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Suma min și max a tuturor sub-serilor de mărime k.

Având în vedere o serie de numere întregi pozitive și negative, sarcina este de a calcula suma elementelor minime și maxime ale tuturor sub-matricei de mărime k.

Exemple: 

Intrare : arr[] = {2 5 -1 7 -3 -1 -2}
K = 4
Ieșire : 18
Explicaţie : Subbarys de dimensiunea 4 sunt:
{2 5 -1 7} min + max = -1 + 7 = 6
{5 -1 7 -3} min + max = -3 + 7 = 4
{-1 7 -3 -1} min + max = -3 + 7 = 4
{7 -3 -1 -2} min + max = -3 + 7 = 4

Lipsesc sub-matrice -

{2 -1 7 -3}
{2 7 -3 -1}
{2 -3 -1 -2}
{5 7 -3 -1}
{5 -3 -1 -2}
si inca putine -- de ce acestea nu au fost luate in considerare??
Având în vedere că rezultatul matricelor lipsă este 27

Suma tuturor min și max = 6 + 4 + 4 + 4 = 18



Această problemă este în principal o extensie a problemei de mai jos. 
Maximul tuturor subgrupurilor de mărime k 

Abordare naivă:

Rulați două bucle pentru a genera toate subbaryurile și apoi alegeți toate subbary-urile de dimensiunea k și găsiți valorile maxime și minime. În cele din urmă returnați suma tuturor elementelor maxime și minime. 

C++
// C++ program to find sum of all minimum and maximum // elements Of Sub-array Size k. #include    using namespace std; // Returns sum of min and max element of all subarrays // of size k int SumOfKsubArray(int arr[] int N int k) {  // To store final answer  int sum = 0;  // Find all subarray  for (int i = 0; i < N; i++) {  // To store length of subarray  int length = 0;  for (int j = i; j < N; j++) {  // Increment the length  length++;  // When there is subarray of size k  if (length == k) {  // To store maximum and minimum element  int maxi = INT_MIN;  int mini = INT_MAX;  for (int m = i; m <= j; m++) {  // Find maximum and minimum element  maxi = max(maxi arr[m]);  mini = min(mini arr[m]);  }  // Add maximum and minimum element in sum  sum += maxi + mini;  }  }  }  return sum; } // Driver program to test above functions int main() {  int arr[] = { 2 5 -1 7 -3 -1 -2 };  int N = sizeof(arr) / sizeof(arr[0]);  int k = 4;  cout << SumOfKsubArray(arr N k);  return 0; } 
Java
// Java program to find sum of all minimum and maximum // elements Of Sub-array Size k. import java.util.Arrays; class GFG {  // Returns sum of min and max element of all subarrays  // of size k  static int SumOfKsubArray(int[] arr int N int k) {  // To store the final answer  int sum = 0;  // Find all subarrays  for (int i = 0; i < N; i++) {  // To store the length of the subarray  int length = 0;  for (int j = i; j < N; j++) {  // Increment the length  length++;  // When there is a subarray of size k  if (length == k) {  // To store the maximum and minimum element  int maxi = Integer.MIN_VALUE;  int mini = Integer.MAX_VALUE;  for (int m = i; m <= j; m++) {  // Find the maximum and minimum element  maxi = Math.max(maxi arr[m]);  mini = Math.min(mini arr[m]);  }  // Add the maximum and minimum element to the sum  sum += maxi + mini;  }  }  }  return sum;  }  // Driver program to test above functions  public static void main(String[] args) {  int[] arr = {2 5 -1 7 -3 -1 -2};  int N = arr.length;  int k = 4;  System.out.println(SumOfKsubArray(arr N k));  } } //This code is contributed by Vishal Dhaygude 
Python
# Returns sum of min and max element of all subarrays # of size k def sum_of_k_subarray(arr N k): # To store final answer sum = 0 # Find all subarrays for i in range(N): # To store length of subarray length = 0 for j in range(i N): # Increment the length length += 1 # When there is a subarray of size k if length == k: # To store maximum and minimum element maxi = float('-inf') mini = float('inf') for m in range(i j + 1): # Find maximum and minimum element maxi = max(maxi arr[m]) mini = min(mini arr[m]) # Add maximum and minimum element to sum sum += maxi + mini return sum # Driver program to test above function def main(): arr = [2 5 -1 7 -3 -1 -2] N = len(arr) k = 4 print(sum_of_k_subarray(arr N k)) if __name__ == '__main__': main() 
C#
using System; class Program {  // Returns sum of min and max element of all subarrays  // of size k  static int SumOfKSubArray(int[] arr int N int k)  {  // To store the final answer  int sum = 0;  // Find all subarrays  for (int i = 0; i < N; i++) {  // To store the length of subarray  int length = 0;  for (int j = i; j < N; j++) {  // Increment the length  length++;  // When there is a subarray of size k  if (length == k) {  // To store the maximum and minimum  // element  int maxi = int.MinValue;  int mini = int.MaxValue;  for (int m = i; m <= j; m++) {  // Find maximum and minimum element  maxi = Math.Max(maxi arr[m]);  mini = Math.Min(mini arr[m]);  }  // Add maximum and minimum element in  // sum  sum += maxi + mini;  }  }  }  return sum;  }  // Driver program to test above functions  static void Main()  {  int[] arr = { 2 5 -1 7 -3 -1 -2 };  int N = arr.Length;  int k = 4;  Console.WriteLine(SumOfKSubArray(arr N k));  } } 
JavaScript
// JavaScript program to find sum of all minimum and maximum // elements of sub-array size k. // Returns sum of min and max element of all subarrays // of size k function SumOfKsubArray(arr N k) {  // To store final answer  let sum = 0;  // Find all subarray  for (let i = 0; i < N; i++) {  // To store length of subarray  let length = 0;  for (let j = i; j < N; j++) {  // Increment the length  length++;  // When there is subarray of size k  if (length === k) {  // To store maximum and minimum element  let maxi = Number.MIN_SAFE_INTEGER;  let mini = Number.MAX_SAFE_INTEGER;  for (let m = i; m <= j; m++) {  // Find maximum and minimum element  maxi = Math.max(maxi arr[m]);  mini = Math.min(mini arr[m]);  }  // Add maximum and minimum element in sum  sum += maxi + mini;  }  }  }  return sum; } // Driver program to test above function const arr = [2 5 -1 7 -3 -1 -2]; const N = arr.length; const k = 4; console.log(SumOfKsubArray(arr N k)); 

