2.2.1. Factors Affecting Performance

In terms of efficiency we have to look at what are the things that determine the complexity of a problem. It is fairly obvious in this particular case that if we have more cities, the problem is more complicated. So the number of cities, which we can call N, is certainly one parameter which determines how complicated the algorithm is going to be, or not how complicated the algorithm is going to be, or rather how long it is going to take to run.

This chunk discusses how the complexity of a problem can be affected by the number of cities connected in a network. The parameter N represents the number of cities. The more cities there are, the more complex the problem becomes, meaning that algorithms will require more time to process and find solutions. Understanding this relationship is crucial for designing efficient algorithms.

Imagine trying to find a route in a small city (like a neighborhood with only a few streets) compared to a large city (like New York City with numerous streets and intersections). In the small city, it’s easy to find a route to your destination because there are fewer choices to consider. In contrast, in the large city, the number of possible routes increases significantly, making the task much harder and requiring more time to figure out the best path.

The other question which determines how complex the network is this how many direct flights there are... Obviously there are fewer flights, there are fewer places, which can be connected and we have to explore fewer possibilities. So from this, it follows that computing the paths depends on both N and F.

Here, we introduce another factor, F, which represents the number of direct flights between cities. This chunk emphasizes that the fewer direct flights available, the simpler the problem becomes, as there are fewer connections to navigate. The complexity of finding paths through the network depends on both the number of cities (N) and the number of flights (F), and these factors must be considered when developing an algorithm to compute these paths.

Consider a travel planner helping you find a route. If you’re looking at a map with a few direct flights connecting various cities, the planner can quickly suggest routes. However, if there are many cities with only a few flights between them, the planner has to explore many more options and potential connections, which takes longer to find the best route.

Now what is this dependency, how does it grow? If N doubles, does our algorithm take two times more times? Does it takes four times more times? If N term grows to factor of ten, does it takes ten times more or hundred times more time? The other question related to this is given this dependency on N and F what realistic size of networks can we handle?

This section raises critical questions about how the complexity of algorithms scales with the size of the network. It asks whether doubling the number of cities (N) results in doubling the processing time or whether it grows more dramatically (like quadratically). Understanding this growth is important for determining the practical limits of the algorithms we develop in real-world applications.

Think of a classroom activity where students need to solve a puzzle. If there are just 10 pieces (like having 10 cities), the task is relatively quick. However, if you increase it to 100 pieces, the time to solve it may not just double; it might take much longer as students need to consider many more possibilities. This illustrates the concept of scaling complexity with size, applicable to travel networks.

And then of course the problem that we have looked at is a very simple problem; can I get from A to B. But very often it is not good enough to get from A to B. You want to get from A to B within some reasonable time frame.

This chunk describes the added complexity of not just finding a path between two locations (A to B) but also considering time constraints. It's important for algorithms to account for how long the journey will take, making the problem more complex. This reflects real-world expectations where travelers need timely results, not just a confirmation of connectivity.

Imagine booking a flight for an important meeting. You don’t just want to know if there’s a flight from your city to your destination; you need to be there on time. Therefore, searching for the fastest, most efficient options becomes crucial, making the problem more challenging.

Suppose, as you would you expect each sector on this thing has a cost. As a passenger, the cost would be the price of ticket.

Here, we discuss the consideration of costs associated with travel, not only in terms of money but also in terms of time. Different passengers have varying priorities — some find the cheapest route most important, while others prioritize travel time. This multiple dimensionality adds complexity to the algorithm used for finding optimal paths.

When choosing between two similar routes, a business traveler might opt for the more expensive ticket that gets them to their meeting faster, while a vacationer might choose a longer route if it saves money. This illustrates how different needs lead to different decisions, complicating the algorithm that determines the best travel path.

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