Min Heap Basics Quiz

In a min heap, the parent node must always be less than or equal to its children to maintain the heap property. This ensures that the smallest element is always at the root, allowing for efficient retrieval of the minimum value and maintaining the structure required for heap operations.

A valid min heap requires that each parent node is less than or equal to its child nodes. In the array [1, 2, 3, 4, 5, 6], each element satisfies this condition, with 1 as the root and all other elements greater than or equal to it. In contrast, [2, 1, 3, 4, 5, 6] violates this rule since 2 is greater than 1.

In a min heap represented as an array, the left child of a node at index i is located at index 2i + 1. This is because, in a complete binary tree structure, the left child is positioned directly below the parent node in the array representation, following the formula for calculating child node indices.

In a min heap, the structure is designed so that the smallest element is always at the root. This property ensures that the first element can be accessed quickly, maintaining the efficiency of operations such as insertion and deletion. Therefore, the minimum element is consistently located at the root position.

In a min heap, when a new element is added, it may violate the heap property by being smaller than its parent. The "bubble up" operation, also known as heapify-up, corrects this by repeatedly swapping the new element with its parent until the heap property is restored, ensuring that every parent node is less than its children.

A min heap is a complete binary tree, which means it is filled from top to bottom and left to right. The height of such a tree is determined by the number of levels it has. Since each level doubles the number of nodes, the height is logarithmic in relation to the number of elements, hence approximately log n.

In a min heap, the property requires that each parent node is less than or equal to its children. However, this does not restrict the parent from being larger than its grandchildren. Thus, a parent node can indeed be larger than its grandchildren while still maintaining the min heap property with respect to its direct children.

In a min heap, the root is the smallest element. When removing it, the last element in the array is moved to the root position to maintain the complete binary tree structure. This element is then sifted down to restore the heap property, ensuring the smallest element is at the root again.

After removing the root of a heap and moving the last element to the top, the heap may violate the heap property. To restore order, the "bubble down" operation, also known as heapify-down, is used to move the element down the tree until the heap property is reestablished.

In a min heap, for any node at index \( i \), the parent node can be found at index \( \left\lfloor \frac{i - 1}{2} \right\rfloor \). For a node at index 5, its parent is at index \( \left\lfloor \frac{5 - 1}{2} \right\rfloor = \left\lfloor \frac{4}{2} \right\rfloor = 2 \).

Inserting an element into a min heap involves adding the new element at the end of the heap and then "bubbling up" or "sifting up" to maintain the heap property. This process can take up to O(log n) time, as the height of the heap is logarithmic relative to the number of elements.

A min heap is defined as a complete binary tree where each node's value is less than or equal to its children's values. This structure ensures efficient access to the minimum element, which is always at the root. Therefore, the property of being a complete binary tree is essential for maintaining the min heap's characteristics.