2.2. Complexity of the Problem

In this case we want to compute what we call a path. That is the sequence of edges going from one city to another city; where of course the direction must be correct. So, you cannot go backwards, across an edge which is flying from A to B. You cannot take this flight B to A, unless there is another flight.

This chunk discusses the idea of 'paths' in a network. A path refers to a sequence of connections or flights that allow travel from one city to another. Importantly, the directionality of these connections matters. If the flight is only from City A to City B, one cannot simply reverse back unless a direct flight from B to A exists. Thus, understanding how to determine valid paths is crucial for solving connection problems.

Think of it like a one-way street in a city. If you're traveling from your home to a store, and the only road available leads one way, you can't just turn around and go back the same way. Instead, you might have to take a longer route that involves getting onto a different street that loops around. Similarly, when planning flights, you need to identify available routes carefully, as not all paths will work due to directional restrictions.

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...

Here, the text explains that the complexity of determining connections between cities is influenced by two main factors: the number of cities (N) and the number of direct flights (F). As N increases, the number of potential paths grows, which generally increases the complexity of the problem. Similarly, if F is high, it means there are more options for direct connections, and thus more combinations to consider. This aspect of problem-solving is essential for understanding algorithm efficiency.

Imagine organizing a large party with many guests (representing cities). If you have ten guests, managing communications and making connections between them is relatively straightforward. But if the party grows to 100 guests, coordinating who speaks to whom becomes much more complex. Similarly, as the number of flights increases, the number of ways the cities can connect increases exponentially, making the problem harder to solve.

Now, in this case our first question is how do we take this picture and put it into a form that we can manipulate using a program or an algorithm. Now, given the way we represent the graph, we need to manipulate it to answer the question that we handle... If N doubles, does our algorithm take two times more time? Does it take four times more time? If N grows to a factor of ten, does it take ten times more or a hundred times more time?

This chunk delves into the core question of how to translate a visual representation of a problem (the graph of cities and flights) into a format that can be processed by an algorithm. It raises critical questions about how changes in the number of cities (N) affect the algorithm's performance. Understanding how algorithms scale with increased inputs is key to developing efficient solutions.

Consider a recipe that serves four people. If the recipe is scaled up to serve eight, you naturally double the ingredients and time needed. However, if you scale to 40 people, you might not just double the ingredients; some might require more water or larger pots, leading to a different scaling ratio. Similarly, in algorithms, understanding how changes in input affect performance can help predict and manage resource needs effectively.

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 section highlights the complexities that arise when additional constraints, such as time and price, are factoring into the problem of finding a path. Simply connecting two cities is not enough; practical considerations often require a more nuanced approach. For example, travelers may prefer shorter routes or less expensive flights. Understanding these factors is critical to developing algorithms that meet real-world needs.

Imagine you're planning a trip and not only want to get to your destination but also wish to arrive before dinner. You might be willing to pay more for a direct flight instead of a cheaper one with long layovers. Similarly, when designing algorithms, it's essential to consider travelers' needs beyond just finding any route; they seek the most efficient, cost-effective solution that fits their schedules.

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