#773Hard~50 min
Sliding Puzzle
Time O((m * n) * (m * n)!) · Space O((m * n) * (m * n)!) · Official statement on LeetCode
Solutions
// Time: O((m * n) * (m * n)!)
// Space: O((m * n) * (m * n)!)
// A* Search Algorithm
class Solution {
public:
int slidingPuzzle(vector<vector<int>>& board) {
const auto& R = board.size(), &C = board[0].size();
vector<int> begin, end;
unordered_map<int, pair<int, int>> expected;
int zero_idx = 0;
for (int i = 0; i < R; ++i) {
for (int j = 0; j < C; ++j) {
auto val = (C * i + j + 1) % (R * C);
expected[val] = {i, j};
if (board[i][j] == 0) {
zero_idx = begin.size();
}
begin.emplace_back(board[i][j]);
end.emplace_back(val);
}
}
int min_steps = heuristic_estimate(begin, R, C, expected);
unordered_set<vector<int>, Hash<vector<int>>> lookup;
vector<pair<int, vector<int>>> closer{make_pair(zero_idx, begin)}, detour;
while (true) {
if (closer.empty()) {
if (detour.empty()) {
return -1;
}
min_steps += 2;
swap(closer, detour);
}
int zero;
vector<int> board;
tie(zero, board) = closer.back(); closer.pop_back();
if (board == end) {
return min_steps;
}
if (!lookup.count(board)) {
lookup.emplace(board);
int r = zero / C;
int c = zero % C;
static const vector<pair<int, int>> directions{{-1, 0}, {1, 0}, {0, -1}, {0, 1}};
for (const auto& direction : directions) {
int i = r + direction.first;
int j = c + direction.second;
if (0 <= i && i < R && 0 <= j && j < C) {
auto new_zero = C * i + j;
auto new_board = board;
swap(new_board[zero], new_board[new_zero]);
int r2, c2;
tie(r2, c2) = expected[board[new_zero]];
int r1 = zero / C;
int c1 = zero % C;
int r0 = new_zero / C;
int c0 = new_zero % C;
bool is_closer = dot({r1 - r0, c1 - c0}, {r2 - r0, c2 - c0}) > 0;
is_closer ? closer.emplace_back(new_zero, new_board) : detour.emplace_back(new_zero, new_board);
}
}
}
}
return min_steps;
}
private:
int heuristic_estimate(const vector<int>& board, int R, int C, const unordered_map<int, pair<int, int>>& expected) {
int result = 0;
for (int i = 0; i < R; ++i) {
for (int j = 0; j < C; ++j) {
const auto& val = board[C * i + j];
if (val == 0) {
continue;
}
int r, c;
tie(r, c) = expected.at(val);
result += abs(r - i) + abs(c - j);
}
}
return result;
}
inline int dot(const pair<int, int>& a, const pair<int, int>& b) {
return a.first * b.first + a.second * b.second;
}
template<typename ContType>
struct Hash {
size_t operator()(const ContType& v) const {
size_t seed = 0;
for (const auto& i : v) {
seed ^= std::hash<typename ContType::value_type>{}(i) + 0x9e3779b9 + (seed<<6) + (seed>>2);
}
return seed;
}
};
};
// Time: O((m * n) * (m * n)! * log((m * n)!))
// Space: O((m * n) * (m * n)!)
