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| from collections import defaultdict
from bisect import bisect_left
class DistanceLimitedPathsExist:
def __init__(self, n: int, edgeList: List[List[int]]):
def find(node):
if parent[node] != node:
parent[node] = find(parent[node])
return parent[node]
def union(x, y):
parent[find(y)] = find(x)
parent = {i: i for i in range(n)} # parent for each node
edgeList.sort(key=lambda x: x[2])
self.connections = []
self.weights = []
for index, (i, j, weight) in enumerate(edgeList): # for the current weight, connect i, j
union(i, j)
if index != len(edgeList) - 1 and weight == edgeList[index + 1][2]:
continue
self.weights.append(weight) # save the weight keys
self.connections.append([find(i) for i in parent]) # save the connection for the current weight
def query(self, p, q, limit):
index = bisect_left(self.weights, limit)
if index == 0:
return False
return self.connections[index - 1][p] == self.connections[index - 1][q]
```python
This involves applying data structures like Union Find, and concepts like sorting and binary search. Here are 10 problems to prepare:
1. **"Disjoint Set Union (DSU) on Trees" (LeetCode Problem #924):** This problem is a good introduction to the application of the DSU concept.
2. **"Union Find" (LeetCode Problem #261):** This problem provides the basic concept of the Union-Find data structure.
3. **"Graph Valid Tree" (LeetCode Problem #261):** This problem involves using Union-Find to check if a graph is a valid tree.
4. **"Most Stones Removed with Same Row or Column" (LeetCode Problem #947):** This problem involves applying Union Find in a 2D grid.
5. **"Number of Islands" (LeetCode Problem #200):** This problem helps understand the application of Union-Find on a grid.
6. **"Find Eventual Safe States" (LeetCode Problem #802):** This problem is about depth-first search on a directed graph.
7. **"Kth Smallest Element in a Sorted Matrix" (LeetCode Problem #378):** This problem introduces you to the concept of binary search on a 2D matrix.
8. **"Merge Intervals" (LeetCode Problem #56):** This problem involves sorting and merging intervals which is a helpful concept for the main problem.
9. **"Path With Minimum Effort" (LeetCode Problem #1631):** This problem involves applying binary search in a grid-based path finding.
10. **"Network Delay Time" (LeetCode Problem #743):** This problem helps understand application of minimum spanning trees and shortest paths in graphs.
These cover Union-Find data structure, sorting, binary search and related techniques needed to solve the main problem.
The problem "1724. Checking Existence of Edge Length Limited Paths II" involves advanced concepts such as union-find data structure, sorting, graph theory and processing queries. Here are some problems to build up to it:
11. [Redundant Connection](https://leetcode.com/problems/redundant-connection/): This problem helps to understand the concept of finding cycles in a graph using the union-find data structure.
12. [Connecting Cities With Minimum Cost](https://leetcode.com/problems/connecting-cities-with-minimum-cost/): This problem is a basic application of Kruskal's algorithm, which is a greedy algorithm in graph theory.
13. [Shortest Bridge](https://leetcode.com/problems/shortest-bridge/): This problem involves graph traversal using depth-first search and breadth-first search algorithms.
14. [Minimum Height Trees](https://leetcode.com/problems/minimum-height-trees/): This problem also involves graph traversal and can help you understand the concept of tree-based graph problems.
15. [Friend Circles](https://leetcode.com/problems/friend-circles/): This problem is about finding connected components in a graph, a concept that is often useful in union-find problems.
16. [Path With Maximum Minimum Value](https://leetcode.com/problems/path-with-maximum-minimum-value/): This problem introduces the concept of finding paths with a certain constraint, which is crucial for "1724. Checking Existence of Edge Length Limited Paths II".
17. [Satisfiability of Equality Equations](https://leetcode.com/problems/satisfiability-of-equality-equations/): This problem also deals with the union-find data structure and helps understand how to process queries in the context of union-find.
