Minimum Path Cost in a Hidden Grid
Time O(m * n * log(m * n)) · Space O(m * n) · Official statement on LeetCode
Solutions
// Time: O(m * n * log(m * n))
// Space: O(m * n)
/**
* class GridMaster {
* public:
* bool canMove(char direction);
* void move(char direction);
* boolean isTarget();
* };
*/
class Solution {
public:
int findShortestPath(GridMaster &master) {
const pair<int, int> start = {MAX_M, MAX_N};
pair<int, int> target = {0, 0};
bool found = false;
unordered_set<int> lookup;
unordered_map<int, unordered_map<int, int>> adj;
dfs(start, &target, &master, &found, &lookup, &adj);
if (!found) {
return -1;
}
return dijkstra(adj, start, target);
}
private:
int index(pair<int, int> p) {
return (2 * MAX_N + 1) * p.first + p.second;
}
int dijkstra(const unordered_map<int, unordered_map<int, int>>& adj,
const pair<int, int>& start,
const pair<int, int>& target) {
unordered_map<int, int> dist = {{index(start), 0}};
priority_queue<pair<int, int>, vector<pair<int, int>>, greater<pair<int, int>>> min_heap;
min_heap.emplace(0, index(start));
while (!empty(min_heap)) {
const auto [curr, u] = min_heap.top(); min_heap.pop();
if (dist[u] < curr) {
continue;
}
for (const auto& [v, w] : adj.at(u)) {
if ((dist.count(v) && dist[v] - curr <= w)) {
continue;
}
dist[v] = curr + w;
min_heap.emplace(curr + w, v);
}
}
return dist.count(index(target)) ? dist[index(target)] : -1;
}
void dfs(const pair<int, int>& pos,
pair<int, int> *target,
GridMaster *master,
bool *found,
unordered_set<int> *lookup,
unordered_map<int, unordered_map<int, int>> *adj) {
static const unordered_map<char, char> rollback = {
{'L', 'R'}, {'R', 'L'}, {'U', 'D'}, {'D', 'U'}
};
static const unordered_map<char, pair<int, int>> directions = {
{'L', {0, -1}},
{'R', {0, 1}},
{'U', {-1, 0}},
{'D', {1, 0}}
};
if (!(*found) && master->isTarget()) {
*found = true;
*target = pos;
}
lookup->emplace(index(pos));
for (const auto& [d, dir] : directions) {
if (!master->canMove(d)) {
continue;
}
pair<int, int> nei = {pos.first + dir.first, pos.second + dir.second};
if ((*adj)[index(pos)].count(index(nei))) {
continue;
}
(*adj)[index(pos)][index(nei)] = master->move(d);
if (!lookup->count(index(nei))) {
dfs(nei, target, master, found, lookup, adj);
}
(*adj)[index(nei)][index(pos)] = master->move(rollback.at(d));
}
}
static const int MAX_M = 100;
static const int MAX_N = 100;
};
Beginner Explanation
What is Minimum Path Cost in a Hidden Grid?
Minimum Path Cost in a Hidden Grid (LeetCode #1810) is a Medium problem that primarily trains graph.
How to think about it
- Restate the goal in your own words before coding.
- Work a tiny example by hand so the invariant becomes obvious.
- Identify the pattern — this problem aligns with dfs backtracking and dijkstras algorithm.
- Only then translate the idea into code.
Why this problem matters
It sits in the sweet spot of interview difficulty: multiple valid approaches, clear trade-offs. Official solution notes mention: DFS, Dijkstra's Algorithm.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Minimum Path Cost in a Hidden Grid
Opening (30–60 seconds)
- Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
- State a brute force so the interviewer knows you can solve it naively.
- Propose the optimal direction tied to dfs backtracking and dijkstras algorithm.
Core solution narrative
- Define the state you track (pointers, DP cell, set membership, stack top, etc.).
- Explain the transition when you process the next element.
- Call out time (O(m * n * log(m * n))) and space (O(m * n)) before coding.
- Code cleanly; narrate variable names.
