# Visual Model of a Problem

Modeling is the art of formulating your application in terms of precisely described well understood problems. You must learn to describe a problem abstractly, in terms of procedures on fundamental structures.

In this phase, we devote the necessary time and critical thinking required to analyze a problem, understand its basic terms and expression. We then carry out some calculations. To carry these calculations out, we often need to develop a model. We have to identify the most important components of the problem and describe them in a meaningful way. This is how we model the problem.

The quality of the solution depends on the quality of the model. The process of solving any real world problem consists of the same two steps:

- Building a model of the problem.
- Using that model to generate a solution.

Modeling a problem is a critical step in the problem-solving process, and often involves abstract thinking and a firm grasp of the problem domain. When we talk about building a conceptual model, we are essentially trying to abstract away from the real-world specifics of a problem and represent it in a more generalized, structured form that we can manipulate and analyze mathematically or algorithmically.

Let’s elaborate this with an example. Suppose we have a real-world problem, such as optimizing the routes of delivery trucks in a city to minimize fuel consumption. The city, its streets, and the trucks are all real-world entities, but they are too complex and too specific to analyze directly.

**Building a model**

To build a model of this problem, we must identify the key components and their relationships:

- The city can be modeled as a graph, where each intersection is a node and each street is an edge connecting two nodes.
- The delivery trucks can be modeled as entities that move along the edges of the graph.
- The fuel consumption can be modeled as a cost associated with each edge in the graph - perhaps based on the length of the street, the speed limit, traffic conditions, etc.

This model strips away unneeded details (like the names of the streets or the color of the delivery trucks) and represents the problem in terms of well-understood structures - in this case, a graph with costs associated with each edge. The problem then becomes a well-known problem in computer science: finding the minimum-cost path in a graph, which can be solved using various algorithms.

**Using the model**

Once the model is built, we can use it to generate a solution to our problem. This involves applying relevant algorithms or mathematical techniques to the model. In our example, we might use Dijkstra’s algorithm to find the shortest path for each delivery truck.

The solution generated from the model can then be interpreted back into the real world. In this case, the shortest path in the graph corresponds to the optimal route for each delivery truck in the city.

The process of problem modeling is a crucial aspect of problem-solving, allowing complex real-world problems to be translated into abstract forms that can be solved using mathematical or algorithmic techniques. This process is fundamental in fields such as computer science, operations research, and engineering.

## Building a Visual Model

How to build a visual model of a real world problem?

Building a visual model of a real-world problem involves several steps, typically including:

**Identify Key Components**: Understand the key elements, entities, or factors in the problem.**Draw Basic Elements**: Start by representing these elements visually. For example, in a problem involving relations between people, you might draw circles for each person.**Establish Relationships**: Identify how these elements interact or relate to one another. This could be expressed through arrows, lines connecting the elements, or by arranging the elements spatially based on their relationships.**Add Details**: If necessary, add more detail to your visual model. For example, you might add labels, color code different types of elements, or use shapes to represent different categories.**Test Your Model**: Use your visual model to walk through potential scenarios or solutions. Does it accurately represent the problem? Does it help you to understand and solve the problem?

For example, if you’re trying to solve a problem about traffic flow, you might draw a map of the roads involved, add symbols for cars, and use arrows to show the direction of traffic. This visual model can then be used to simulate different scenarios, such as adding a new traffic light or changing a two-way road into a one-way road. By visualizing the problem, you can gain new insights and potentially identify solutions more easily.

Remember, the purpose of a visual model is to simplify complex real-world problems into a form that is easier to understand and manipulate, enabling problem-solving.

There are standard ways of ordering information on paper for certain classes of problems. These methods are not always obvious. For instance, for matrix problems, we put the information into a table or matrix. Graph is the visual model for path problems.

There are ways we can organize and visualize problem-related information to facilitate the solution. Different classes of problems may require different representation methods. These methods help in simplifying and clarifying the problem, allowing us to understand its structure better and develop suitable solutions.

For instance, when dealing with matrix problems, it’s natural and often beneficial to represent the data in a table or matrix format. Matrix problems often involve multidimensional data, so a two-dimensional grid-like structure (i.e., a matrix or a table) provides a visually intuitive way to organize and manipulate the data.

