Check if There is a Valid Path in a Grid

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class Solution:
    def hasValidPath(self, grid: List[List[int]]) -> bool:
        if not grid:
            return true
        directions = {1: [(0,-1),(0,1)],
                      2: [(-1,0),(1,0)],
                      3: [(0,-1),(1,0)],
                      4: [(0,1),(1,0)],
                      5: [(0,-1),(-1,0)],
                      6: [(0,1),(-1,0)]}
        visited = set()
        goal = (len(grid)-1, len(grid[0]) - 1)
        def dfs(i, j):
            visited.add((i,j))
            if (i,j) == goal:
                return True
            for d in directions[grid[i][j]]:
                ni, nj = i+d[0], j+d[1]
                is_valid_row = 0 <= ni < len(grid)
                is_valid_column = 0 <= nj < len(grid[0])
                is_not_visited = (ni, nj) not in visited

                if is_valid_row and is_valid_column and is_not_visited:
                    is_direction_valid = (-d[0], -d[1]) in directions[grid[ni][nj]]
                    if is_direction_valid:
                        if dfs(ni, nj):
                            return True
            return False

        return dfs(0,0)

10 Prerequisite LeetCode Problems

For this, the following are a good preparation:

“200. Number of Islands” - Understanding how to traverse connected components in a grid.
“130. Surrounded Regions” - Another problem about traversing through a grid and understanding how to define a connected path.
“79. Word Search” - Helps in understanding the concept of moving in multiple directions within a grid.
“547. Friend Circles” - While this problem is about adjacency matrix and not a grid, the idea of finding connected components is very similar.
“695. Max Area of Island” - Another problem for understanding how to traverse connected components in a grid.
“994. Rotting Oranges” - A grid problem where movement is key to the solution, provides a different kind of traversal strategy (BFS).
“286. Walls and Gates” - This problem gives a good understanding about grid traversal with certain restrictions.
“207. Course Schedule” - This problem, while not directly related to grids, is about determining a valid path within a graph, which is similar to the problem in question.
“127. Word Ladder” - This problem is about finding a valid path, just like the problem in question, although it’s in the context of a graph represented as words.
“417. Pacific Atlantic Water Flow” - This problem is all about movement within a grid, based on the surrounding cells, similar to our original problem.

These cover grid traversal and the idea of finding valid paths or connected components, which are crucial for problem 1391.

Problem Classification

Based on the problem statement, this is a path finding problem in the domain of graphs and algorithms.

The key ‘What’ components are:

An m x n grid representing a graph with cells as nodes and cell values indicating edge connections
A starting node at (0, 0)
An ending node at (m-1, n-1)
Valid paths must follow edge connections from start to end
Return a boolean indicating if a valid path exists

Given these components, this problem can be further classified as:

Graph theory - The grid forms a graph structure with nodes and edges
Path finding - Finding if a valid path exists between two nodes
Grid-based - The graph is represented as a 2D grid structure
Connectivity - Valid paths rely on connectivity between nodes via edges
Decision problem - Returns a boolean indicating path existence rather than the path itself

So in summary, this is a grid-based graph connectivity and path finding decision problem. The core challenge is using the grid/graph structure and edge connections to determine if a valid path can be formed from start to end.

Clarification Questions

Here are some clarification questions to ask:

Is the grid directed or undirected? (i.e. can we move in both directions along an edge or only one?)
Can we visit the same cell multiple times in a path?
Are all cells in the grid connected to some edge? Or can some cells be isolated with no connections?
Can we move diagonally or only orthogonally (up, down, left, right)?
What happens if there are cycles in the graph? Can we traverse them?
For cells on the edge of the grid, do they “wrap around” to the other side? Or are their edges blocked?
Is an empty path from start to end considered valid?
Is finding the shortest valid path a requirement? Or just any valid path?
Do all edges have the same “weight” or cost to traverse?
Are there any constraints on the total path length or nodes visited?
What values can the cells contain? Just 1-6 or anything?
What kind of data structure should the output path be in? List, matrix, etc?

These would clarify the rules, constraints, and edge cases of the problem so we can better understand the requirements and solve it correctly.

Problem Analysis and Key Insights

Here are some key insights from analyzing the problem statement:

The grid represents a graph with cells as nodes and cell values indicating edge connections. This allows us to leverage graph algorithms and concepts.
Valid paths must follow the edge connections. So we need to check if each cell on the path has an edge leading to the next cell.
The edge connections are undirected (no indication edges are one-way). So we can traverse edges in both directions.
All cells seem to be connected to some edge based on the cell value ranges. So no isolated nodes.
Diagonal movements are not mentioned. So likely only orthogonal moves are allowed.
No indication of cycles or looping paths. Likely each cell can only be visited once.
Grid edges are blocked, no wrapping around.
Empty paths are invalid since we must start at (0,0) and end at (m-1, n-1).
Shortest path not required, just connectivity.
All edges have equal weight/cost.
No other constraints mentioned on path length, nodes visited etc.
Only values 1-6 have meaning, others are likely invalid.
Output is a boolean, so no need to construct the actual path.

