Evaluate Division
To solve this problem, you can create a graph representing the relationships between the variables. Then, you can use a Depth First Search (DFS) to traverse the graph and find the values for the queries.
Here’s the code to achieve this:


This code constructs a graph based on the equations and values and then uses Depth First Search to traverse this graph to find the required values for each query.
The dfs
function is a recursive function to find the value for the query, and the main function iterates through all the queries to find the required values.
Given the constraints, the code should run efficiently for the provided inputs.
Identifying Problem Isomorphism
“Evaluate Division” involves dealing with a graph of ratios and performing depthfirst or breadthfirst searches to evaluate the division queries. Hence, it can be related to problems that involve graph traversal and pathfinding.
An approximate mapping is “Network Delay Time” (LeetCode #743). This problem asks for the time it will take for a signal to reach all nodes in a network (graph). The nodes receive a signal at a certain time, which can be compared to a ratio (or rate) in the “Evaluate Division” problem. In both cases, a kind of depthfirst or breadthfirst search is needed to traverse the graph and find the answer.
This mapping is an approximate isomorphism as the “Network Delay Time” problem also introduces an element of finding the node that takes the maximum time to receive the signal (i.e., finding the maximum time), while “Evaluate Division” focuses on finding the correct path to evaluate each division query. Furthermore, “Evaluate Division” requires handling of queries that may not be possible to evaluate (returning 1.0), while in “Network Delay Time”, it’s guaranteed that a path exists.
However, the core task in both problems – traversing the graph in a specific way to find a solution to the problem – is similar, and the skills and methods used in solving one could be very helpful in solving the other.
10 Prerequisite LeetCode Problems
Before tackling the “Evaluate Division” problem, which requires the knowledge of graph theory and particularly applying depthfirst search (DFS) or unionfind to solve it, get comfortable with these problems:
Number of Islands (LeetCode 200): This problem is a good introduction to depthfirst search in a 2D grid.
Course Schedule (LeetCode 207): This problem helps to understand topological sorting and DFS in the context of graph.
Course Schedule II (LeetCode 210): This is a followup to Course Schedule I, giving you more practice with topological sorting and DFS.
Flood Fill (LeetCode 733): Another problem to practice DFS on a 2D grid.
Clone Graph (LeetCode 133): This problem gives you practice with graph traversal, specifically depthfirst search and breadthfirst search, and how to handle visited nodes.
Graph Valid Tree (LeetCode 261): This problem gives you practice with both DFS and unionfind algorithm.
Redundant Connection (LeetCode 684): This problem allows you to practice UnionFind, a critical algorithm for the “Evaluate Division” problem.
Accounts Merge (LeetCode 721): This problem provides practice in using DFS and UnionFind on a real world problem.
Network Delay Time (LeetCode 743): This problem introduces Dijkstra’s algorithm, which is a useful graph algorithm for finding shortest paths.
Friend Circles (LeetCode 547): This problem is an application of DFS and UnionFind, similar to “Evaluate Division”.


Problem Classification
You are given an array of variable pairs equations and an array of real numbers values, where equations[i] = [Ai, Bi] and values[i] represent the equation Ai / Bi = values[i]. Each Ai or Bi is a string that represents a single variable.
You are also given some queries, where queries[j] = [Cj, Dj] represents the jth query where you must find the answer for Cj / Dj = ?.
Return the answers to all queries. If a single answer cannot be determined, return 1.0.
Note: The input is always valid. You may assume that evaluating the queries will not result in division by zero and that there is no contradiction.
Note: The variables that do not occur in the list of equations are undefined, so the answer cannot be determined for them.
