Sum of Perfect Square Ancestors
Time precompute: O(r) runtime: O(nlogx) · Space O(r + n) · Official statement on LeetCode
Solutions
// Time: precompute: O(r)
// runtime: O(nlogx)
// Space: O(r + n)
vector<int> linear_sieve_of_eratosthenes(int n) { // Time: O(n), Space: O(n)
vector<int> spf(n + 1, -1);
vector<int> primes;
for (int i = 2; i <= n; ++i) {
if (spf[i] == -1) {
spf[i] = i;
primes.emplace_back(i);
}
for (const auto& p : primes) {
if (i * p > n || p > spf[i]) {
break;
}
spf[i * p] = p;
}
}
return spf;
}
const int MAX_NUMS = 1e5;
const auto& SPF = linear_sieve_of_eratosthenes(MAX_NUMS);
// number theory, iterative dfs, freq table
class Solution {
public:
long long sumOfAncestors(int n, vector<vector<int>>& edges, vector<int>& nums) {
const auto& prime_factors = [](int x) {
int result = 1;
for (; x != 1; x /= SPF[x]) {
if (result % SPF[x] == 0) {
result /= SPF[x];
} else {
result *= SPF[x];
}
}
return result;
};
vector<vector<int>> adj(n);
for (const auto& e : edges) {
adj[e[0]].emplace_back(e[1]);
adj[e[1]].emplace_back(e[0]);
}
unordered_map<int, int> cnt;
const auto& iter_dfs = [&]() {
int64_t result = 0;
using P = tuple<int, int, int, int, int>;
vector<P> stk = {{1, 0, -1, -1, 0}};
while (!empty(stk)) {
const auto [step, u, p, i, x] = stk.back(); stk.pop_back();
if (step == 1) {
const auto& x = prime_factors(nums[u]);
result += cnt[x]++;
stk.emplace_back(3, -1, -1, -1, x);
stk.emplace_back(2, u, p, 0, 0);
} else if (step == 2) {
if (i == size(adj[u])) {
continue;
}
stk.emplace_back(2, u, p, i + 1, 0);
const auto& v = adj[u][i];
if (v == p) {
continue;
}
stk.emplace_back(1, v, u, -1, 0);
} else if (step == 3) {
--cnt[x];
}
}
return result;
};
return iter_dfs();
}
};
// Time: precompute: O(r)
// runtime: O(nlogx)
// Space: O(r + n)
// number theory, dfs, freq table
class Solution2 {
public:
long long sumOfAncestors(int n, vector<vector<int>>& edges, vector<int>& nums) {
const auto& prime_factors = [](int x) {
int result = 1;
for (; x != 1; x /= SPF[x]) {
if (result % SPF[x] == 0) {
result /= SPF[x];
} else {
result *= SPF[x];
}
}
return result;
};
vector<vector<int>> adj(n);
for (const auto& e : edges) {
adj[e[0]].emplace_back(e[1]);
adj[e[1]].emplace_back(e[0]);
}
unordered_map<int, int> cnt;
const function<int64_t (int, int)> dfs = [&](int u, int p) {
const auto& x = prime_factors(nums[u]);
int64_t result = cnt[x]++;
for (const auto& v : adj[u]) {
if (v == p) {
continue;
}
result += dfs(v, u);
}
--cnt[x];
return result;
};
return dfs(0, -1);
}
};
Beginner Explanation
What is Sum of Perfect Square Ancestors?
Sum of Perfect Square Ancestors (LeetCode #3715) is a Hard problem that primarily trains tree.
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 tree and dfs backtracking.
- 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: Tree, Number Theory, Linear Sieve of Eratosthenes, Freq Table.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Sum of Perfect Square Ancestors
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 tree and 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 (precompute: O(r) runtime: O(nlogx)) and space (O(r + 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 precompute: O(r) runtime: O(nlogx) time and O(r + n) space.
Pattern focus: tree and dfs backtracking
Use the pattern as a checklist:
- tree — confirm the invariant holds after each step
- 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 | precompute: O(r) runtime: O(nlogx) |
| Space | O(r + 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 Sum of Perfect Square Ancestors
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for tree and 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 tree and 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: tree.
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 Sum of Perfect Square Ancestors in a second language (cpp, python).
- Drill 3–5 more problems tagged tree.
- 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 tree and 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
Sum of Perfect Square Ancestors (#3715) — Hard. Pattern: tree and dfs backtracking. Complexity: precompute: O(r) runtime: O(nlogx) time / O(r + n) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Sum of Perfect Square Ancestors?+
The reference solutions aim for precompute: O(r) runtime: O(nlogx) time and O(r + n) space. Always re-derive complexity from the code you write in the interview.
What pattern does Sum of Perfect Square Ancestors use?+
It primarily maps to tree and dfs backtracking, within the broader topic of tree.
Is Sum of Perfect Square Ancestors 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/sum-of-perfect-square-ancestors/