These insights allow us to narrow the scope of the problem and identify the key aspects we need to solve it - mainly using the grid as a graph and checking connectivity through valid edge connections. The other insights help avoid unnecessary work that doesn’t contribute to the core problem.

Problem Boundary

Based on the problem statement and analysis, the scope of this problem is:

Input space - A 2D grid/matrix of size m x n, with integer values 1-6 representing different edge connections
Output - A boolean value indicating if a valid path exists from the start cell (0,0) to the end cell (m-1, n-1)
Rules/Constraints - Movements are only orthogonal (no diagonal), each cell can only be visited once, all edge connections are undirected and have equal weight, grid edges are bounded
Objective - Find if there exists a valid path satisfying the movement and edge connection rules from start to end
Non-goals - Finding the actual path, minimizing path length, handling cycles/loops, wrapping around grid edges

So in summary, the scope is limited to:

Determining connectivity rather than constructing the path
Using a 2D grid as an undirected unweighted graph
Checking if edges allow traversing start to end following rules
Returning a boolean for path existence

Anything beyond this simple connectivity check like finding the actual optimal path, handling edge weights, minimizing path length etc. is out of scope. The core focus is just on using the provided grid/graph structure correctly to determine connectivity.

Here are some ways to establish the boundary and scope of this problem:

Inputs:

The input is restricted to a 2D grid/matrix of size m x n. No other input types or shapes.
Cell values are limited to integers from 1 to 6 representing different edge connections. No other data types or values are supported.

Preprocessing:

No preprocessing of the grid is required before determining connectivity. We use the grid as-is.

Processing:

Movement is restricted to orthogonal directions only (no diagonals).
Each cell can be visited only once. No cyclic or looping paths.
The objective is simply checking connectivity, not finding the actual path or minimizing distance.
Only valid edge connections represented by cell values can be used. No other implicit edges exist.

Outputs:

The only output is a boolean value indicating if a path exists or not.
We do not need to return the actual path or path length.
No other metadata like visited nodes, total distance etc. is required.

State:

No additional state persistence or change is required. Each test case can be handled independently.

So in summary, the bounds are established by the input grid structure and cell values, orthogonal move-based processing rules, boolean-only output, and stateless test cases. Anything outside these limits is out of scope for the problem.

Distilling the Problem to Its Core Elements

This problem is based on the fundamental concept of graph connectivity and path finding.

At its core, it is checking if two nodes in a graph are connected based on the edges between nodes. I would describe it simply as:

“Given a grid that represents a map with connected streets, can you get from point A to point B only moving through the streets?”

The core problem is determining connectivity, not actually finding the path. We can simplify the problem statement to:

“Given a grid representing a connected graph, determine if node A is connected to node B based on grid edges.”

The key components are:

Representing the grid as a graph
Identifying the start and end nodes
Checking each step for a valid edge to traverse
Tracking visited nodes
Returning true/false for connectivity

The minimal set of operations is:

Convert the grid to a graph representation
Mark start and end nodes
Use graph traversal algorithm (DFS/BFS) to visit connected nodes
Check if end node was visited
Return true if visited, false otherwise

So in summary, this problem can be simplified to its fundamental graph theory concepts and minimal operations needed to determine connectivity. The rest are details that build on top of this core problem.

Visual Model of the Problem

Here are some ways we could visualize the problem statement:

Show a sample m x n grid representing the input, with start and end nodes marked. Use colors/symbols to indicate the different edge values.
Animate a path traversing the grid, highlighting allowed and disallowed moves based on edge values. Show both a successful full path case and a failed case.
Illustrate the grid as a graph with nodes and edge connections. Highlight the nodes and edges relevant to a sample path.
Use a flow chart to show the logic and steps involved - converting grid to graph, running traversal, checking visited nodes, returning boolean value.
Create small example grids and show the valid and invalid paths on them visually.
Show grids with edge cases - isolated nodes, edges that lead to dead ends, disconnected components. Demonstrate positive and negative path outcomes.
Visualize the algorithm logic itself as flow charts, animations or illustrations rather than just the grid.
Use visualization libraries like D3.js to animated traversals on interactive grid/graph representations.

The goal is to create engaging visuals that demonstrate the core concepts and logical flow in an accessible way. Both the input structures and procedural logic can be visualized through samples, animations and diagrams. Interactive visuals could even let users experiment with test cases.

Problem Restatement

Here’s how I would paraphrase the problem statement in my own words:

We’re given a 2D grid of streets represented as numbers that show how each street cell connects to its neighbors. Our goal is to find out if it’s possible to traverse a path from the top-left starting point of the grid all the way to the bottom-right endpoint by moving only through connected streets.

Some key points:

The grid is like a map with streets as cells. The numbers show street connections.
We start at the top-left and want to reach the bottom-right.
We can only move through streets that are connected based on the number values.
We can’t modify the streets or connections. They are fixed.
The path should not repeat streets already visited.
Diagonal moves are not allowed, only up/down/left/right.
We don’t need to find the actual path, just determine if one exists.
The result should be a boolean value - true if there is a valid path, false otherwise.

In summary, this is a graph connectivity problem where we need to check if a path exists between two nodes in a grid-based graph by following valid edge connections representing streets. The core challenge is using the grid structure correctly to model connectivity.

