Path With Minimum Effort
This is a binary search combined with depthfirst search (DFS) or breadthfirst search (BFS) problem. The idea is to use binary search to search for the maximum absolute difference and use DFS or BFS to check if it is possible to reach the end with the current maximum difference.
We can start the binary search from 0 to the maximum possible difference in the matrix, which is heights[i][j]  1
(where i
and j
are the coordinates of the maximum and minimum heights in the grid respectively).
During each iteration of the binary search:
 We take the mid value as the effort.
 We then use BFS/DFS to see if we can travel from the topleft to the bottomright with the current effort.
 If we can, we update the right boundary of the binary search. Otherwise, we update the left boundary.
Python solution:


Here, we used BFS for traversal. The can_cross
function will determine whether we can travel from topleft to bottomright using BFS with the given effort.
10 Prerequisite LeetCode Problems
For this, the following are a good preparation:
“200. Number of Islands”: This problem is about exploring a grid in 2D space similar to the hiking terrain, it introduces the idea of Depth First Search (DFS) on a grid.
“994. Rotting Oranges”: This problem also deals with exploring a grid but with Breadth First Search (BFS) which could be useful for traversing the terrain.
“127. Word Ladder”: The concept of finding the shortest transformation sequence could be mapped to finding the minimum effort path in problem 1631.
“207. Course Schedule”: This problem introduces topological sort and cycle detection in a directed graph which is a more advanced concept in graph theory.
“286. Walls and Gates”: This problem has similar logic with 1631 that also looks for the shortest path in a grid.
“490. The Maze”: This problem involves navigating a maze which could be helpful for understanding how to navigate the grid in problem 1631.
“695. Max Area of Island”: This problem also requires traversing a 2D grid and it helps in understanding how to keep track of the visited cells in problem 1631.
“752. Open the Lock”: This problem involves finding the shortest path in a graph, similar to finding the path with minimum effort in problem 1631.
“787. Cheapest Flights Within K Stops”: This problem introduces the concept of Djikstra’s algorithm for finding shortest paths in a graph, which can be modified to find the path with minimum effort in problem 1631.
“1306. Jump Game III”: The idea of jumping through an array could help in understanding how to move through the grid in problem 1631.
These cover graph theory and depth/breadth first searches, which are crucial in solving problem 1631.
Problem Classification
This is a path finding problem in the domain of graph theory and algorithms.
The ‘What’ components are:
 Grid of cells with height values (graph vertices)
 Start and end vertices
 Ability to move up/down/left/right (graph edges)
 Cost metric based on consecutive cell height difference (edge weights)
 Goal is minimum effort path (shortest path)
I would categorize this problem as:
 Graph problem, since it involves vertices and edges
 Shortest path problem, as we want to find the minimum cost path
 More specifically, a gridbased shortest path problem, since the graph is a 2D grid
 Can also view it as a variation of Dijkstra’s algorithm, which is commonly used for shortest paths
The core of the problem is finding the shortest path on a graph with weighted edges. The specifics of it being a grid, the height difference cost metric, and allowing only vertical/horizontal moves make it a specialized case of the broader shortest path category.
Clarification Questions
Here are some clarification questions I would ask about this minimum effort path problem:
Are the heights always positive integers or can they be 0 or negative?
Can we move diagonally or only up/down/left/right?
Is each cell guaranteed to be reachable from the start cell? Or could there potentially be disconnected groups of cells?
For the effort calculation, do we consider the difference between the start and end cells of the path or just consecutive cells along the path?
Is the effort minimized for the entire path or just between consecutive cells?
Can we revisit the same cell multiple times in the path?
Do we need to return the actual minimum effort path or just the effort value?
Is the grid sparse (mostly empty cells) or dense (most cells filled)?
What are the time and space complexity requirements?
Are the dimensions of the grid fixed or variable?
Can we assume the input is valid or do we need to validate it?
Does the solution need to be optimal or is a good heuristic solution acceptable?
Is the grid jagged (different row lengths) or rectangular?
Getting clarity on these aspects would allow me to better understand ambiguous requirements, validate assumptions, handle edge cases properly, and design an optimal solution fitting the problem constraints.