Common operations in matrix problems include traversing rows and columns, swapping elements, performing calculations on specific cell values, etc. By visualizing the data in a table or matrix, we can more easily see patterns, understand relationships, and plan efficient ways to perform these operations.

Similarly, for problems related to paths, connectivity, and networks - typically solved with graph theory - a graph is a more appropriate and intuitive visual model. In a graph, we represent items (nodes) and the connections between them (edges), which can simplify understanding the relationships and pathways within the problem space.

For example, if you’re trying to find the shortest path between two cities with various possible routes, visually representing the cities as nodes and the routes as edges on a graph can help you better understand and solve the problem.

Both of these are instances of how problem representation, matching the nature of the problem, can greatly aid in problem-solving. However, the best way to organize information is not always obvious and might require some problem-solving experience and deep understanding of the problem’s nature.

## Simplicity vs Accuracy of a Model

We are finding a solution to the model of a problem. If the model accurately represents the problem, then the solution will be meaningful. Conversely, if the model is inaccurate or has vague assumptions and rough approximations, then the solution may be meaningless.

For this reason, it is critical to make the distinction between the model and the problem, because the model we work with is not the same as the problem. It is only a representation of the real problem. Furthermore, every model leaves something out. Otherwise it would be as complicated as the real world. As a result, we always end up working with simplifications of real things.

When we create a model for a problem, what we’re essentially doing is abstracting the complex elements of the real-world problem into a simpler, more manageable representation. A model is a kind of bridge between the concrete real-world problem and the abstract mathematical or algorithmic methods we use to solve it. It’s crucial to understand that the model isn’t the problem itself; it’s our interpretation or representation of the problem.

Let’s delve a bit deeper into this using the example of the delivery trucks and city routes from before.

The real problem involves the specifics of the city: the street layouts, the number of delivery trucks, traffic rules, fuel consumption rates, and many other factors. Our model, however, is a simplified version of this: it represents the city as a graph and the fuel consumption as cost values on each edge.

The accuracy of our model is vital for the solution to make sense in the real world. If we ignore important aspects (e.g., we don’t account for one-way streets or peak traffic times), our model may lead to a solution that doesn’t work in reality (e.g., suggesting a route that goes the wrong way on a one-way street or that gets stuck in peak hour traffic).

Every model makes some simplifications, and that’s okay—it’s actually necessary, because an exact model of the real world would be as complex as the real world itself, and we wouldn’t gain anything by creating it. However, we have to be careful about what we leave out when we simplify. The key is to include enough detail for the model to be accurate for the problem at hand, but not so much detail that the model becomes unwieldy or impossible to solve.

So, modeling is a delicate balance between simplicity and accuracy. A good model provides just enough abstraction to make the problem solvable while preserving enough detail to ensure that the solution is meaningful and applicable to the original, real-world problem.

## Ways to Visualize

There are various ways to organize and visualize problem-related information depending on the nature of the problem. Some common methods include:

**Arrays and Lists:**These are used for problems dealing with sequences or collections of items. They are especially helpful when there’s a need to iterate over elements in a specific order, or when the problem involves finding a specific item within a collection.**Trees:**These are used in problems that involve hierarchical relationships or when problems can be broken down into smaller subproblems. Examples of such problems include parsing expressions, managing sorted data, and handling queries efficiently.**Stacks and Queues:**These are used in problems that require specific order of processing. Stacks are often used in problems involving nested or recursive structures (like parsing parentheses in an expression), while queues are useful in breadth-first search algorithms and in simulations of real-world processes (like processing tasks in order of their arrival).**Hash Tables:**These are used when problems require constant-time lookups, or when there is a need to track unique elements, count occurrences, or store data for quick lookups.**Heap:**Heaps are used in problems requiring efficient maximum or minimum comparisons, like finding the top k elements in a collection.**Graphs:**As mentioned before, graphs are extremely useful for problems that involve relationships and connections, like finding the shortest path, or detecting cycles.**2D Grids:**They are typically used to visually represent two-dimensional space, making them ideal for problems involving physical layouts, such as mazes, or board games like chess.**Point plots:**These are used to visualize individual elements on a two or three dimensional plane, useful for problems dealing with spatial coordinates.**Venn Diagrams:**They are used for problems involving sets and their intersections and unions.**Timelines and Schedules:**Used for problems involving sequences of events, intervals, or scheduling issues.

