Shortest Path in a Hidden Grid


Identifying Problem Isomorphism
“Shortest Path in a Hidden Grid” can be approximately mapped to “Walls and Gates”.
Here’s why:
In “Shortest Path in a Hidden Grid”, the goal is to determine the shortest path in a hidden grid which can only be navigated using a robot. The grid is hidden meaning that the robot does not know the layout of the grid and must discover it as it goes.
“Walls and Gates” shares a similar idea, you are given a 2D grid, and you need to find the shortest distance to each gate (0) from each empty room (Infinity). Both problems can be solved with similar breadthfirst search (BFS) techniques, which is particularly efficient in solving shortest path problems on grids.
However, the mapping is not exact. In “Shortest Path in a Hidden Grid”, you are dealing with a hidden grid, so the robot needs to explore the grid first before it can find the shortest path, which adds an extra layer of complexity. In “Walls and Gates”, you already have the full view of the grid and can directly apply the BFS.
“Walls and Gates” is simpler due to its straightforward application of BFS on a known grid. “Shortest Path in a Hidden Grid” is more complex due to the requirement of exploration before finding the shortest path.
10 Prerequisite LeetCode Problems
This involves depthfirst search (DFS) and breadthfirst search (BFS), along with interactive problem solving. Here are 10 problems to understand these concepts:
LeetCode 200: Number of Islands
 This problem helps you to understand basic concept of depthfirst search in a grid.
LeetCode 127: Word Ladder
 This problem will introduce you to the concept of BFS in the form of finding transformations with minimum steps, which is very similar to the problem at hand.
LeetCode 279: Perfect Squares
 This problem provides an understanding of BFS to find minimum number of perfect square numbers which sum up to a given target number.
LeetCode 130: Surrounded Regions
 This problem further helps in understanding depthfirst search in a grid.
LeetCode 994: Rotting Oranges
 This problem is an excellent introduction to BFS in a grid, where you have to find minimum time to rot all oranges.
LeetCode 286: Walls and Gates
 This problem is also similar to the problem at hand where BFS is used to update each room’s distance to its nearest gate.
LeetCode 542: 01 Matrix
 This problem uses BFS to compute shortest distance from all zeros in the grid.
LeetCode 785: Is Graph Bipartite?
 This problem will help in understanding BFS for graph traversal and coloring the graph.
LeetCode 417: Pacific Atlantic Water Flow
 This problem uses DFS to find cells from which water can flow to both Pacific and Atlantic oceans.
LeetCode 490: The Maze
 This problem helps understand how to navigate a maze using DFS, which is a very similar concept to finding the shortest path in a hidden grid.
These cover DFS and BFS in a grid, which is key to solving the “Shortest Path in a Hidden Grid” problem.
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:This is an interactive problem.
There is a robot in a hidden grid, and you are trying to get it from its starting cell to the target cell in this grid. The grid is of size m x n, and each cell in the grid is either empty or blocked. It is guaranteed that the starting cell and the target cell are different, and neither of them is blocked.
You want to find the minimum distance to the target cell. However, you do not know the grid’s dimensions, the starting cell, nor the target cell. You are only allowed to ask queries to the GridMaster object.
Thr GridMaster class has the following functions:
boolean canMove(char direction) Returns true if the robot can move in that direction. Otherwise, it returns false. void move(char direction) Moves the robot in that direction. If this move would move the robot to a blocked cell or off the grid, the move will be ignored, and the robot will remain in the same position. boolean isTarget() Returns true if the robot is currently on the target cell. Otherwise, it returns false. Note that direction in the above functions should be a character from {‘U’,‘D’,‘L’,‘R’}, representing the directions up, down, left, and right, respectively.
Return the minimum distance between the robot’s initial starting cell and the target cell. If there is no valid path between the cells, return 1.
Custom testing:
The test input is read as a 2D matrix grid of size m x n where:
grid[i][j] == 1 indicates that the robot is in cell (i, j) (the starting cell). grid[i][j] == 0 indicates that the cell (i, j) is blocked. grid[i][j] == 1 indicates that the cell (i, j) is empty. grid[i][j] == 2 indicates that the cell (i, j) is the target cell. There is exactly one 1 and 2 in grid. Remember that you will not have this information in your code.
Example 1:
Input: grid = [[1,2],[1,0]] Output: 2 Explanation: One possible interaction is described below: The robot is initially standing on cell (1, 0), denoted by the 1.
 master.canMove(‘U’) returns true.
 master.canMove(‘D’) returns false.
 master.canMove(‘L’) returns false.
 master.canMove(‘R’) returns false.
 master.move(‘U’) moves the robot to the cell (0, 0).
 master.isTarget() returns false.
 master.canMove(‘U’) returns false.
 master.canMove(‘D’) returns true.
 master.canMove(‘L’) returns false.
 master.canMove(‘R’) returns true.
 master.move(‘R’) moves the robot to the cell (0, 1).
 master.isTarget() returns true. We now know that the target is the cell (0, 1), and the shortest path to the target cell is 2.
Example 2:
Input: grid = [[0,0,1],[1,1,1],[2,0,0]] Output: 4 Explanation: The minimum distance between the robot and the target cell is 4. Example 3:
Input: grid = [[1,0],[0,2]] Output: 1 Explanation: There is no path from the robot to the target cell.
Constraints:
1 <= n, m <= 500 m == grid.length n == grid[i].length grid[i][j] is either 1, 0, 1, or 2. There is exactly one 1 in grid. There is exactly one 2 in grid.
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 can I recognize these properties by drawing tables or diagrams?
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.