Greedy Algorithms General Structure

A greedy algorithm solves problems by making the best choice at each step. Instead of looking at all possible solutions, it focuses on the option that seems best right now.

Problem structure:

Most of the problems where greedy algorithms work follow these two properties:

1). Greedy Choice Property:- This property states that choosing the best possible option at each step will lead to the best overall solution. If this is not true, a greedy approach may not work.

2). Optimal Substructure:- This means that you can break the problem down into smaller parts, and solving these smaller parts by making greedy choices helps solve the overall problem.

How to Identify Greedy Problems:

There are two major ways to detect greedy problems -

1). Can we break the problem into smaller parts? If so, and solving those parts helps us solve the main problem, it probably would be solved using greedy approach. For example - In activity selection problem, once we have selected a activity then remaining subproblem is to choose those activities that start after the selected activity.

2). Will choosing the best option at each step lead to the best overall solution? If yes, then a greedy algorithm could be a good choice. For example - In Dijkstra’s shortest path algorithm, choosing the minimum-cost edge at each step guarantees the shortest path.

Difference between Greedy and Dynamic Programming:

1). Greedy algorithm works when the problem has Greedy Choice Property and Optimal Substructure, Dynamic programming also works when a problem has optimal substructure but it also requires Overlapping Subproblems.

2). In greedy algorithm each local decision leads to an optimal solution for the entire problem whereas in dynamic programming solution to the main problem depends on the overlapping subproblems.

Some common ways to solve Greedy Problems:

1). Sorting

Job Sequencing:- In order to maximize profits, we prioritize jobs with higher profits. So we sort them in descending order based on profit. For each job, we try to schedule it as late as possible within its deadline to leave earlier slots open for other jobs with closer deadlines.

Activity Selection:- To maximize the number of non-overlapping activities, we prioritize activities that end earlier, which helps us to select more activities. Therefore, we sort them based on their end times in ascending order. Then, we select the first activity and continue adding subsequent activities that start after the previous one has ended.

Disjoint Intervals:- The approach for this problem is exactly similar to previous one, we sort the intervals based on their start or end times in ascending order. Then, select the first interval and continue adding next intervals that start after the previous one ends.

Fractional Knapsack:- The basic idea 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).

Kruskal Algorithm:- To find the Minimum Spanning Tree (MST), we prioritize edges with the smallest weights to minimize the overall cost. We start by sorting all the edges in ascending order based on their weights. Then, we iteratively add edges to the MST while ensuring that adding an edge does not form a cycle.

2). Using Priority Queue or Heaps

Dijkstra Algorithm:- To find the shortest path from a source node to all other nodes in a graph, we prioritize nodes based on the smallest distance from the source node. We begin by initializing the distances and using a min-priority queue. In each iteration, we extract the node with the minimum distance from the priority queue and update the distances of its neighboring nodes. This process continues until all nodes have been processed, ensuring that we find the shortest paths efficiently.

Connect N ropes:- In this problem, the lengths of the ropes picked first are counted multiple times in the total cost. Therefore, the strategy is to connect the two smallest ropes at each step and repeat the process for the remaining ropes. To implement this, we use a min-heap to store all the ropes. In each operation, we extract the top two elements from the heap, add their lengths, and then insert the sum back into the heap. We continue this process until only one rope remains.

Huffman Encoding:- To compress data efficiently, we assign shorter codes to more frequent characters and longer codes to less frequent ones. We start by creating a min-heap that contains all characters and their frequencies. In each iteration, we extract the two nodes with the smallest frequencies, combine them into a new node, and insert this new node back into the heap. This process continues until there is only one node left in the heap.

3). Arbitrary

Minimum Number of Jumps To Reach End:- In this problem we maintain a variable to store maximum reachable position at within the current jump's range and increment the jump counter when the current jump range has been traversed. We stop this process when the maximum reachable position at any point is greater than or equal to the last index value.