Does this help summarize and clarify the essence and constraints of the problem? Let me know if any part needs more explanation or I missed anything important.

Abstract Representation of the Problem

Here is one way to formulate an abstract representation of this problem:

We can model the grid as a graph G = (V, E) where:

V is the set of vertices representing the cells in the grid
E is the set of edges between vertices representing the connections between cells

Let s ∈ V be the source vertex (start cell) Let t ∈ V be the target vertex (end cell)

The problem is then equivalent to finding if there exists a path P from s to t in G, such that:

P contains no repeated vertices (cells)
Consecutive vertices v1, v2 in P have an edge e(v1, v2) ∈ E between them

In other words, the core problem is finding if there is a simple path between two vertices s and t in a graph G, following only the edges E.

We can further abstract this as:

Given:

Graph G = (V, E)
Source vertex s
Target vertex t

Determine:

If there exists a simple path P from s to t using only edges E

Return:

True if such path exists, False otherwise

This captures the essence of the problem in graph theoretical terms without referring to streets, grid, cell values etc. The key elements are the graph connectivity structure and simple path determination between two nodes.

Does this help provide an abstract representation that could apply to problems beyond just grids and streets? Let me know if you would modify or clarify the abstraction in any way.

Terminology

Here are some key terms and concepts for understanding this problem:

Graph - A mathematical structure used to model pairwise relations between objects. It consists of vertices (nodes) connected by edges.

Grid/Matrix - A 2D structure of rows and columns used to represent a graph visually.Cells are nodes, cell values indicate edges.

Connectivity - Refers to the linkage between nodes in a graph based on edge connections. Checking connectivity is the main goal.

Path - A sequence of vertices connected by edges that allows traversing a graph from one node to another. Finding a valid path is key.

Orthogonal - Movement only in 4 compass directions - up, down, left right. No diagonal moves allowed in this problem.

Undirected Graph - Edges between vertices don’t have a direction associated with them. Connections are bi-directional.

Unweighted Graph - Edges do not have an associated cost or weight. All connections have equal priority.

DFS/BFS - Depth/Breadth First Search algorithms used to traverse a graph checking connectivity. Useful here.

Simple Path - A path with no repeated vertices. Needed to prevent cycles/loops.

Some other relevant concepts are adjacency matrix, edge lists, graph theory, traversal, directed vs undirected graphs, shortest paths, optimization etc. But the terms above capture the key concepts for this specific problem.

Problem Simplification and Explanation

Here’s one way to break down this problem into simpler concepts and an analogy:

Key Concepts:

Graph - The grid represents a graph with cells as nodes and cell values as edges
Connectivity - We need to check if two nodes are connected based on edge links
Pathfinding - Finding a walkable path between two nodes
Traversal - Navigating the graph in a systematic way to analyze connectivity
Validation - Ensuring each step taken follows the rules and constraints

Analogy:

Think of the grid as a city map with intersections as nodes and streets as edges. We want to navigate only via connected streets from our start point (home) to the destination (work).

The core challenge is checking if we can actually get from home to work based on existing streets. We don’t need to find the shortest or fastest route - just determine if we can ultimately get there by following valid streets.

It’s like asking “Can I drive from my house to the office given the current city map?” We check street connectivity iteratively until we either reach the destination or get stuck, then return true/false.

The key is modeling the grid as a graph and using graph traversal techniques to incrementally validate connectivity between nodes. Let me know if this helps explain the concepts and overall objective!

Constraints

Here are some characteristics and conditions from the problem that could help optimize or simplify the solution:

Small grid size - Max 300 x 300 cells means brute force approaches may be feasible.
Limited edge types - Only 6 possible edge connection types. Can encode these values for fast validation.
No weights - Uniform edge costs simplifies shortest path logic. Only need to check connectivity.
Undirected edges - Bidirectional traversal without one-way constraints.
No diagonals - Reduces number of neighbor checks to 4 directions per node.
Discrete grid - Allows encoding nodes as (row, col) pairs for fast indexing.
Single start/end nodes - Avoid more complex multiple source/sink logic.
No cycles - Nodes can only be visited once, avoiding repeating work.
Output is binary - Returns true/false for path existence. No need to reconstruct path.
Stateless - Each test case is independent. No shared state across cases.

These constraints eliminate a lot of potential complexity around large graphs, weighted edges, reconstructing optimal paths etc. The limited discrete input space and simple output allow optimizations like bit-encoding, memoization, pruning, and other techniques to speed up the baseline traversal approach.

Here are some key insights from analyzing the constraints of the problem:

Small grid size allows brute force approaches like iterating through all possible paths. We likely don’t need sophisticated optimizations for such small grids.
Limited edge types means we can encode connections as simple numeric values for fast validation checks during traversal.
No edge weights and undirected edges simplify shortest path logic. We can focus just on connectivity rather than optimal path finding.
Disallowing diagonal movements reduces the branching factor during traversal, speeding up search.
Discrete grid coordinates enable fast hashing and indexing structures to track visited nodes.
Single start and end points avoid handling multiple sources and sinks.
Forbidding cycles prevents repeated work and lets us stop search early if no unvisited neighbors exist.
Binary output means we can return as soon as we find any valid path, rather than finding all possible paths.
Stateless test cases mean we don’t have to optimize for dynamic updates or shared state between calls.

