Hva er Heap?

En haug er et komplett binært tre, og det binære treet er et tre der noden kan ha maksimalt to barn. Før du vet mer om haugen Hva er et komplett binært tre?

Et komplett binært tre er en binært tre der alle nivåene unntatt det siste nivået, dvs. bladnoden skal være fullstendig fylt, og alle nodene skal venstrejusteres.

La oss forstå gjennom et eksempel.

I figuren ovenfor kan vi observere at alle interne noder er fullstendig fylt bortsett fra bladnoden; derfor kan vi si at treet ovenfor er et komplett binært tre.

Figuren ovenfor viser at alle interne noder er fullstendig fylt bortsett fra bladnoden, men bladnodene er lagt til i høyre del; derfor er ikke treet ovenfor et fullstendig binært tre.

Merk: Heaptreet er en spesiell balansert binær tredatastruktur der rotnoden sammenlignes med sine underordnede og ordnede deretter.

Hvordan kan vi ordne nodene i treet?

Det er to typer hauger:

• Min haug
• Maks haug

Min haug: Verdien til den overordnede noden skal være mindre enn eller lik ett av dens underordnede node.

Eller

programvareutvikling livssyklus

Med andre ord, min-heapen kan defineres som at for hver node i, verdien av node i er større enn eller lik dens overordnede verdi bortsett fra rotnoden. Matematisk kan det defineres som:

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

La oss forstå min-haugen gjennom et eksempel.

I figuren ovenfor er 11 rotnoden, og verdien til rotnoden er mindre enn verdien til alle de andre nodene (venstre underordnede eller et høyre underordnede).

endre legg til kolonne orakel

Maks haug: Verdien til overordnet node er større enn eller lik dens underordnede node.

Eller

Med andre ord, maks haugen kan defineres som for hver node i; verdien av node i er mindre enn eller lik dens overordnede verdi bortsett fra rotnoden. Matematisk kan det defineres som:

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

Treet ovenfor er et maks haugtre da det tilfredsstiller egenskapen til maks haugen. La oss nå se matriserepresentasjonen av den maksimale haugen.

Tidskompleksitet i Max Heap

Det totale antallet sammenligninger som kreves i den maksimale haugen er i henhold til høyden på treet. Høyden på det komplette binære treet er alltid logget; derfor vil tidskompleksiteten også være O(logn).

Algoritme for innsatsoperasjon i maks. haug.

 // 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])>