Pyramid Transition Matrix
Time O(a^b) · Space O(a^b) · Official statement on LeetCode
Solutions
// Time: O((a^(b+1)-a)/(a-1)) = O(a^b) , a is the size of allowed,
// b is the length of bottom
// Space: O((a^(b+1)-a)/(a-1)) = O(a^b)
// dfs solution
class Solution {
public:
bool pyramidTransition(string bottom, vector<string>& allowed) {
vector<vector<vector<int>>> edges(7, vector<vector<int>>(7));
unordered_set<string> lookup;
for (const auto& s: allowed) {
edges[s[0] - 'A'][s[1] - 'A'].emplace_back(s[2] - 'A');
}
return pyramidTransitionHelper(bottom, edges, &lookup);
}
private:
bool pyramidTransitionHelper(const string& bottom, const vector<vector<vector<int>>>& edges,
unordered_set<string> *lookup) {
if (bottom.size() == 1) {
return true;
}
for (int i = 0; i < bottom.size() - 1; ++i) {
if (edges[bottom[i] - 'A'][bottom[i + 1] - 'A'].empty()) {
return false;
}
}
if (lookup->count(bottom)) {
return false;
}
lookup->emplace(bottom);
string new_bottom(bottom.size() - 1, 'A');
return dfs(bottom, edges, &new_bottom, 0, lookup);
}
bool dfs(const string& bottom, const vector<vector<vector<int>>>& edges, string *new_bottom, int idx,
unordered_set<string> *lookup) {
if (idx == bottom.size() - 1) {
return pyramidTransitionHelper(*new_bottom, edges, lookup);
}
for (const auto& i : edges[bottom[idx] - 'A'][bottom[idx + 1] - 'A']) {
(*new_bottom)[idx] = i + 'A';
if (dfs(bottom, edges, new_bottom, idx + 1, lookup)) {
return true;
}
}
return false;
}
};
Beginner Explanation
What is Pyramid Transition Matrix?
Pyramid Transition Matrix (LeetCode #756) is a Medium problem that primarily trains depth 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 dfs backtracking.
- 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.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Pyramid Transition Matrix
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.
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(a^b)) and space (O(a^b)) 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(a^b) time and O(a^b) space.
Pattern focus: dfs backtracking
Use the pattern as a checklist:
- dfs backtracking — 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(a^b) |
| Space | O(a^b) |
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 Pyramid Transition Matrix
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for dfs backtracking — 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:
- 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: depth 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 Pyramid Transition Matrix in a second language (cpp, python).
- Drill 3–5 more problems tagged depth 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
Study checklist
- Read the official problem statement on LeetCode
- Solve on paper / whiteboard first
- Implement the dfs backtracking 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
Pyramid Transition Matrix (#756) — Medium. Pattern: dfs backtracking. Complexity: O(a^b) time / O(a^b) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Pyramid Transition Matrix?+
The reference solutions aim for O(a^b) time and O(a^b) space. Always re-derive complexity from the code you write in the interview.
What pattern does Pyramid Transition Matrix use?+
It primarily maps to dfs backtracking, within the broader topic of depth first search.
Is Pyramid Transition Matrix 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/pyramid-transition-matrix/