Question

Problem background

I am currently developing a framework of Ant Colony System algorithms. I thought I’d start out by trying them on the first problem they were applied to: Travelling Salesman Problem (TSP). I will be using C# for the task.

All TSP instances will consist of a complete undirected graph with 2 different weights associated with each edge.

Question

Until now I’ve only used adjacency-list representations but I’ve read that they are recommended only for sparse graphs. As I am not the most knowledgeable of persons when it comes to data structures I was wondering what would be the most efficient way to implement an undirected complete graph?

I can provide additional details if required.

Thank you for your time.

UPDATE

Weight clarification. Each edge will have the two values associated with them:

1. distance between two cities ( d(i,j) = d(j,i) same distance in both directions)
2. amount of pheromone deposited by ants on that particular edge

Operations. Small summary of the operations I will be doing on the graph:

• for each node, the ant on that particular node will have to iterate through the values associated with all incident edges

Problem clarification

Ant Colony Optimization algorithms can “solve” TSP as this is where they were first applied to . I say “solve” because they are part of a family of algorithms called metaheuristics optimizations, thus they never guarantee to return the optimal solution.

Regarding the problem at hand:

• ants will know how to complete a tour because each ant will have a memory.
• each time an ant visits a city it will store that city in its memory.
• each time an ant considers visiting a new city it will search in its memory and pick an outgoing edge only if that edge will not lead it to an already visited city.
• when there are no more edges the ant can choose it has complete a tour; at this point we can retrace the tour created by the ant by backtracking through its memory.

Research article details: Ant Colony System article

Efficiency considerations

I am more worried about run time (speed) than memory.

Answer 1

First, there’s a general guide to adjacency lists vs matrices here. That’s a pretty low-level, non-specific discussion, though, so it might not tell you anything you don’t already know.

The takeaway, I think, is this: If you often find yourself needing to answer the question, “I need to know the distance or the pheromone level of the edge between exactly node i and node j,” then you probably want the matrix form, as that question can be answered in O(1) time.

You do mention needing to iterate over the edges adjacent to a node– here is where some cleverness and subtlety may come in. If you don’t care about the order of the iteration, then you don’t care about the data structure. If you care deeply about the order, and you know the order up front, and it never changes, you can probably code this directly into an adjacency list. If you find yourself always wanting, e.g., the largest or smallest concentration of pheromones, you may want to try something even more structured, like a priority queue. It really depends on what kind of operations you’re doing.

Finally, I know you mention that you’re more interested in speed than memory, but it’s not clear to me how many graph representations you’ll be using. If only one, then you truly don’t care. But, if each ant is building up its own representation of the graph as it goes along, you might care more than you think, and the adjacency list will let you carry around incomplete graph representations; the flip side of that is that it will take time to build the representations up when the ant is exploring on its frontier.

Finally finally, I know you say you’re dealing with complete graphs and TSP, but it is worth thinking about whether you will ever need to adapt these routines to some other problem on possibly graphs, and if so, what then.

I lean toward adjacency lists and/or even more structure, but I don’t think you will find a clean, crisp answer.