of the cities as the first one and apply that strategy.

As happened in previous examples, we can always build a disposition of the cities in a way that the greedy strategy finds the worst possible solution.

In this section, we have seen that a greedy strategy could lead us to disaster. But there are problems in which such an approach can approximate the optimal solution quite well.

When being greedy is not that bad

We have seen that a greedy strategy can become as bad as we want for some problems. This means that we cannot use it to obtain the optimal solution nor even a good approximation of it.

But there are some examples in which greedy algorithms provide us with very good approximations! In these cases, the greedy approach is very useful because it tends to be cheaper and easier to implement.

The vertex cover of a graph is the minimum set of vertices such that every edge of the graph has at least one of its endpoints in the set.

This is a very hard problem. Actually, there isn't any efficient and exact solution for it. But the good news is that we can make a good approximation with a greedy algorithm.

We select any edge <u, v> from the graph, and add u and v to the set. Then, we remove all the edges that have u or v as one of their endpoints, and we repeat the previous process while the remaining graph had edges.

This might be pseudocode of the previous algorithm:

As you can see, this is a simple and relatively fast algorithm. But the best part is that the solution will always be less than or equal to two times the optimal solution! We'll never obtain a set that is bigger than two times the smaller vertex cover, no matter how the input graph was built.

I'm not going to include the demonstration of this statement in this post, but you can prove it by noticing that for every edge <u, v> that we add to the vertex cover, either u or v are in the optimal solution (that is, in the smaller vertex cover).

Many computer scientists are working to find more of these approximations. There are more examples, but I'm going to stop here.

This is an interesting and very active research field in Computer Science and Applied Mathematics. With these approximations, we can get very good solutions for very hard problems by implementing pretty simple algorithms.

Conclusions

In this post, I gave you a shallow introduction to greedy algorithms. We saw examples of problems that can be solved using the greedy strategy. Then, I talk about some problems for which the greedy strategy is a bad option. And finally, we saw an example of a greedy algorithm that'll get you an approximated solution to a hard problem.

Sometimes we can solve a problem using a greedy approach but it is hard to come up with the right strategy. And demonstrating the correctness of greedy algorithms (for exact or approximated solutions) can be very difficult. So, there are a lot of things we can discuss about greedy algorithms!

If you enjoyed this post and want me to keep this type of content coming, let me know by sharing it and tagging me. You can also follow me on Twitter for more CS-related content.

Computer Scientist. Teaching Probabilities and Statistics at the University of Havana. Also researching in the field of Combinatorial Optimization.

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Greedy algorithms are a popular and powerful technique used in problem-solving and optimization. This class of algorithms focuses on making the best possible decision at each step while considering the current state of the problem. In many cases, these algorithms can be simple to implement and provide a quick, efficient solution to a wide range of real-world problems.

In this blog post, we'll explore the concept of greedy algorithms, their applications, and provide examples to help you understand how they work. We'll also look at the strengths and limitations of these algorithms and answer some frequently asked questions. So, let's dive in!

  • What are Greedy Algorithms?

Greedy algorithms are a type of algorithm that make decisions based on the current state of the problem, aiming to find the best possible solution at each step. The basic idea is to make a locally optimal choice, hoping that these choices will lead to a globally optimal solution. Greedy algorithms are not always optimal, but they can often provide near-optimal solutions relatively quickly.

  • Key Components of a Greedy Algorithm

There are three main components to a greedy algorithm:

  • Selection policy: Determines the best candidate for the solution at the current stage.
  • Feasibility check: Ensures that the chosen candidate can be part of the solution.
  • Solution check: Verifies if the current solution is complete, or if more steps are needed.
  • Advantages and Disadvantages of Greedy Algorithms
  • Greedy algorithms are often simple to implement.
  • They can provide quick, efficient solutions for certain problems.
  • In some cases, they can find an optimal solution.

Disadvantages

  • Greedy algorithms do not always guarantee an optimal solution.
  • They may fail to consider long-term consequences, leading to suboptimal solutions.
  • In some cases, they can become stuck in local optima, preventing them from finding better solutions.
  • Real-World Applications of Greedy Algorithms

Let's look at some real-world applications where greedy algorithms are used:

1. Kruskal's Algorithm for Minimum Spanning Tree

Kruskal's algorithm is a classic greedy algorithm used to find the minimum spanning tree (MST) of a connected, undirected graph with weighted edges. A minimum spanning tree connects all the vertices in the graph while minimizing the total weight of the edges.