Ieșire
18

Complexitatea timpului: PE2*k) pentru că două bucle pentru a găsi toate subbarray și o buclă pentru a găsi elementele maxime și minime din subbarra de dimensiunea k
Spațiu auxiliar: O(1) deoarece nu a fost folosit spațiu suplimentar

Metoda 2 (folosind MultiSet):

Ideea este de a utiliza structura de date Multiset și conceptul de fereastră glisantă.

  • În primul rând, creăm un multiset de pereche de {numberindex} deoarece indexul ne-ar ajuta să eliminăm elementul i și să trecem la următoarea fereastră de dimensiune k .
  • În al doilea rând avem i şi j care sunt indicatorul din spate și din față utilizate pentru a menține o fereastră.
  • Traversați prin matrice și inserați în perechea multiset de {numberindex} și, de asemenea, verificați pentru windowSize odată ce aceasta devine egală cu k începeți obiectivul dvs. principal, adică să găsiți suma elementelor maxime și minime.
  • Apoi ștergeți al i-lea număr de index din set și mutați al i-lea indicator la următoarea locație, adică o nouă fereastră de dimensiune k.

Implementare:

C++
// C++ program to find sum of all minimum and maximum // elements Of Sub-array Size k. #include    using namespace std; // Returns sum of min and max element of all subarrays // of size k int SumOfKsubArray(int arr[] int n int k) {  int sum = 0; // to store our final sum  // multiset because nos. could be repeated  // multiset pair is {numberindex}  multiset<pair<int int> > ms;  int i = 0; // back pointer  int j = 0; // front pointer  while (j < n && i < n) {  ms.insert(  { arr[j] j }); // inserting {numberindex}  // front pointer - back pointer + 1 is for checking  // window size  int windowSize = j - i + 1;  // Once they become equal start what we need to do  if (windowSize == k) {  // extracting first since set is always in  // sorted ascending order  int mini = ms.begin()->first;  // extracting last element aka beginning from  // last (numbers extraction)  int maxi = ms.rbegin()->first;  // adding summation of maximum & minimum element  // of each subarray of k into final sum  sum += (maxi + mini);  // erasing the ith index element from set as it  // won't appaer in next window of size k  ms.erase({ arr[i] i });  // increasing back pointer for next window of  // size k;  i++;  }  j++; // always increments front pointer  }  return sum; } // Driver program to test above functions int main() {  int arr[] = { 2 5 -1 7 -3 -1 -2 };  int n = sizeof(arr) / sizeof(arr[0]);  int k = 4;  cout << SumOfKsubArray(arr n k);  return 0; } 

Ieșire
18

Complexitatea timpului: O(nlogk)
Spațiu auxiliar: O(k)

Metoda 3 (Eficientă folosind retragerea la coadă):

Ideea este să folosiți structura datelor Dequeue și conceptul de fereastră glisante. Creăm două cozi goale dublu de dimensiunea k ("S" "G") care stochează doar indici ai elementelor ferestrei curente care nu sunt inutile. Un element este inutil dacă nu poate fi maxim sau minim al următoarelor subgrupuri. 