Problem Analysis and Key Insights
Here are some key insights from analyzing this minimum effort path problem:
Modeling the grid as a graph with cells as vertices and movements as edges allows leveraging graph algorithms.
Effort minimization means we want shortest path, suggests Dijkstra’s or A* algorithm.
Effort definition based on consecutive cell height difference informs edge weight calculation.
Restricted to only vertical/horizontal movements, no diagonals, simplifies edge definitions.
Start and end cells given allows singlesource shortest path search.
Grid is 2D rectangle, so we can use 2D array and coordinate system.
Reasonable grid size limit allows simpler brute force approaches.
No clarification on repeated cell visits or disconnected grids means we should assume simplest connected case.
No optimality criteria given, so fast heuristic solution may suffice over slower optimal algorithms.
No need to reconstruct actual path, only return effort value.
In summary, modeling as a graph, using shortest path algorithms, mapping effort to edge weights, leveraging 2D grid structure, and implementing a reasonable heuristic approach seem like good fits based on the insights from the problem statement analysis.
Problem Boundary
Here are some key points about the scope of this minimum effort path problem:
The input is a 2D grid (matrix) of height values. Reasonably small size of up to 100x100.
Heights are positive integers from 1 to 1 million. No special values need handling.
The scope is focused just on finding the optimal path and its effort value.
Constructing the actual path sequence is out of scope. Only the effort is needed.
Movement is limited to the 4 cardinal directions. No diagonals allowed.
It’s assumed all cells are reachable and connected into one group. Disjoint groups not handled.
Repeated visits to cells are implicitly allowed based on problem description.
The optimal substructure property likely holds, allowing dynamic programming.
No clarification on time/space complexity limits, so can try slower optimal algorithms first.
The problem is selfcontained. No interaction with external data sources or state.
So in summary, the scope is limited to finding the optimal effort value on a reasonably sized, fully connected grid graph with only vertical/horizontal movements using standard algorithms. Construction of the actual path sequence is not needed.
Here are some ways to establish the boundaries for this minimum effort path problem:
Input Space
 Grid dimensions: 1x1 to 100x100
 Height values: 1 to 1,000,000
 Data types: Integer grid
 Allowed movements: Up, down, left, right (no diagonals)
 Start: Top left cell
 End: Bottom right cell
Output Space
 Output type: Integer effort value
 Minimum output: 0 (no effort path exists)
 Maximum output: 1,000,000 (max height difference)
Algorithm Space
 Optimality: Guaranteed optimal solution not required
 Time complexity: No established limit, can try exponential brute force
 Space complexity: No set limit
Problem Rules
 All cells connected and reachable
 Multiple visits allowed
 Only vertical and horizontal movements
 Effort calculated from consecutive cell differences
Out of Scope
 Reconstructing full path sequence
 Handling disjoint cell groups
 Finding all possible minimum effort paths
Establishing these clear boundaries on the input, output, algorithms, and rules provides a solid problem specification to work within.
Distilling the Problem to Its Core Elements
This minimum effort path problem is based on the core concept of finding the shortest path in a graph.
In simplest terms, I would describe it as:
“Finding the easiest route to get from point A to point B on a map with varying difficulty levels across it.”
The core problem is finding the path between two points that minimizes some cost metric. We can simplify it to:
“Find the lowest effort path between two points on a grid.”
The key components are:
 Grid of points with defined difficulty
 Start and end points
 Ability to move between adjacent points
 Effort calculated from difficulty differences
 Goal is minimum total effort path
The minimal operations needed are:
 Model grid as graph
 Define effort cost function
 Search for lowest effort path
 Return total effort of best path
Fundamentally, it involves modeling a spatial network as a graph and leveraging shortest path algorithms to find the optimal route. The domain specifics add constraints, but the underlying concept is determining the shortest path based on defined edge costs.
Visual Model of the Problem
Here are some ways to visualize the minimum effort path problem statement:
Draw a simple 5x5 grid representing the 2D array of heights. Label the axes and mark the start and end points.
Show a path traversing from start to end, using arrows to indicate the path direction. Annotate the height values along the path.
Highlight the maximum height difference between two consecutive cells on the path to illustrate the effort calculation.
Draw multiple paths with different maximum effort values. Indicate the path with minimum effort visually.
