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A heap is a specialized form of binary search tree question 1 options


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Priority Queue is an extension of the choices queue with the subsequent properties:  Every object has a concern related to it.An element with high priority is dequeued before an detail with low priority.If elements have the choices identical precedence, they may be served in line with their order in the queue.A Binary Heap is a Binary Tree with the following houses:  It is a Complete Tree. This assets of Binary Heap makes them suitable to be saved in an array.A Binary Heap is either Min Heap or Max Heap.In a Min Binary Heap, the choices key at the root need to be minimum amongst all keys present in Binary Heap. The equal belongings need to be recursively real for all nodes in Binary Tree.Similarly, in a Max Binary Heap, the key at the choices root must be maximum among all keys found in Binary Heap. The equal property ought to be recursively real for all nodes in Binary Tree.Operation on Binary Heap insert(p): Inserts a new element with precedence p.extractMax(): Extracts an element with maximum priority.cast off(i): Removes an detail pointed by using an iterator i.getMax(): Returns an element with maximum precedence.changePriority(i, p): Changes the priority of an detail pointed with the aid of i to p.Example of A Binary Max Heap Suppose below is the given Binary Heap that follows all of the houses of Binary Max Heap.  Now a node with value 32 need to be insert inside the above heap: To insert an element, connect the brand new element to any leaf. For Example A node with priority 32 can be added to the leaf of the choices node eleven. But this violates the choices heap assets. To preserve the choices heap property, shift up the new node 32.  Shift Up Operation get node with 32 at the proper role: Swap the choices incorrectly positioned node with its discern till the heap belongings is glad. For Example: As node 11 is much less than node 32 so, switch node 11 and node 32. Then, switch node 14 and node 32. And at last, change node 31 and node 32.  ExtractMax: The maximum value is stored at the root of the tree. But the choices root of the choices tree can’t be directly eliminated. First, it is replaced with any one of the choices leaves after which eliminated. For Example: To put off Node forty five, it’s far first replaced with node 11. But this violates the heap property, so pass the choices changed node down. For that, use shift down operation.  ShiftDown operation: Swap the choices incorrectly placed node with a larger infant till the heap belongings is satisfied. For Example: Node eleven is swapped with node 32 then, with node 31 and in ultimate it’s far swapped with node 14.  ChangePriority: Let the choices changed detail shift up or down relying on whether its precedence reduced or extended. For Example: Change the priority of nodes eleven to 35, because of this modification the choices node has to shift up the node so one can keep the choices heap assets.Remove: To get rid of an detail, trade its priority to a price larger than the choices modern most, then shift it up, and then extract it the use of extract max. Find the choices contemporary most using getMax.GetMax: The max value is stored at the choices root of the tree. To getmax, just go back the value at the root of the tree.Array Representation of Binary HeapSince the choices heap is maintained in shape of a entire binary tree, because of this reality the choices heap may be represented inside the shape of an array. To hold the tree whole and shallow, while placing a new detail insert it in the leftmost vacant role within the remaining stage i.e., at the end of our array. Similarly, whilst extracting most update the root with the choices last leaf at the last degree i.e., the choices last element of the array. Below is the choices instance of the choices identical:  Below is the software to put into effect Priority Queue using Binary Heap:  C++ // C++ code to enforce priority-queue// the use of array implementation of// binary heap #include using namespace std; int H[50];int length = -1; // Function to return the index of the choices// discern node of a given nodeint figure(int i)     go back (i – 1) / 2; // Function to return the index of the // left child of the given nodeint leftChild(int i)     return ((2 * i) + 1); // Function to return the choices index of