❓ What is a Greedy Algorithm : definition, examples of use.

Contents of content show

What is Greedy Algorithm?

A greedy algorithm is a problem-solving method that makes the best choice at each step, hoping that these local solutions will lead to a global optimal solution. It is popular in artificial intelligence for optimization problems where the best immediate option is taken without considering the overall problem.

Main Formulas in Greedy Algorithms

1. Greedy Choice Property

At each step: choose x ∈ S such that f(x) is optimal and x satisfies constraints

The algorithm makes a locally optimal choice x from the set S based on a greedy function f(x), without reconsidering previous decisions.

2. Activity Selection Problem

Sort activities by finish time  
Select activity i if start[i] ≥ finish[last_selected]

This greedy strategy maximizes the number of non-overlapping activities.

3. Fractional Knapsack (Greedy Value Density)

Value Density = value / weight  
Select items in decreasing order of value density

The algorithm prioritizes items with the highest value-to-weight ratio for maximum profit.

4. Huffman Coding (Greedy Merge Cost)

Total Cost = ∑ (frequency × code length)  
Merge two lowest-frequency nodes iteratively

Huffman’s algorithm builds an optimal prefix code by repeatedly merging the two least frequent symbols.

5. Minimum Spanning Tree (Prim’s and Kruskal’s Criteria)

Prim: Add minimum-weight edge connecting visited to unvisited nodes  
Kruskal: Add minimum-weight edge that does not form a cycle

Both Prim’s and Kruskal’s algorithms build a minimum spanning tree using greedy choices based on edge weights.

How Greedy Algorithm Works

The greedy algorithm works by making a sequence of choices that seem the best at each step. For example, in the coin change problem, the algorithm will pick the highest denomination coin available, then repeat this with the remaining amount. This is done until the total is achieved, effectively solving the problem step by step based on local optimizations.

Types of Greedy Algorithm

Prim’s Algorithm. Prim’s algorithm finds the minimum spanning tree in a graph by selecting edges with the lowest weight that connect new vertices.
Kruskal’s Algorithm. Kruskal’s algorithm is another method to find the minimum spanning tree by sorting the edges and adding them one by one while avoiding cycles.
Dijkstra’s Algorithm. Dijkstra’s algorithm is used for finding the shortest path between nodes in a graph, focusing on paths with minimum total weight.
Huffman Coding. This greedy algorithm is used for data compression, constructing a binary tree to represent frequencies of characters to minimize the total number of bits used.
Activity Selection Problem. This algorithm focuses on selecting the maximum number of activities that don’t overlap, by always picking the activity that ends first.

Algorithms Used in Greedy Algorithm

Backtracking. Backtracking is an algorithmic technique that builds a solution incrementally and abandons solutions that do not satisfy the problem constraints.
Dynamic Programming. This approach breaks down problems into simpler subproblems, storing solutions to avoid redundancy, often used alongside greedy methods for optimization.
Branch and Bound. A method that systematically explores the branches of a solution space, giving better bounds for optimization problems.
Divide and Conquer. An algorithm design paradigm that divides a problem into smaller instances, solves them independently, and combines the results.
Binary Search. This algorithm finds an element in a sorted array by repeatedly dividing the search interval in half, ensuring efficient search.

Industries Using Greedy Algorithm

Finance. Financial institutions use greedy algorithms for portfolio selection and asset allocation to optimize returns.
Telecommunications. Companies use greedy algorithms for routing and frequency allocation to effectively manage resources.
Travel and Transportation. Greedy algorithms assist in optimizing routes and schedules, reducing travel time and costs.
Computer Networking. Greedy algorithms help in load balancing and resource allocation across networks, enhancing performance and efficiency.
Manufacturing. Industries utilize greedy methods for production scheduling, optimizing the use of materials and reducing waste.

Practical Use Cases for Businesses Using Greedy Algorithm

Taxi Dispatching. Greedy algorithms are applied to quickly assign the nearest taxi to a passenger, enhancing customer satisfaction and reducing wait times.
Job Scheduling. Companies use greedy methods to schedule jobs on machines, maximizing output by selecting the most beneficial job first.
Data Compression. Greedy algorithms optimize the size of files through effective encoding, streamlining data storage and transmission.
Network Traffic Management. By prioritizing data packets that require immediate handling, greedy algorithms streamline network traffic to improve service quality.
Supply Chain Optimization. Businesses use greedy approaches to efficiently manage inventory levels, ensuring cost-effective operations.