Remember, visual models are tools to help problem solving. Different problems or even the same problem can benefit from different visualizations. Sometimes combining multiple methods can lead to a better understanding of the problem.

All the ten methods are relevant to solving algorithmic problems. The method you would use depends on the nature of the problem at hand. Let’s take a closer look:

**Arrays and Lists:**Almost all problems involving data manipulation, searching, and sorting can use arrays and lists. They are foundational data structures in algorithm design.**Trees:**Problems related to hierarchical data, such as file systems, or binary search trees for efficient search and sort operations, use trees.**Stacks and Queues:**These are commonly used in problems involving sequences, such as evaluating expressions or managing process execution.**Hash Tables:**Any problem requiring quick access to data, such as phone book lookups, caching, or counting distinct elements, benefits from hash tables.**Heap:**Problems requiring efficient maximum or minimum comparisons, or those that need to maintain a sorted structure, can use a heap.**Graphs:**Used in a wide variety of problems ranging from network routing, social network analysis, recommendation systems to pathfinding in games and maps.**2D Grids:**Useful in many 2D simulation problems, maze problems, game-state representation in AI (like chess or tic-tac-toe).**Point plots:**Useful in problems related to geometry, clustering, or any problem where spatial relationships between points is important.**Venn Diagrams:**Used in set theory and probability problems, these diagrams can help in problems requiring understanding of unions, intersections, and complements.**Timelines and Schedules:**Useful for problems involving intervals, such as meeting room scheduling, or for visualizing algorithms over time (like sorting or scheduling algorithms).

It’s important to note that these are not exclusive categories. Many complex problems may require the use of multiple data structures or visualizations for different parts of the problem.

## Creating an Effective Model

A good model should satisfy two requirements:

- It should be general enough, so that irrelevant details of the problem are hidden.
- It should be specific enough, so that we can derive a meaningful solution.

If all aspects of the real world problem are represented in the model, then the model would be as complex as the real world. On the other hand, we have to preserve the important characteristics of the problem. Otherwise the derived solution might be meaningless.

A piece of information should be included in the model if it is important for deriving a solution to the problem.

Creating a useful model for a problem is an art in itself. We have to find a balance between abstraction and specificity. The model should be general enough to hide unnecessary details, yet specific enough to include all the important characteristics that influence the problem’s solution.

Let’s think of a model as a map. If a map included every single detail of a real geographical area – every tree, every rock, every small path – it would be too complicated to use for general navigation. But if the map was too simplified, leaving out major roads or landmarks, it would not help users to reach their destination. A good map (model) contains just the right amount of detail: enough to help users navigate effectively but not so much that it becomes confusing or overwhelming.

In terms of problem-solving, a model that is too general may not allow us to find a meaningful solution because it leaves out important factors. For example, if we were trying to optimize a delivery route for a fleet of trucks, and our model didn’t account for the variable of traffic congestion at different times of day, our solution may be unrealistic or inefficient in the real world.

Conversely, a model that is too specific may be too complicated to find a solution, or it may create a solution that is overly complex and difficult to implement. Using the same example, if we were to account for every possible variable (e.g., weather conditions, individual driving styles, mechanical condition of the trucks), our model may become too complex, making it difficult to derive a practical solution.

Thus, when building a model, we should aim to include the essential information that directly affects the problem’s outcome and ignore the less influential details. This is where domain knowledge and critical thinking play a crucial role. They guide us in identifying which pieces of information are important for the problem and which ones are not.

In conclusion, creating an effective model is about achieving a balance. It involves understanding the essence of the problem and representing it in a simplified yet sufficiently detailed way. This facilitates deriving meaningful, practical, and implementable solutions.

## Efficient Problem Solving

Solid calculations and reasoning are more meaningful when you build a model of the problem by defining its variables, constraints and objectives. To build a model of the problem, we need to have a very good understanding of the problem. We need to identify the variables, constraints and objectives from the problem statement.

We are building a model of the problem to make solid calculations. Finding a solution requires reasoning within the model. It may mean applying mathematical principles, narrowing the domains of the variables, applying an optimization technique, interpreting data sets etc.

