Collect Coins in a Tree

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from collections import defaultdict, deque

class Solution:
    def collectTheCoins(self, coins: List[int], edges: List[List[int]]) -> int:
        n = len(coins)
        g = defaultdict(set)

        for e in edges:
            g[e[0]].add(e[1])
            g[e[1]].add(e[0])

        leaves = deque()
        totalEdges = len(edges) * 2
        deleted = 0
        
        # Add leaves that don't have any coins
        for i in range(n):
            if len(g[i]) == 1 and coins[i] == 0:
                leaves.append(i)

        while leaves:
            leaf = leaves.popleft()
            if leaf in g and g[leaf]:
                p = g[leaf].pop()
                g[p].remove(leaf)
                deleted += 2

                if len(g[p]) == 1 and coins[p] == 0:
                    leaves.append(p)

        # Add leaves that have coins
        for i in range(n):
            if len(g[i]) == 1:
                leaves.append(i)

        step = 2  # 1 for parent and 1 for children
        while step > 0 and leaves:
            sz = len(leaves)
            while sz > 0:
                leaf = leaves.popleft()
                if leaf in g and g[leaf]:
                    p = g[leaf].pop()
                    g[p].remove(leaf)
                    deleted += 2

                    if len(g[p]) == 1:
                        leaves.append(p)
                sz -= 1
            step -= 1

        return totalEdges - deleted

10 Prerequisite LeetCode Problems

“Collect Coins in a Tree” involves tree traversal, post-order traversal specifically, and dynamic programming. Here are ten simpler problems to get ready for this one:

Maximum Depth of Binary Tree: This is a basic tree traversal problem to get you started.
Symmetric Tree: This problem requires understanding of tree symmetry and traversal.
Binary Tree Level Order Traversal: Helps understand level-order traversal, which is a form of breadth-first search.
Binary Tree Paths: This problem will help you to understand how to traverse from a root to a leaf.
Binary Tree Postorder Traversal: It’s a good practice for post-order traversal.
Path Sum: This problem will help you understand how to keep track of sum while traversing a tree.
Diameter of Binary Tree: This problem also requires tree traversal and can help understand depth calculations.
House Robber: This problem helps understand the concept of dynamic programming which is needed in “Collect Coins in a Tree”.
Climbing Stairs: Another problem to understand dynamic programming.
Coin Change: This problem is relevant for understanding the concept of working with coins, which is necessary for the main problem.

You can strengthen your understanding of binary trees and dynamic programming, which are both necessary to solve the “Collect Coins in a Tree” problem.

Problem Classification

Problem Statement:There exists an undirected and unrooted tree with n nodes indexed from 0 to n - 1. You are given an integer n and a 2D integer array edges of length n - 1, where edges[i] = [ai, bi] indicates that there is an edge between nodes ai and bi in the tree. You are also given an array coins of size n where coins[i] can be either 0 or 1, where 1 indicates the presence of a coin in the vertex i.

Initially, you choose to start at any vertex in the tree. Then, you can perform the following operations any number of times:

Collect all the coins that are at a distance of at most 2 from the current vertex, or Move to any adjacent vertex in the tree. Find the minimum number of edges you need to go through to collect all the coins and go back to the initial vertex.

Note that if you pass an edge several times, you need to count it into the answer several times.

Example 1:

Input: coins = [1,0,0,0,0,1], edges = [[0,1],[1,2],[2,3],[3,4],[4,5]] Output: 2 Explanation: Start at vertex 2, collect the coin at vertex 0, move to vertex 3, collect the coin at vertex 5 then move back to vertex 2.

Example 2:

Input: coins = [0,0,0,1,1,0,0,1], edges = [[0,1],[0,2],[1,3],[1,4],[2,5],[5,6],[5,7]] Output: 2 Explanation: Start at vertex 0, collect the coins at vertices 4 and 3, move to vertex 2, collect the coin at vertex 7, then move back to vertex 0.

Constraints:

n == coins.length 1 <= n <= 3 * 104 0 <= coins[i] <= 1 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.

Clarification Questions

What are the clarification questions we can ask about this problem?

Identifying Problem Isomorphism

The problem “Collect Coins in a Tree” can be mapped to “House Robber III”.

Both problems deal with optimizing the collection of values (coins or rob amounts) in a tree structure, where certain conditions must be met to ensure the maximum collection.

In “Collect Coins in a Tree”, the task is to move coins around to balance the tree, while optimizing the number of moves. In “House Robber III”, the task is to maximize the amount robbed, given the constraint that no two direct-linked houses can be robbed simultaneously.

The isomorphism lies in the use of a depth-first search approach to traverse the tree and the need for maintaining states for optimal substructure. However, it’s an approximate mapping because the specific conditions and objectives in the two problems are not exactly the same.

Regarding complexity, “House Robber III” is simpler as it only requires optimizing the rob amount under a certain condition. In contrast, “Collect Coins in a Tree” involves not only the optimization of coin moves but also the need to balance the tree, which adds complexity to the problem.

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.