Array Representation
Array Representation
A complete binary tree has a simple array representation. Suppose we number the nodes from left to right, beginning at the top and ending at the bottom. Then we can store the various data items in the corresponding elements of an array. For example
can be represented by the array
This in fact corresponds to the level order enumeration of the tree. Note that we only use an initial segment of the array. Provided the array is long enough, and we know the number of tree nodes, it doesn't matter how many unused components there are at the end.
* length[A]: the size of the array * heap-size[A]: the number of items stored into the array A * Note: heap-size[A] <= length[A] * The root of the tree is at A[1], i.e., the indexing typically begins at index 1 (not 0). A[0] can be reserved for the variable heap-size[A].
Heap is implemented as an array, but its operations can be grasped more easily by looking at the binary tree representation. The mapping between the array representation and binary tree representation is unambiguous. The array representation can be achieved by traversing the binary tree in level order.
Figure 1: Binary tree and array representation for the MaxHeap containing elements (that has the priorities) [16,14,10,8,7,9,3,2,4,1].
2.1 Routines to access the array
Lets consider the i:th node in a Heap that has the value A[i] PARENT(i) = i/2 | Return the index of the father node | LEFT(i) = 2i | Return the index of the left child | RIGHT(i) = 2i+1 | Return the index of the right child |
Example 1: In Figure 1, node i=3 (A[3]=10) has father at index PARENT(3) = 3/2 = 1 (A[1] = 16). In addition, its left and right children are at LEFT(3) = 2*3 = 6 and RIGHT(3) = 2*3 + 1 = 7 (A[6] = 9 andA[7] = 3,
A complete binary tree has a simple array representation. Suppose we number the nodes from left to right, beginning at the top and ending at the bottom. Then we can store the various data items in the corresponding elements of an array. For example
can be represented by the array
This in fact corresponds to the level order enumeration of the tree. Note that we only use an initial segment of the array. Provided the array is long enough, and we know the number of tree nodes, it doesn't matter how many unused components there are at the end.
* length[A]: the size of the array * heap-size[A]: the number of items stored into the array A * Note: heap-size[A] <= length[A] * The root of the tree is at A[1], i.e., the indexing typically begins at index 1 (not 0). A[0] can be reserved for the variable heap-size[A].
Heap is implemented as an array, but its operations can be grasped more easily by looking at the binary tree representation. The mapping between the array representation and binary tree representation is unambiguous. The array representation can be achieved by traversing the binary tree in level order.
Figure 1: Binary tree and array representation for the MaxHeap containing elements (that has the priorities) [16,14,10,8,7,9,3,2,4,1].
2.1 Routines to access the array
Lets consider the i:th node in a Heap that has the value A[i] PARENT(i) = i/2 | Return the index of the father node | LEFT(i) = 2i | Return the index of the left child | RIGHT(i) = 2i+1 | Return the index of the right child |
Example 1: In Figure 1, node i=3 (A[3]=10) has father at index PARENT(3) = 3/2 = 1 (A[1] = 16). In addition, its left and right children are at LEFT(3) = 2*3 = 6 and RIGHT(3) = 2*3 + 1 = 7 (A[6] = 9 andA[7] = 3,