Count Ways to Build Rooms in an Ant Colony
Time O(n) · Space O(n) · Official statement on LeetCode
Solutions
// Time: O(n)
// Space: O(n)
class Solution {
public:
int waysToBuildRooms(vector<int>& prevRoom) {
vector<vector<int>> adj(size(prevRoom));
for (int i = 1; i < size(prevRoom); ++i) {
adj[prevRoom[i]].emplace_back(i);
}
return dfs(adj, 0).first;
}
private:
pair<int, int> dfs(const vector<vector<int>>& adj, int curr) {
static const int MOD = 1e9 + 7;
int total_ways = 1, total_cnt = 0;
for (const auto& child : adj[curr]) {
const auto& [ways, cnt] = dfs(adj, child);
total_cnt += cnt;
total_ways = mulmod(mulmod(total_ways, ways, MOD), nCr(total_cnt, cnt, MOD), MOD);
}
return {total_ways, total_cnt + 1};
}
int nCr(int n, int k, uint32_t mod) {
while (size(inv_) <= n) { // lazy initialization
fact_.emplace_back(mulmod(fact_.back(), size(inv_), mod));
inv_.emplace_back(mulmod(inv_[mod % size(inv_)], mod - mod / size(inv_), mod)); // https://cp-algorithms.com/algebra/module-inverse.html
inv_fact_.emplace_back(mulmod(inv_fact_.back(), inv_.back(), mod));
}
return mulmod(mulmod(fact_[n], inv_fact_[n - k], mod), inv_fact_[k], mod);
}
uint32_t addmod(uint32_t a, uint32_t b, uint32_t mod) { // avoid overflow
a %= mod, b %= mod;
if (mod - a <= b) {
b -= mod; // relied on unsigned integer overflow in order to give the expected results
}
return a + b;
}
// reference: https://stackoverflow.com/questions/12168348/ways-to-do-modulo-multiplication-with-primitive-types
uint32_t mulmod(uint32_t a, uint32_t b, uint32_t mod) { // avoid overflow
a %= mod, b %= mod;
uint32_t result = 0;
if (a < b) {
swap(a, b);
}
while (b > 0) {
if (b % 2 == 1) {
result = addmod(result, a, mod);
}
a = addmod(a, a, mod);
b /= 2;
}
return result;
}
static const int MOD = 1e9 + 7;
vector<int> fact_ = {1, 1};
vector<int> inv_ = {0, 1};
vector<int> inv_fact_ = {1, 1};
};
Beginner Explanation
What is Count Ways to Build Rooms in an Ant Colony?
Count Ways to Build Rooms in an Ant Colony (LeetCode #1916) is a Hard problem that primarily trains math.
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 tree.
- 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: DFS, Tree.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Count Ways to Build Rooms in an Ant Colony
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 tree.
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(n)) and space (O(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(n) time and O(n) space.
Pattern focus: dfs backtracking and tree
Use the pattern as a checklist:
- dfs backtracking — confirm the invariant holds after each step
- tree — 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(n) |
| Space | O(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 Count Ways to Build Rooms in an Ant Colony
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for dfs backtracking and tree — 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 tree:
- 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: math.
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 Count Ways to Build Rooms in an Ant Colony in a second language (cpp, python).
- Drill 3–5 more problems tagged math.
- 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 tree 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
Count Ways to Build Rooms in an Ant Colony (#1916) — Hard. Pattern: dfs backtracking and tree. Complexity: O(n) time / O(n) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Count Ways to Build Rooms in an Ant Colony?+
The reference solutions aim for O(n) time and O(n) space. Always re-derive complexity from the code you write in the interview.
What pattern does Count Ways to Build Rooms in an Ant Colony use?+
It primarily maps to dfs backtracking and tree, within the broader topic of math.
Is Count Ways to Build Rooms in an Ant Colony 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/count-ways-to-build-rooms-in-an-ant-colony/