For larger grids, show a zoomin of a 3x3 region to visualize the cell height differences and effort at a granular level.
Visualize effort as a third dimension coming out of the grid. Show peaks and valleys based on cell heights. Illustrate path traversing along this surface.
Use a heat map to represent the grid, with hotter colors for higher cell heights. Shows how heights vary across the grid.
Illustrate the problem as a character navigating through an obstacle course maze from start to finish. Effort relates to obstacle difficulty.
Use realworld analogies like hiking elevation or construction grading to explain effort values.
These types of diagrams, graphs and illustrations could help provide an intuitive sense of the core concepts of the problem, including the grid, heights, path and effort calculation. Visuals make the problem more accessible.
Problem Restatement
Here’s how I would paraphrase the problem statement in my own words:
We are given a 2D grid or matrix of cells, where each cell has a height value. We start at the topleft cell and want to find a path to the bottomright cell. We can only move up, down, left or right  no diagonal moves allowed.
When moving between cells, we calculate the “effort” required as the maximum absolute difference between the heights of the consecutive cells along the path. Our goal is to find a path from start to end that minimizes this maximum effort value.
We don’t need to return the actual sequence of steps, only the minimum effort value for an optimal path. The grid dimensions can be up to 100x100 cells, and height values are positive integers up to 1 million.
To summarize, we need to find the lowest effort path from topleft to bottomright of a grid by optimizing the maximum height difference between consecutive cells. Our output is simply the minimum effort value, not the path itself. The grid size and height values are constrained to reasonable limits.
Let me know if I’m missing anything major or misinterpreting any aspects of the problem based on this paraphrasing! Getting the problem statement clearly framed helps align our thinking.
Abstract Representation of the Problem
Here is one way to formulate an abstract representation of this problem:
We can model the 2D grid as a undirected weighted graph G = (V, E).
The set of vertices V represents the cells in the grid.
The set of edges E represents allowable movements between adjacent cells (up/down/left/right).
Each edge (u, v) has a weight w(u, v) equal to the absolute difference in heights between vertices u and v.
We designate two special vertices:
 s  the start vertex
 t  the target vertex
The goal is to find the path P from s to t that minimizes the maximum edge weight along P. More formally:
min(max(w(u,v) : (u,v) in P))
This abstracts away the specifics of a grid and heights into a standard graph shortest path problem with a minmax objective function. The key aspects are the graph structure, weighted edges based on height differences, specified start/end vertices, and optimizing the bottleneck “effort” edge rather than total path length.
Let me know if I should clarify or expand on any part of this abstract representation! Extracting the core structure helps simplify and generalize the problem.
Terminology
Here are some key terms and concepts for understanding this minimum effort path problem:
Graph  Mathematical structure used to model pairwise relationships between objects. Used here to model the grid as vertices and movements as edges.
Vertex  Fundamental unit that makes up a graph. Represents a cell in the grid graph.
Edge  Connection between two vertices in a graph. Represents allowable movements between cells.
Weighted graph  Graph with numeric costs assigned to edges. Used here to model effort values.
Shortest path  Optimal path between two vertices that minimizes total edge weights. Solving this finds the minimum effort route.
Effort  Max absolute height difference between consecutive cells. Serves as the edge weight metric.
Adjacency matrix  Grid data structure recording connectivity of a graph. Can represent the grid graph.
Greedy algorithm  Optimization approach that takes locally optimal choices to find global optimum. Can help find the shortest path.
Dynamic programming  Method to solve problems by breaking into subproblems and storing solutions. Helps optimize effort calculation.
Understanding these graph theory and algorithm concepts is key to modeling and solving this problem efficiently. The definitions provide clarity on how core techniques can be applied.
Problem Simplification and Explanation
Here’s one way I could break down this minimum effort path problem into simpler terms:
Imagine you’re hiking through a mountainous area from your base camp to a lake (metaphor). There are different trails you can take that go up and down the mountain slopes.
The key concepts are:
The map of the area is like a grid of points. Each point is a location.
The elevation or height at each point is like the difficulty of traveling through. Higher means harder.
You need to calculate the effort or energy needed to move between adjacent points. This depends on the change in elevation.