the // proper toddler of the given nodeint rightChild(int i)     go back ((2 * i) + 2); // Function to shift up the node so as// to hold the choices heap propertyvoid shiftUp(int i)    even as (i > zero && H[parent(i)] < H[i])          // Swap determine and present day node        switch(H[parent(i)], H[i]);         // Update i to discern of i        i = figure(i);     // Function to shift down the node in// order to preserve the heap propertyvoid shiftDown(int i)    int maxIndex = i;     // Left Child    int l = leftChild(i);     if (l H[maxIndex])         maxIndex = l;         // Right Child    int r = rightChild(i);     if (r H[maxIndex])         maxIndex = r;         // If i no longer same as maxIndex    if (i != maxIndex)         swap(H[i], H[maxIndex]);        shiftDown(maxIndex);     // Function to insert a new element// in the Binary Heapvoid insert(int p)    length = length + 1;    H[size] = p;     // Shift Up to preserve heap assets    shiftUp(size); // Function to extract the choices element with// maximum priorityint extractMax()    int result = H[0];     // Replace the price at the root    // with the choices remaining leaf    H[0] = H[size];    size = size – 1;     // Shift down the choices changed detail    // to keep the choices heap assets    shiftDown(0);    return result; // Function to exchange the priority// of an elementvoid changePriority(int i, int p)    int oldp = H[i];    H[i] = p;     if (p > oldp)         shiftUp(i);        else         shiftDown(i);     // Function to get value of the choices current// most elementint getMax()     return H[0]; // Function to dispose of the element// placed at given indexvoid dispose of(int i)    H[i] = getMax() + 1;     // Shift the node to the choices root    // of the heap    shiftUp(i);     // Extract the choices node    extractMax(); // Driver Codeint foremost()     /*         forty five            /                 31      14          /      /           13  20  7   eleven        /         12   7    Create a priority queue shown in    instance in a binary max heap form.    Queue might be represented within the    shape of array as:    45 31 14 thirteen 20 7 eleven 12 7 */     // Insert the detail to the     // precedence queue    insert(45);    insert(20);    insert(14);    insert(12);    insert(31);    insert(7);    insert(eleven);    insert(13);    insert(7);     int i = 0;     // Priority queue before extracting max    cout << "Priority Queue : ";    whilst (i <= length)         cout << H[i] << " ";        i++;         cout << "n";     // Node with most priority    cout << "Node with most priority : "         << extractMax() << "n";     // Priority queue after extracting max    cout << "Priority queue after "         << "extracting most : ";    int j = zero;    whilst (j <= size)         cout << H[j] << " ";        j++;         cout << "n";     // Change the concern of element    // present at index 2 to forty nine    changePriority(2, 49);    cout << "Priority queue after "         << "priority exchange : ";    int ok = zero;    while (ok <= length)         cout << H[k] << " ";        ok++;         cout << "n";     // Remove detail at index three    get rid of(3);    cout << "Priority queue after "         << "casting off the choices element : ";    int l = 0;    even as (l <= length)         cout << H[l] < zero &&         H[parent(i)] < H[i])      // Swap parent and modern node    swap(figure(i), i);     // Update i to discern of i    i = discern(i);   // Function to shift down the choices node in// order to maintain the heap propertystatic void shiftDown(int i)  int maxIndex = i;   // Left Child  int l = leftChild(i);   if (l H[maxIndex])      maxIndex = l;     // Right Child  int r = rightChild(i);   if (r H[maxIndex])      maxIndex = r;     // If i now not identical as maxIndex  if (i != maxIndex)      switch(i, maxIndex);    shiftDown(maxIndex);   // Function to insert a// new detail in// the choices Binary Heapstatic void insert(int p)  length = length + 1;  H[size] = p;   // Shift Up to hold  // heap property  shiftUp(size); // Function to extract// the choices detail with// maximum prioritystatic int extractMax()  int end result = H[0];   // Replace the fee  // at the root with  // the closing leaf  H[0] = H[size];  size = length – 1;   // Shift down the choices changed  // detail to hold the choices  // heap belongings  shiftDown(zero);  go back result; // Function to trade the priority// of an elementstatic void changePriority(int