Pattern this teaches:Shortest Path
Path With Minimum Effort
A medium Graph interview guide. Task statement, worked examples, intuition, and step-by-step solutions.
- Topic
- Graph
- Sheets
- 1
- Core for
- 12 roles
The problem
Given an m x n matrix of non-negative integers where each value represents the elevation at that cell, find a path from the top-left to the bottom-right with the minimum possible effort. The effort of a path is the maximum absolute difference in heights between consecutive cells.
Example 1
- Input
- heights = [[2,4,3],[5,1,4],[3,2,6]]
- Output
- 2
- Why
- Path 2->4->3->4->6 has max step difference 2, which is minimal.
Example 2
- Input
- heights = [[2,3,1],[4,2,5],[3,1,4]]
- Output
- 3
- Why
- Path 2->3->1->5->4 has max step difference 3, which is minimal.
Constraints
- m == heights.length
- n == heights[i].length
- 1 <= m, n <= 100
- 0 <= heights[i][j] <= 10^6
How to think about it
Updated 2026-09-29Path effort is defined as the maximum step difference encountered along the journey, not the sum of differences. This minimax path property still satisfies greedy shortest path substructure: the cost to extend a path to neighbor (nr, nc) is `max(currentEffort, abs(height[r][c] - height[nr][nc]))`. Dijkstra's priority queue finds the minimal bottleneck path directly.
Approaches, worst first
Binary search the effort, BFS
time O(m * n * log(maxHeight)) · space O(m * n)
Binary search the answer threshold between 0 and 10^6. For each mid, test whether a path exists from (0, 0) to (m-1, n-1) using only steps with height difference <= mid via BFS or DFS.
Modified Dijkstra with minimax effort heapWrite this one
time O(m * n log(m * n)) · space O(m * n)
Maintain an `effort[r][c]` matrix initialized to infinity with `effort[0][0] = 0`. Pop `(effort, r, c)` from a min-heap. For each 4-directional neighbor, calculate `nextEffort = max(effort, abs(heights[r][c] - heights[nr][nc]))`. If `nextEffort < effort[nr][nc]`, update and push to heap.
Where people lose marks · 3
- A 1x1 grid starts and ends at (0, 0); the required effort is 0 because no step is taken.
- Summing height differences along the path instead of taking the maximum consecutive difference across the path.
- Skipping stale heap extractions without verifying `currEffort > effort[r][c]` leads to redundant evaluations.
The theory behind it
Graph — the ground this problem stands on. All Graph problems
What Graph is
A graph is a network of individual points, called vertices or nodes, connected by lines called edges. Think of a subway transit map, an electrical circuit, or a web of social friends. Unlike a tree, a graph has no designated top node and no parent-child hierarchy. Connections can run one-way or both ways, and paths can loop back on themselves to form closed cycles.
When to reach for it
Reach for graph algorithms when inputs describe relationships, networks, flights between cities, course prerequisites, or clone networks. Signals include finding the shortest route across unweighted connections, ordering tasks that depend on earlier tasks, counting isolated clusters, or checking whether a path contains an infinite loop. Whenever problems present pairs of related entities and ask for reachability, distances, or dependencies, graph representations apply.
How the pattern works
First convert edge lists into an adjacency list, mapping each node to an array of its neighbors. Choose your exploration strategy based on the goal: use a queue and breadth-first search to find the shortest path in unweighted networks, or use recursion and depth-first search to explore full paths and detect cycles. Because graphs can have loops, always track visited nodes in a set or boolean array. Add nodes to the visited set at the moment they enter the queue so they are never visited twice.
More Graph problems
Target Roles
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