Valid Arrangement of Pairs

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class Solution:
    def validArrangement(self, pairs: List[List[int]]) -> List[List[int]]:    
        adjacency_list = defaultdict(list)
        degree = Counter()
        for u, v in pairs:
            adjacency_list[u].append(v)
            degree[u] += 1
            degree[v] -= 1

        for start in adjacency_list:
            if degree[start] == 1:
                break

        answer = []
        def dfs(node):
            while adjacency_list[node]:
                dfs(neighbor := adjacency_list[node].pop())
                answer.append([node, neighbor])
        dfs(start)
        return answer[::-1]

Identifying Problem Isomorphism

“Valid Arrangement of Pairs” has an approximate isomorphism: “Reconstruct Itinerary”.

In “Reconstruct Itinerary”, you are given a list of flights represented as pairs of strings, where the first string is the departure airport and the second string is the arrival airport. The task is to reconstruct the itinerary in such a way that all flights are used and the itinerary begins at ‘JFK’. If there are multiple valid itineraries, you should return the itinerary that has the smallest lexical order.

In “Valid Arrangement of Pairs”, you are given pairs of integers and asked to arrange them in a valid order such that for every index i where 1 <= i < pairs.length, we have endi-1 == starti.

The mapping can be visualized: The ‘start’ and ’end’ elements of the pairs in “Valid Arrangement of Pairs” correspond to the ‘departure’ and ‘arrival’ airports in “Reconstruct Itinerary”. The condition endi-1 == starti is similar to following the itinerary from one airport to the next in “Reconstruct Itinerary”.

This is an approximate mapping. “Reconstruct Itinerary” involves lexical ordering, whereas “Valid Arrangement of Pairs” doesn’t. “Reconstruct Itinerary” requires to start from a specific point (‘JFK’), while the starting point can be any in “Valid Arrangement of Pairs”.

“Reconstruct Itinerary” is more complex because it has an extra requirement of lexical order and a fixed starting point.

Although these two problems are not exactly isomorphic, their solutions share a common strategy, which is to traverse the pairs in a certain order to create a ‘path’.

10 Prerequisite LeetCode Problems

This involves graph theory, and more specifically, Eulerian paths. You could view each pair as an edge in a graph, and you are trying to find a path that visits each edge exactly once, such that the end of one edge matches the start of the next.

Here are 10 LeetCode problems to prepare for this problem:

LeetCode 207. Course Schedule
- This problem is a good introduction to the concept of detecting cycles in a directed graph using depth-first search (DFS).
LeetCode 210. Course Schedule II
- This problem extends the previous problem by asking for a valid order of courses, which introduces the idea of topological sort.
LeetCode 269. Alien Dictionary
- This problem also involves topological sorting and the creation of a directed graph from given conditions.
LeetCode 399. Evaluate Division
- This problem involves creating a graph and then finding paths between certain nodes, which can be a good introduction to the type of graph traversal you will need to do in your target problem.
LeetCode 743. Network Delay Time
- This problem introduces the concept of shortest path algorithms in a weighted graph, which is a fundamental graph theory concept.
LeetCode 787. Cheapest Flights Within K Stops
- This problem requires finding the shortest path in a graph with a constraint on the number of edges in the path, introducing a level of complexity that might be helpful for your target problem.
LeetCode 332. Reconstruct Itinerary
- This problem involves finding an Eulerian path in a graph, which is very similar to what you’re trying to do in your target problem. It’s a key problem to understand.
LeetCode 797. All Paths From Source to Target
- This problem asks you to find all paths from the source to the target in a directed acyclic graph (DAG), which is an important skill for understanding graph traversals.
LeetCode 684. Redundant Connection
- This problem involves finding cycles in an undirected graph using Union-Find, which is a key algorithm in graph theory.
LeetCode 200. Number of Islands
- This problem involves using depth-first search (DFS) or breadth-first search (BFS) to count the number of connected components in a grid.

Problem Classification

Problem Statement:You are given a 0-indexed 2D integer array pairs where pairs[i] = [starti, endi]. An arrangement of pairs is valid if for every index i where 1 <= i < pairs.length, we have endi-1 == starti. Return any valid arrangement of pairs. Note: The inputs will be generated such that there exists a valid arrangement of pairs.

Example 1:

Input: pairs = [[5,1],[4,5],[11,9],[9,4]] Output: [[11,9],[9,4],[4,5],[5,1]] Explanation: This is a valid arrangement since endi-1 always equals starti. end0 = 9 == 9 = start1 end1 = 4 == 4 = start2 end2 = 5 == 5 = start3 Example 2:

Input: pairs = [[1,3],[3,2],[2,1]] Output: [[1,3],[3,2],[2,1]] Explanation: This is a valid arrangement since endi-1 always equals starti. end0 = 3 == 3 = start1 end1 = 2 == 2 = start2 The arrangements [[2,1],[1,3],[3,2]] and [[3,2],[2,1],[1,3]] are also valid. Example 3:

Input: pairs = [[1,2],[1,3],[2,1]] Output: [[1,2],[2,1],[1,3]] Explanation: This is a valid arrangement since endi-1 always equals starti. end0 = 2 == 2 = start1 end1 = 1 == 1 = start2

Constraints:

1 <= pairs.length <= 105 pairs[i].length == 2 0 <= starti, endi <= 109 starti != endi No two pairs are exactly the same. There exists a valid arrangement of pairs.

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.