i,                           int p)  int oldp = H[i];  H[i] = p;   if (p > oldp)      shiftUp(i);    else      shiftDown(i);   // Function to get fee of// the choices present day most elementstatic int getMax()  return H[0]; // Function to dispose of the choices detail// placed at given indexstatic void cast off(int i)  H[i] = getMax() + 1;   // Shift the choices node to the root  // of the heap  shiftUp(i);   // Extract the choices node  extractMax();   static void change(int i, int j)  int temp= H[i];  H[i] = H[j];  H[j] = temp; // Driver Codepublic static void major(String[] args)   /*           forty five            /                   31      14          /      /           13  20  7   eleven        /         12   7    Create a concern queue shown in    example in a binary max heap shape.    Queue will be represented in the    form of array as:    45 31 14 13 20 7 11 12 7 */   // Insert the element to the   // priority queue  insert(45);  insert(20);  insert(14);  insert(12);  insert(31);  insert(7);  insert(11);  insert(13);  insert(7);   int i = 0;   // Priority queue before extracting max  System.out.print(“Priority Queue : “);  at the same time as (i <= size)      System.out.print(H[i] + " ");    i++;     System.out.print("n");   // Node with most precedence  System.out.print("Node with most priority : " +                    extractMax() + "n");   // Priority queue after extracting max  System.out.print("Priority queue after " +                   "extracting most : ");  int j = zero;  while (j <= length)      System.out.print(H[j] + " ");    j++;     System.out.print("n");   // Change the priority of detail  // present at index 2 to forty nine  changePriority(2, 49);  System.out.print("Priority queue after " +                   "priority trade : ");  int ok = zero;  even as (okay <= length)      System.out.print(H[k] + " ");    k++;     System.out.print("n");   // Remove detail at index three  do away with(3);  System.out.print("Priority queue after " +                   "removing the element : ");  int l = zero;  whilst (l zero and H[parent(i)] < H[i]) :                   # Swap parent and modern node        change(discern(i), i)               # Update i to discern of i        i = figure(i)         # Function to shift down the node in# order to keep the heap propertydef shiftDown(i) :     maxIndex = i           # Left Child    l = leftChild(i)           if (l H[maxIndex]) :             maxIndex = l           # Right Child    r = rightChild(i)           if (r H[maxIndex]) :             maxIndex = r           # If i not equal as maxIndex    if (i != maxIndex) :             switch(i, maxIndex)        shiftDown(maxIndex)         # Function to insert a # new detail in # the choices Binary Heapdef insert(p) :         global size    size = size + 1    H[size] = p           # Shift Up to preserve     # heap property    shiftUp(length)   # Function to extract # the detail with# maximum prioritydef extractMax() :         global length    result = H[0]           # Replace the choices price     # at the choices root with     # the choices ultimate leaf    H[0] = H[size]    size = length – 1           # Shift down the changed     # detail to keep the      # heap property    shiftDown(zero)    go back end result   # Function to alternate the concern# of an elementdef changePriority(i,p) :     oldp = H[i]    H[i] = p           if (p > oldp) :             shiftUp(i)      else :             shiftDown(i)   # Function to get fee of # the choices cutting-edge maximum elementdef getMax() :      return H[0]   # Function to take away the detail# placed at given indexdef Remove(i) :     H[i] = getMax() + 1           # Shift the node to the choices root    # of the choices heap    shiftUp(i)           # Extract the node    extractMax()   def switch(i, j) :         temp = H[i]    H[i] = H[j]    H[j] = temp     # Insert the choices detail to the choices# priority queueinsert(forty five)insert(20)insert(14)insert(12)insert(31)insert(7)insert(11)insert(13)insert(7)   i = 0   # Priority queue before extracting maxprint(“Priority Queue : “, end = “”)while (i <= size) :     print(H[i], give up = " ")    i += 1   print()   # Node with maximum priorityprint("Node with maximum priority :" ,  extractMax())   # Priority queue after extracting maxprint("Priority queue after extracting most : ", cease = "")j = 0while (j <= length) :     print(H[j], end = " ")    j += 1   print()   # Change the concern of detail# gift at