Modeling a problem in a structured manner provides a foundation for clear thinking, precise calculation, and efficient problem-solving. It involves defining the variables, constraints, and objectives, which form the key components of any problem model. Let’s break this down:

**Variables**: These are the elements that can change within the problem. They represent the unknowns we are trying to solve for. For example, in a logistics problem, the variables could be the number of trucks, their routes, and their departure times.**Constraints**: These are the limitations or conditions that the solution must meet. They define the feasible region of the problem. For instance, in the logistics problem, constraints could include the maximum load a truck can carry, the working hours of drivers, or the requirement to deliver all goods within a certain timeframe.**Objectives**: This represents what we’re trying to achieve or optimize. In other words, it’s the goal of our problem-solving process. In our logistics example, the objective might be to minimize total delivery time or cost.

Building a model of a problem involves translating the real-world problem into an abstract representation that captures these elements. Once we have this model, we can perform calculations and reasoning within its structure.

For instance, we could apply mathematical principles or algorithms to find the optimal solution. For instance, if we’re dealing with a scheduling problem, we might apply graph theory to find the most efficient sequence of tasks. Alternatively, we could use optimization techniques to find the best set of variable values that minimize or maximize our objective, subject to the constraints.

The process may also involve narrowing the domain of the variables, i.e., reducing the set of possible values they can take based on the problem constraints. For example, if we know a delivery truck can only operate within a certain geographical area, we can exclude routes that go beyond this area.

Finally, the process of solving the problem might involve interpreting datasets. This is particularly true in data-driven problems, where the solution depends on patterns or trends in the data.

In summary, building a model of a problem sets the stage for clear, structured problem-solving. It allows us to use mathematical and computational tools effectively and derive meaningful, actionable solutions.

## Map as a Model

Consider the map as a model used for the purpose of travel. We construct the model of the problem by specifying:

- Variables
- Constraints
- Objective

The formulation of the objective produces an equation relating the variables. The constraint can be represented as an equation. The questions to ask yourself to formulate the mathematical model for the problem are:

- Is the model adequate for the problem?
- Did we include all the relevant variables, constraints and the objective?
- Is the model precise enough?

Complex models are more difficult to solve. For any given problem we can build several different models of varying complexity. For example, we can create many different maps of varying precision.

Creating a model of a problem, much like creating a map for travel, involves defining the core components that allow us to navigate through the problem space. These components are:

**Variables**: These are the dynamic elements within the problem. In a map, this could be the current location, destination, or the various routes one could take. In other problems, these could be quantities we wish to optimize or elements we need to control.**Constraints**: These are the limitations within which we must operate. On a map, these might be roads that are under construction or areas that are inaccessible. In other problems, constraints can represent physical, budgetary, or temporal limitations, among other things.**Objective**: This is the goal we aim to achieve. In terms of a map, the objective would be to reach the destination in the shortest time or with the least distance. In other problems, the objective could be to maximize profit, minimize waste, optimize performance, etc.

Once we’ve identified the variables, constraints, and objective, we can formulate them into mathematical equations that define our model. For example, in a traveling salesman problem, the objective (minimizing total travel distance) can be represented as a function of the variables (the order of cities to visit), subject to constraints (visiting each city only once).

However, we must ensure that the model we build is both adequate and precise.

**Adequacy** relates to whether our model represents all relevant aspects of the problem. Have we included all necessary variables and constraints? Does our objective accurately reflect what we’re trying to achieve? If any critical element is missing, the model may lead to flawed solutions.

**Precision**, on the other hand, deals with the level of detail included in our model. A highly precise model includes a high level of detail, which can make it more accurate but also more complex and difficult to solve. Conversely, a less precise model might be easier to work with but may not capture all nuances of the problem.

Just as a map can be drawn at different scales, a problem can be modeled at varying levels of complexity. Choosing the right level of complexity depends on the problem at hand, the available resources, and the precision required in the solution.

In summary, modeling a problem involves careful identification and representation of variables, constraints, and objectives, striking a balance between the model’s adequacy and precision to ensure a meaningful and feasible solution.

## Extracting Key Insights

There is a tradeoff between the quality of the model and cost of finding a solution. The more complex and precise the model, the harder it is to find a solution. Some problems are already formulated in terms of a model. For example, factorial and Fibonacci. For some problems, we have to derive the mathematical model, for example the number of ways to climb stairs.