Bigger changes in elevation require more effort. Small changes need less effort.
You want to find the path from your camp to the lake that requires the minimal effort. This means optimizing the biggest elevation changes along the way.
The goal is the lowest maximum effort, not the total effort over the path. Just minimize the hardest part.
So in essence, it’s like finding the most efficient hiking path by minimizing the largest change in elevation between two points, rather than worrying about the total up and down. Hopefully this makes the core ideas more intuitive! Let me know if any part needs more clarification.
Constraints
Here are some characteristics of this problem that could help optimize the solution:
The grid dimensions are reasonably small, at most 100x100. This means simpler, less optimized algorithms may be feasible.
The height values are positive integers in a reasonable range from 1 to 1 million. We don’t have to worry about negative differences or overflow.
Effort depends only on the maximum difference, not the total sum. This allows greedy approaches that prioritize the bottleneck.
Only four movement directions are allowed. We can optimize data structures and operations for this restricted movement.
The effort calculation is straightforward using the absolute height difference. This is simple to compute.
There are no obstacles or barriers stated, so likely we can assume all cells are reachable.
Multiple visits to the same cell are probably allowed, opening up cyclic paths.
No clarification on whether an optimal solution is required. Heuristics may be acceptable.
We only need to return the effort, not reconstruct the full path. This simplifies output.
Overall, the small discrete input space, simple effort calculation, lack of obstacles, and output requirements allow optimization opportunities such as greedy heuristics, fewer data structures, simplified computations and output.
Here are some key insights gained from analyzing the constraints:
The limited grid size allows brute force approaches without needing complex optimization algorithms. This simplifies implementation.
Having heights as positive integers eliminates handling negative differences or overflow errors when calculating effort.
Optimizing for the maximum effort instead of total encourages greedy heuristics that prioritize the bottleneck at each step.
Restricted movements in only 4 directions saves computation compared to allowing diagonals.
Effort depending only on consecutive cells means we can focus optimization locally. Global shortest path not needed.
Single fully connected group of cells implies basic graph search algorithms are sufficient without handling disjoint sets.
No need to reconstruct path places emphasis on optimizing effort calculation, not path tracking.
Undefined optimality criteria provides flexibility to use simpler heuristics in place of complete algorithms.
Reasonable height range bounds effort metric to a tractable integer scale.
In summary, the key insights are the constraints allow simplification of the graph, effort computation, search scope, and output while retaining flexibility in the solution approach. Together these present optimization opportunities.
Case Analysis
Here are some additional test cases covering different scenarios:
Basic case:
Input:
[[1, 2, 3],
[2, 5, 1],
[3, 1, 2]]
Output: 2
Reasoning: Has variation in heights, basic case to test effort calculation.
All equal heights:
Input:
[[1, 1, 1],
[1, 1, 1],
[1, 1, 1]]
Output: 0
Reasoning: Edge case with no effort required.
Disconnected regions:
Input:
[[1, 2, 2, 2],
[1, 1, 1, 1],
[2, 2, 2, 2]]
Output: Undefined
Reasoning: Disjoint sets not handled by problem statement.
Large grid:
Input: 100 x 100 grid of random heights
Output: Expected reasonable effort
Reasoning: Stress test performance with largest input size.
Zigzag path:
Input: Grid where optimal zigzag path is better than straight.
Output: Nontrivial effort based on zigzag optimization.
Reasoning: Optimal path may not be straightforward.
Analyzing these cases helps validate correctness, spot edge cases, and verify reasonable performance on different input types.
Here are some ideas for visualizing these test cases:
Basic case:
 Grid with numbered heights
 Path drawn from start to end
 Max difference labeled along path
All equal heights:
 Grid of all 1s
 Straight path from start to end
 Label 0 effort along path
Disconnected regions:
 Grid with groups of different values
 Start and end in separate groups
 Show inability to connect groups
Large grid:
 Sparse 10x10 grid representation
 Indicate it continues to 100x100
Zigzag path:
 Grid with consecutive high differences on straight path
 Zigzag path with smaller differences shown
In general:
 3D surface plot to visualize heights
 Heat maps to indicate effort levels
 Animated agent traversing optimal path
Visualization provides intuition about key aspects like optimality, differences in height, bottlenecks, etc. Plots help validate expected output.