Here's a high-level description of Kruskal's algorithm:

  • Sort all the edges in the graph by their weight.
  • Initialize an empty set to store the edges of the MST.
  • Iterate through the sorted edges, adding each edge to the MST if it doesn't form a cycle.
  • Continue until all vertices are connected.

Example: Kruskal's Algorithm in Python

2. dijkstra's algorithm for shortest path.

Dijkstra's algorithm is another famous greedy algorithm used to find the shortest path between a source vertex and all other vertices in a weighted graph. The algorithm maintains a set of unvisited vertices and calculates the tentative distance from the source to each vertex.

Here's a high-level description of Dijkstra's algorithm:

  • Assign a tentative distance value to every vertex: set it to zero for the source vertex and infinity for all other vertices.
  • Set the source vertex as the current vertex.
  • For the current vertex, consider all its unvisited neighbors and calculate their tentative distances. If the calculated distance is less than the current tentative distance, update it.
  • Mark the current vertex as visited.
  • Select the unvisited vertex with the smallest tentative distance and set it as the new current vertex. Repeat steps 3-5 until all vertices have been visited or the smallest tentative distance is infinity.

Example: Dijkstra's Algorithm in Python

  • Frequently Asked Questions

Q: When should I use a greedy algorithm?

A: Greedy algorithms are best suited for problems that have an optimal substructure and satisfy the greedy-choice property. In these cases, a greedy algorithm can provide a quick and efficient solution.

Q: Can greedy algorithms always guarantee an optimal solution?

A: No, greedy algorithms do not always guarantee an optimal solution. However, they often providenear-optimal solutions relatively quickly. In some cases, they may even find the optimal solution. It's essential to analyze the problem and determine if a greedy approach is suitable before implementing a greedy algorithm.

Q: What are some common examples of greedy algorithms?

A: Some well-known greedy algorithms include Kruskal's algorithm for minimum spanning trees, Dijkstra's algorithm for shortest paths, Huffman coding for data compression, and the Fractional Knapsack Problem.

Q: How can I determine if a greedy algorithm is suitable for my problem?

A: To determine if a greedy algorithm is suitable for your problem, first check if the problem has an optimal substructure and satisfies the greedy-choice property. If these conditions are met, a greedy algorithm may be a suitable approach. However, it's important to analyze the problem thoroughly and consider any potential pitfalls of using a greedy algorithm.

Q: What are some limitations of greedy algorithms?

A: Greedy algorithms have several limitations, including not always guaranteeing an optimal solution, potentially becoming stuck in local optima, and failing to consider long-term consequences, which may lead to suboptimal solutions.

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Fractional Knapsack Problem: Greedy algorithm with Example

Matthew Martin

What is Greedy Strategy?

Greedy algorithms are like dynamic programming algorithms that are often used to solve optimal problems (find best solutions of the problem according to a particular criterion).

Greedy algorithms implement optimal local selections in the hope that those selections will lead to an optimal global solution for the problem to be solved. Greedy algorithms are often not too hard to set up, fast (time complexity is often a linear function or very much a second-order function). Besides, these programs are not hard to debug and use less memory. But the results are not always an optimal solution.

Greedy strategies are often used to solve the combinatorial optimization problem by building an option A. Option A is constructed by selecting each component Ai of A until complete (enough n components). For each Ai, you choose Ai optimally. In this way, it is possible that at the last step you have nothing to select but to accept the last remaining value.

There are two critical components of greedy decisions:

  • Way of greedy selection. You can select which solution is best at present and then solve the subproblem arising from making the last selection. The selection of greedy algorithms may depend on previous selections. But it cannot depend on any future selection or depending on the solutions of subproblems. The algorithm evolves in a way that makes selections in a loop, at the same time shrinking the given problem to smaller subproblems.
  • Optimal substructure. You perform the optimal substructure for a problem if the optimal solution of this problem contains optimal solutions to its subproblems.