cel mai frumos zâmbet din lume
C++
// C++ program to find sum of all minimum and maximum // elements Of Sub-array Size k. #include   using namespace std; // Returns sum of min and max element of all subarrays // of size k int SumOfKsubArray(int arr[]  int n  int k) {  int sum = 0; // Initialize result  // The queue will store indexes of useful elements  // in every window  // In deque 'G' we maintain decreasing order of  // values from front to rear  // In deque 'S' we maintain increasing order of  // values from front to rear  deque< int > S(k) G(k);  // Process first window of size K  int i = 0;  for (i = 0; i < k; i++)  {  // Remove all previous greater elements  // that are useless.  while ( (!S.empty()) && arr[S.back()] >= arr[i])  S.pop_back(); // Remove from rear  // Remove all previous smaller that are elements  // are useless.  while ( (!G.empty()) && arr[G.back()] <= arr[i])  G.pop_back(); // Remove from rear  // Add current element at rear of both deque  G.push_back(i);  S.push_back(i);  }  // Process rest of the Array elements  for ( ; i < n; i++ )  {  // Element at the front of the deque 'G' & 'S'  // is the largest and smallest  // element of previous window respectively  sum += arr[S.front()] + arr[G.front()];  // Remove all elements which are out of this  // window  if ( !S.empty() && S.front() == i - k)  S.pop_front();  if ( !G.empty() && G.front() == i - k)  G.pop_front();  // remove all previous greater element that are  // useless  while ( (!S.empty()) && arr[S.back()] >= arr[i])  S.pop_back(); // Remove from rear  // remove all previous smaller that are elements  // are useless  while ( (!G.empty()) && arr[G.back()] <= arr[i])  G.pop_back(); // Remove from rear  // Add current element at rear of both deque  G.push_back(i);  S.push_back(i);  }  // Sum of minimum and maximum element of last window  sum += arr[S.front()] + arr[G.front()];  return sum; } // Driver program to test above functions int main() {  int arr[] = {2 5 -1 7 -3 -1 -2} ;  int n = sizeof(arr)/sizeof(arr[0]);  int k = 4;  cout << SumOfKsubArray(arr n k) ;  return 0; } 
Java
// Java program to find sum of all minimum and maximum  // elements Of Sub-array Size k.  import java.util.Deque; import java.util.LinkedList; public class Geeks {  // Returns sum of min and max element of all subarrays   // of size k   public static int SumOfKsubArray(int arr[]  int k)   {   int sum = 0; // Initialize result     // The queue will store indexes of useful elements   // in every window   // In deque 'G' we maintain decreasing order of   // values from front to rear   // In deque 'S' we maintain increasing order of   // values from front to rear   Deque<Integer> S=new LinkedList<>()G=new LinkedList<>();  // Process first window of size K   int i = 0;   for (i = 0; i < k; i++)   {   // Remove all previous greater elements   // that are useless.   while ( !S.isEmpty() && arr[S.peekLast()] >= arr[i])   S.removeLast(); // Remove from rear     // Remove all previous smaller that are elements   // are useless.   