Here are some key insights gained from analyzing these test cases:
The basic case proves the core algorithm is working as expected by verifying effort calculation.
The equal heights case reveals an edge case that produces a minimum possible effort of 0.
Disconnected groups demonstrate the need to handle invalid inputs or make connectivity assumptions clear.
Scaling to large grids shows if performance decays gracefully and where optimizations are needed.
Zigzag examples prove shortest straight line path isn’t always optimal, algorithms must explore broadly.
Visualizations provide intuition for height variations, path optimality and effort levels across the grid.
A wide variety of test cases validates the robustness of the solution across diverse inputs.
Analyzing expected outputs guides development of the effort calculation logic.
Testing boundary and edge cases early prevents bugs by exposing faulty assumptions.
Overall, these insights show that careful testing is crucial to develop a complete understanding of the problem space, illuminate subtle details, strengthen the solution, and gain confidence that diverse scenarios are handled properly.
Identification of Applicable Theoretical Concepts
Here are some mathematical and algorithmic concepts that could help optimize this problem:
Graph theory  Model the grid as a graph to leverage graph algorithms.
Shortest/lowest cost path algorithms  Dijkstra’s algorithm, A* search could efficiently find the optimal path.
Dynamic programming  Optimal substructure of problem suits memoization to speed up effort calculations.
Heuristics  Relaxing optimality allows greedy heuristics to quickly get a good solution.
Euclidean distance  Can accelerate neighbor searching in the grid graph.
Bounding box pruning  Use slack values to prune suboptimal paths.
Game theory  Formulate as a game by modeling cost, reward, actions.
Linear programming  Relax into LP form and solve for lower bound on optimal solution.
Parallelization  Effort checks for each path can be parallelized.
Hill climbing  Incrementally build path choosing locally optimal neighbor.
Leveraging relevant concepts from graph theory, algorithms, optimization, and related mathematical fields can enhance performance and simplify the overall solution.
Simple Explanation
Here’s how I would explain this minimum effort path problem in simple nontechnical terms:
Let’s imagine you’re hiking in a mountainous area from your base camp to a lake (metaphor). There are different trails you can take to get there, some trails go uphill and some go downhill.
The higher up in elevation the trail is, the more effort it takes to hike it. So going steeply uphill requires a lot of effort!
Now we want to find a path from the base camp to the lake that is easiest to hike. We do this by choosing a path that minimizes the biggest elevation change between two points along the path.
For example, if one path has a 100 foot uphill segment and the rest is flat, while another path has a 50 foot uphill segment but the rest is hilly, we would choose the second path since 50 feet is easier to hike than 100 feet.
The goal is to find the path that makes the hardest part  the biggest elevation change  as easy as possible. This allows us to hike the whole trail with the minimal effort overall.
Does this help explain the core idea in simple realworld terms? Let me know if any part needs more clarification!
Problem Breakdown and Solution Methodology
Here is a detailed stepbystep approach I would take to solve this minimum effort path problem:
Model the 2D grid as a graph. Each cell is a vertex, movements between cells are edges. This allows us to leverage graph algorithms.
Define a function to calculate effort as the absolute height difference between two adjacent vertices. This will represent the edge weights.
Use a graph search algorithm like Dijkstra’s to find the shortest path from start to end based on effort. Maintain min effort as we go.
To reduce repeated effort calculations, memoize computed efforts using dynamic programming. Store in a grid or map.
For efficiency, we can use a min heap priority queue to track minimum effort paths, since we want to optimize the bottleneck.
Iterate until we reach the end vertex. Return the minimum effort seen throughout the search.
Increasing grid size would slow things down, but we can use heuristics to prune obviously suboptimal paths to help scale. Using memoization helps minimize redundant effort calculations on overlapping paths.
Let’s walk through a simple 3x3 grid:
 Calculate effort values for each edge
 Explore possible paths with Dijkstra’s
 Get minimum effort path: down, right with effort 2
This converts the grid to a graph representation we can search efficiently using standard algorithms and optimization techniques.
Inference of ProblemSolving Approach from the Problem Statement
Here are the key terms and how they guide the solution approach:
Grid  Represents 2D layout of cells, suggests modeling as graph with grid coordinates.
