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DowngradedOur downstream service providers are currently experiencing outages, and our engineering team is actively working on a resolution. Some services—including the Solver, Partner, and Tools—are temporarily degraded with higher latency and lower bandwidth. Rest assured, Intervipedia, Solutions, and the Question Bank features are not impacted and remain fully operational.DowngradedOur downstream service providers are currently experiencing outages, and our engineering team is actively working on a resolution. Some services—including the Solver, Partner, and Tools—are temporarily degraded with higher latency and lower bandwidth. Rest assured, Intervipedia, Solutions, and the Question Bank features are not impacted and remain fully operational.DowngradedOur downstream service providers are currently experiencing outages, and our engineering team is actively working on a resolution. Some services—including the Solver, Partner, and Tools—are temporarily degraded with higher latency and lower bandwidth. Rest assured, Intervipedia, Solutions, and the Question Bank features are not impacted and remain fully operational.DowngradedOur downstream service providers are currently experiencing outages, and our engineering team is actively working on a resolution. Some services—including the Solver, Partner, and Tools—are temporarily degraded with higher latency and lower bandwidth. Rest assured, Intervipedia, Solutions, and the Question Bank features are not impacted and remain fully operational.
The Question
Coding

Evaluate Division Ratios

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 representing a single variable. You are also given some queries, where queries[j] = [Cj, Dj] represents the j-th query where you must find the value of 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. Constraints: - 1 <= equations.length <= 20 - equations[i].length == 2 - values.length == equations.length - 0.0 < values[i] <= 20.0 - 1 <= queries.length <= 20 - queries[i].length == 2 - Each Ai, Bi, Cj, Dj consists of lower case English letters and has length between 1 and 5.
Java
DFS
Union-Find
Adjacency List
Questions & Insights

Clarifying Questions

Graph Consistency: Can we assume that the input equations are always consistent (e.g., we won't get a/b = 2.0 and a/b = 3.0 simultaneously)?
Variable Constraints: Are variable names guaranteed to be non-empty strings, and are the division values always positive?
Handling Unknowns: If a query involves a variable that never appeared in the equations list, should we return -1.0?
Disconnected Components: Is it possible for the variables to form multiple disconnected sub-graphs? (e.g., a/b is given, and c/d is given, but no relation between and c).

Thinking Process

Graph Representation: Treat each variable as a node in a directed graph. An equation a/b = v creates two edges: a directed edge from a to b with weight v, and a reverse edge from b to a with weight 1/v.
Pathfinding Logic: To solve a query x/y, find a path from x to y in the graph. The result of the query is the product of the weights along that path (e.g., if x \to a \to y has weights w_1 and w_2, then x/y = w_1 \times w_2).
Search Strategy: Since we need to find any valid path in a potentially cyclic (but consistent) graph, a Depth-First Search (DFS) with a "visited" set is efficient to prevent infinite loops and explore reachable nodes.
Optimization: For a Staff-level approach, one might consider Union-Find with path compression and weight tracking, which allows queries to be answered in near O(1) after a one-time O(N) processing of equations.
Implementation Breakdown

Problem Set

Input:
List<List<String>> equations: pairs of variables.
double[] values: corresponding values for equations.
List<List<String>> queries: pairs of variables to evaluate.
Output: double[] containing the results of queries.
Constraints:
1 \le equations.length \le 20
1 \le queries.length \le 20
Strings are 1-5 characters long.

Approach

Algorithm: Depth-First Search (DFS)
Data Structure: Adjacency List (Map of Maps)
Complexity:
Time: O(E + Q \cdot V), where E is the number of equations, Q is the number of queries, and V is the number of unique variables.
Space: O(V + E) to store the graph and the recursion stack.

Implementation

Wrap Up

Advanced Topics

Union-Find with Weights: In a production system with a massive number of queries and infrequent updates, a "Weighted Union-Find" (Disjoint Set Union) is superior. Each node stores its parent and its ratio relative to the parent. Finding a/b becomes a lookup of their roots: if they share a root, the ratio is (ratio(a \to root) / ratio(b \to root)). This reduces query time to O(\alpha(V)).
Parallelization: If the query set is enormous, the DFS for each query can be executed in parallel since they are read-only operations on the graph.
Pre-calculation (Floyd-Warshall): If the number of variables V is very small, we could use a variation of the Floyd-Warshall algorithm to pre-calculate all-pairs shortest paths (ratios) in O(V^3).