A greedy algorithm has five components:

  • A set of candidates, from which to create solutions.
  • A selection function, to select the best candidate to add to the solution.
  • A feasible function is used to decide if a candidate can be used to build a solution.
  • An objective function, fixing the value of a solution or an incomplete solution.
  • An evaluation function, indicating when you find a complete solution.

The Idea of Greedy One

With the first idea, you have the following steps of Greedy One:

  • Sort in non-increasing order of values.
  • In turn consider the ordered packages, put the considering package into knapsack if the remaining capacity of the knapsack is enough to contain it (which means that the total weight of the packages that have been put into the knapsack and weight of considering packages do not exceed the capacity of the knapsack).

However, this greedy algorithm does not always give the optimal solution. Here you have a counter-example:

  • The parameters of the problem are: n = 3; M = 19.
  • The packages: {i = 1; W[i] = 14; V[i] = 20}; {i = 2; W[i] = 6; V[i] = 16}; {i = 3; W[i] = 10; V[i] = 8} -> Great value but also great weight.
  • The algorithm will select package 1 with a total value of 20, while the optimal solution of the problem is selected (package 2, package 3) with a total value of 24.

The Idea of Greedy Two

With the second idea, you have the following steps of Greedy Two:

  • Sort in non-decreasing order of weights.
  • The parameters of the problem are: n = 3; M = 11.
  • The packages: {i = 1; W[i] = 5; V[i] = 10}; {i = 2; W[i] = 6; V[i] = 16}; {i = 3; W[i] = 10; V[i] = 28} -> Light weight but the value is also very light.
  • The algorithm will select (package 1, package 2) with a total value of 26, while the optimal solution of the problem is (package 3) with a total value of 28.

The Idea of Greedy Three

With the third idea, you have the following steps of Greedy Three. In fact, this is the most widely used algorithm.

The Idea of Greedy Three

Way to select the packages:

  • Consider the array of unit costs. You select packages according to decreasing unit costs.

The Idea of Greedy Three

  • Suppose you found a partial solution: (x 1 , …, x i ).
  • The value of the knapsack is obtained:

The Idea of Greedy Three

  • Corresponding to the weight of packages that have been put into the knapsack:

The Idea of Greedy Three

  • Therefore, the remaining weight limit of the knapsack is:

The Idea of Greedy Three

Steps of Algorithm

You see this is a problem of finding max. The list of packages is sorted in descending order of unit costs to consider branching.

  • Step 1 : Node root represents the initial state of the knapsack, where you have not selected any package.
  • TotalValue = 0.
  • The upper bound of the root node UpperBound = M * Maximum unit cost.
  • Step 2 : Node root will have child nodes corresponding to the ability to select the package with the largest unit cost. For each node, you re-calculate the parameters:
  • TotalValue = TotalValue (old) + number of selected packages * value of each package.
  • M = M (old) – number of packages selected * weight of each package.
  • UpperBound = TotalValue + M (new) * The unit cost of the packaced to be considered next.
  • Step 3 : In child nodes, you will prioritize branching for the node having the larger upper bound. The children of this node correspond to the ability of selecting the next package having large unit cost. For each node, you must re-calculate the parameters TotalValue, M, UpperBound according to the formula mentioned in step 2.
  • Step 4 : Repeat Step 3 with the note: for nodes with upper bound is lower or equal values to the temporary maximum cost of an option found, you do not need to branch for that node anymore.
  • Step 5 : If all nodes are branched or cut off, the most expensive option is the one to look for.

Pseudo code for the algorithm:

The complexity of the algorithm:

  • If using a simple sort algorithm (selection, bubble…) then the complexity of the whole problem is O(n2).
  • If using quick sort or merge sort then the complexity of the whole problem is O(nlogn).

Java code for Greedy Three

Greedy Three

  • You then create a function to perform the algorithm Greedy Three.

Function knapsackGreProc() in Java

Explanation of code:

  • Initialize weight and value for each knapsack package.
  • Sort knapsack packages by cost with descending order.
  • If select package i.
  • If select the number of package i is enough.
  • Stop when browsing all packages.

