TechCodeview

Kaj je Heap?

Kup je popolno binarno drevo, binarno drevo pa je drevo, v katerem ima lahko vozlišče največ dva otroka. Preden izvemo več o kopici Kaj je popolno binarno drevo?

Popolno binarno drevo je a binarno drevo, v katerem morajo biti vse ravni razen zadnje ravni, tj. listnega vozlišča popolnoma zapolnjene, vsa vozlišča pa poravnana levo.

Razumejmo skozi primer.

Na zgornji sliki lahko opazimo, da so vsa notranja vozlišča popolnoma zapolnjena, razen listnega vozlišča; zato lahko rečemo, da je zgornje drevo popolno binarno drevo.

Zgornja slika prikazuje, da so vsa notranja vozlišča popolnoma zapolnjena, razen listnega vozlišča, vendar so listna vozlišča dodana na desnem delu; zato zgornje drevo ni popolno binarno drevo.

Opomba: Drevo kopice je posebna uravnotežena podatkovna struktura binarnega drevesa, kjer se korensko vozlišče primerja s svojimi podrejenimi in ustrezno razporedi.

Kako lahko razporedimo vozlišča v drevesu?

Obstajata dve vrsti kopice:

• Najmanjša kopica
• Največji kup

Najmanjša kopica: Vrednost nadrejenega vozlišča mora biti manjša ali enaka kateremu koli od njegovih podrejenih vozlišč.

oz

Z drugimi besedami, minimalno kopico lahko definiramo tako, da je za vsako vozlišče i vrednost vozlišča i večja ali enaka njegovi nadrejeni vrednosti, razen korenskega vozlišča. Matematično se lahko opredeli kot:

A[Parent(i)]<= a[i]< strong>

Razumejmo min-heap na primeru.

Na zgornji sliki je 11 korensko vozlišče in vrednost korenskega vozlišča je manjša od vrednosti vseh drugih vozlišč (levega podrejenega ali desnega podrejenega).

Največji kup: Vrednost nadrejenega vozlišča je večja ali enaka njegovim podrejenim vozliščem.

oz

matriko v jeziku c

Z drugimi besedami, največjo kopico lahko definiramo kot za vsako vozlišče i; vrednost vozlišča i je manjša ali enaka njegovi nadrejeni vrednosti, razen korenskega vozlišča. Matematično se lahko opredeli kot:

A[Parent(i)] >= A[i]

Zgornje drevo je drevo največjega kupa, saj izpolnjuje lastnost največjega kupa. Zdaj pa poglejmo matrično predstavitev največjega kupa.

Časovna kompleksnost v Max Heapu

Skupno število zahtevanih primerjav v največjem kupu je odvisno od višine drevesa. Višina celotnega binarnega drevesa je vedno logn; zato bi bila tudi časovna kompleksnost O(logn).

Algoritem delovanja vstavljanja v največji kup.

 // algorithm to insert an element in the max heap. insertHeap(A, n, value) { n=n+1; // n is incremented to insert the new element A[n]=value; // assign new value at the nth position i = n; // assign the value of n to i // loop will be executed until i becomes 1. while(i&gt;1) { parent= floor value of i/2; // Calculating the floor value of i/2 // Condition to check whether the value of parent is less than the given node or not if(A[parent] <a[i]) { swap(a[parent], a[i]); i="parent;" } else return; < pre> <p> <strong>Let&apos;s understand the max heap through an example</strong> .</p> <p>In the above figure, 55 is the parent node and it is greater than both of its child, and 11 is the parent of 9 and 8, so 11 is also greater than from both of its child. Therefore, we can say that the above tree is a max heap tree.</p> <p> <strong>Insertion in the Heap tree</strong> </p> <p> <strong>44, 33, 77, 11, 55, 88, 66</strong> </p> <p>Suppose we want to create the max heap tree. To create the max heap tree, we need to consider the following two cases:</p> <ul> <li>First, we have to insert the element in such a way that the property of the complete binary tree must be maintained.</li> <li>Secondly, the value of the parent node should be greater than the either of its child.</li> </ul> <p> <strong>Step 1:</strong> First we add the 44 element in the tree as shown below:</p> <img src="//techcodeview.com/img/ds-tutorial/89/heap-data-structure-5.webp" alt="Heap Data Structure"> <p> <strong>Step 2:</strong> The next element is 33. As we know that insertion in the binary tree always starts from the left side so 44 will be added at the left of 33 as shown below:</p> <img src="//techcodeview.com/img/ds-tutorial/89/heap-data-structure-6.webp" alt="Heap Data Structure"> <p> <strong>Step 3:</strong> The next element is 77 and it will be added to the right of the 44 as shown below:</p> <img src="//techcodeview.com/img/ds-tutorial/89/heap-data-structure-7.webp" alt="Heap Data Structure"> <p>As we can observe in the above tree that it does not satisfy the max heap property, i.e., parent node 44 is less than the child 77. So, we will swap these two values as shown below:</p> <img src="//techcodeview.com/img/ds-tutorial/89/heap-data-structure-8.webp" alt="Heap Data Structure"> <p> <strong>Step 4:</strong> The next element is 11. The node 11 is added to the left of 33 as shown below:</p> <img src="//techcodeview.com/img/ds-tutorial/89/heap-data-structure-9.webp" alt="Heap Data Structure"> <p> <strong>Step 5:</strong> The next element is 55. To make it a complete binary tree, we will add the node 55 to the right of 33 as shown below:</p> <img src="//techcodeview.com/img/ds-tutorial/89/heap-data-structure-10.webp" alt="Heap Data Structure"> <p>As we can observe in the above figure that it does not satisfy the property of the max heap because 33<55, so we will swap these two values as shown below:< p> <img src="//techcodeview.com/img/ds-tutorial/89/heap-data-structure-11.webp" alt="Heap Data Structure"> <p> <strong>Step 6:</strong> The next element is 88. The left subtree is completed so we will add 88 to the left of 44 as shown below:</p> <img src="//techcodeview.com/img/ds-tutorial/89/heap-data-structure-12.webp" alt="Heap Data Structure"> <p>As we can observe in the above figure that it does not satisfy the property of the max heap because 44<88, so we will swap these two values as shown below:< p> <p>Again, it is violating the max heap property because 88&gt;77 so we will swap these two values as shown below:</p> <p> <strong>Step 7:</strong> The next element is 66. To make a complete binary tree, we will add the 66 element to the right side of 77 as shown below:</p> <p>In the above figure, we can observe that the tree satisfies the property of max heap; therefore, it is a heap tree.</p> <p> <strong>Deletion in Heap Tree</strong> </p> <p>In Deletion in the heap tree, the root node is always deleted and it is replaced with the last element.</p> <p> <strong>Let&apos;s understand the deletion through an example.</strong> </p> <p> <strong>Step 1</strong> : In the above tree, the first 30 node is deleted from the tree and it is replaced with the 15 element as shown below:</p> <p>Now we will heapify the tree. We will check whether the 15 is greater than either of its child or not. 15 is less than 20 so we will swap these two values as shown below:</p> <p>Again, we will compare 15 with its child. Since 15 is greater than 10 so no swapping will occur.</p> <p> <strong>Algorithm to heapify the tree</strong> </p> <pre> MaxHeapify(A, n, i) { int largest =i; int l= 2i; int r= 2i+1; while(lA[largest]) { largest=l; } while(rA[largest]) { largest=r; } if(largest!=i) { swap(A[largest], A[i]); heapify(A, n, largest); }} </pre> <hr></88,></p></55,></p></a[i])>