while ( !G.isEmpty() && arr[G.peekLast()] <= arr[i])   G.removeLast(); // Remove from rear     // Add current element at rear of both deque   G.addLast(i);   S.addLast(i);   }     // Process rest of the Array elements   for ( ; i < arr.length; i++ )   {   // Element at the front of the deque 'G' & 'S'   // is the largest and smallest   // element of previous window respectively   sum += arr[S.peekFirst()] + arr[G.peekFirst()];     // Remove all elements which are out of this   // window   while ( !S.isEmpty() && S.peekFirst() <= i - k)   S.removeFirst();   while ( !G.isEmpty() && G.peekFirst() <= i - k)   G.removeFirst();     // remove all previous greater element that are   // useless   while ( !S.isEmpty() && arr[S.peekLast()] >= arr[i])   S.removeLast(); // Remove from rear     // remove all previous smaller that are elements   // are useless   while ( !G.isEmpty() && arr[G.peekLast()] <= arr[i])   G.removeLast(); // Remove from rear     // Add current element at rear of both deque   G.addLast(i);   S.addLast(i);   }     // Sum of minimum and maximum element of last window   sum += arr[S.peekFirst()] + arr[G.peekFirst()];     return sum;   }   public static void main(String args[])   {  int arr[] = {2 5 -1 7 -3 -1 -2} ;   int k = 4;   System.out.println(SumOfKsubArray(arr k));  } } //This code is contributed by Gaurav Tiwari 
Python
# Python3 program to find Sum of all minimum and maximum  # elements Of Sub-array Size k. from collections import deque # Returns Sum of min and max element of all subarrays # of size k def SumOfKsubArray(arr n  k): Sum = 0 # Initialize result # The queue will store indexes of useful elements # in every window # In deque 'G' we maintain decreasing order of # values from front to rear # In deque 'S' we maintain increasing order of # values from front to rear S = deque() G = deque() # Process first window of size K for i in range(k): # Remove all previous greater elements # that are useless. while ( len(S) > 0 and arr[S[-1]] >= arr[i]): S.pop() # Remove from rear # Remove all previous smaller that are elements # are useless. while ( len(G) > 0 and arr[G[-1]] <= arr[i]): G.pop() # Remove from rear # Add current element at rear of both deque G.append(i) S.append(i) # Process rest of the Array elements for i in range(k n): # Element at the front of the deque 'G' & 'S' # is the largest and smallest # element of previous window respectively Sum += arr[S[0]] + arr[G[0]] # Remove all elements which are out of this # window while ( len(S) > 0 and S[0] <= i - k): S.popleft() while ( len(G) > 0 and G[0] <= i - k): G.popleft() # remove all previous greater element that are # useless while ( len(S) > 0 and arr[S[-1]] >= arr[i]): S.pop() # Remove from rear # remove all previous smaller that are elements # are useless while ( len(G) > 0 and arr[G[-1]] <= arr[i]): G.pop() # Remove from rear # Add current element at rear of both deque G.append(i) S.append(i) # Sum of minimum and maximum element of last window Sum += arr[S[0]] + arr[G[0]] return Sum # Driver program to test above functions arr=[2 5 -1 7 -3 -1 -2] n = len(arr) k = 4 print(SumOfKsubArray(arr n k)) # This code is contributed by mohit kumar 
C#
// C# program to find sum of all minimum and maximum  // elements Of Sub-array Size k.  using System; using System.Collections.Generic; class Geeks {  // Returns sum of min and max element of all subarrays   // of size k   public static int SumOfKsubArray(int []arr  int k)   {   int sum = 0; // Initialize result   // The queue will store indexes of useful elements   // in every window   // In deque 'G' we maintain decreasing order of   // values from front to rear   // In deque 'S' we maintain increasing order of   // values from front to rear   List<int> S = new List<int>();  List<int> G = new List<int>();  // Process first window of size K   int i = 0;   for (i = 0; i < k; i++)   {   // Remove all previous greater elements   // that are useless.   