In this tutorial, you have two examples. Here is java code to run the above program with two examples:

You have the output:

  • First Example:
  • Second Example:

Analyze the first example:

  • The parameters of the problem are: n = 4; M = 37.
  • The packages: {i = 1; W[i] = 15; V[i] = 30; Cost = 2.0}; {i = 2; W[i] = 10; V[i] = 25; Cost = 2.5}; {i = 3; W[i] = 2; V[i] = 4; Cost = 1.0}; {i = 4; W[i] = 4; V[i] = 6; Cost = 1.5}.
  • You sort packages in the order of no increasing of the value of unit costs. You have: {i = 2} -> {i = 1} -> {i = 4} -> {i = 3}.

Steps for applying algorithm for the first example:

  • Define x1, x2, x3, x4 is the number of each selected package, corresponding to package {i = 2} -> {i = 1} -> {i = 4} -> {i = 3}.
  • M = 37 (as proposed).
  • UpperBound = 37 * 2.5 = 92.5, of which 37 is M and 2.5 is the unit cost of package {i = 2}.
  • With package {i = 2}, you have 4 possibilities: select 3 package {i = 2} (x1 = 3); select 2 package {i = 2} (x1 = 2); select 1 package {i = 2} (x1 = 1) and not select package {i = 2} (x1 = 0). In accordance with these 4 possibilities, you branch the root node N to 4 children N[1], N[2], N[3] and N[4].
  • TotalValue = 0 + 3 * 25 = 75, where 3 is the number of package {i = 2} selected and 25 is the value of each package {i = 2}.
  • M = 37 – 3 * 10 = 7, where 37 is the initial quantity of the knapsack, 3 is the number of package {i = 2}, 10 is the weight of each package {i = 2}.
  • UpperBound = 75 + 7 * 2 = 89, where 75 is TotalValue, 7 is the remaining weight of the knapsack and 2 is the unit cost of the package {i = 1}.
  • Similarly, you can calculate the parameters for nodes N[2], N[3] and N[4], in which the UpperBound is 84, 79 and 74 respectively.
  • Among nodes N[1], N[2], N[3] and N[4], node N[1] has the largest UpperBound, so you will branch node N[1] first in the hope that there will be a good plan from this direction.
  • From node N[1], you have only one child node N[1-1] corresponding to x2 = 0 (due to the remaining weight of the backpack is 7, while the weight of each package {i = 1} is 15). After determining the parameters for the N[1-1] button you have the UpperBound of N[1-1] is 85.5.
  • You continue branching node N[1-1]. Node N[1-1] has 2 children N[1-1-1] and N[1-1-2] corresponding to x3 = 1 and x3 = 0. After determining the parameters for these two nodes, you see that the UpperBoundary of N[1-1-1] is 84 and that of N[1-1-2] is 82, so you continue branching node N[1-1-1].
  • Node N[1-1-1] has two children, N[1-1-1-1] and N[1-1-1-2], corresponding to x4 = 1 and x4 = 0. These are two leaf nodes (representing the option) because for each node the number of packages has been selected. In which node N[1-1-1-1] represents the option x1 = 3, x2 = 0, x3 = 1 and x4 = 1 for 83, while node N[1-1-1-2] represents the option x1 = 3, x2 = 0, x3 = 1 and x4 = 01 at 81. So the temporary maximum value here is 83.
  • Turning back to node N[1-1-2], you see that the UpperBound of N[1-1-2] is 82 < 83, so you trim node N[1-1-2].
  • Turning back to node N2, you see that the UpperBound of N2 is 84 > 83, so you continue branching node N2. The node N2 has two children N[2-1] and N[2-2] corresponding to x2 = 1 and x2 = 0. After calculating the parameters for N[2-1] and N[2-2], you see the UpperBound of N[2-1] is 83 and that of N[2-2] is 75.25. Neither of these values is greater than 83 so both nodes are trimmed.
  • Finally, nodes N3 and N4 are also trimmed.
  • So all the nodes on the tree are branched or trimmed so the best temporary solution is the one to look for. Accordingly, you need to select 3 packages {i = 2}, 1 package {i = 4} and one package {i = 3} with total value of 83, total weight is 36.

With the same analysis of the second example, you have the result: select package 4 (3 times) and package 5 (3 times).