Overall, these constraints allow us to narrow our focus to just efficient connectivity checking with simple data structures like grids, sets and maps. We can use brute force exhaustive search approaches without worrying about scalability over large graph sizes or optimal path criteria. The problem is simplified to traversal while validating edges.

Case Analysis

Here are some additional test cases covering different aspects of the problem:

Minimal grid (2x2)

Input: [[1,2],[2,1]]

Output: True

Analysis:

Smallest possible valid grid
Tests logic works on minimal viable input
Important edge case

Disconnected grid

Input: [[1,0],[0,1]]

Output: False

Analysis:

No path exists from start to end
Tests logic handles unconnected components correctly
Important edge case

Longer linear path

Input: [[1,2,3,4,5]]

Output: True

Analysis:

Simple straight line path
Tests logic scales to larger grids
Main simple case

Partially disconnected

Input: [[1,2,0],[3,4,5]]

Output: False

Analysis:

Some connectivity but no full path
Tests partial connectivity is handled properly
Edge case

Many options

Input: [[1,2,1],[2,3,2],[1,2,1]]

Output: True

Analysis:

Multiple possible paths
Tests earliest path found is returned
Main case

Blocked endpoints

Input: [[0,2,3],[4,5,6],[7,8,0]]

Output: False

Analysis:

Tests blocked endpoints are detected
Important edge case

Wraparound

Input: [[1,2,3],[4,5,6],[1,7,8]]

Output: False

Analysis:

Wraparound is invalid connectivity
Ensures boundaries are respected
Important bad case

The key edge cases are the smallest valid grid, disconnected components, and blocked endpoints. Other cases help cover a range of connectivity options and grid sizes.

Here are some ideas for visualizing these test cases:

Minimal grid

Show 2x2 grid with start and end nodes marked. Draw path between them.

Disconnected grid

Show 2x2 grid with no path between start and end nodes. Indicate disconnect.

Longer linear path

Illustrate a single straight path across a 1xN grid.

Partially disconnected

Draw path from start that dead-ends before reaching end.

Many options

Show grid with multiple possible paths between start and end.

Blocked endpoints

Illustrate walls or missing edges blocking endpoints of path.

Wraparound

Indicate path wrapping around edges/boundaries incorrectly.

In general:

Use grids with start (S) and end (E) nodes marked
Show valid paths with bold green arrows, invalid in dashed red.
Indicate connectivity and disconnects visually.
Keep examples small for clarity.
Animate traversals and connectivity checks.
Show counter-examples of wrong behavior.

The goal is to make each case visually clear and distinct for better understanding. Animation can help illuminate the traversal logic. Counter-examples are useful to show what not to do.

Here are some key insights gained from analyzing these test cases:

The minimal grid shows the bare minimum needed for a valid test case and path. Useful for testing core logic and edge conditions.
Disconnected and blocked cases emphasize the need to properly detect and handle lack of connectivity early. This is a key edge condition.
Varying grid sizes and dimensions ensures the solution scales properly in simple connected cases.
Partial connectivity situations highlights the need to fully traverse the graph rather than stopping early.
Multiple paths showcase that only one valid path needs to be found, optimization is not needed.
Wraparound invalidates assumptions about graph boundaries and connectivity. Important to handle boundaries properly.
Simple linear paths, though unrealistic, are useful as a minimal baseline before adding complexity.
Negative test cases are just as important, to confirm robust handling of edge conditions.
Smaller examples are preferable for understanding, debugging and testing. Production data may be larger but harder to analyze manually.

Overall, these cases validate that the solution works correctly across a spectrum from minimal to edge cases, simple to complex connectivity, and invalid to valid paths. Together they expose and prevent many potential pitfalls.

Identification of Applicable Theoretical Concepts

Here are some mathematical and algorithmic concepts that could help simplify and solve this problem:

Graph theory - Representing the grid as a graph leverages existing graph algorithms and properties. Metrics like adjacency matrix, edge lists apply.
Pathfinding algorithms - BFS, DFS and Dijkstra’s algorithm provide efficient ways to traverse graphs and find paths. Useful for the core traversal logic.
Hash tables - Can map grid coordinates to visited status for fast lookup during traversal. More efficient than nested lists.
Bitmasking - The limited edge types can be binary encoded for fast validation checks when traversing.
Recursive backtracking - DFS traversal can be implemented recursively, allowing early pruning of dead ends.
Dynamic programming - Memoization can cache intermediate results and visited nodes to avoid recomputation.
Shortest path optimizations - Properties like optimal substructure can speed up traversal, despite not needing the absolute shortest path.

So concepts around graph theory, efficient data structures like hash tables and bitmasks, and recursive/dynamic programming approaches could help create a optimized algorithm. The problem lends itself well to computer science techniques despite being framed in a real-world context.

Simple Explanation

Here is how I would explain this problem in simple non-technical terms:

Imagine a map of city streets laid out in a grid pattern. Each intersection represents a city block. Some streets connect north-south, some east-west.

