Hvad er Heap?
En heap er et komplet binært træ, og det binære træ er et træ, hvor noden kan have højst to børn. Før du ved mere om dyngen Hvad er et komplet binært træ?
Et komplet binært træ er et binært træ, hvor alle niveauer undtagen det sidste niveau, dvs. bladknude, skal være fuldstændigt udfyldt, og alle noder skal venstrejusteres.
Lad os forstå gennem et eksempel.
I ovenstående figur kan vi observere, at alle de indre knudepunkter er fuldstændigt fyldte undtagen bladknuden; derfor kan vi sige, at ovenstående træ er et komplet binært træ.
Ovenstående figur viser, at alle de indre knuder er fuldstændigt fyldte undtagen bladknuden, men bladknuderne er tilføjet i højre del; derfor er ovenstående træ ikke et komplet binært træ.
Bemærk: Dyngetræet er en speciel balanceret binær trædatastruktur, hvor rodknuden sammenlignes med dens børn og arrangeres derefter.
Hvordan kan vi arrangere noderne i træet?
Der er to typer af bunken:
- Min bunke
- Max bunke
Min bunke: Værdien af den overordnede node skal være mindre end eller lig med en af dens børn.
Eller
git rebase
Med andre ord kan min-heapen defineres som, for hver node i, værdien af node i er større end eller lig med dens overordnede værdi undtagen rodknuden. Matematisk kan det defineres som:
A[Forælder(i)]<= a[i]< strong>
Lad os forstå min-heapen gennem et eksempel.
I ovenstående figur er 11 rodnoden, og værdien af rodknuden er mindre end værdien af alle de andre knudepunkter (venstre underordnede eller et højre underordnede).
java-listen er tom
Max Heap: Værdien af den overordnede node er større end eller lig med dens børn.
Eller
Med andre ord kan den maksimale heap defineres som for hver node i; værdien af node i er mindre end eller lig med dens overordnede værdi undtagen rodnoden. Matematisk kan det defineres som:
A[Forælder(i)] >= A[i]
Ovenstående træ er et max-dyngetræ, da det opfylder egenskaben for max-bunken. Lad os nu se array-repræsentationen af den maksimale heap.
Tidskompleksitet i Max Heap
Det samlede antal sammenligninger, der kræves i den maksimale bunke, afhænger af træets højde. Højden af det komplette binære træ er altid logn; derfor ville tidskompleksiteten også være O(logn).
Algoritme for indsætningsoperation i den maksimale heap.
// 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>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'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>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'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])>