What interviewers listen for
- Correctness on edge cases
- Complexity honesty
- Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)
Follow-up questions they may ask
- Can you solve it with less memory?
- What if the input stream is infinite / doesn't fit in RAM?
- How would tests look for adversarial inputs?
Optimized Approach
Optimized solution notes
The reference solutions on AlgoForge target O(m * n * log(m * n)) time and O(m * n) space.
Pattern focus: dfs backtracking and dijkstras algorithm
Use the pattern as a checklist:
- dfs backtracking — confirm the invariant holds after each step
- dijkstras algorithm — confirm the invariant holds after each step
Multiple methods appear in the source solutions — compare them and explain when each is preferable.
Implementation tips
- Prefer readable names over micro-optimizations in interviews.
- Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
- After AC-level logic, re-scan for off-by-one and null checks.
Complexity Analysis
Complexity
| Measure | Bound |
|---|---|
| Time | O(m * n * log(m * n)) |
| Space | O(m * n) |
How to justify this in an interview
- Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
- Space: include hash maps, recursion stack, and output allocation when the problem asks for it.
If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.
Common Mistakes
Common mistakes on Minimum Path Cost in a Hidden Grid
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for dfs backtracking and dijkstras algorithm — updating state too early or too late.
- Mutating input unexpectedly when the problem forbids it.
- Off-by-one in windows, ranges, or binary search bounds.
- Ignoring overflow / precision for integer arithmetic problems.
- Overengineering — jumping to an advanced structure when a simpler approach works.
Alternative Approaches
Alternatives
The source file includes more than one method. Compare:
- Primary optimized path — best complexity for typical interviews.
- Secondary approach — often brute force, sorting-based, or space-optimized variant.
Practice articulating when you would pick each (constraints, readability, follow-ups).
Edge Cases
Edge cases checklist
- Minimum input size
- Maximum input size / time limits
- Duplicates and already-sorted input
- Negative numbers / zeros (if applicable)
- Disconnected structures (graphs/trees)
- Single path vs branching recursion depth
Pattern Recognition
Spotting this pattern
Signal phrases that point to dfs backtracking and dijkstras algorithm:
- Sorted input or ability to sort without changing the answer class
- Need for contiguous subarray / substring → consider sliding window
- Need for O(1) membership → hash set/map
- Optimal substructure + overlapping subproblems → DP
- Connectivity / components → graph DFS/BFS or Union-Find
Primary topics: graph.
Follow-up Interview Questions
Follow-ups
- How does the solution change if the input is a stream?
- Can you solve it in-place?
- What if duplicates must be handled differently?
- How would you parallelize the approach?
- Design tests that would break a buggy implementation.
Practice Recommendations
What to practice next
- Re-solve Minimum Path Cost in a Hidden Grid in a second language (cpp, python).
- Drill 3–5 more problems tagged graph.
- Teach the solution out loud in under 5 minutes.
- Add this problem to your revision calendar in 3 days and 14 days.
Visualization
Study checklist
- Read the official problem statement on LeetCode
- Solve on paper / whiteboard first
- Implement the dfs backtracking and dijkstras algorithm approach
- Verify edge cases from the checklist
- State time and space complexity aloud
- Compare with the AlgoForge reference solution
- Schedule a revision session
Revision notes
Minimum Path Cost in a Hidden Grid (#1810) — Medium. Pattern: dfs backtracking and dijkstras algorithm. Complexity: O(m * n * log(m * n)) time / O(m * n) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Minimum Path Cost in a Hidden Grid?+
The reference solutions aim for O(m * n * log(m * n)) time and O(m * n) space. Always re-derive complexity from the code you write in the interview.
What pattern does Minimum Path Cost in a Hidden Grid use?+
It primarily maps to dfs backtracking and dijkstras algorithm, within the broader topic of graph.
Is Minimum Path Cost in a Hidden Grid good for interviews?+
Yes — as a Medium problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.
Where can I read the official statement?+
Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/minimum-path-cost-in-a-hidden-grid/