We want to go from the top-left intersection to the bottom-right one. But we can only travel along connected streets - no cutting diagonally across blocks!

The map shows which directions you can travel from each block. Our goal is to figure out if there is some path from start to finish using just the connected streets shown.

I wouldn’t need the full route planned out - just whether it’s possible or not to get there. If the streets let you eventually travel through the grid to reach the end, it’s possible. If there’s no way to get there through valid streets, it’s impossible.

It’s like asking a friend if they can drive from their house to the mall based on the roads in your city, without a GPS giving the exact directions. You just want to know, based on the connections, is there some path that would get them there?

The map is the grid, intersections are blocks, streets are connections. We try different paths until finding one that reaches the destination or getting stuck, then know if it’s possible or not. No need to map the exact route!

Problem Breakdown and Solution Methodology

Here is a step-by-step explanation of how I would approach solving this problem:

Represent the grid as a graph

Each cell is a node, cell values are edges between nodes
Creates a mental model using graph terminology and properties

Mark start and end nodes

Keep reference to start and end index positions for traversal

Traverse grid with DFS/BFS

Use recursion or queue to visit connected nodes incrementally
Check cell value allows movement before visiting next node

Track visited nodes

Use a hash table or set to mark visited node indexes
Avoid visiting same node multiple times

Check if end node visited

If end node was visited, return true, path exists
If not, return false, no path

It’s like exploring a maze with some blocked paths. We take one step at a time checking for valid openings until we either reach the exit or get stuck. Tracking visited cells avoids going in circles.

The traversal order can be tuned - BFS vs DFS, prioritizing certain directions. As long as all connected nodes are eventually visited, the end result is the same.

If the grid was much larger, I may optimize data structures and heuristics to prune the search space. But the overall approach of iterative traversal with validation remains the same.

Let’s walk through an example grid:

[[1,2],[3,4]]

Represent as graph
Mark (0,0) start and (1,1) end
Traverse - check (0,0) to (0,1) is valid
Check (0,1) to (1,1) is valid
Reached end at (1,1), return true

This incremental validation during traversal allows checking connectivity for any size grid.

Inference of Problem-Solving Approach from the Problem Statement

Here are some key terms and how they guide my approach:

Grid - Represents the input structure as a 2D array or matrix. Leads to modeling as a graph.
Graph - Tells me to use graph algorithms and traversals like BFS/DFS.
Connectivity - Indicates the core problem is checking reachability between nodes, not finding shortest path.
Path - Means I need an incremental validation approach to verify a full start to end path exists.
Orthogonal - Restricts movement to 4 directions. Simplifies neighbor checking during traversal.
Undirected - Edges are bidirectional so I can traverse efficiently without direction constraints.
Unweighted - Removing edge weights simplifies shortest path logic. Only connectivity matters.
Matrix - Data structure to represent the grid for efficient row/column indexing.
Traversal - Search algorithms like DFS/BFS that visit connected nodes efficiently.
Simple Path - Avoid repeated nodes. Use a visited set to track nodes seen.
Boolean - Output is binary existence. Can return immediately upon finding any valid path.

The core terms of grid, graph, connectivity, traversal, path and simple path tell me this fundamentally uses graph algorithms and incremental validation to check reachability. I focus on efficiently searching the structure rather than optimizing the path.

We can visualize some key properties and aspects of this problem using tables and diagrams:

Sample Grid:

| 1 | 2 | 0 |
| 3 | 0 | 5 | | 6 | 7 | 8 |

Show the grid structure as a table with start and end nodes labeled

Graph Model:

(0,0) — (0,1) | (1,0) — (1,2) | (2,0) — (2,1) — (2,2)

Illustrate graph representation with vertices as coordinates and edges as lines between connected nodes

Connectivity Checking:

      (0,1)
      /   \

(0,0) — (1,0) — (1,2) \ / (2,1)

Use directional arrows to show traversal logic and connectivity checking between nodes

Tracking Visited:

(0,0) —> (0,1) | (1,0) — (1,2) | (2,0) — (2,1) — (2,2) end

Mark visited nodes differently, here with *

Negative Case:

(0,0) — (0,1) dead end | (1,0) — (1,2) | (2,0) — (2,1) — (2,2) end

Show cases with no connectivity, leading to dead ends

These simple diagrams help visualize key aspects like graph structure, traversal logic, tracking state, and connectivity vs lack of connectivity.

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?

Here is one way we could break this down into more granular steps:

High-level approach:

Model grid as graph
Traverse graph checking for path

Refined steps:

Represent grid as adjacency matrix or edge list
Mark start and end vertices
Choose traversal algorithm (BFS or DFS)
Initialize empty visited set
Write traversal function:
1. Check current vertex against visited
2. If not visited, mark visited
3. Check all neighbors
4. Recursively call traversal on valid unvisited neighbors
Call traversal from start vertex
Check if end vertex was visited
Return true if visited, false otherwise

Independent parts:

Graph representation
Traversal algorithm selection
Tracking visited vertices

Repeatable patterns:

Checking node neighbors
Validating edge connectivity
Marking visited status
Recursive traversal call on each node

This breaks the problem down into smaller building blocks - representing the input graph, choosing a traversal algorithm, tracking state, and the core traversal logic. Independent parts can be solved modularly and combined. Repeated validation patterns emerge in the traversal function.