Height  Vertex property, informs edge weight calculation as height difference.
Effort  Key optimization goal, maps to edge weight in graph.
Path  Sequence of edges, finding optimal path means shortest path algorithm.
Minimum  Optimization criteria, aim to minimize effort guides techniques like Dijkstra’s.
Movements  Grid navigation, translates to graph edge definitions.
Adjacent  Neighbor access implies local graph edge connectivity.
Absolute difference  How effort is measured, technique for weight function.
Start and end  Defined nodes, allows singlesource/singletarget shortest path search.
The core terms like grid, height, effort, path and optimization signal converting the grid to a graph and finding the shortest path using standard algorithms like Dijkstra’s or A*. The effort definition guides how edge weights are calculated.
Here are some ways to visualize the properties and concepts of this problem using diagrams:
Grid to Graph Mapping
 Draw grid cells as graph vertices
 Connect vertices with edges representing movements
 Label edges with effort weights based on height differences
Shortest Path Search
 Animate Dijkstra’s/A* search expanding from start vertex
 Show edge relaxations and effort updates
 Indicate optimal path once end is reached
Dynamic Programming Memoization
 Grid showing cells colored based on minimized effort values
 New searches consult cache to avoid recomputing efforts
Min Heap Priority Queue
 Show min heap tree structure prioritizing lowest effort paths
 Effort values decrease deeper into heap
Height Differences
 Plot 3D graph with peaks and valleys representing heights
 Show steep edges require high effort to traverse
These types of visual models reinforce the concepts and methods used in the solution, including mapping grids to graphs, searching techniques, memoization, data structures, and effort calculation.
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
Here is a stepwise refinement for solving the minimum effort path problem:
Highlevel approach: Use graph search algorithm to find shortest path based on effort
Break down into steps:
 Model grid as graph with cells as vertices
 Calculate edge weights as height differences
 Initialize search from start vertex
 Use priority queue to track minimum effort path
 Expand search to neighboring vertices
 Update effort if new path is better
 Repeat until reaching end vertex
 Return minimum effort
 Independent subproblems:
 Calculating effort for a given edge
 Initializing graph from grid
 Priority queue operations
 Repeatable patterns:
 Main search loop, expanding optimal path each iteration
 Updating effort if improved path found
 Priority queue insert/extract min operations
The key is modeling the grid as a graph to enable using standard shortest path algorithms. The effort calculation, graph initialization, and priority queue usage can be handled independently. The main search loop has a very repetitive pattern of expanding best paths and updating efforts.
Solution Approach and Analysis
Here is a detailed stepbystep approach to solving the minimum effort path problem:
Model the grid as a graph. Each cell is a vertex, edges connect adjacent cells. This allows using graph algorithms.
Define a function to calculate effort as the absolute height difference between two connected vertices. This will represent the edge weights.
Use Dijkstra’s shortest path algorithm to incrementally find the lowest effort path from start to end. Maintain the running minimum effort.
Use a minheap priority queue to efficiently track minimum effort paths for expansion. Prioritize vertices with lowest effort.
For each vertex, check all unvisited neighbors. If new path has lower effort, update the vertex effort and priority in queue.
Continue until we reach the end vertex or priority queue is empty. Return the final minimum effort value.
Increasing the grid size would slow things down. We can mitigate this by pruning obviously suboptimal paths and memoizing previously computed efforts. Optimized priority queue operations also help improve performance.
Let’s walk through a 3x3 grid:
 Calculate edge efforts: (0,0)(0,1)=3, (0,1)(0,2)=1, etc.
 Run Dijkstra’s using min heap queue to prioritize smallest effort path.
 Find (0,0)(0,1)(0,2) path with effort 3.
This leverages Dijkstra’s shortest path algorithm, using the priority queue for efficiency and modeling the grid as a graph to frame it as a wellstudied problem.
Identify Invariant
The invariant in this minimum effort path problem is:
At each step of the algorithm, the path with the current minimum effort value is guaranteed to be optimal among all paths explored so far.
This means as we are searching for the optimal path, the path currently stored with the minimum effort must be the lowest effort path considering all paths inspected up to that point.
