Evaluate division leetcode 399 python

okay, let's dive into the "evaluate division" problem on leetcode (problem 399). this problem involves graph traversal and requires a good understanding of data structures like dictionaries and algorithms like depth-first search (dfs) or breadth-first search (bfs).

**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]`. you are also given an array of `queries` where `queries[j] = [cj, dj]`.

for each query `queries[j] = [cj, dj]`, you should **evaluate** the equation `cj / dj` and **return** the result. if you cannot find the answer to a query, return `-1.0`.

**note:**

* the input consists of equations that represent some relationships between variables. you need to use these relationships to answer the queries.
* the `values` array corresponds to the `equations` array.
* the variables are represented as strings (e.g., "a", "b", "c").
* you can assume that the input equations are consistent (no contradictions).
* division by zero is not allowed.

**example**

**core idea & approach**

the key to solving this problem is to model the equations as a graph. here's the breakdown:

1. **graph representation:**

* each variable (e.g., "a", "b", "c") will be a node in the graph.
* an equation like "a / b = 2.0" represents a directed edge from node "a" to node "b" with a weight of 2.0. we also add the reverse edge from "b" to "a" with a weight of 1.0 / 2.0.

2. **query evaluation:**

* for each query `[cj, dj]`, we need to find a path from node `cj` to node `dj` in the graph.
* if a path exists, the result of the query (i.e., the value of `cj / dj`) is the product of the weights along that path.
* if no path exists, the result is `-1.0`. we also handle the case where either `cj` or `dj` are not present in the graph.

3. **algorithm (depth-first search - dfs):**

* we'l ...

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Evaluate Division
LeetCode 399
Python
Graph Algorithm
DFS
Union-Find
Equations
Variable Division
Weight Calculation
Cycle Detection
Data Structures
Problem Solving
Coding Challenge
Mathematical Expression
Complexity Analysis