Solution Approach and Analysis

Here is a step-by-step explanation of how I would solve this problem:

Represent the grid as a graph

Convert the 2D grid input into a graph representation. Each cell is a node, cell values indicate edges.
This allows us to leverage graph algorithms and terminologies.

Mark the start and end nodes

Keep references to the start and end cell coordinates to initialize and terminate traversal.

Select a graph traversal algorithm

Use a search like breadth-first search (BFS) or depth-first search (DFS) to traverse the graph.
Check that each edge to a neighbor is valid before visiting the node.

Track visited nodes

Use a set or other data structure to track nodes already visited.
Avoid visiting the same node multiple times to prevent infinite loops.

Check if end node was reached

After traversal, check if the search reached the end node.
If so, return true, else return false.

This incrementally builds up connectivity by checking edges outwards from the start node until the end is reached or no more unvisited nodes exist.

If the grid was much larger, I may optimize the traversal and visited data structures. But the approach remains the same.

Example on 2x2 Grid:

[[1, 2], [3, 4]]
Start: (0, 0), End: (1, 1)
Use DFS traversal
Mark (0, 0) as visited
Check edge from (0, 0) to (0, 1) is valid
Visit (0, 1), mark as visited
Check edge from (0, 1) to (1, 1) is valid
Visit (1, 1), mark as visited
Reached end node, return true

This shows how the approach incrementally checks connectivity node-by-node until the end is reached.

Identify Invariant

An invariant in this problem is a condition or statement that remains true throughout the execution of the solution. Some potential invariants are:

The grid structure and cell values do not change. The grid input remains constant.
The start and end nodes remain fixed as the provided input coordinates.
Visited nodes, once marked visited, remain visited for the duration. Their visited status doesn’t change.
Any node can be reached by some path from the start node if the grid is connected. The overall connectivity won’t be altered.
The output is always a boolean value indicating connectivity. The output type remains the same.
Edges have consistent unambiguous definitions based on the fixed cell values. An edge exists or doesn’t exist - this edge validity doesn’t change.
The path finding rules remain constant, like no diagonals and no repeated nodes. These constraints are always enforced.
Intermediate traversal state may change, but the final output depends only on the original input and consistent rules/constraints.

So in summary, key invariants are the unchanged grid structure, fixed start and end points, consistent edge validity rules, and the boolean output. These establish boundaries within which our implementation operates.

Identify Loop Invariant

The loop invariant in this problem is a condition that remains true before and after each iteration of the main traversal loop. Here are some possibilities:

The grid structure remains unchanged. The grid input does not get modified.
The start and end nodes stay the same. Their coordinates are fixed.
Any visited nodes remain visited between iterations. Their visited status is preserved.
The output variable (True/False) keeps its initialized value until explicitly set.
The traversal algorithm (BFS/DFS) follows the same rules and constraints.
Edges between nodes maintain consistent definitions based on cell values.
The set of visited nodes only grows or stays the same. No nodes become unvisited.
If the end node was unreachable before an iteration, it remains unreachable after. Connectivity doesn’t change.

So in essence, the key loop invariants are:

Unchanged input grid
Fixed start and end points
Monotonically growing set of visited nodes
Consistent edge validity rules
Invariant final output variable

The loop cannot alter these core aspects of the problem - they must remain constant to allow the traversal to incrementally build up connectivity in a sound manner.

For this specific problem, the invariant and loop invariant are essentially the same concepts:

The invariant refers to a condition or property that remains true for the entire problem or algorithm.

The loop invariant is a condition that remains true before and after each iteration of a loop within the algorithm.

For this graph traversal problem, the invariant properties are:

The input grid does not change
The start and end vertices are fixed
Visited nodes remain visited
Valid edge definitions are consistent
The final output variable starts false

These same properties also need to be maintained as loop invariants before and after each traversal loop iteration.

For example, the set of visited nodes can only grow within the traversal loop. The grid edges remain fixed. And the output variable is not set to true until the very end.

So the invariants that describe the overall problem constraints and graph structure also need to be upheld by the traversal loop. Therefore, for this specific problem, the invariants and loop invariants are identical.

The key difference is that invariants apply globally while loop invariants only apply to the body of a loop. But the properties we need to remain constant throughout this problem are the same at both levels.

Thought Process

Here is how I would think through solving this step-by-step:

The problem statement cues:

Grid of values representing connections suggests modeling this as a graph
Goal is to find a path from start to end following valid connections
Path finding on a graph lends itself to graph traversal algorithms like BFS or DFS

With that insight, the steps would be:

Represent the grid as a graph by converting values to edges between nodes
Mark the starting and ending nodes
Implement BFS traversal:
- Initialize queue with start node
- Track visited nodes in a set
- Loop while queue not empty:
  - Pop node from queue
  - Check if node is end node, return True if so
  - Else add unvisited neighbors to queue
- Return False if end never reached
Return result of BFS traversal

This translates the grid into a graph representation we can traverse using standard BFS, incrementally checking connectivity until the end is reached or all nodes are visited.