The problem "1724. Checking Existence of Edge Length Limited Paths II" deals with concepts related to graphs, path existence, and disjoint set/union-find. Here are some problems that can help you prepare for this:
18. **LeetCode 547. Number of Provinces**
- This problem involves counting disconnected components in a graph using the union-find data structure.
19. **LeetCode 721. Accounts Merge**
- This problem involves merging connected components, for which the union-find algorithm can be applied.
20. **LeetCode 785. Is Graph Bipartite?**
- This problem involves checking a condition for all connected components in a graph.
21. **LeetCode 305. Number of Islands II**
- This problem requires keeping track of the number of connected components in a dynamic graph, which can be solved using the union-find algorithm.
22. **LeetCode 1319. Number of Operations to Make Network Connected**
- This problem involves counting the minimum number of operations to connect all nodes in a graph.
23. **LeetCode 839. Similar String Groups**
- This problem can be solved using the union-find data structure to group similar strings together.
24. **LeetCode 323. Number of Connected Components in an Undirected Graph**
- This problem involves finding the number of connected components in an undirected graph.
25. **LeetCode 399. Evaluate Division**
- This problem involves finding the connected component for a specific node in a graph.
By practicing these problems, you'll gain a good understanding of the union-find data structure and how it can be used to solve problems related to connected components in a graph, preparing you to tackle "1724. Checking Existence of Edge Length Limited Paths II".
## 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, and the graph may not be connected.
Implement the DistanceLimitedPathsExist class:
DistanceLimitedPathsExist(int n, int[][] edgeList) Initializes the class with an undirected graph.
boolean query(int p, int q, int limit) Returns true if there exists a path from p to q such that each edge on the path has a distance strictly less than limit, and otherwise false.
Example 1:
Input
["DistanceLimitedPathsExist", "query", "query", "query", "query"]
[[6, [[0, 2, 4], [0, 3, 2], [1, 2, 3], [2, 3, 1], [4, 5, 5]]], [2, 3, 2], [1, 3, 3], [2, 0, 3], [0, 5, 6]]
Output
[null, true, false, true, false]
Explanation
DistanceLimitedPathsExist distanceLimitedPathsExist = new DistanceLimitedPathsExist(6, [[0, 2, 4], [0, 3, 2], [1, 2, 3], [2, 3, 1], [4, 5, 5]]);
distanceLimitedPathsExist.query(2, 3, 2); // return true. There is an edge from 2 to 3 of distance 1, which is less than 2.
distanceLimitedPathsExist.query(1, 3, 3); // return false. There is no way to go from 1 to 3 with distances strictly less than 3.
distanceLimitedPathsExist.query(2, 0, 3); // return true. There is a way to go from 2 to 0 with distance < 3: travel from 2 to 3 to 0.
distanceLimitedPathsExist.query(0, 5, 6); // return false. There are no paths from 0 to 5.
Constraints:
2 <= n <= 104
0 <= edgeList.length <= 104
edgeList[i].length == 3
0 <= ui, vi, p, q <= n-1
ui != vi
p != q
1 <= disi, limit <= 109
At most 104 calls will be made to query.
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 ?
## Stepwise Refinement
1. Could you please provide a stepwise refinement of our approach to solving this problem?
2. How can we take the high-level solution approach and distill it into more granular, actionable steps?
3. Could you identify any parts of the problem that can be solved independently?
4. 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.
1. What are the high-level problem-solving strategies or techniques being used by this code?
2. If you had to explain the purpose of this code to a non-programmer, what would you say?
3. Can you identify the logical elements or constructs used in this code, independent of any programming language?
4. Could you describe the algorithmic approach used by this code in plain English?
5. What are the key steps or operations this code is performing on the input data, and why?
6. 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.
1. 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.
2. 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.
3. 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.
1. 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.
2. 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.
3. 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 problem-solving approaches as the provided problem. Please also give a brief reason as to why you think each problem is similar to the given problem.
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