index 2 to 49changePriority(2, 49)print("Priority queue after precedence exchange : ", give up = "")ok = 0while (ok <= size) :     print(H[k], stop = " ")    okay += 1   print()   # Remove detail at index 3Remove(three)print("Priority queue after casting off the detail : ", cease = "")l = 0while (l 0 &&           H[parent(i)] < H[i])                     // Swap figure and modern node        switch(determine(i), i);             // Update i to parent of i        i = parent(i);     // Function to shift down the choices node in// order to hold the choices heap propertystatic void shiftDown(int i)    int maxIndex = i;         // Left Child    int l = leftChild(i);         if (l H[maxIndex])            maxIndex = l;             // Right Child    int r = rightChild(i);         if (r H[maxIndex])            maxIndex = r;             // If i not same as maxIndex    if (i != maxIndex)            change(i, maxIndex);        shiftDown(maxIndex);     // Function to insert a// new detail in// the Binary Heapstatic void insert(int p)    length = size + 1;    H[size] = p;         // Shift Up to keep    // heap assets    shiftUp(length); // Function to extract// the choices element with// maximum prioritystatic int extractMax()    int end result = H[0];         // Replace the cost    // at the root with    // the choices remaining leaf    H[0] = H[size];    size = size – 1;         // Shift down the choices replaced    // detail to keep the choices    // heap belongings    shiftDown(0);    return end result; // Function to alternate the concern// of an elementstatic void changePriority(int i,                           int p)    int oldp = H[i];    H[i] = p;         if (p > oldp)            shiftUp(i);        else            shiftDown(i);     // Function to get cost of// the modern most elementstatic int getMax()    return H[0]; // Function to do away with the choices element// placed at given indexstatic void Remove(int i)    H[i] = getMax() + 1;         // Shift the choices node to the choices root    // of the heap    shiftUp(i);         // Extract the choices node    extractMax(); static void switch(int i, int j)    int temp = H[i];    H[i] = H[j];    H[j] = temp; // Driver Codepublic static void Main(String[] args) /*              45            /                31      14          /      /         13  20   7   11       /       12  7    Create a priority queue proven in    example in a binary max heap form.    Queue will be represented in the    shape of array as:    forty five 31 14 13 20 7 11 12 7 */     // Insert the detail to the     // precedence queue    insert(forty five);    insert(20);    insert(14);    insert(12);    insert(31);    insert(7);    insert(11);    insert(13);    insert(7);         int i = zero;         // Priority queue before extracting max    Console.Write(“Priority Queue : “);    whilst (i <= length)            Console.Write(H[i] + " ");        i++;             Console.Write("n");         // Node with most precedence    Console.Write("Node with maximum priority : " +                   extractMax() + "n");         // Priority queue after extracting max    Console.Write("Priority queue after " +                  "extracting most : ");    int j = zero;    whilst (j <= size)            Console.Write(H[j] + " ");        j++;             Console.Write("n");         // Change the concern of element    // gift at index 2 to forty nine    changePriority(2, 49);    Console.Write("Priority queue after " +                  "precedence change : ");    int ok = zero;    whilst (ok <= length)            Console.Write(H[k] + " ");        okay++;             Console.Write("n");         // Remove element at index 3    Remove(3);    Console.Write("Priority queue after " +                  "disposing of the choices detail : ");    int l = zero;    whilst (l <= size)            Console.Write(H[l] + " ");        l++;     // This code is contributed by way of Amit KatiyarOutput: Priority Queue : 45 31 14 thirteen 20 7 11 12 7  Node with most precedence : forty five  Priority queue after extracting maximum : 31 20 14 13 7 7 eleven 12  Priority queue after precedence exchange : forty nine 20 31 thirteen 7 7 11 12  Priority queue after getting rid of the choices detail : forty nine 20 31 12 7 7 eleven Time Complexity: The time complexity of all of the operation is O(log N) except for GetMax() which has time complexity of O(1). Auxiliary Space: O(N) Attention reader! 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