In Python:

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def hasValidPath(grid):

  start = (0, 0)
  end = (len(grid)-1, len(grid[0])-1)
  
  visited = set()
  
  q = Queue()
  q.put(start)
  
  while not q.empty():
    node = q.get()
    
    if node == end:
      return True
    
    visited.add(node)
    
    for neighbor in getNeighbors(node, grid):
      if neighbor not in visited:
        q.put(neighbor)
        
  return False

The key is recognizing we can model this as graph traversal, then applying that algorithmic approach. We incrementally build up connectivity from start to end.

Here is one way to implement a solution to check if a valid path exists in the given grid:

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class Solution:
    def hasValidPath(self, grid: List[List[int]]) -> bool:
        
        # Dictionary to map cell value to direction
        DIR = {
            1: (0, 1), # Right
            2: (0, -1), # Left 
            3: (-1, 0), # Up
            4: (1, 0), # Down
            5: (0, -1), # Left
            6: (1, 0) # Right
        }
        
        # Perform DFS traversal
        def dfs(row, col):
            
            # Check if out of bounds or already visited
            if not (0 <= row < len(grid) and 0 <= col < len(grid[0])) or (row, col) in visited:
                return False
            
            # Check if reached end
            if row == len(grid) - 1 and col == len(grid[0]) - 1:
                return True
            
            # Mark cell as visited
            visited.add((row, col))
            
            # Get next cell based on direction mapping
            next_row, next_col = row + DIR[grid[row][col]][0], col + DIR[grid[row][col]][1]
            
            # Recursive DFS on next cell
            return dfs(next_row, next_col)
            
        
        visited = set()
        return dfs(0, 0) # Start DFS from top-left corner 

The key steps are:

Map cell values to directions
Use DFS traversal to visit nodes
Check bounds and visited status
Return true if reach end
Mark visited and recursively DFS to next cell

This incrementally checks connectivity cell-by-cell through valid directions until reaching the end or visiting all cells.

CLAUDE GENERATED CODE IS WRONG

Establishing Preconditions and Postconditions

Here are the details for the hasValidPath method:

Parameters:

grid: 2D list of ints - represents the grid of cell values
Each element is an int from 1 to 6 - indicates cell connectivity

Preconditions:

grid must not be empty
grid only contains ints from 1 to 6
grid is a valid rectangular matrix

Method Functionality:

Models grid as a graph with cells as nodes and values as edges
Does DFS traversal checking if each edge is valid
Returns true if node (m-1, n-1) can be reached from (0,0)
Returns false if end node cannot be reached

Postconditions:

grid is unchanged
Returns bool indicating if path exists from start to end

Error Handling:

If grid is invalid, raise ValueError
If preconditions fail, raise RuntimeError
Handle out of bounds with false return in DFS

So in summary, this takes a 2D grid input, checks if a valid connected path exists from start to end using DFS traversal, and returns a boolean without modifying the original grid.

Problem Decomposition

Here is how I would break down this problem:

Problem Understanding: Determine if a path exists from the top-left to bottom-right of a 2D grid following valid directional connections between cells.
Initial Breakdown:

Represent grid as a graph
Implement graph traversal to check connectivity
Validate movements during traversal
Identify if end node was reached

Subproblem Refinement:

Represent grid:
- Convert grid to adjacency matrix
- Encode cell values as edges
Traversal:
- Implement BFS or DFS search
- Enqueue/recurse to neighboring nodes
Validation:
- Check cell value allows movement
- Track visited nodes
- Check within grid bounds
Check end:
- If end node visited, return true
- Else return false

Task Identification:

Grid to graph conversion
BFS/DFS traversal framework
Neighbor enumeration
Cell validation
Tracking visited status

Task Abstraction:

Grid representation
Graph traversal
Node expansion
Movement validation
Visitation tracking

Method Naming:

convertToGraph
bfsTraversal / dfsTraversal
getValidNeighbors
checkMove
trackVisited

Subproblem Interactions:

Convert grid before traversal
Traversal relies on neighbor expansion and validation
Tracking visited nodes enables cycle detection
Check end after full traversal completes

So in summary, break into representing the grid, traversing it through valid expansions, and tracking state - then refine and generalize these tasks.

From Brute Force to Optimal Solution

Here is a brute force solution and how we can optimize it:

Brute force:

Generate all possible paths from start to end
Check if each path only moves through valid edges
Return true if any valid path found

This tries every single path exhaustively.

Complexity: Time - O(n!), explores n! permutations Space - O(n), store path

Optimizations:

Use DFS/BFS traversal instead of brute force path generation

This builds up connectivity incrementally instead of trying all paths.

Complexity: Time - O(mn) Space - O(mn)

Add memoization to avoid repeated exploration

Store visited nodes in a hashmap. Don’t re-explore visited nodes.

Complexity: Time - O(mn) Space - O(mn)

Prioritize certain directions during traversal

Heuristics can guide traversal to likely better paths first.

Complexity: Time - O(mn) Space - O(mn)

Bitmask cell values for constant time validation

Encode cell values as bitmasks for O(1) edge checking.

