4. When i>A.heap-size/2, what will be the result of calling MAX-HEAPIFY(A, i)?
In a word:
When i > A.heap-size/2, calling MAX-HEAPIFY(A, i) swaps A[i] with the largest element in its subtree, treating A[i] as the new root node. This operation will make the subtree with A[i] as the root node satisfy the property of the maximum heap, that is, the root node is larger than its left and right children.
Specifically, assuming that the left child of A[i] is A[2i], and the right child is A[2i+1], the operation process of MAX-HEAPIFY(A, i) is as follows:
1. Exchange A[i] with the largest element in its subtree, that is, exchange with A[2i] or A[2i+1].
2. Recursively call MAX-HEAPIFY(A, 2i) to operate on the left subtree, and exchange the maximum element of the left subtree to A[2i].
3. Recursively call MAX-HEAPIFY(A, 2i+1) to operate on the right subtree, and exchange the maximum element of the right subtree to A[2i+1].
Through these operations, the subtree with A[i] as the root node will satisfy the property of the maximum heap, that is, A[i] is larger than its left and right children. At the same time, since i > A.heap-size/2, the subtree with A[i] as the root node is the left or right half of the entire heap, while the other half of the subtree satisfies the property of the maximum heap. Therefore, the entire heap still satisfies the property of the max heap.
Xunfei Spark:
When i > A.heap-size/2, calling MAX-HEAPIFY(A,i) compares A[i] with A[A.heap-size/2] and compares the smaller value with A[ A.heap-size/2] Exchange. Then, MAX-HEAPIFY(A,A.heap-size/2) will be called recursively until i <= A.heap-size/2. This process will reorganize the heap structure so that A[A.heap-size/2] becomes the new root node. This preserves the property of the heap that each child node has a value less than or equal to its parent node's value.
Huawei Pangu:
360 Smart Brain:
If i>A.heap-size/2 and the value of A.heap[i] is smaller than A.heap[A.heap-size/2], then a maximum heap adjustment is required. The purpose of the maximum heap adjustment is to move A.heap[i] to the correct position, so that the entire heap is still the maximum heap. In the process of adjusting the maximum heap, if the size of the heap is odd, then the current element A.heap[i] will be placed on the far right of the heap; if the size of the heap is even, the current element A.heap[i] will be is placed on the left half of the heap.
Specifically, suppose the size of the current heap is n=2m+1, where m is a positive integer, and A.heap[i]=x<y, and both x and y are non-negative integers. Now it is necessary to adjust the maximum heap, that is, move x to the leftmost of the heap, so that A.heap still maintains the nature of the maximum heap. According to the properties of the maximum heap, we can draw the following conclusions:
1. When x=0, A.heap is still the largest heap.
2. When x<0, since A.heap[A.heap-size/2]=y>x, a maximum heap adjustment is required, that is, move x to the leftmost of the heap. At this point, A.heap becomes the following form: [y, x, A.heap[A.heap-size/2], A.heap[A.heap-size/2]+k*A.heap-size , A.heap[A.heap-size/2]], where k is a positive integer.
3. When x>0, since A.heap[A.heap-size/2]=y>x, a maximum heap adjustment is required, that is, move x to the leftmost of the heap. At this point, A.heap becomes the following form: [y, A.heap[A.heap-size/2], k A.heap-size+x, A.heap[A.heap-size/2]+ k A.heap-size, A.heap[A.heap-size/2]], where k is a positive integer.
To sum up, when i>A.heap-size/2, calling MAX-HEAPIFY(A,i) will change A.heap into the following form: [y, x, A.heap[A.heap-size /2], A.heap[A.heap-size/2]+k*A.heap-size, A.heap[A.heap-size/2]], where x and y are non-negative integers, k is a positive integer.
chatglm：
When i>A.heap-size/2, calling MAX-HEAPIFY(A,i) will exchange the maximum value in the heap to the position with index i in the array.
Specifically, the function of the MAX-HEAPIFY(A,i) function is to exchange the node with subscript i in the heap (assuming that the current heap is represented by a complete binary tree) with the maximum value of its two child nodes. If i is the highest bit of the heap (ie i=A.heap-size-1), it only needs to be exchanged with the left child node of i; otherwise, it needs to be exchanged with the maximum value of the left child node and the right child node of i exchange. After the exchange, the position of i is still the maximum value in the heap.
If i>A.heap-size/2, it means that i is the middle position of the heap. At this time, it needs to be exchanged with the maximum value of i's child nodes to maintain the properties of the heap. After the exchange, the position of i is still the maximum value in the heap, but the exchanged node may affect the balance properties of the heap, so it may be necessary to continue to adjust other nodes of the heap to keep the heap in balance.
chatgpt：
i > A.heap-size/2When , MAX-HEAPIFY(A, i)the function call will have no result. This is because in the binary tree representation of the heap, for any node i, the subscript of its left child node is 2i, and the subscript of its right child node is 2i+1. Since the heap is a complete binary tree, all its leaf nodes are located A.heap-size/2at positions with subscripts greater than . Therefore, it i > A.heap-size/2means that the node iis a leaf node, it has no child nodes, so no heap adjustment (ie call MAX-HEAPIFY) is needed.
Summarize
Chatgpt won this question.
When i>A.heap-size/2, there is 2*i>A.heap-size, which exceeds the boundary range of the array, and the elements will not be transferred when MAX-HEAPIFY(A,i) is executed.
Because when i>a.heap-size/2, i is the subscript of the leaf node, so the original array will not change.