Minimize the Total Price of the Trips

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class Solution:
    def minimumTotalPrice(self, n: int, edges: List[List[int]], price: List[int], trips: List[List[int]]) -> int:
        graph = defaultdict(list)
        for a,b in edges:
            graph[a].append(b)
            graph[b].append(a)

        count = Counter()
        totalCost = 0

        def dfs(node,par,end):
            nonlocal count
            nonlocal totalCost
            if node == end:
                return True

            for nei in graph[node]:
                if nei != par:
                    if dfs(nei, node, end):
                        count[nei] += 1
                        totalCost += price[nei]
                        return True
            return False

        for start,end in trips:
            count[start] += 1
            totalCost += price[start]
            dfs(start,None,end)

        @cache
        def dp(node, par, canReduce):
            if canReduce:
                res = (price[node]//2)*count[node]
            else:
                res = 0
            red = 0
            for nei in graph[node]:
                if nei != par:
                    if canReduce:
                        cur = dp(nei, node, False)
                    else:
                        cur = max(dp(nei, node, False), dp(nei, node, True))
                    red += cur
            return res + red

        reduce = 0
        for i in range(n):
            reduce = max( reduce, dp(i, None, True), dp(i, None, False) )

        return totalCost - reduce

Prerequisite LeetCode Problems

This involves Graph Theory, Dynamic Programming, and Greedy Algorithms and sorting. Here are 10 problems:

217. Contains Duplicate: Basic problem on working with arrays.
1046. Last Stone Weight: A problem that involves sorting and working with arrays. This will also introduce you to the idea of greedy algorithms.
121. Best Time to Buy and Sell Stock: This is a problem that involves dynamic programming and can help you understand how to break down complex problems into simpler subproblems.
322. Coin Change: Another dynamic programming problem. It will help you practice creating and reasoning about state transitions.
787. Cheapest Flights Within K Stops: This problem has a graph theory component and will help you understand how to model and solve problems using graphs.
743. Network Delay Time: Another graph problem, this one involving calculating shortest paths in a graph.
139. Word Break: A dynamic programming problem that involves working with strings. It will help you understand how to create and use a dp array.
198. House Robber: This problem involves dynamic programming and can provide insight into how to handle different cases for the problem at hand.
406. Queue Reconstruction by Height: This problem involves sorting and greedy algorithms. It can help you understand how to apply greedy thinking to solve problems.
134. Gas Station: This problem involves circular routes and can give you practice in handling edge cases.
1029. Two City Scheduling: This problem requires a greedy approach and sorting, which may be relevant for your target problem.
455. Assign Cookies: This problem also involves a greedy approach and sorting, similar to problem 1029.
647. Palindromic Substrings: This problem is a different type of dynamic programming problem and can help expand your understanding of the technique.
376. Wiggle Subsequence: This problem is another dynamic programming problem that requires careful thought about the states you need to track.
110. Balanced Binary Tree: This problem is not directly related to your target problem but can help improve your problem-solving skills and familiarity with tree-based data structures.
53. Maximum Subarray: This is a classic dynamic programming problem that is simpler than your target problem but still relevant.
121. Best Time to Buy and Sell Stock: This problem is another optimization problem that can be solved with dynamic programming.
139. Word Break: This problem is a slightly more complex dynamic programming problem that can help prepare you for your target problem.
406. Queue Reconstruction by Height: This problem requires a combination of sorting and dynamic programming, similar to your target problem.

Problem Classification

Problem Statement: There exists an undirected and unrooted tree with n nodes indexed from 0 to n - 1. You are given the 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. Each node has an associated price. You are given an integer array price, where price[i] is the price of the ith node. The price sum of a given path is the sum of the prices of all nodes lying on that path. Additionally, you are given a 2D integer array trips, where trips[i] = [starti, endi] indicates that you start the ith trip from the node starti and travel to the node endi by any path you like. Before performing your first trip, you can choose some non-adjacent nodes and halve the prices. Return the minimum total price sum to perform all the given trips.

Example 1:

Input: n = 4, edges = [[0,1],[1,2],[1,3]], price = [2,2,10,6], trips = [[0,3],[2,1],[2,3]] Output: 23 Explanation: The diagram above denotes the tree after rooting it at node 2. The first part shows the initial tree and the second part shows the tree after choosing nodes 0, 2, and 3, and making their price half. For the 1st trip, we choose path [0,1,3]. The price sum of that path is 1 + 2 + 3 = 6. For the 2nd trip, we choose path [2,1]. The price sum of that path is 2 + 5 = 7. For the 3rd trip, we choose path [2,1,3]. The price sum of that path is 5 + 2 + 3 = 10. The total price sum of all trips is 6 + 7 + 10 = 23. It can be proven, that 23 is the minimum answer that we can achieve. Example 2:

Input: n = 2, edges = [[0,1]], price = [2,2], trips = [[0,0]] Output: 1 Explanation: The diagram above denotes the tree after rooting it at node 0. The first part shows the initial tree and the second part shows the tree after choosing node 0, and making its price half. For the 1st trip, we choose path [0]. The price sum of that path is 1. The total price sum of all trips is 1. It can be proven, that 1 is the minimum answer that we can achieve.

Constraints:

1 <= n <= 50 edges.length == n - 1 0 <= ai, bi <= n - 1 edges represents a valid tree. price.length == n price[i] is an even integer. 1 <= price[i] <= 1000 1 <= trips.length <= 100 0 <= starti, endi <= n - 1

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?

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.