Operations on Tree

We need to create a class LockingTree to manage the locking system in a tree structure. Let’s start by understanding the different parts of the problem and then move towards the solution.

Initialization

We need to initialize the data structure with the parent array, which gives the parent of each node.

Lock

This function will lock a given node for a user if it’s currently unlocked.

Unlock

This function will unlock a given node for a user if it’s currently locked by that user.

Upgrade

This function will lock a given node for a user and unlock all of its descendants, given some conditions. It’s only valid if the node is unlocked, has at least one locked descendant, and does not have any locked ancestors.

Here’s the implementation:

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class LockingTree:

    def __init__(self, parent: List[int]):
        self.parent = parent
        self.locked_by = [None] * len(parent) # Keeps track of who locked each node
        self.children = [[] for _ in parent]
        for child, parent_node in enumerate(parent[1:], 1):
            self.children[parent_node].append(child)

    def lock(self, num: int, user: int) -> bool:
        if self.locked_by[num] is None: # Check if the node is unlocked
            self.locked_by[num] = user
            return True
        return False

    def unlock(self, num: int, user: int) -> bool:
        if self.locked_by[num] == user: # Check if the node is locked by this user
            self.locked_by[num] = None
            return True
        return False

    def upgrade(self, num: int, user: int) -> bool:
        if self.locked_by[num] is not None: # Check if the node is unlocked
            return False
        if not self.has_locked_descendants(num): # Check if the node has any locked descendants
            return False
        ancestor = self.parent[num]
        while ancestor != -1: # Check if any ancestors are locked
            if self.locked_by[ancestor] is not None:
                return False
            ancestor = self.parent[ancestor]
        # Perform the upgrade
        self.locked_by[num] = user
        self.unlock_descendants(num)
        return True

    def has_locked_descendants(self, num: int) -> bool:
        for child in self.children[num]:
            if self.locked_by[child] is not None or self.has_locked_descendants(child):
                return True
        return False

    def unlock_descendants(self, num: int):
        for child in self.children[num]:
            self.locked_by[child] = None
            self.unlock_descendants(child)

Example:

The given example will work as described in the prompt.

Key Takeaways

We need to maintain information about the locked status of each node and who has locked it.
The upgrade function is a bit more complex as it needs to check multiple conditions involving both ancestors and descendants. Recursion helps in checking conditions for descendants.
The overall design must consider the tree structure to efficiently manage lock, unlock, and upgrade operations.

10 Prerequisite LeetCode Problems

This involves understanding trees, depth-first search (DFS), and designing classes with complex methods. Here are some problems to build up these skills:

104. Maximum Depth of Binary Tree: This is a straightforward problem to get you started with tree traversal, specifically depth-first search.
226. Invert Binary Tree: This problem continues to build on the concept of tree traversal.
617. Merge Two Binary Trees: This problem adds a bit of complexity, requiring you to traverse and manipulate two trees at once.
98. Validate Binary Search Tree: This problem will help you understand the properties of a specific type of tree, the binary search tree (BST), which could be useful in more complex tree problems.
101. Symmetric Tree: It’s another tree traversal problem which is slightly more complex than the previous ones.
112. Path Sum: This problem adds the element of tracking a sum as you traverse the tree.
105. Construct Binary Tree from Preorder and Inorder Traversal: This problem takes the concept of tree construction, and adds the complexity of doing so from different traversal orders.
133. Clone Graph: While this problem is about graphs, the depth-first search concept it uses is directly applicable to trees.
208. Implement Trie (Prefix Tree): This problem is about implementing a different type of tree, a Trie, which might help you think about trees in a different way.
211. Design Add and Search Words Data Structure: This problem will help you build skills in designing a class with complex methods, which is a crucial part of the “Operations on Tree” problem.

Solving these problems should provide a solid foundation in tree-based problems, and help prepare you for tackling the “Operations on Tree” problem.

This problem involves implementing a data structure that supports locking and unlocking operations on a tree, and it requires an understanding of tree traversal techniques and the ability to keep track of state information (such as the locked status and the user who locked a node) in a data structure.