Python3 code for Greedy Three

  • Firstly, you define class KnapsackPackage.

Function knapsackGreProc() in Python

Here is Python3 code to run the above program with the first example:

C# code for Greedy Three

Function KnapsackGreProc() in C#

Here is C# code to run the above program with the first example:

Counter-example of Greedy Three

The algorithm of Greedy Three resolves quickly and can also be optimal in some cases. However, in some special cases, it does not give the optimal solution. Here you have a counter-example:

  • The parameters of the problem are: n = 3; M = 10.
  • The packages: {i = 1; W[i] = 7; V[i] = 9; Cost = 9/7}; {i = 2; W[i] = 6; V[i] = 6; Cost = 1}; {i = 3; W[i] = 4; V[i] = 4; Cost = 1}.
  • The algorithm will select (package 1) with a total value of 9, while the optimal solution of the problem is (package 2, package 3) with a total value of 10.

Here is java code to run the above program with the counter-example:

Here is the result:

That’s all to Fractional Knapsack problem.

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Standard Greedy algorithms

Activity selection problem | greedy algo-1.

  • Job Sequencing Problem
  • Huffman Coding | Greedy Algo-3
  • Huffman Decoding
  • Water Connection Problem
  • Greedy Algorithm for Egyptian Fraction
  • Policemen catch thieves
  • Fitting Shelves Problem
  • Assign Mice to Holes

Greedy algorithm on Array

  • Minimum product subset of an array
  • Maximize array sum after K negations using Sorting
  • Minimum sum of product of two arrays
  • Minimum sum of absolute difference of pairs of two arrays
  • Minimum increment/decrement to make array non-Increasing
  • Sorting array with reverse around middle
  • Sum of Areas of Rectangles possible for an array
  • Largest lexicographic array with at-most K consecutive swaps
  • Partition into two subsets of lengths K and (N - k) such that the difference of sums is maximum

Greedy algorithm on Operating System

  • Program for First Fit algorithm in Memory Management
  • Program for Best Fit algorithm in Memory Management
  • Program for Worst Fit algorithm in Memory Management
  • Program for Shortest Job First (or SJF) CPU Scheduling | Set 1 (Non- preemptive)
  • Job Scheduling with two jobs allowed at a time
  • Optimal Page Replacement Algorithm

Greedy algorithm on Graph

  • Prim’s Algorithm for Minimum Spanning Tree (MST)
  • Boruvka's algorithm | Greedy Algo-9
  • Dial's Algorithm (Optimized Dijkstra for small range weights)
  • Minimum cost to connect all cities
  • Number of single cycle components in an undirected graph

Approximate Greedy algorithm for NP complete problems

  • Greedy Approximate Algorithm for Set Cover Problem
  • Bin Packing Problem (Minimize number of used Bins)
  • Graph Coloring Using Greedy Algorithm
  • Greedy Approximate Algorithm for K Centers Problem
  • Approximate solution for Travelling Salesman Problem using MST

Greedy algorithm for special cases of DP

  • Fractional Knapsack Problem
  • Greedy Algorithm to find Minimum number of Coins

Some easy problems on Greedy algorithm

  • Split n into maximum composite numbers
  • Buy Maximum Stocks if i stocks can be bought on i-th day
  • Find the minimum and maximum amount to buy all N candies
  • Find maximum equal sum of every three stacks
  • Divide cuboid into cubes such that sum of volumes is maximum
  • Maximum number of customers that can be satisfied with given quantity
  • Minimum rotations to unlock a circular lock
  • Minimum rooms for m events of n batches with given schedule
  • Minimum cost to make array size 1 by removing larger of pairs
  • Minimum cost for acquiring all coins with k extra coins allowed with every coin
  • Minimum increment by k operations to make all elements equal
  • Find minimum number of currency notes and values that sum to given amount
  • Smallest subset with sum greater than all other elements

Some medium level problems on Greedy algorithm

  • Maximum trains for which stoppage can be provided
  • Minimum Fibonacci terms with sum equal to K
  • Divide 1 to n into two groups with minimum sum difference
  • Paper Cut into Minimum Number of Squares
  • Minimum difference between groups of size two
  • Minimum Number of Platforms Required for a Railway/Bus Station
  • Minimum initial vertices to traverse whole matrix with given conditions
  • Largest palindromic number by permuting digits
  • Find smallest number with given number of digits and sum of digits
  • Lexicographically largest subsequence such that every character occurs at least k times

