# Relationship Analysis in Problem Solving

## Look for Relationships

Read the problem statement and ask yourself:

- Are the inputs related to each other?
- Is there a relationship between input and output?
- Can you express the relationship as an equation?
- Is there an implicit variable that can be derived from the input and output?

Think about how the outputs are related to the inputs. Look for any relationships in the input and variables. These relationships may be expressed as formulas. You may have to solve the problem by hand to notice the relationships. It might also require you to have basic mathematical knowledge to express the relationships.

This process can be referred to as “Pattern Recognition” or “Relationship Analysis” in problem-solving. It involves studying the inputs and outputs to identify patterns, connections, and potential relationships that can lead to an efficient algorithmic solution. It’s a critical step in the process of problem decomposition and abstraction, helping to translate a real-world problem into a more mathematical or computational format that’s easier to manage and solve.

Start by exploring possible connections between different parts of the problem. Understanding these relationships can often be the key to discovering an efficient solution.

In an algorithmic problem, we usually have some form of input data, and we want to compute some kind of output from this data. To do this effectively, we need to understand the relationship between the input and the output. This relationship could be direct (like finding the sum of a list of numbers) or more complex (like determining if a given graph contains a cycle).

Express the relationship between input and output as an equation, if possible. This can be an extremely useful technique, especially in problems involving numeric data. A mathematical formula can often provide a clear, efficient way of transforming input into output.

The “implicit variable” can be thought of as a hidden piece of data that isn’t directly given in the problem statement, but can be derived from the given input and output. For example, in a problem where you need to sort a list of numbers, the “implicit variable” might be the number of inversions (pairs of elements that are out of order) in the list. Recognizing and understanding these implicit variables can often be crucial for coming up with an effective solution.

Finally, it’s worth noting that this kind of analysis often requires basic mathematical knowledge and intuition, as well as a willingness to experiment and explore the problem by hand. This kind of hands-on exploration can often reveal patterns and relationships that aren’t immediately obvious from just reading the problem statement.

## Examples

An example for how the inputs are related to each other is the relationship between the number of vertices and the edges in a graph. The number of edges is one less than the number of nodes if they are connected to each other without forming any cycle. We can use this relationship to relate how the output depends on the inputs. For instance if the expression relating the number of edges and vertices is not satisfied, we can return false, since we cannot reach all the nodes in the graph.

A good example for deriving a value for an implicit input variable is the problem where we are given an array of elements and the number of subsets that must add up to the same amount. In this case, we donâ€™t know the value of the amount that the elements in the subset add up to. However, we can sum all the elements in the array and divide by the number of subsets to derive the amount for each subset.

These are great examples of pattern recognition and relationship analysis in problem-solving!

In the case of the graph, you’re leveraging a well-known property of trees, which are a type of graph. In a tree, the number of edges is always one less than the number of vertices. This relationship can help you quickly determine whether a given graph is a tree without having to perform a more complex analysis. In essence, the graph’s structure (its number of vertices and edges) provides a pattern that we can use to derive useful information.

The subset sum example demonstrates how you can use the given inputs (the array and the number of subsets) to calculate an implicit variable (the target sum for each subset). This calculated variable is essential to solving the problem, but it’s not explicitly provided in the problem statement. By understanding the relationship between the array, the number of subsets, and the desired subset sum, you’re able to derive this missing piece of information.

This process of identifying relationships between inputs and using them to solve a problem is a fundamental skill in algorithmic problem solving and is crucial in fields like computer science and mathematics. It can dramatically simplify the problem-solving process and lead to more efficient and effective solutions.

## Pattern Recognition

Pattern recognition and relationship analysis are fundamental skills in algorithmic problem-solving. Here are some common patterns and relationships found in different categories of problems:

**Array Problems**:- Identifying arithmetic or geometric progressions.
- Recognizing sorting-related problems, such as looking for pairs or triplets that satisfy a certain condition.
- Detecting overlapping or non-overlapping subarray problems, such as maximum/minimum subarray sum or length.

**String Problems**:- Recognizing palindromes, anagrams, or other patterns in strings.
- Problems involving prefixes, suffixes, or substrings.
- Problems that can be solved using sliding window technique.

**Tree and Graph Problems**:- Depth-first search (DFS) and breadth-first search (BFS) patterns in tree and graph traversal.
- Spotting problems related to topological sorting, shortest paths, or cycles.
- In trees, if the problem involves traversing a path or the entire tree, recursion is a common approach.

**Number Theory Problems**:- Problems involving prime numbers, factors, multiples, or divisibility rules.
- Euler’s totient function problems, where you have to count the number of integers less than ’n’ that are coprime with ’n'.
- Problems based on modular arithmetic.

**Dynamic Programming Problems**:- Identifying overlapping subproblems.
- Spotting the need for memoization or tabulation to avoid recomputation.
- Recognizing problems that involve choices, where at each step, you have a choice to include or exclude an element.

**Greedy Problems**:- Spotting problems that require locally optimal choices at each step.
- Interval scheduling problems, where you have to choose the maximum number of activities given start and end times.
- Problems where sorting is the first step, such as arranging jobs by finish time.

In all these categories, it’s important to identify relationships between inputs and use them to derive more information or simplify the problem. This process is integral to the development of efficient and effective solutions.