Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree

The problem of finding critical and pseudo-critical edges in a minimum spanning tree (MST) can be solved using disjoint set (also known as union-find) data structure.

Here’s the solution that includes the code to find both critical and pseudo-critical edges:

Critical Edges: An edge is considered critical if it appears in all possible MSTs. To find these edges, we need to calculate the minimum spanning tree weight without the edge, and if the MST weight increases, it means the edge is critical.
Pseudo-Critical Edges: An edge is considered pseudo-critical if it appears in some, but not all, MSTs. To find these edges, we need to calculate the minimum spanning tree weight with the edge, and if the MST weight doesn’t change, it means the edge is pseudo-critical.

Here’s the code:

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class Solution:
    def findCriticalAndPseudoCriticalEdges(self, n: int, edges: List[List[int]]) -> List[List[int]]:
        # Add edge index to each edge
        for i, edge in enumerate(edges):
            edge.append(i)

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

        # Disjoint set union-find function
        def find(u, parent):
            if u != parent[u]:
                parent[u] = find(parent[u], parent)
            return parent[u]

        # Calculate MST weight function
        def mstWeight(n, edges, ignore, include):
            parent = [i for i in range(n)]
            weight = 0

            # If there's an edge to include
            if include != -1:
                weight += edges[include][2]
                u, v = edges[include][:2]
                parent[find(u, parent)] = find(v, parent)

            # Iterate through all edges
            for i, (u, v, w, _) in enumerate(edges):
                # Ignore the edge if it is in the ignore list
                if i == ignore or i == include:
                    continue

                # If the vertices are not connected
                if find(u, parent) != find(v, parent):
                    parent[find(u, parent)] = find(v, parent)
                    weight += w

            # Check if all vertices are connected
            if len(set(find(i, parent) for i in range(n))) == 1:
                return weight
            return float('inf')

        # Find the normal MST weight without any restrictions
        baseWeight = mstWeight(n, edges, -1, -1)

        critical = []
        pseudoCritical = []

        # Check for critical and pseudo-critical edges
        for i in range(len(edges)):
            # If the MST weight increases when the edge is ignored, it's critical
            if mstWeight(n, edges, i, -1) > baseWeight:
                critical.append(edges[i][3])
            # If the MST weight doesn't change when the edge is included, it's pseudo-critical
            elif mstWeight(n, edges, -1, i) == baseWeight:
                pseudoCritical.append(edges[i][3])

        return [critical, pseudoCritical]

Explanation:

find(u, parent) is a helper function to find the root of a vertex using path compression.
mstWeight(n, edges, ignore, include) calculates the MST weight while ignoring a specific edge or including a specific edge.
baseWeight is the weight of the original MST without any restrictions.
The results are gathered in the critical and pseudoCritical lists and returned as the final answer.

Identifying Problem Isomorphism

“Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree” can be mapped to “Redundant Connection II”.

Both problems involve working with a graph structure and dealing with edges in some way. In “Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree”, you are given a weighted graph and have to identify critical and pseudo-critical edges in relation to the minimum spanning tree of that graph.

In “Redundant Connection II”, you are given a directed graph that is almost a tree except for an additional edge and you have to find that edge.

These two problems share a focus on understanding graph structure and analyzing the importance of individual edges. The first is a more complex scenario, with both a weighted graph and the need to distinguish between two different categories of edge.

“Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree” is a more complex problem, as it involves more specific concepts related to graph theory (Minimum Spanning Tree) and requires a deeper understanding of the properties and roles of edges in a graph.

10 Prerequisite LeetCode Problems

“1489. Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree” involves minimum spanning trees (MST), Union-Find and Kruskal’s algorithm. Here are some problems to prepare for this:

“684. Redundant Connection”:
- Basic application of Union-Find.
“261. Graph Valid Tree”:
- This problem will help you understand when a graph is a valid tree and when it forms a cycle.
“1135. Connecting Cities With Minimum Cost”:
- This problem is about finding the MST of a graph. You can use Kruskal’s or Prim’s algorithm.
“778. Swim in Rising Water”:
- This problem is similar to finding an MST with some variations. Understanding this problem will help you think creatively about MST problems.
“1102. Path With Maximum Minimum Value”:
- A variation on the MST problem, it’ll help you think about how to solve problems with similar concepts but different requirements.
“959. Regions Cut By Slashes”:
- This problem can be solved using Union-Find and it will also require you to think about how to transform the problem into a graph problem.
“200. Number of Islands”:
- This problem is about finding connected components in a grid. It will give you more practice on Union-Find.
“305. Number of Islands II”:
- This problem adds a dynamic component to the classic Number of Islands problem. It’s a good way to test your understanding of the Union-Find data structure.
“323. Number of Connected Components in an Undirected Graph”:
- More practice on finding connected components.
“1202. Smallest String With Swaps”:
- This problem can be solved with Union-Find and it’ll force you to think about how to use the Union-Find data structure in novel ways.

These problems will help you understand the concepts and give you enough practice on Union-Find and MSTs to tackle “1489. Find Critical and Pseudo-Critical Edges in Minimum Spanning Tree”.

Problem Classification

Problem Statement: 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.

Distilling the Problem to Its Core Elements

In order to have me distill a problem to its core, you could ask questions that prompt for a deeper analysis of the problem, understanding of the underlying concepts, and simplification of the problem’s essence. Here are some examples of such prompts:

Can you identify the fundamental concept or principle this problem is based upon? Please explain.
What is the simplest way you would describe this problem to someone unfamiliar with the subject?
What is the core problem we are trying to solve? Can we simplify the problem statement?
Can you break down the problem into its key components?
What is the minimal set of operations we need to perform to solve this problem?

These prompts guide the discussion towards simplifying the problem, stripping it down to its essential elements, and understanding the core problem to be solved.

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.

Provide names by categorizing these cases

What are the key insights from analyzing the different cases?

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?

Code Explanation and Design Decisions

Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

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.