Here are ten problems that might help you build up the necessary skills for this problem:

116. Populating Next Right Pointers in Each Node: A tree problem that requires you to maintain state information for each node.
117. Populating Next Right Pointers in Each Node II: Similar to the previous problem, but with a binary tree that is not perfect.
297. Serialize and Deserialize Binary Tree: This problem involves serialization and deserialization of a binary tree, which requires you to encode state information about the tree.
208. Implement Trie (Prefix Tree): This is not a tree problem per se, but implementing a trie will help you practice managing state information in a data structure.
1448. Count Good Nodes in Binary Tree: This problem requires maintaining state information during tree traversal.
1026. Maximum Difference Between Node and Ancestor: This problem requires keeping track of the maximum and minimum values seen during a tree traversal.
199. Binary Tree Right Side View: This problem requires a depth-first search and keeping track of the depth of each node.
662. Maximum Width of Binary Tree: This problem requires tracking the position of each node in the tree.
124. Binary Tree Maximum Path Sum: In this problem, you have to keep track of the maximum path sum at each node during the tree traversal.
863. All Nodes Distance K in Binary Tree: This problem requires tracking the parent of each node and then performing a breadth-first search.

These problems all involve tracking state information during tree traversal, and solving them should help prepare you to tackle the problem at hand.

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Problem Classification

Problem Statement:You are given a tree with n nodes numbered from 0 to n - 1 in the form of a parent array parent where parent[i] is the parent of the ith node. The root of the tree is node 0, so parent[0] = -1 since it has no parent. You want to design a data structure that allows users to lock, unlock, and upgrade nodes in the tree.

The data structure should support the following functions:

Lock: Locks the given node for the given user and prevents other users from locking the same node. You may only lock a node using this function if the node is unlocked. Unlock: Unlocks the given node for the given user. You may only unlock a node using this function if it is currently locked by the same user. Upgrade: Locks the given node for the given user and unlocks all of its descendants regardless of who locked it. You may only upgrade a node if all 3 conditions are true: The node is unlocked, It has at least one locked descendant (by any user), and It does not have any locked ancestors. Implement the LockingTree class:

LockingTree(int[] parent) initializes the data structure with the parent array. lock(int num, int user) returns true if it is possible for the user with id user to lock the node num, or false otherwise. If it is possible, the node num will become locked by the user with id user. unlock(int num, int user) returns true if it is possible for the user with id user to unlock the node num, or false otherwise. If it is possible, the node num will become unlocked. upgrade(int num, int user) returns true if it is possible for the user with id user to upgrade the node num, or false otherwise. If it is possible, the node num will be upgraded.

Example 1:

Input [“LockingTree”, “lock”, “unlock”, “unlock”, “lock”, “upgrade”, “lock”] [[[-1, 0, 0, 1, 1, 2, 2]], [2, 2], [2, 3], [2, 2], [4, 5], [0, 1], [0, 1]] Output [null, true, false, true, true, true, false]

Explanation LockingTree lockingTree = new LockingTree([-1, 0, 0, 1, 1, 2, 2]); lockingTree.lock(2, 2); // return true because node 2 is unlocked. // Node 2 will now be locked by user 2. lockingTree.unlock(2, 3); // return false because user 3 cannot unlock a node locked by user 2. lockingTree.unlock(2, 2); // return true because node 2 was previously locked by user 2. // Node 2 will now be unlocked. lockingTree.lock(4, 5); // return true because node 4 is unlocked. // Node 4 will now be locked by user 5. lockingTree.upgrade(0, 1); // return true because node 0 is unlocked and has at least one locked descendant (node 4). // Node 0 will now be locked by user 1 and node 4 will now be unlocked. lockingTree.lock(0, 1); // return false because node 0 is already locked.

Constraints:

n == parent.length 2 <= n <= 2000 0 <= parent[i] <= n - 1 for i != 0 parent[0] == -1 0 <= num <= n - 1 1 <= user <= 104 parent represents a valid tree. At most 2000 calls in total will be made to lock, unlock, and upgrade.

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

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?

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 edge 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?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

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

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method.

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

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

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

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

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.

Establishing Preconditions and Postconditions

Parameters:
- What are the inputs to the method?
- What types are these parameters?
- What do these parameters represent in the context of the problem?
Preconditions:
- Before this method is called, what must be true about the state of the program or the values of the parameters?
- Are there any constraints on the input parameters?
- Is there a specific state that the program or some part of it must be in?
Method Functionality:
- What is this method expected to do?
- How does it interact with the inputs and the current state of the program?
Postconditions:
- After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
- What does the return value represent or indicate?
- What side effects, if any, does the method have?
Error Handling:
- How does the method respond if the preconditions are not met?
- Does it throw an exception, return a special value, or do something else?

Problem Decomposition

Problem Understanding:
- Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
- Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
- For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
- Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
- For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
- Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
- How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

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.