Our abstract model must allow us to calculate as well as reason about the solution and make sense. We gradually build the model of the problem. Clarify any issues during the process. For many problems, the process of building a model can lead to important insights and consequently to a quick solution.

The model is an abstraction of the real world. It expresses the mathematical functions that represent the behavior of the system. We abstract by concentrating on the dominant variables that control the behavior of the real system.

Just making observations about a system is not sufficient to come up with solutions. We have to distill our observations to key insights that we can use to solve the problem and come up with elegant solutions.

Modeling a problem accurately is a balancing act. On one hand, we need the model to be as close to the real problem as possible to ensure that the solution is relevant and useful. This means including all the important variables, constraints, and relationships between them. On the other hand, the more complex and detailed the model, the more challenging and resource-intensive it is to find a solution.

For some problems, like calculating factorial or Fibonacci numbers, the models are already well-defined. These problems have clear mathematical definitions that we can directly translate into algorithms.

For other problems, like determining the number of ways to climb a set of stairs given that you can climb 1, 2, or 3 steps at a time, we need to develop a model. In this case, we might note that the number of ways to climb ’n’ steps is the sum of the number of ways to climb ’n-1’, ’n-2’, and ’n-3’ steps. This observation, derived from the problem, forms our mathematical model and provides a clear approach for finding a solution.

Creating a useful model is an iterative process. As we start defining variables, constraints, and objectives, we might uncover more nuances of the problem. We might need to revise our model, add more variables, or redefine constraints. The process of building and refining a model often leads to a deeper understanding of the problem and can reveal novel approaches to a solution.

The goal of a model is to simplify the real-world problem by focusing on the most important variables and their relationships, creating an abstract representation that is manageable yet maintains the essential characteristics of the problem. We essentially create a mathematical ‘world’ in which we can perform calculations, draw conclusions, and predict outcomes.

Observing a system or a problem is just the first step. To move from observations to solutions, we need to extract the key insights from our observations, construct an accurate model, and then use this model to reason about the problem. This process can lead to innovative and elegant solutions that might not be immediately apparent from the raw observations of the problem.

## Examine the Problem

Understand the problem and its terminology. Finding a solution would require some reasoning within our model. The three primary tools we can use are:

- Applying mathematical principles
- Narrowing the domains of the variables
- Apply optimization technique

The model determines the usefulness of the solution and the amount of time it takes to find a solution. The key question is: how can we build models that are quick to build and easy to use? It is necessary to investigate the problem for special characteristics. What are these special characteristics? We already saw how we can look for clues in the problem space and the table showed some of the clues and the approaches we can take.

Understanding the problem and its specific terminology is a crucial first step in solving any algorithmic problem. This step entails getting familiar with the given data and constraints and identifying the key elements of the problem.

Once the problem is well understood, we proceed to build a model of the problem. This model will then be used for problem-solving by applying the following primary tools:

**Applying mathematical principles:**Mathematics provides the language and structure for describing and solving problems. It allows us to precisely express relationships between variables and to derive solutions using various mathematical principles such as arithmetic, algebra, calculus, statistics, etc. For example, we might use combinatorics (the study of counting, arrangement, and combination) to solve problems about permutations or combinations.**Narrowing the domains of the variables:**This involves focusing on a subset of all possible values for the variables based on the problem’s constraints. By doing so, we reduce the size of the problem space and hence the computation required. For example, if we are looking for a solution in an array of numbers and we know that the solution is positive, we can ignore all negative numbers.**Applying optimization techniques:**These are methods used to find the best (optimal) solution among a set of possible solutions. Optimization techniques help to identify the most efficient solution in terms of resource usage (like time and space). For instance, we might use dynamic programming to find the shortest path in a graph.

The usefulness and efficiency of the solution heavily depend on the model’s quality. A well-constructed model can significantly reduce the time and effort required to find a solution. But how can we ensure our models are efficient and easy to use?

We need to examine the problem thoroughly for any special characteristics or patterns. These characteristics can serve as clues to guide our modeling and problem-solving approach. For instance, the presence of a sorted array might hint at using binary search, the need for frequent insertions and deletions may suggest a linked list, and so forth.

The key is to remain observant and adaptable, modifying our model as we gain a deeper understanding of the problem. This process ensures that our model is tailor-made for the problem at hand, increasing its effectiveness and utility.