We can maintain this invariant by:
 Initializing the minimum effort to infinity
 Only updating the minimum effort when a new path with lower effort is found
 Using optimal graph search algorithms like Dijkstra’s or A* that guarantee lowest cost paths are prioritized first
This ensures that at every iteration, we are expanding the most promising path that can lead to the global minimum. The final path with the minimum effort after the full search must be optimal over the entire graph.
The invariant allows an incremental optimized search by ensuring locally optimal decisions lead to the global optimum. We can be certain the current best path holds the minimum effort value so far.
Identify Loop Invariant
What is the loop invariant in this problem?
Is invariant and loop invariant the same for this problem?
Thought Process
Here is the thought process to solve this minimum effort path problem:
The problem statement cues:
 Grid of cells with heights
 Start and end points
 Movements between adjacent cells
 Effort calculated from height differences
 Goal is minimum effort path
This suggests:
 Model grid as graph
 Start/end vertices
 Connected vertices for adjacent cells
 Edge weights from height differences
 Apply shortest path algorithm
Steps:
 Represent grid as graph
 Initialize start and end vertices
 Calculate edge weights as absolute height differences
 Use Dijkstra’s algorithm to find shortest path by effort
 Return minimum effort seen
Python code:


This implements the thought process and systematic steps to solve the problem.
Here is an implementation of the minimum effort path algorithm in Python:


This implements Dijkstra’s algorithm using a minheap priority queue to efficiently find the minimum effort path from start to end based on maximum absolute height differences between adjacent cells.
Establishing Preconditions and Postconditions
Here are the details for the minEffortPath() method:
 Parameters:
 grid: 2D list of integers  The height grid
 start: Tuple  The start cell coordinates
 end: Tuple  The end cell coordinates
 Preconditions:
 grid is a valid 2D grid of heights
 start and end are within grid bounds
 Method Functionality:
 Finds the minimum effort path from start to end
 Returns the total minimum effort value
 Postconditions:
 The minimum effort path is found
 Minimum effort integer value is returned
 Error Handling:
 Invalid grid shape raises ValueError
 Out of bounds start/end raises IndexError
 Unreachable end raises RuntimeError
 On other errors, raises RuntimeError
This method takes a grid and start/end points, calculates the optimal path by effort, and returns the minimum total effort value, while validating inputs and handling potential errors.
Problem Decomposition
Here’s how I would break down this minimum effort path problem:
We need to find the lowest effort path between cells in a grid based on height differences.
Major parts are:
 Model grid as graph
 Calculate effort values
 Find shortest path
 Track minimum effort
Breakdown:
 Build graph from grid
 Write effort calculation function
 Initialize search
 Relax edges and update efforts
 Maintain min effort seen
 Stop when end reached
General tasks:
 Graph construction
 Effort computation
 Shortest path search
 Minimum value tracking
Abstraction looks good
Naming looks clear
Order:
 Build graph
 Initialize search
 Compute efforts
 Relax edges
 Update minimum
 Repeat until end vertex
 Return minimum effort
The key is modeling as a graph and leveraging standard shortest path algorithms. Effort calculation and min tracking can be done independently. Overall, clear structured breakdown with reusable components.
From Brute Force to Optimal Solution
Here is a brute force approach and optimizations:
Brute force:
 Try all possible paths from start to end
 For each path, compute the maximum effort
 Return the path with minimum maximum effort
This evaluates an exponential number of paths in the worst case.
Optimizations:
 Prune infeasible paths:
 Discard paths that go out of bounds or form cycles
 Eliminates invalid paths early
 Memoize computed efforts:
 Cache effort values already computed
 Avoid recomputing efforts for shared subpaths
 Prioritize most promising paths:
 Use a min heap queue to expand paths with lowest effort first
 Find optimal quickly without checking all paths
 Iterative deepening:
 Incrementally increase max search depth
 Find optimal but prune deeper paths
 Heuristics:
 Use a heuristic that underestimates effort to guide search
 Further accelerate finding the optimal path
These optimizations improve the worst case time complexity from O(n!) to O(E log V) by pruning the search space, caching redundant work, and prioritizing high quality paths.
Space complexity is reduced from O(n) to O(V) by eliminating storage of all paths.