Some hard problems on Greedy algorithm

  • Maximum elements that can be made equal with k updates
  • Minimize Cash Flow among a given set of friends who have borrowed money from each other
  • Minimum Cost to cut a board into squares
  • Minimum cost to process m tasks where switching costs
  • Find minimum time to finish all jobs with given constraints
  • Minimize the maximum difference between the heights
  • Minimum edges to reverse to make path from a source to a destination
  • Find the Largest Cube formed by Deleting minimum Digits from a number
  • Rearrange characters in a String such that no two adjacent characters are same
  • Rearrange a string so that all same characters become d distance away
Greedy is an algorithmic paradigm that builds up a solution piece by piece, always choosing the next piece that offers the most obvious and immediate benefit. Greedy algorithms are used for optimization problems.  An optimization problem can be solved using Greedy if the problem has the following property:   At every step, we can make a choice that looks best at the moment, and we get the optimal solution to the complete problem. 

If a Greedy Algorithm can solve a problem, then it generally becomes the best method to solve that problem as the Greedy algorithms are in general more efficient than other techniques like Dynamic Programming. But Greedy algorithms cannot always be applied. For example, the Fractional Knapsack problem can be solved using Greedy, but 0-1 Knapsack cannot be solved using Greedy.

Following are some standard algorithms that are Greedy algorithms:

1) kruskal’s minimum spanning tree (mst) :  .

In Kruskal’s algorithm, we create an MST by picking edges one by one. The Greedy Choice is to pick the smallest weight edge that doesn’t cause a cycle in the MST constructed so far

2) Prim’s Minimum Spanning Tree :  

In Prim’s algorithm also, we create a MST by picking edges one by one. We maintain two sets: a set of the vertices already included in MST and the set of the vertices not yet included. The Greedy Choice is to pick the smallest weight edge that connects the two sets

3) Dijkstra’s Shortest Path : 

Dijkstra’s algorithm is very similar to Prim’s algorithm. The shortest-path tree is built up, edge by edge. We maintain two sets: a set of the vertices already included in the tree and a set of the vertices not yet included. The Greedy Choice is to pick the edge that connects the two sets and is on the smallest weight path from the source to the set that contains not yet included vertices

4) Huffman Coding :  

Huffman Coding is a loss-less compression technique. It assigns variable-length bit codes to different characters. The Greedy Choice is to assign the least bit length code to the most frequent character.

The greedy algorithms are sometimes also used to get an approximation for Hard optimization problems. For example, Traveling Salesman Problem is an NP-Hard problem. A Greedy choice for this problem is to pick the nearest unvisited city from the current city at every step. These solutions don’t always produce the best optimal solution but can be used to get an approximately optimal solution.

Here let us see one such problem that can be solved using Greedy algorithm

You are given n activities with their start and finish times. Select the maximum number of activities that can be performed by a single person, assuming that a person can only work on a single activity at a time. 

Examples:  

Input: start[]  =  {10, 12, 20}, finish[] =  {20, 25, 30} Output: 0 2 Explanation: A person can perform at most two activities. The  maximum set of activities that can be executed  is {0, 2} [ These are indexes in start[] and finish[] ] Input: start[]  =  {1, 3, 0, 5, 8, 5}, finish[] =  {2, 4, 6, 7, 9, 9}; Output: 0 1 3 4 Explanation: A person can perform at most four activities. The  maximum set of activities that can be executed  is {0, 1, 3, 4} [ These are indexes in start[] and finish[]

Approach : To solve the problem follow the below idea:

The greedy choice is to always pick the next activity whose finish time is the least among the remaining activities and the start time is more than or equal to the finish time of the previously selected activity. We can sort the activities according to their finishing time so that we always consider the next activity as the minimum finishing time activity

Follow the given steps to solve the problem:

  • Sort the activities according to their finishing time 
  • Select the first activity from the sorted array and print it 
  • If the start time of this activity is greater than or equal to the finish time of the previously selected activity then select this activity and print it

Note: In the implementation, it is assumed that the activities are already sorted according to their finish time, otherwise the time complexity will rise to O(N*log(N)) and Auxiliary Space will rise to O(N), as we have to create a 2-D array for storing the start and finish times together.