Example 1:
Input: equations = [[“a”,“b”],[“b”,“c”]], values = [2.0,3.0], queries = [[“a”,“c”],[“b”,“a”],[“a”,“e”],[“a”,“a”],[“x”,“x”]] Output: [6.00000,0.50000,1.00000,1.00000,1.00000] Explanation: Given: a / b = 2.0, b / c = 3.0 queries are: a / c = ?, b / a = ?, a / e = ?, a / a = ?, x / x = ? return: [6.0, 0.5, 1.0, 1.0, 1.0 ] note: x is undefined => 1.0
Example 2:
Input: equations = [[“a”,“b”],[“b”,“c”],[“bc”,“cd”]], values = [1.5,2.5,5.0], queries = [[“a”,“c”],[“c”,“b”],[“bc”,“cd”],[“cd”,“bc”]] Output: [3.75000,0.40000,5.00000,0.20000]
Example 3:
Input: equations = [[“a”,“b”]], values = [0.5], queries = [[“a”,“b”],[“b”,“a”],[“a”,“c”],[“x”,“y”]] Output: [0.50000,2.00000,1.00000,1.00000]
Constraints:
1 <= equations.length <= 20 equations[i].length == 2 1 <= Ai.length, Bi.length <= 5 values.length == equations.length 0.0 < values[i] <= 20.0 1 <= queries.length <= 20 queries[i].length == 2 1 <= Cj.length, Dj.length <= 5 Ai, Bi, Cj, Dj consist of lower case English letters and digits.
Language Agnostic Coding Drills
Understanding and using data structures: This includes dictionaries (or maps), sets, lists, and understanding their usage.
Defining and calling functions: Understanding how to define a function with or without parameters, how to call it, and passing parameters between different functions.
Recursion and depthfirst search (DFS): Understanding how recursion works, how to implement it, and specifically the concept of DFS, which is a specific type of recursion used in graph traversal.
Looping over data structures: This includes iterating over lists and dictionaries, accessing elements and using them in computations.
Conditional Statements: Using ifelse conditionals to control program flow based on certain conditions.
Class and objectoriented programming: Understanding how to define a class, create methods within a class, and instantiate an object of the class to call its methods.
Type hints and annotations: Although optional, type hints are a good way to document your code and make it more understandable.
Error Handling: This includes managing unexpected or undesired outputs, such as a missing value in a dictionary.
Graph Theory: Understanding of basic graph concepts, how to represent a graph in code (like adjacency lists), and how to traverse them.
The drills should be ordered in this way, starting with understanding basic data structures and moving towards more complex concepts like depthfirst search and graph theory.
Targeted Drills in Python
Understanding and using data structures:
Create and manipulate data structures like dictionaries, sets, and lists.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
# Dictionary dict_sample = {'a': 1, 'b': 2, 'c': 3} print(dict_sample['a']) # Accessing a value dict_sample['d'] = 4 # Adding a new keyvalue pair # Set set_sample = {1, 2, 3} set_sample.add(4) # Adding a new value print(4 in set_sample) # Checking if a value exists # List list_sample = [1, 2, 3] list_sample.append(4) # Adding a new value print(list_sample[2]) # Accessing a value
Defining and calling functions:
Define a function that squares a number and call it with various inputs.
1 2 3 4 5
def square_number(n): return n**2 print(square_number(4)) # Output: 16 print(square_number(9)) # Output: 81
Recursion and depthfirst search (DFS):
Implement a recursive function that calculates the factorial of a number.
1 2 3 4 5 6 7
def factorial(n): if n == 1: return 1 else: return n * factorial(n1) print(factorial(5)) # Output: 120
Looping over data structures:
Loop over a dictionary and print each keyvalue pair.
1 2 3
dict_sample = {'a': 1, 'b': 2, 'c': 3} for key, value in dict_sample.items(): print(f"Key: {key}, Value: {value}")
Conditional Statements:
Implement a function that checks if a number is positive, negative, or zero.
1 2 3 4 5 6 7 8 9
def check_number(n): if n > 0: return "Positive" elif n < 0: return "Negative" else: return "Zero" print(check_number(5)) # Output: Positive
Class and objectoriented programming:
Define a simple class with a couple of methods and create an instance of that class.
1 2 3 4 5 6 7 8 9
class SampleClass: def __init__(self, value): self.value = value def get_value(self): return self.value sample_object = SampleClass(5) print(sample_object.get_value()) # Output: 5
Type hints and annotations:
Implement a function with type hints.
1 2 3 4
def add_numbers(a: int, b: int) > int: return a + b print(add_numbers(3, 4)) # Output: 7
Error Handling:
Implement error handling for a division operation.
1 2 3 4 5 6 7
def divide_numbers(a, b): try: return a / b except ZeroDivisionError: return "Cannot divide by zero." print(divide_numbers(10, 0)) # Output: Cannot divide by zero.
