Heaps & Heapsort

5. Practice Problems

1. Find the k-th smallest element of an array using a max-heap of size k.
   size-k max-heap দিয়ে k-th smallest বের করুন।
   ✨ Show Answer (উত্তর দেখুন)

   a1.cpp

   #include <bits/stdc++.h>
   using namespace std;
   int main() {
       vector<int> a = {7,10,4,3,20,15}; int k = 3;
       priority_queue<int> pq;
       for (int x : a) {
           pq.push(x);
           if ((int)pq.size() > k) pq.pop();
       }
       cout << pq.top();
   }

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2. Connect ropes to minimise total cost (always join the two shortest).
   Rope-গুলো এমনভাবে join করুন যাতে মোট cost সর্বনিম্ন হয়।
   ✨ Show Answer (উত্তর দেখুন)

   a2.cpp

   #include <bits/stdc++.h>
   using namespace std;
   int main() {
       vector<int> r = {4,3,2,6};
       priority_queue<int, vector<int>, greater<>> pq(r.begin(), r.end());
       long long cost = 0;
       while (pq.size() > 1) {
           int a = pq.top(); pq.pop();
           int b = pq.top(); pq.pop();
           cost += a + b;
           pq.push(a + b);
       }
       cout << cost;
   }

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3. Running median of a stream of integers using two heaps (max-heap + min-heap).
   দুটি heap ব্যবহার করে stream-এর running median বের করুন।
   ✨ Show Answer (উত্তর দেখুন)

   a3.cpp

   #include <bits/stdc++.h>
   using namespace std;
   int main() {
       priority_queue<int> lo;                                // max-heap (lower half)
       priority_queue<int, vector<int>, greater<>> hi;     // min-heap (upper half)
       for (int x : {1,3,5,2,4,6,7}) {
           lo.push(x);
           hi.push(lo.top()); lo.pop();
           if (hi.size() > lo.size()) { lo.push(hi.top()); hi.pop(); }
           double med = lo.size() == hi.size() ? (lo.top() + hi.top()) / 2.0 : lo.top();
           cout << med << " ";
       }
   }

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4. Sort a nearly-sorted array (each element at most k positions away from its sorted slot) in O(n log k).
   প্রায়-sorted array (k দূরত্বের মধ্যে সব) — O(n log k)-এ sort করুন।
   ✨ Show Answer (উত্তর দেখুন)

   Approach: push first k+1 elements into a min-heap. For each remaining element, pop one (the smallest of the window) into the output, push the new one. After all input is exhausted, drain the heap.

5. Top-k frequent words in a list (lexicographic tie-break).
   Top-k frequent word — tie-break dictionary order-এ।
   ✨ Show Answer (উত্তর দেখুন)

   a5.cpp

   #include <bits/stdc++.h>
   using namespace std;
   int main() {
       vector<string> w = {"i","love","leetcode","i","love","coding"};
       int k = 2;
       map<string,int> cnt;
       for (auto& s : w) cnt[s]++;
       auto cmp = [](auto& a, auto& b) {
           return a.second != b.second ? a.second < b.second : a.first > b.first;
       };
       priority_queue<pair<string,int>, vector<pair<string,int>>, decltype(cmp)> pq(cmp);
       for (auto& p : cnt) pq.push(p);
       while (k--) { cout << pq.top().first << " "; pq.pop(); }
   }

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6. Use std::make_heap / push_heap / pop_heap on a vector to confirm the STL heap behaves the same.
   STL heap functions দিয়ে নিজেদের impl-এর সাথে আউটপুট মিলিয়ে দেখুন।
   ✨ Show Answer (উত্তর দেখুন)

   a6.cpp

   #include <bits/stdc++.h>
   using namespace std;
   int main() {
       vector<int> v = {3,10,1,7,15,9};
       make_heap(v.begin(), v.end());
       while (!v.empty()) {
           cout << v.front() << " ";
           pop_heap(v.begin(), v.end());
           v.pop_back();
       }
   }

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7. Why is heapsort O(n log n) worst-case but quicksort isn't? When would you choose heapsort over quicksort?
   Heapsort worst-case-এও O(n log n) কেন? কখন quicksort-এর বদলে heapsort বেছে নেবেন?
   ✨ Show Answer (উত্তর দেখুন)

   Answer: heapsort always extracts log-n levels exactly n times — the worst case is the average case. Quicksort's worst-case is O(n²) on adversarial inputs (sorted with bad pivot). Choose heapsort when worst-case guarantees matter (real-time systems, kernel) or memory must be O(1) (heapsort is in-place; quicksort uses O(log n) stack).