Below is the implementation of the above approach.

Time Complexity: O(N) Auxiliary Space: O(1)

How does Greedy Choice work for Activities sorted according to finish time?  

Let the given set of activities be S = {1, 2, 3, …n}, and activities are sorted by finish time. The greedy choice is to always pick activity 1. How come activity 1 always provides one of the optimal solutions?

 We can prove it by showing that if there is another solution B with the first activity other than 1, then there is also a solution A of the same size as activity 1 as the first activity. Let the first activity selected by B be k, then there always exist A = {B – {k}} U {1}.

Note: The activities in B are independent and k has the smallest finishing time among all. Since k is not 1, finish(k) >= finish(1))

How to implement when given activities are not sorted?  

We create a structure/class for activities. We sort all activities by finish time (Refer sort in C++ STL ). Once we have the activities sorted, we apply the same algorithm.

Below image is an illustration of the above approach: 

Approach for Solving Activity Selection Problem

Below is the implementation of the above approach:

Time Complexity: O(N log N), If input activities may not be sorted. It takes O(n) time when it is given that input activities are always sorted. Auxiliary Space: O(1)

Activity Selection Problem using Priority-Queue :

We can use Min-Heap to get the activity with minimum finish time. Min-Heap can be implemented using priority-queue
  • Create a priority queue (Min-Heap) and push the activities into it.
  • Push the top of the priority queue into the answer vector and set the variable start to the start time of the first activity and end to the finish time of the activity
  • Take the top of the priority queue and check
  • If the start time of this activity is greater than or equal to the finish time of the last chosen activity then push this activity into the answer vector
  • Else ignore it
  • Print the activities chosen, stored in the answer vector

Time Complexity: O(N * log N) Auxiliary Space: O(N)

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COMMENTS

  1. Greedy Algorithm

    A greedy algorithm is an approach for solving a problem by selecting the best option available at the moment. It doesn't worry whether the current best result will bring the overall optimal result. The algorithm never reverses the earlier decision even if the choice is wrong. It works in a top-down approach.

  2. What is a Greedy Algorithm? Examples of Greedy Algorithms

    In computer science, a greedy algorithm is an algorithm that finds a solution to problems in the shortest time possible. It picks the path that seems optimal at the moment without regard for the overall optimization of the solution that would be formed.

  3. Greedy Algorithms

    Greedy is an algorithmic paradigm that builds up a solution piece by piece, always choosing the next piece that offers the most obvious and immediate benefit. So the problems where choosing locally optimal also leads to global solution are the best fit for Greedy. For example consider the Fractional Knapsack Problem.

  4. Greedy Algorithms

    A greedy algorithm is a simple, intuitive algorithm that is used in optimization problems. The algorithm makes the optimal choice at each step as it attempts to find the overall optimal way to solve the entire problem.

  5. Greedy Algorithms (General Structure and Applications)

    The greedy technique is used for optimization problems (where we have to find the maximum or minimum of something). The Greedy technique is best suited for looking at the immediate situation. All greedy algorithms follow a basic structure: declare an empty result = 0.

  6. Introduction to Greedy Algorithm

    Greedy Algorithm is defined as a method for solving optimization problems by taking decisions that result in the most evident and immediate benefit irrespective of the final outcome. It works for cases where minimization or maximization leads to the required solution. Characteristics of Greedy algorithm

  7. Learn Greedy Algorithms and Solve Coding Challenges

    A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. We just published a course on the freeCodeCamp.org YouTube channel that will teach you how to use greedy algorithms to solve coding challenges. Tanishq Chaudhary developed this course.

  8. Greedy Algorithms Explained with Examples

    A Greedy algorithm makes greedy choices at each step to ensure that the objective function is optimized. The Greedy algorithm has only one shot to compute the optimal solution so that it never goes back and reverses the decision. Greedy algorithms have some advantages and disadvantages:

  9. Greedy algorithm

    A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. [1] In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time.