// A* Search Algorithm
class Solution2 {
public:
int slidingPuzzle(vector<vector<int>>& board) {
const auto& R = board.size(), &C = board[0].size();
vector<int> begin, end;
unordered_map<int, pair<int, int>> expected;
int zero_idx = 0;
for (int i = 0; i < R; ++i) {
for (int j = 0; j < C; ++j) {
auto val = (C * i + j + 1) % (R * C);
expected[val] = {i, j};
if (board[i][j] == 0) {
zero_idx = begin.size();
}
begin.emplace_back(board[i][j]);
end.emplace_back(val);
}
}
vector<int> end_wrong(end);
swap(end_wrong[end_wrong.size() - 2], end_wrong[end_wrong.size() - 3]);
using P = tuple<int, int, int, vector<int>>;
priority_queue<P, vector<P>, greater<P>> min_heap;
min_heap.emplace(make_tuple(0, 0, zero_idx, begin));
unordered_map<vector<int>, int, Hash<vector<int>>> lookup;
lookup[begin] = 0;
while (!min_heap.empty()) {
int f, g, zero;
vector<int> board;
tie(f, g, zero, board) = min_heap.top(); min_heap.pop();
if (board == end) {
return g;
}
if (board == end_wrong) {
return -1;
}
if (f > lookup[board]) {
continue;
}
int r = zero / C;
int c = zero % C;
static const vector<pair<int, int>> directions{{-1, 0}, {1, 0}, {0, -1}, {0, 1}};
for (const auto& direction : directions) {
int i = r + direction.first;
int j = c + direction.second;
if (0 <= i && i < R && 0 <= j && j < C) {
auto new_zero = C * i + j;
auto new_board = board;
swap(new_board[zero], new_board[new_zero]);
f = g + 1 + heuristic_estimate(new_board, R, C, expected);
if (!lookup.count(new_board) || f < lookup[new_board])
lookup[new_board] = f;
min_heap.emplace(make_tuple(f, g + 1, new_zero, new_board));
}
}
}
}
return -1;
}
private:
int heuristic_estimate(const vector<int>& board, int R, int C, const unordered_map<int, pair<int, int>>& expected) {
int result = 0;
for (int i = 0; i < R; ++i) {
for (int j = 0; j < C; ++j) {
const auto& val = board[C * i + j];
if (val == 0) {
continue;
}
int r, c;
tie(r, c) = expected.at(val);
result += abs(r - i) + abs(c - j);
}
}
return result;
}
template<typename ContType>
struct Hash {
size_t operator()(const ContType& v) const {
size_t seed = 0;
for (const auto& i : v) {
seed ^= std::hash<typename ContType::value_type>{}(i) + 0x9e3779b9 + (seed<<6) + (seed>>2);
}
return seed;
}
};
};
Beginner Explanation
What is Sliding Puzzle?
Sliding Puzzle (LeetCode #773) is a Hard problem that primarily trains breadth first search.
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 a search algorithm.
- Only then translate the idea into code.
Why this problem matters
Hard problems force you to combine patterns and prove complexity carefully — interview gold. Official solution notes mention: A* Search Algorithm.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Sliding Puzzle
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 a search 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) * (m * n)!)) and space (O((m * n) * (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) * (m * n)!) time and O((m * n) * (m * n)!) space.
Pattern focus: a search algorithm
Use the pattern as a checklist:
- a search 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) * (m * n)!) |
| Space | O((m * n) * (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 Sliding Puzzle
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for a search 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 a search 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: breadth first search.
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 Sliding Puzzle in a second language (cpp, python).
- Drill 3–5 more problems tagged breadth first search.
- Teach the solution out loud in under 5 minutes.
- Add this problem to your revision calendar in 3 days and 14 days.
Visualization
Conceptual diagram for Sliding Puzzle: show input structure (breadth first search), highlight the moving parts of the a search algorithm approach, and annotate each step with the maintained invariant and complexity.
Study checklist
- Read the official problem statement on LeetCode
- Solve on paper / whiteboard first
- Implement the a search 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
Sliding Puzzle (#773) — Hard. Pattern: a search algorithm. Complexity: O((m * n) * (m * n)!) time / O((m * n) * (m * n)!) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Sliding Puzzle?+
The reference solutions aim for O((m * n) * (m * n)!) time and O((m * n) * (m * n)!) space. Always re-derive complexity from the code you write in the interview.
What pattern does Sliding Puzzle use?+
It primarily maps to a search algorithm, within the broader topic of breadth first search.
Is Sliding Puzzle good for interviews?+
Yes — as a Hard 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/sliding-puzzle/