while ( S.Count != 0 && arr[S[S.Count - 1]] >= arr[i])   S.RemoveAt(S.Count - 1); // Remove from rear   // Remove all previous smaller that are elements   // are useless.   while ( G.Count != 0 && arr[G[G.Count - 1]] <= arr[i])   G.RemoveAt(G.Count - 1); // Remove from rear   // Add current element at rear of both deque   G.Add(i);   S.Add(i);   }   // Process rest of the Array elements   for ( ; i < arr.Length; i++ )   {   // Element at the front of the deque 'G' & 'S'   // is the largest and smallest   // element of previous window respectively   sum += arr[S[0]] + arr[G[0]];   // Remove all elements which are out of this   // window   while ( S.Count != 0 && S[0] <= i - k)   S.RemoveAt(0);   while ( G.Count != 0 && G[0] <= i - k)   G.RemoveAt(0);   // remove all previous greater element that are   // useless   while ( S.Count != 0 && arr[S[S.Count-1]] >= arr[i])   S.RemoveAt(S.Count - 1 ); // Remove from rear   // remove all previous smaller that are elements   // are useless   while ( G.Count != 0 && arr[G[G.Count - 1]] <= arr[i])   G.RemoveAt(G.Count - 1); // Remove from rear   // Add current element at rear of both deque   G.Add(i);   S.Add(i);   }   // Sum of minimum and maximum element of last window   sum += arr[S[0]] + arr[G[0]];   return sum;   }   // Driver code  public static void Main(String []args)   {  int []arr = {2 5 -1 7 -3 -1 -2} ;   int k = 4;   Console.WriteLine(SumOfKsubArray(arr k));  } } // This code is contributed by gauravrajput1  
JavaScript
// JavaScript program to find sum of all minimum and maximum // elements Of Sub-array Size k. // Returns sum of min and max element of all subarrays // of size k function SumOfKsubArray(arr  k) {  let sum = 0; // Initialize result  // The queue will store indexes of useful elements  // in every window  // In deque 'G' we maintain decreasing order of  // values from front to rear  // In deque 'S' we maintain increasing order of  // values from front to rear  let S = [];  let G = [];  // Process first window of size K  let i = 0;  for (i = 0; i < k; i++)  {  // Remove all previous greater elements  // that are useless.  while ( S.length != 0 && arr[S[S.length - 1]] >= arr[i])  S.pop(); // Remove from rear  // Remove all previous smaller that are elements  // are useless.  while ( G.length != 0 && arr[G[G.length - 1]] <= arr[i])  G.pop(); // Remove from rear  // Add current element at rear of both deque  G.push(i);  S.push(i);  }  // Process rest of the Array elements  for ( ; i < arr.length; i++ )  {  // Element at the front of the deque 'G' & 'S'  // is the largest and smallest  // element of previous window respectively  sum += arr[S[0]] + arr[G[0]];  // Remove all elements which are out of this  // window  while ( S.length != 0 && S[0] <= i - k)  S.shift(0);  while ( G.length != 0 && G[0] <= i - k)  G.shift(0);  // remove all previous greater element that are  // useless  while ( S.length != 0 && arr[S[S.length-1]] >= arr[i])  S.pop(); // Remove from rear  // remove all previous smaller that are elements  // are useless  while ( G.length != 0 && arr[G[G.length - 1]] <= arr[i])  G.pop(); // Remove from rear  // Add current element at rear of both deque  G.push(i);  S.push(i);  }  // Sum of minimum and maximum element of last window  sum += arr[S[0]] + arr[G[0]];  return sum; } // Driver code  let arr = [2 5 -1 7 -3 -1 -2];  let k = 4;  console.log(SumOfKsubArray(arr k)); // This code is contributed by _saurabh_jaiswal 

Ieșire
18

Complexitatea timpului: O(n)
Spațiu auxiliar: O(k)

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