Examples of Applying Greedy Algorithm Formulas

Example 1: Activity Selection Problem

Given activities with start times [1, 3, 0, 5, 8, 5] and finish times [2, 4, 6, 7, 9, 9], sort by finish time and select non-overlapping activities.

Sorted by finish:  
Activities = [(1, 2), (3, 4), (0, 6), (5, 7), (5, 9), (8, 9)]  

Select (1, 2)  
Next valid: (3, 4) → start ≥ 2  
Next valid: (5, 7) → start ≥ 4  
Next valid: (8, 9) → start ≥ 7

Selected activities: (1, 2), (3, 4), (5, 7), (8, 9) — maximum of 4 non-overlapping intervals.

Example 2: Fractional Knapsack Problem

Items: values = [60, 100, 120], weights = [10, 20, 30], knapsack capacity = 50. Use value density.

Value Density = [6, 5, 4]  
Pick item 1 fully: 10 kg → 60  
Pick item 2 fully: 20 kg → 100  
Pick 20 kg of item 3: (20 / 30) × 120 = 80  

Total value = 60 + 100 + 80 = 240

The greedy selection by value density yields a maximum profit of 240 within the weight limit.

Example 3: Huffman Coding Cost

Frequencies of characters: A = 5, B = 9, C = 12, D = 13, E = 16, F = 45.

Step 1: Combine 5 (A) + 9 (B) = 14  
Step 2: Combine 12 (C) + 13 (D) = 25  
Step 3: Combine 14 + 16 (E) = 30  
Step 4: Combine 25 + 30 = 55  
Step 5: Combine 45 (F) + 55 = 100  

Total Huffman cost = 224

Huffman’s greedy merge approach minimizes the total encoding cost of the character set.

Software and Services Using Greedy Algorithm Technology

Software	Description	Pros	Cons
Google Maps	Uses greedy algorithms to provide the shortest routes and real-time traffic updates.	Highly efficient and user-friendly.	Dependent on current traffic data.
Uber	Utilizes greedy algorithms to optimize ride-sharing by assigning drivers to the nearest riders.	Improves response times for customers.	Can lead to uneven distribution of driver workload.
Klout	Employs greedy algorithms to analyze social media influence and user engagement metrics.	Accurate measurements of social media impact.	Data dependency might skew results.
Ad Placement Software	Uses greedy strategies to determine the most profitable ad placements for maximizing revenue.	Increases click-through rates.	Limited to historical data patterns.
E-commerce Recommendation Systems	Adopts greedy algorithms to provide personalized product recommendations.	Enhances user experience and increases sales.	Limited to user data available.

Future Development of Greedy Algorithm Technology

As technology advances, greedy algorithms will become more sophisticated, incorporating machine learning techniques to better evaluate choices based on vast datasets. They will provide enhanced decision-making capabilities for businesses across various sectors, leading to improved efficiencies and reduced operational costs. Their adaptability will ensure relevance in dynamic environments.

Greedy Algorithm: Frequently Asked Questions

How does a greedy algorithm make decisions during execution?

A greedy algorithm makes the best possible decision at each step based on a local criterion, without considering the global outcome or backtracking. This choice is final and influences all subsequent decisions.

When is a greedy algorithm guaranteed to find an optimal solution?

Greedy algorithms produce optimal solutions when the problem exhibits the greedy-choice property and optimal substructure. Classic examples include Huffman coding and activity selection.

Why is greedy strategy not suitable for 0/1 knapsack problem?

In the 0/1 knapsack problem, items must be taken entirely or not at all. A greedy approach based on value or weight may fail to explore better combinations, while dynamic programming considers all possibilities for optimal selection.

How do greedy algorithms compare to dynamic programming?

Greedy algorithms make one-pass decisions and are faster but not always correct. Dynamic programming stores subproblem solutions and explores multiple paths to ensure global optimality, but it is usually more computationally intensive.

Which real-world problems are solved effectively using greedy algorithms?

Greedy algorithms are well-suited for problems like minimum spanning tree construction, scheduling tasks, Huffman encoding, and Dijkstra’s shortest path in graphs with non-negative weights.

Conclusion

The greedy algorithm is a powerful technique in artificial intelligence, offering practical solutions across multiple industries. Its ability to make quick, optimal choices makes it suitable for real-time applications. While it may not always yield the absolute best result, its efficiency and ease of implementation continue to drive its popularity.