Largest Color Value in a Directed Graph

To solve this problem, you can use depth-first search (DFS) and dynamic programming. Here’s a simple step-by-step breakdown of the algorithm:

Build the Graph: Construct an adjacency list to represent the given edges.
Cycle Detection: Track the state of each node during DFS to detect cycles. Use three states: unvisited, visiting, and visited.
Dynamic Programming (DP) Table: Create a 2D table dp to store the maximum color value for each node and each color in the graph.
DFS Function: Write a recursive DFS function that updates the DP table by visiting the neighbors of the current node. It returns the maximum color value for the current node.
Result Computation: Iterate over all nodes, and for each node, compute the maximum color value using the DFS function. Keep track of the largest value.

Here’s the code:

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class Solution:
    def largestPathValue(self, colors: str, edges: List[List[int]]) -> int:
        n = len(colors)
        graph = [[] for _ in range(n)]
        for a, b in edges:
            graph[a].append(b)

        # dp[i][j] stores the maximum color value for node i and color j
        dp = [[0] * 26 for _ in range(n)]

        # States: 0 = unvisited, 1 = visiting, 2 = visited
        states = [0] * n

        # DFS function to compute the maximum color value for a node
        def dfs(node):
            if states[node] == 2:
                return
            if states[node] == 1:
                # Cycle detected
                raise Exception("Cycle detected")
            states[node] = 1

            # Visit neighbors and update the DP table
            for neighbor in graph[node]:
                dfs(neighbor)
                for c in range(26):
                    dp[node][c] = max(dp[node][c], dp[neighbor][c])

            # Increment the count for the color of the current node
            color_idx = ord(colors[node]) - ord('a')
            dp[node][color_idx] += 1

            states[node] = 2

        try:
            # Run DFS for all nodes
            for i in range(n):
                dfs(i)
            return max(max(row) for row in dp)
        except:
            return -1

The function returns the largest color value of any valid path in the given graph, or -1 if the graph contains a cycle.

This solution has a time complexity of (O(n + m)), where (n) is the number of nodes and (m) is the number of edges. It also has a space complexity of (O(n)) for the DFS call stack and (O(n \times 26)) for the DP table.

Identifying Problem Isomorphism

“Largest Color Value in a Directed Graph” can be mapped to “Course Schedule II”.

Both share the underlying structure of a directed graph and require a depth-first search (DFS) or breadth-first search (BFS) to traverse the graph.

“Largest Color Value in a Directed Graph” is about finding the maximum color value in a path, which in essence is finding the path with the maximum sum of values.

“Course Schedule II” also requires traversing the graph to find a valid order in which courses can be taken, which can be mapped to finding a valid path through the graph.

“Largest Color Value in a Directed Graph” is a more complex problem due to the additional requirement of calculating color values. “Course Schedule II”, on the other hand, only requires determining if a valid order exists, making it a simpler problem.

This is an approximate mapping based on the structural similarity and the shared algorithmic approach (DFS or BFS) between the two problems, not an exact one-to-one mapping due to the differing problem details.

10 Prerequisite LeetCode Problems

“1857. Largest Color Value in a Directed Graph” combines the usage of graph theory and dynamic programming. This requires familiarity with concepts like depth-first search (DFS), topological sort, handling directed cycles, and dynamic programming. Here are some problems to prepare:

“Course Schedule” (LeetCode Problem #207): This problem helps understand the basics of topological sort and how to detect cycles in a directed graph.
“Course Schedule II” (LeetCode Problem #210): This problem is an extension of “Course Schedule” where you have to return one correct ordering. This will help reinforce your understanding of topological sorting.
“Longest Increasing Path in a Matrix” (LeetCode Problem #329): This problem is about finding the longest path in a directed graph, which can be solved using depth-first search and dynamic programming.
“Network Delay Time” (LeetCode Problem #743): This problem requires understanding how to traverse weighted directed graphs.
“Reconstruct Itinerary” (LeetCode Problem #332): This problem involves finding a path in a directed graph using depth-first search.
“Longest Path in Directed Acyclic Graph” (LeetCode Problem #444): This problem will provide further practice on finding paths in a DAG using depth-first search.
“Clone Graph” (LeetCode Problem #133): This problem requires understanding how to traverse and copy a graph, which are useful skills for handling graphs.
“Path with Maximum Probability” (LeetCode Problem #1514): This problem involves finding a path with maximum probability which can be solved using depth-first search.
“Number of Islands” (LeetCode Problem #200): This problem helps you understand the basics of depth-first search in a grid which can be treated as a graph.
“Redundant Connection” (LeetCode Problem #684): This problem helps in understanding union-find which can be used for cycle detection in a graph.

Problem Classification

Problem Statement: There is a directed graph of n colored nodes and m edges. The nodes are numbered from 0 to n - 1.

You are given a string colors where colors[i] is a lowercase English letter representing the color of the ith node in this graph (0-indexed). You are also given a 2D array edges where edges[j] = [aj, bj] indicates that there is a directed edge from node aj to node bj.

A valid path in the graph is a sequence of nodes x1 -> x2 -> x3 -> … -> xk such that there is a directed edge from xi to xi+1 for every 1 <= i < k. The color value of the path is the number of nodes that are colored the most frequently occurring color along that path.

Return the largest color value of any valid path in the given graph, or -1 if the graph contains a cycle.

Example 1:

Input: colors = “abaca”, edges = [[0,1],[0,2],[2,3],[3,4]] Output: 3 Explanation: The path 0 -> 2 -> 3 -> 4 contains 3 nodes that are colored “a” (red in the above image). Example 2:

Input: colors = “a”, edges = [[0,0]] Output: -1 Explanation: There is a cycle from 0 to 0.

Constraints:

n == colors.length m == edges.length 1 <= n <= 105 0 <= m <= 105 colors consists of lowercase English letters. 0 <= aj, bj < n

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 ?

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.