Graph Theory:
Represent a graph as a dictionary and implement a simple function to find neighbors.
1 2 3 4 5 6 7 8 9 10 11 12 13
graph = { 'A': ['B', 'C'], 'B': ['A', 'D', 'E'], 'C': ['A', 'F'], 'D': ['B'], 'E': ['B', 'F'], 'F': ['C', 'E'] } def find_neighbors(node): return graph[node] print(find_neighbors('A')) # Output: ['B', 'C']
10 Prerequisite LeetCode Problems
The “399. Evaluate Division” problem involves evaluating expressions given in the form of a list of equations. You are to calculate division equations as queries.
Here are 10 simpler problems to understand concepts involved in this problem:
207. Course Schedule: This problem asks to find if it’s possible to finish all courses given some prerequisites, a good exercise for graph traversal.
210. Course Schedule II: This is a similar problem to the previous one but this time you need to return a sequence of courses that you should take to finish all courses.
261. Graph Valid Tree: This problem involves detecting cycles in an undirected graph.
127. Word Ladder: In this problem, you have to find the shortest transformation sequence from one word to another.
200. Number of Islands: This problem involves depthfirst search (DFS) in a 2D grid.
133. Clone Graph: This problem involves creating a copy of a given graph.
684. Redundant Connection: This problem involves detecting a cycle in an undirected graph and removing an edge to break the cycle.
547. Number of Provinces: In this problem, you are given a matrix representing connections between cities, and you need to find out how many provinces there are.
785. Is Graph Bipartite?: Here, you need to determine if a given graph is bipartite.
323. Number of Connected Components in an Undirected Graph: You have to determine the number of connected components in a graph.
These problems involve graph traversal and cycle detection which are important concepts for solving the “Evaluate Division” problem. Understand the solutions to these problems and how depthfirst search and breadthfirst search can be applied to solve problems involving graphs.
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 an array of variable pairs equations and an array of real numbers values, where equations[i] = [Ai, Bi] and values[i] represent the equation Ai / Bi = values[i]. Each Ai or Bi is a string that represents a single variable. You are also given some queries, where queries[j] = [Cj, Dj] represents the jth query where you must find the answer for Cj / Dj = ?. Return the answers to all queries. If a single answer cannot be determined, return 1.0.
Note: The input is always valid. You may assume that evaluating the queries will not result in division by zero and that there is no contradiction.
Example 1:
Input: equations = [[“a”,“b”],[“b”,“c”]], values = [2.0,3.0], queries = [[“a”,“c”],[“b”,“a”],[“a”,“e”],[“a”,“a”],[“x”,“x”]] Output: [6.00000,0.50000,1.00000,1.00000,1.00000] Explanation: Given: a / b = 2.0, b / c = 3.0 queries are: a / c = ?, b / a = ?, a / e = ?, a / a = ?, x / x = ? return: [6.0, 0.5, 1.0, 1.0, 1.0 ]
Example 2:
Input: equations = [[“a”,“b”],[“b”,“c”],[“bc”,“cd”]], values = [1.5,2.5,5.0], queries = [[“a”,“c”],[“c”,“b”],[“bc”,“cd”],[“cd”,“bc”]] Output: [3.75000,0.40000,5.00000,0.20000]
Example 3:
Input: equations = [[“a”,“b”]], values = [0.5], queries = [[“a”,“b”],[“b”,“a”],[“a”,“c”],[“x”,“y”]] Output: [0.50000,2.00000,1.00000,1.00000]
Constraints:
1 <= equations.length <= 20 equations[i].length == 2 1 <= Ai.length, Bi.length <= 5 values.length == equations.length 0.0 < values[i] <= 20.0 1 <= queries.length <= 20 queries[i].length == 2 1 <= Cj.length, Dj.length <= 5 Ai, Bi, Cj, Dj consist of lower case English letters and digits.
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 realworld 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 nontechnical 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 ProblemSolving 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?
Stepwise Refinement
Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the highlevel 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
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.
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 highlevel problemsolving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a nonprogrammer, 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 languageagnostic 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 problemsolving 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 stepbystep 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 Pythonbased 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 problemspecific 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.
Q&A
Similar Problems
Can you suggest 10 problems from LeetCode that require similar problemsolving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem.