Number of Good Paths


10 Prerequisite LeetCode Problems
“Number of Good Paths” is related to Graph Theory and Dynamic Programming. Here are 10 problems as preparation:
“Number of Islands” (LeetCode Problem #200): This problem involves traversing a grid, which can be considered a basic form of a graph.
“Flood Fill” (LeetCode Problem #733): This problem involves performing a depthfirst search on a grid, similar to a graph.
“Find the Town Judge” (LeetCode Problem #997): This problem introduces the concept of indegrees and outdegrees in a graph.
“Course Schedule” (LeetCode Problem #207): This problem requires understanding of topological sort, a key concept in graph theory.
“All Paths From Source to Target” (LeetCode Problem #797): This problem introduces the concept of pathfinding in a directed graph.
“Minimum Path Sum” (LeetCode Problem #64): This problem introduces pathfinding in a grid with additional complexity of finding the minimum path sum.
“Shortest Path in Binary Matrix” (LeetCode Problem #1091): This problem practices finding shortest paths in a grid which is a form of a graph.
“Clone Graph” (LeetCode Problem #133): This problem practices deep copying of a graph, which helps in understanding graph traversal.
“Path with Maximum Gold” (LeetCode Problem #1219): This problem requires finding a path in a grid, and introduces the concept of calculating paths with certain properties (maximum gold in this case).
“Graph Valid Tree” (LeetCode Problem #261): This problem practices understanding and identifying properties of a graph.
These cover basic graph theory and dynamic programming, which are key to solving the “Number of Good Paths” problem.
Problem Classification
Problem Statement: There is a tree (i.e. a connected, undirected graph with no cycles) consisting of n nodes numbered from 0 to n  1 and exactly n  1 edges.
You are given a 0indexed integer array vals of length n where vals[i] denotes the value of the ith node. You are also given a 2D integer array edges where edges[i] = [ai, bi] denotes that there exists an undirected edge connecting nodes ai and bi.
A good path is a simple path that satisfies the following conditions:
The starting node and the ending node have the same value. All nodes between the starting node and the ending node have values less than or equal to the starting node (i.e. the starting node’s value should be the maximum value along the path). Return the number of distinct good paths.
Note that a path and its reverse are counted as the same path. For example, 0 > 1 is considered to be the same as 1 > 0. A single node is also considered as a valid path.
Example 1:
Input: vals = [1,3,2,1,3], edges = [[0,1],[0,2],[2,3],[2,4]] Output: 6 Explanation: There are 5 good paths consisting of a single node. There is 1 additional good path: 1 > 0 > 2 > 4. (The reverse path 4 > 2 > 0 > 1 is treated as the same as 1 > 0 > 2 > 4.) Note that 0 > 2 > 3 is not a good path because vals[2] > vals[0]. Example 2:
Input: vals = [1,1,2,2,3], edges = [[0,1],[1,2],[2,3],[2,4]] Output: 7 Explanation: There are 5 good paths consisting of a single node. There are 2 additional good paths: 0 > 1 and 2 > 3. Example 3:
Input: vals = [1], edges = [] Output: 1 Explanation: The tree consists of only one node, so there is one good path.
Constraints:
n == vals.length 1 <= n <= 3 * 104 0 <= vals[i] <= 105 edges.length == n  1 edges[i].length == 2 0 <= ai, bi < n ai != bi edges represents a valid tree.
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?
Alternatively, if you’re working on a specific problem, you might ask something like:
Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific realworld 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 ProblemSolving Approach from the Problem Statement
How did you infer from the problem statement that this problem can be solved using ?
Stepwise Refinement
Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the highlevel 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
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.
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 highlevel problemsolving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a nonprogrammer, 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 languageagnostic 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 problemsolving 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 stepbystep 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 Pythonbased 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 problemspecific 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.
Q&A
Similar Problems
Given the problem [provide the problem], identify and list down 10 similar problems on LeetCode. These should cover similar concepts or require similar problemsolving approaches as the provided problem. Please also give a brief reason as to why you think each problem is similar to the given problem.