Complexity: Time - O(m*n) Space - O(1)

In summary, we optimized from brute force permutation checking to graph traversal, pruning via memoization, heuristic search, and bitmask encoding. This improved the complexity from factoral time to linear time and space.

Code Explanation and Design Decisions

Here is how I would explain the reasoning behind the code:

The initial parameters are the input grid and visited set. The grid represents the graph to traverse and visited tracks state.
The main loop performs traversal on the graph, iterating over adjacent unvisited nodes. Each iteration expands the reachable set of nodes from the start.
Branches check if the node is within bounds and unvisited. This prevents traversing invalid nodes and avoids infinite cycles.
Visited is updated to include the current node. This tracks state to prune the traversal and enforce no repeated nodes.
An invariant is that visited only grows, and grid/rules are static. This ensures a sound traversal.
The final output indicates reachability between start and end nodes. True means a connecting path exists per the problem constraints.

In summary, the code traverses a graph represented by the grid, using branching and state tracking to incrementally build connectivity between nodes. The output satisfies the requirement of determining this connectivity. The rationale ties back to graph algorithms and sound state management.

Coding Constructs

Here are the high-level explanations for this code:

It uses graph traversal (DFS) to incrementally check connectivity between nodes. Also uses visited set for state tracking.
The code checks if you can navigate from the top-left to bottom-right of a grid map only moving through connected paths.
The logical constructs are a recursive traversal function, tracking visited nodes, validating moves, and returning true if the target is reached.
It does a depth-first search, checking each possible path of nodes, marking visited nodes, and returning true if the destination is found.
The key steps are recursively exploring neighbors, tracking visited status, validating moves, and terminating when the goal is reached or no moves remain. This builds up connectivity.
The core algorithmic strategies are recursive graph traversal with state tracking and incremental validation. This allows exploring all connected nodes in a structured way.

In summary, the code leverages common graph algorithms like DFS traversal and visited sets to incrementally validate connectivity between start and end points on a grid. The explanations focus on the high-level reasoning rather than language-specific implementation details.

Language Agnostic Coding Drills

Here is how I would break down this code into learning units:

Concepts in increasing difficulty:

Variable initialization - Create needed variables like visited set, directions map, start and end points. Easy setup step.
Function definition - Define the dfs recursive function signature. Basic function creation.
Data structure update - Add to visited set. Simple data mutation.
Base case handling - Check if current node is end. Key recursion concept.
Neighbor generation - Use directions to get adjacent nodes. Core of traversal.
Data validation - Check bounds and visited. Key to preventing errors.
Recursive call - Call dfs on each valid neighbor. Recursion driver.
Return logic - Return true if end found. Propagate result up.

The problem solving approach would be:

Initialize needed data structures
Define recursive dfs function
In dfs function:
- Check if current node is end
- Generate valid neighbors
- Validate bounds and visited
- Call dfs on each valid neighbor
Call dfs starting from start node
Return result of dfs call

Each coding drill builds up functionality needed for the graph traversal. We start with setup, handle the base case, iteratively expand nodes, filter invalids, recurse, and finally propagate the result. Combining these smaller skills solves the overall problem.

Targeted Drills in Python

Here are Python coding drills for each concept:

Variable initialization

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# Initialize needed variables
visited = set() 
directions = {1: [(0,-1), (0,1)], 2: [(-1,0), (1,0)], ...}
start = (0, 0)
end = (m-1, n-1)

Function definition

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# Define DFS function
def dfs(i, j):

  # Function body
  pass

Data structure update

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visited.add((i, j)) # Add node to visited set

Base case handling

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if (i, j) == end: # Check if current node is end
  return True # Base case

Neighbor generation

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neighbors = []
for d in directions[grid[i][j]]:
  ni, nj = i + d[0], j + d[1]
  neighbors.append((ni, nj))

Data validation

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is_valid = 0 <= ni < m and 0 <= nj < n # Bounds check
is_unvisited = (ni, nj) not in visited # Check visited

Recursive call

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if is_valid and is_unvisited: 
  if dfs(ni, nj):
    return True

Return logic

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return False # If no path found

We can integrate these by:

Initializing data
Defining dfs function
In dfs function body:
- Updating visited
- Checking base case
- Generating neighbors
- Validating nodes
- Calling dfs recursively
- Returning result
Calling dfs on start node

Check if There is a Valid Path in a Grid

10 Prerequisite LeetCode Problems

Problem Classification

Clarification Questions

Problem Analysis and Key Insights

Problem Boundary

Distilling the Problem to Its Core Elements

Visual Model of the Problem

Problem Restatement

Abstract Representation of the Problem

Terminology

Problem Simplification and Explanation

Constraints

Case Analysis

Identification of Applicable Theoretical Concepts

Simple Explanation

Problem Breakdown and Solution Methodology

Inference of Problem-Solving Approach from the Problem Statement

Simple Explanation of the Proof

Stepwise Refinement

Solution Approach and Analysis

Identify Invariant

Identify Loop Invariant

Thought Process

Establishing Preconditions and Postconditions

Problem Decomposition

From Brute Force to Optimal Solution

Code Explanation and Design Decisions

Coding Constructs

Language Agnostic Coding Drills

Targeted Drills in Python

Q&A

Similar Problems