We optimize using informed search strategies, reuse prior work, and leverage problem structure to massively reduce the solution space.
Code Explanation and Design Decisions
Here is an analysis of the logic and reasoning behind this minimum effort path code:
Initial parameters are the grid heights, start/end points, and graph data structures to represent the grid. These model the problem domain.
The main loop pops vertices from the priority queue and relaxes their neighboring edges. This expands the search outward iteratively.
The if condition checks if we reached the end, to stop search early. This meets the objective.
The dist array is updated when a lower effort path is found. This maintains optimal values.
The invariant is that dist array always holds the currently lowest efforts. This ensures paths are optimized.
The final min_effort is the smallest possible to reach the end. It satisfies the goal of minimizing effort.
Overall, the logic models the grid, expands the search strategically using the priority queue, prunes early if end reached, maintains optimal effort values throughout, and returns the final minimized effort  aligning directly with the problem statement requirements.
Coding Constructs
Here is an analysis of the highlevel logic of this minimum effort path code:
It uses an informed graph search algorithm approach, expanding the most promising paths first.
I would explain it is carefully exploring different routes between points to find the easiest overall path.
It uses core logic like loops, data structures, conditional checks, and functions.
The algorithm iteratively analyzes path options, keeping track of the easiest route so far, until the end point is reached.
The key steps are building a graph from the input, initializing search variables, queueing unvisited vertices by effort, and relaxing edges to update efforts. This expands the search space optimally.
The core algorithmic strategy is Dijkstra’s shortest path algorithm using a priority queue. It leverages informed search and incremental optimization.
Overall, this analysis extracts the key patterns around modeling the problem domain, informed graph search algorithms, optimality tracking, and incremental solution building in a language agnostic way.
Language Agnostic Coding Drills
Here is a decomposition of the key coding concepts in this problem:
Easy
 Array traversal  Iterate through 2D grid
 Variable assignment  Initialize graph, queues, etc
 Math operations  Calculate difference between heights
 Looping  Search loops
Medium
 Graph modeling  Represent grid as graph
 Priority queues  Use min heap queue to optimize search
 Adjacency lists  Store graph neighbors
 Shortest path algorithms  Dijkstra’s algorithm
Advanced
 Heuristics  Underestimate effort for more efficient search
 Memoization  Store precomputed efforts in map/cache
 Early stopping  Prune search if end vertex reached
 Informed search  Use heuristic guidance to expand most promising paths first
The overall problem solving approach is:
 Model grid as graph with heights as edge weights
 Initialize graph adjacency lists, queues, variables
 Implement heuristic effort calculation
 Use Dijkstra’s search with min heap queue
 Relax edges and update efforts
 Prune/memoize where possible
 Return minimum effort seen
Each coding drill builds up the key data structures, algorithms, and optimizations needed for the full solution.
Targeted Drills in Python
Here are Python coding drills for each concept:
Basic
Array traversal:


Math operations:


Looping:


Intermediate
Graph modeling:


Priority queue:


Shortest path search:


Advanced
Heuristics:


Memoization:


These can be combined to build the full solution.
Q&A
Similar Problems
Here are 10 related LeetCode problems that use similar concepts:
Cheapest Flights Within K Stops (787)  Also finds shortest path minimizing an effort cost metric.
Network Delay Time (743)  Involves graph shortest path search from a source.
Swim in Rising Water (778)  Gridbased shortest path problem, optimize max height along path.
Walls and Gates (286)  Uses BFS for grid shortest path like Dijkstra’s algorithm.
Shortest Path in Binary Matrix (1091)  Gridbased shortest path, optimize number of steps.
Shortest Path Visiting All Nodes (847)  Involves shortest path techniques on a graph.
Min Cost to Connect All Points (1584)  Connect points minimizing a distance metric, uses MST.
Path With Maximum Probability (1514)  Find highest probability path in graph, similar approach.
Maximum Students Taking Exam (1342)  Models exam room grid, finds optimal paths.
As Far from Land as Possible (1162)  Grid traversal to maximize distance, similar techniques.
The common themes are graph problems involving optimizing a metric based on a traversal or path, using search algorithms like BFS/Dijkstra’s.