Divide & Conquer Sorting: Merge & Quick

7. Practice Problems

Try them on paper first; then run the answer.

1. Count the number of inversions in {2, 4, 1, 3, 5}.
   {2, 4, 1, 3, 5} array-তে inversion সংখ্যা গণনা করুন।
   ✨ Show Answer (উত্তর দেখুন)

   inv.cpp

   #include <bits/stdc++.h>
   using namespace std;
   long long solve(vector<int>& a){
       long long inv=0;
       for(int i=0;i<(int)a.size();i++)
           for(int j=i+1;j<(int)a.size();j++)
               if(a[i]>a[j]) inv++;
       return inv;
   }
   int main(){
       vector<int> a={2,4,1,3,5};
       cout<<solve(a)<<"\n";
   }

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2. Find the k-th smallest using quick-select (partial quicksort).
   quick-select দিয়ে k-তম ক্ষুদ্রতম বের করুন।
   ✨ Show Answer (উত্তর দেখুন)

   qselect.cpp

   #include <bits/stdc++.h>
   using namespace std;
   int part(vector<int>& a,int l,int r){
       int p=a[r], i=l-1;
       for(int j=l;j<r;j++) if(a[j]<p) swap(a[++i],a[j]);
       swap(a[i+1],a[r]); return i+1;
   }
   int qsel(vector<int>& a,int l,int r,int k){
       if(l==r) return a[l];
       int p=part(a,l,r);
       if(p==k) return a[p];
       return p>k? qsel(a,l,p-1,k) : qsel(a,p+1,r,k);
   }
   int main(){
       vector<int> a={7,10,4,3,20,15};
       cout<<qsel(a,0,a.size()-1,2)<<"\n"; // 3rd smallest (0-indexed)
   }

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3. Sort an array of only 0s, 1s and 2s in one pass (Dutch National Flag).
   শুধু 0, 1, 2 দিয়ে গঠিত array এক pass-এ sort করুন (Dutch National Flag)।
   ✨ Show Answer (উত্তর দেখুন)

   dnf.cpp

   #include <bits/stdc++.h>
   using namespace std;
   int main(){
       vector<int> a={2,0,1,2,1,0,0,2,1};
       int lo=0, mid=0, hi=a.size()-1;
       while(mid<=hi){
           if(a[mid]==0) swap(a[lo++],a[mid++]);
           else if(a[mid]==1) mid++;
           else swap(a[mid],a[hi--]);
       }
       for(int x:a) cout<<x<<" ";
       cout<<"\n";
   }

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4. Implement merge sort on a singly linked list (no extra arrays).
   singly linked list-এ merge sort করুন (অতিরিক্ত array ছাড়া)।
   ✨ Show Answer (উত্তর দেখুন)

   ll_merge.cpp

   #include <bits/stdc++.h>
   using namespace std;
   struct Node { int v; Node* nx; };
   Node* merge(Node* a, Node* b){
       Node dummy{0,nullptr}, *t=&dummy;
       while(a&&b){
           if(a->v<=b->v){t->nx=a; a=a->nx;} else {t->nx=b; b=b->nx;}
           t=t->nx;
       }
       t->nx = a? a : b;
       return dummy.nx;
   }
   Node* msort(Node* h){
       if(!h || !h->nx) return h;
       Node *slow=h, *fast=h->nx;
       while(fast && fast->nx){slow=slow->nx; fast=fast->nx->nx;}
       Node* mid=slow->nx; slow->nx=nullptr;
       return merge(msort(h), msort(mid));
   }
   int main(){
       int v[]={4,2,5,1,3}; Node* h=nullptr;
       for(int i=4;i>=0;i--) h=new Node{v[i],h};
       h=msort(h);
       for(auto p=h;p;p=p->nx) cout<<p->v<<" ";
       cout<<"\n";
   }

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5. Merge k sorted arrays into one sorted array (k-way merge).
   k-টি sorted array merge করে একটি sorted array তৈরি করুন।
   ✨ Show Answer (উত্তর দেখুন)

   kway.cpp

   #include <bits/stdc++.h>
   using namespace std;
   int main(){
       vector<vector<int>> arr={{1,5,9},{2,6,7},{3,4,8}};
       priority_queue<tuple<int,int,int>, vector<tuple<int,int,int>>, greater<>> pq;
       for(int i=0;i<(int)arr.size();i++) pq.push({arr[i][0],i,0});
       vector<int> out;
       while(!pq.empty()){
           auto[v,i,j]=pq.top(); pq.pop();
           out.push_back(v);
           if(j+1<(int)arr[i].size()) pq.push({arr[i][j+1],i,j+1});
       }
       for(int x:out) cout<<x<<" ";
       cout<<"\n";
   }

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6. Implement 3-way quick sort for arrays with many duplicates.
   duplicate-বহুল array-এর জন্য 3-way quick sort বানান।
   ✨ Show Answer (উত্তর দেখুন)

   three_way.cpp

   #include <bits/stdc++.h>
   using namespace std;
   void qs3(vector<int>& a,int lo,int hi){
       if(lo>=hi) return;
       int lt=lo, gt=hi, i=lo, p=a[lo];
       while(i<=gt){
           if(a[i]<p) swap(a[lt++],a[i++]);
           else if(a[i]>p) swap(a[i],a[gt--]);
           else i++;
       }
       qs3(a,lo,lt-1); qs3(a,gt+1,hi);
   }
   int main(){
       vector<int> a={3,5,3,1,5,3,1,5,3};
       qs3(a,0,a.size()-1);
       for(int x:a) cout<<x<<" ";
       cout<<"\n";
   }

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7. Given two sorted arrays, find the median of the merged array in O(log(min(n,m))).
   দুটি sorted array দেওয়া; merge করার পরের median O(log(min(n,m)))-এ বের করুন।
   ✨ Show Answer (উত্তর দেখুন)

   median2.cpp

   #include <bits/stdc++.h>
   using namespace std;
   double median(vector<int>& A, vector<int>& B){
       if(A.size()>B.size()) return median(B,A);
       int n=A.size(), m=B.size(), lo=0, hi=n;
       while(lo<=hi){
           int i=(lo+hi)/2, j=(n+m+1)/2-i;
           int Lx = i==0?INT_MIN:A[i-1], Rx = i==n?INT_MAX:A[i];
           int Ly = j==0?INT_MIN:B[j-1], Ry = j==m?INT_MAX:B[j];
           if(Lx<=Ry && Ly<=Rx){
               if((n+m)%2) return max(Lx,Ly);
               return (max(Lx,Ly)+min(Rx,Ry))/2.0;
           }
           if(Lx>Ry) hi=i-1; else lo=i+1;
       }
       return 0;
   }
   int main(){
       vector<int> A={1,3}, B={2,4,5};
       cout<<median(A,B)<<"\n";
   }

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