  10. Greedy algorithm

    Greedy algorithms are among the simplest types of algorithms; as such, they are among the first examples taught when demonstrating the subject. They have the advantage of being ruthlessly efficient, when correct, and they are usually among the most natural approaches to a problem. For this reason, they are often referred to as "naïve methods".

  11. Basics of Greedy Algorithms Tutorials & Notes

    A Greedy algorithm makes greedy choices at each step to ensure that the objective function is optimized. The Greedy algorithm has only one shot to compute the optimal solution so that it never goes back and reverses the decision. Greedy algorithms have some advantages and disadvantages:

  12. Greedy Algorithms

    Greedy Algorithms. A greedy algorithm is a type of algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage with the hope of finding a global optimum. While it may not find the global optimum, greedy algorithms are often simpler and faster while being not too far from the global optimum.

  13. What is Greedy Algorithm: Example, Applications and More

    A Greedy algorithm is an approach to solving a problem that selects the most appropriate option based on the current situation. This algorithm ignores the fact that the current best result may not bring about the overall optimal result. Even if the initial decision was incorrect, the algorithm never reverses it.

  14. Greedy Algorithm with Example: What is, Method and Approach

    To solve a problem based on the greedy approach, there are two stages Scanning the list of items Optimization These stages are covered parallelly in this Greedy algorithm tutorial, on course of division of the array. To understand the greedy approach, you will need to have a working knowledge of recursion and context switching.

  15. Basics of Greedy Algorithms Practice Problems

    ATTEMPTED BY: 131 SUCCESS RATE: 36% LEVEL: Hard. SOLVE NOW. 1 2 3. Solve practice problems for Basics of Greedy Algorithms to test your programming skills. Also go through detailed tutorials to improve your understanding to the topic. | page 1.

  16. When to Use Greedy Algorithms

    One of the most popular greedy algorithms is Dijkstra's algorithm that finds the path with the minimum cost from one vertex to the others in a graph. This algorithm finds such a path by always going to the nearest vertex. That's why we say it is a greedy algorithm. This is pseudocode for the algorithm.

  17. What are Greedy Algorithms? Real-World Applications and Examples

    Greedy algorithms are a popular and powerful technique used in problem-solving and optimization. This class of algorithms focuses on making the best possible decision at each step while considering the current state of the problem.

  18. When to use Greedy Algorithms in Problem Solving

    A greedy algorithm is an algorithmic paradigm that follows the problem-solving heuristic of making the locally optimal choice at each stage with the hope of finding a global optimum. Advantages of ...

  19. Fractional Knapsack Problem

    Naive Approach: To solve the problem follow the below idea: Try all possible subsets with all different fractions. Time Complexity: O (2 N) Auxiliary Space: O (N) Fractional Knapsack Problem using Greedy algorithm: An efficient solution is to use the Greedy approach.

  20. Fractional Knapsack Problem: Greedy algorithm with Example

    Greedy strategies are often used to solve the combinatorial optimization problem by building an option A. Option A is constructed by selecting each component Ai of A until complete (enough n components). For each Ai, you choose Ai optimally.

  21. Top 7 Greedy Algorithm Problems

    Dec 20, 2018 A greedy algorithm is an algorithmic paradigm that follows the problem-solving heuristic of making the locally optimal choice at each stage with the hope of finding a global...

  22. LeetCode Pattern: 19 Tips & Strategies for Solving Greedy Algorithms

    Here are tips and strategies for solving problems using greedy algorithms: I. TIPS & STRATEGIES FOR SOLVING GREEDY ALGORITHMS PROBLEMS: 1. Understand the Problem: ... Problem Type: Optimize tree-related problems using greedy algorithms. # Problem: Minimum Cost to Merge Stones # There are N piles of stones arranged in a row.

  23. Activity Selection Problem

    If a Greedy Algorithm can solve a problem, then it generally becomes the best method to solve that problem as the Greedy algorithms are in general more efficient than other techniques like Dynamic Programming. But Greedy algorithms cannot always be applied.

  24. Dhruvi khandelwal

    Understanding data structures and algorithms is crucial for writing efficient code These are top 3 data structures and algorithms projects (Sa...