Checking Existence of Edge Length Limited Paths

To solve this problem, we can utilize the Union-Find data structure to create a Minimum Spanning Tree (MST) where we only connect edges that are smaller than the given limits for the queries. By sorting the edges in ascending order of distance and the queries in ascending order of limits, we can efficiently process the graph and determine whether a path exists between the nodes specified in each query.

Here’s a step-by-step explanation of the approach:

Sort Edges: Sort the edgeList in ascending order of distance.
Preprocess Queries: Add the index of each query in the queries list, then sort the queries based on the limitj.
Initialize Union-Find: Create a Union-Find data structure to keep track of the connected components of the graph.
Iterate Through Queries: Iterate through the sorted queries. For each query, add all edges that have a distance less than the current query’s limit to the Union-Find structure. Then, check if the nodes specified in the query are in the same connected component.
Create Result: For each query, add True to the result if the nodes are connected, False otherwise.

Python solution:

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class Solution:
    def distanceLimitedPathsExist(self, n: int, edgeList: List[List[int]], queries: List[List[int]]) -> List[bool]:
        # Union-Find functions
        def find(x):
            if x != parents[x]:
                parents[x] = find(parents[x])
            return parents[x]

        def union(x, y):
            parents[find(x)] = find(y)

        # Sort edges by distance
        edgeList.sort(key=lambda x: x[2])

        # Add indices to queries and sort by limit
        queries = [[p, q, l, i] for i, (p, q, l) in enumerate(queries)]
        queries.sort(key=lambda x: x[2])

        parents = [i for i in range(n)]
        res = [False] * len(queries)
        edge_idx = 0

        # Iterate through sorted queries
        for p, q, limit, idx in queries:
            # Add edges with distance < limit
            while edge_idx < len(edgeList) and edgeList[edge_idx][2] < limit:
                u, v, _ = edgeList[edge_idx]
                union(u, v)
                edge_idx += 1

            # Check if nodes are connected
            if find(p) == find(q):
                res[idx] = True

        return res

The time complexity of this solution is O(E * log(E) + Q * log(Q) + E + Q), where E is the length of edgeList and Q is the length of queries. This complexity is due to the sorting steps and the iterations through the edges and queries. The space complexity is O(n), where n is the number of nodes, as we need to keep track of the parents in the Union-Find structure.

“Checking Existence of Edge Length Limited Paths” requires an understanding of graph theory, specifically union-find data structure and sorting. Here are 10 problems to prepare:

“Number of Islands” (LeetCode Problem #200): This problem helps you understand basic graph traversal using depth-first search (DFS) which is a fundamental concept in graph theory.
“Friend Circles” (LeetCode Problem #547): This problem also uses DFS, but in the context of undirected graphs, which can help you understand how to traverse such graphs.
“Redundant Connection” (LeetCode Problem #684): This problem introduces you to the concept of Union-Find, which is used in the “Checking Existence of Edge Length Limited Paths” problem.
“Flood Fill” (LeetCode Problem #733): This is a simple problem that will help you understand how to implement DFS in a 2D grid.
“Accounts Merge” (LeetCode Problem #721): This problem also uses Union-Find data structure, but in a slightly more complex setting than “Redundant Connection”.
“Path With Maximum Minimum Value” (LeetCode Problem #1102): This problem requires a similar kind of sorting to “Checking Existence of Edge Length Limited Paths”, but in a simpler context.
“Course Schedule” (LeetCode Problem #207): This problem helps you to understand basic topological sorting in graphs.
“Network Delay Time” (LeetCode Problem #743): This problem gives you exposure to the Dijkstra’s algorithm, a key graph theory algorithm.
“Graph Valid Tree” (LeetCode Problem #261): This problem provides more practice on the Union-Find data structure in the context of checking whether a graph forms a valid tree.
“Find the City With the Smallest Number of Neighbors at a Threshold Distance” (LeetCode Problem #1334): This problem is a bit more complex, but it helps you understand how to manipulate graphs based on edge lengths, which is also a key component of the “Checking Existence of Edge Length Limited Paths” problem.

Problem Classification

Problem Statement: An undirected graph of n nodes is defined by edgeList, where edgeList[i] = [ui, vi, disi] denotes an edge between nodes ui and vi with distance disi. Note that there may be multiple edges between two nodes.

Given an array queries, where queries[j] = [pj, qj, limitj], your task is to determine for each queries[j] whether there is a path between pj and qj such that each edge on the path has a distance strictly less than limitj .

Return a boolean array answer, where answer.length == queries.length and the jth value of answer is true if there is a path for queries[j] is true, and false otherwise.

Example 1:

Input: n = 3, edgeList = [[0,1,2],[1,2,4],[2,0,8],[1,0,16]], queries = [[0,1,2],[0,2,5]] Output: [false,true] Explanation: The above figure shows the given graph. Note that there are two overlapping edges between 0 and 1 with distances 2 and 16. For the first query, between 0 and 1 there is no path where each distance is less than 2, thus we return false for this query. For the second query, there is a path (0 -> 1 -> 2) of two edges with distances less than 5, thus we return true for this query.

Example 2:

Input: n = 5, edgeList = [[0,1,10],[1,2,5],[2,3,9],[3,4,13]], queries = [[0,4,14],[1,4,13]] Output: [true,false] Explanation: The above figure shows the given graph.

Constraints:

2 <= n <= 105 1 <= edgeList.length, queries.length <= 105 edgeList[i].length == 3 queries[j].length == 3 0 <= ui, vi, pj, qj <= n - 1 ui != vi pj != qj 1 <= disi, limitj <= 109 There may be multiple edges between two nodes.

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 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.