Probability of a Two Boxes Having The Same Number of Distinct Balls
Time O(k^3 * n^2) · Space O(k^2 * n) · Official statement on LeetCode
Solutions
// Time: O(k^3 * n^2)
// Space: O(k^2 * n)
class Solution {
public:
double getProbability(vector<int>& balls) {
unordered_map<pair<int, int>, uint64_t, PairHash<int>> dp;
dp[pair(0, 0)] = 1; // dp[i, j] is the ways of number difference i and color difference j
for (const auto& n : balls) { // O(k) times
unordered_map<pair<int, int>, uint64_t, PairHash<int>> new_dp;
for (const auto& kvp : dp) { // O(k^2 * n) times
const auto& [ndiff, cdiff] = kvp.first;
for (int k = 0, new_count = 1; k <= n; ++k, new_count *= n - k + 1, new_count /= k) { // O(n) times
const auto& new_ndiff = ndiff + (k - (n - k));
const auto& new_cdiff = (k == 0) ? cdiff - 1 : ((k == n) ? cdiff + 1 : cdiff);
new_dp[pair(new_ndiff, new_cdiff)] += kvp.second * new_count;
}
}
dp = move(new_dp);
}
const auto& total = accumulate(cbegin(balls), cend(balls), 0);
return double(dp[pair(0, 0)]) / nCr(total, total / 2);
}
private:
uint64_t nCr(int n, int r) { // Time: O(n), Space: O(1)
if (n - r < r) {
return nCr(n, n - r);
}
uint64_t c = 1;
for (int k = 1; k <= r; ++k) {
c *= n - k + 1;
c /= k;
}
return c;
}
template <typename T>
struct PairHash {
size_t operator()(const pair<T, T>& p) const {
size_t seed = 0;
seed ^= std::hash<T>{}(p.first) + 0x9e3779b9 + (seed<<6) + (seed>>2);
seed ^= std::hash<T>{}(p.second) + 0x9e3779b9 + (seed<<6) + (seed>>2);
return seed;
}
};
};
Beginner Explanation
What is Probability of a Two Boxes Having The Same Number of Distinct Balls?
Probability of a Two Boxes Having The Same Number of Distinct Balls (LeetCode #1467) is a Hard problem that primarily trains dynamic programming.
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 dynamic programming.
- 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: Binomial Coefficients.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Probability of a Two Boxes Having The Same Number of Distinct Balls
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 dynamic programming.
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(k^3 * n^2)) and space (O(k^2 * 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(k^3 * n^2) time and O(k^2 * n) space.
Pattern focus: dynamic programming
Use the pattern as a checklist:
- dynamic programming — 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(k^3 * n^2) |
| Space | O(k^2 * 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 Probability of a Two Boxes Having The Same Number of Distinct Balls
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for dynamic programming — 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 dynamic programming:
- 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: dynamic programming.
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 Probability of a Two Boxes Having The Same Number of Distinct Balls in a second language (cpp, python).
- Drill 3–5 more problems tagged dynamic programming.
- 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 dynamic programming 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
Probability of a Two Boxes Having The Same Number of Distinct Balls (#1467) — Hard. Pattern: dynamic programming. Complexity: O(k^3 * n^2) time / O(k^2 * n) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Probability of a Two Boxes Having The Same Number of Distinct Balls?+
The reference solutions aim for O(k^3 * n^2) time and O(k^2 * n) space. Always re-derive complexity from the code you write in the interview.
What pattern does Probability of a Two Boxes Having The Same Number of Distinct Balls use?+
It primarily maps to dynamic programming, within the broader topic of dynamic programming.
Is Probability of a Two Boxes Having The Same Number of Distinct Balls 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/probability-of-a-two-boxes-having-the-same-number-of-distinct-balls/