Path With Minimum Effort

This is a binary search combined with depth-first search (DFS) or breadth-first 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 top-left to the bottom-right with the current effort.
If we can, we update the right boundary of the binary search. Otherwise, we update the left boundary.

Python solution:

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from typing import List
from collections import deque

class Solution:
    def minimumEffortPath(self, heights: List[List[int]]) -> int:
        rows, cols = len(heights), len(heights[0])

        # Check if a cell is valid and within bounds
        def is_valid(x, y):
            return 0 <= x < rows and 0 <= y < cols

        # BFS function to check if we can reach bottom-right with current effort
        def can_cross(effort):
            visited = [[0 for _ in range(cols)] for _ in range(rows)]
            directions = [(0, 1), (1, 0), (0, -1), (-1, 0)]
            queue = deque([(0, 0)])
            visited[0][0] = 1

            while queue:
                x, y = queue.popleft()
                for dx, dy in directions:
                    nx, ny = x + dx, y + dy
                    if is_valid(nx, ny) and not visited[nx][ny]:
                        diff = abs(heights[nx][ny] - heights[x][y])
                        if diff <= effort:
                            visited[nx][ny] = 1
                            queue.append((nx, ny))

            return visited[rows-1][cols-1]

        # Binary search for the effort
        l, r, ans = 0, max(max(row) for row in heights) - 1, 0

        while l <= r:
            mid = (l + r) // 2
            if can_cross(mid):
                ans = mid
                r = mid - 1
            else:
                l = mid + 1

        return ans

Here, we used BFS for traversal. The can_cross function will determine whether we can travel from top-left to bottom-right 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 grid-based 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 single-source 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 self-contained. 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 zoom-in 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 real-world 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 top-left cell and want to find a path to the bottom-right 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 top-left to bottom-right 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 min-max 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.

Zig-zag path:

Input: Grid where optimal zig-zag path is better than straight.

Output: Non-trivial effort based on zig-zag 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

Zig-zag path:

Grid with consecutive high differences on straight path
Zig-zag 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.
Zig-zag 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 non-technical 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 real-world terms? Let me know if any part needs more clarification!

Problem Breakdown and Solution Methodology

Here is a detailed step-by-step 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 Problem-Solving 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 single-source/single-target 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?

Here is a stepwise refinement for solving the minimum effort path problem:

High-level 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 sub-problems:

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 step-by-step 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 min-heap 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 well-studied 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:

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import heapq

grid = [
  [1, 2, 3],
  [4, 5, 6],
  [7, 8, 9] 
]

start = (0, 0)
end = (2, 2)

# Build graph 
graph = {}
for i in range(len(grid)):
  for j in range(len(grid[0])):
    if (i, j) == start:
      set_start = True 
    if (i, j) == end:
      set_end = True
    
    graph[(i,j)] = []
    
    if i > 0:
      graph[(i,j)].append(((i-1,j), abs(grid[i][j] - grid[i-1][j])))
      
    if j > 0:  
      graph[(i, j)].append(((i, j-1), abs(grid[i][j] - grid[i][j-1])))

# Dijkstra's algorithm
heap = [(0, start)] # (effort, vertex)
effort = {}

while heap:
  curr_effort, curr_vert = heapq.heappop(heap)
  
  if curr_vert in effort:
    continue
  
  effort[curr_vert] = curr_effort
  
  if curr_vert == end:
    print("Min effort", curr_effort)
    break
    
  for neigh, weight in graph[curr_vert]:
    new_effort = curr_effort + weight
    heapq.heappush(heap, (new_effort, neigh))

This implements the thought process and systematic steps to solve the problem.

Here is an implementation of the minimum effort path algorithm in Python:

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class Solution:
  def minimumEffortPath(self, heights: List[List[int]]) -> int:
    
    rows, cols = len(heights), len(heights[0])
    dist = [[float("inf")] * cols for _ in range(rows)]
    min_effort = float("inf")

    def neighbors(r, c):
      dirs = [[-1, 0], [1, 0], [0, -1], [0, 1]]
      for dr, dc in dirs:
        nr, nc = r + dr, c + dc
        if 0 <= nr < rows and 0 <= nc < cols:
          yield nr, nc

    q = [(0, 0, 0)] # effort, row, col
    dist[0][0] = 0

    while q:
      effort, r, c = heapq.heappop(q)
      if r == rows - 1 and c == cols - 1:
        min_effort = min(effort, min_effort)
        break
        
      for nr, nc in neighbors(r, c):
        new_effort = max(effort, abs(heights[r][c] - heights[nr][nc]))
        if dist[nr][nc] > new_effort:
          dist[nr][nc] = new_effort
          heapq.heappush(q, (new_effort, nr, nc))

    return min_effort 

This implements Dijkstra’s algorithm using a min-heap 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 high-level 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:

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grid = [[1,2,3], [4,5,6]]
for i in range(len(grid)):
  for j in range(len(grid[0])):
    print(grid[i][j]) 

Math operations:

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a = 5
b = 3
print(a + b)
print(abs(a - b)) 

Looping:

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for i in range(5):
  print(i)

Intermediate

Graph modeling:

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graph = {
  0: [(1, 6), (3, 2)],
  1: [(0, 6), (2, 5)],
}

Priority queue:

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import heapq

heap = []
heapq.heappush(heap, (5, 'A'))
heapq.heappush(heap, (3, 'B'))

print(heapq.heappop(heap))

Shortest path search:

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# Dijkstra's algorithm template

dist = {vertex: float('inf') for vertex in graph}
dist[source] = 0

queue = [(0, source)]

while queue:
  dist_u, u = heapq.heappop(queue)
  
  for v, weight in graph[u]:
    alt = dist_u + weight
    if alt < dist[v]:
      dist[v] = alt
      heapq.heappush(queue, (alt, v))

Advanced

Heuristics:

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def heuristic(a, b):
  return abs(a[0] - b[0]) # Underestimate compared to Manhattan

Memoization:

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cache = {}

def fn(n):
  if n not in cache:
    cache[n] = compute(n)
  
  return cache[n]

These can be combined to build the full solution.

Path With Minimum Effort

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