heap insertion algorithm

The heap insertion algorithm inserts a new value into a heap, maintaining the heap property. Let $H$ be a heap, storing $n$ elements over which the relation (http://planetmath.org/Relation) $⪯$ imposes a total ordering. Insertion of a value $x$ consists of initially adding $x$ to the bottom of the tree, and then sifting it upwards until the heap property is regained.

Sifting consists of comparing $x$ to its parent $y$. If $x⪯y$ holds, then the heap property is violated. If this is the case, $x$ and $y$ are swapped and the operation is repeated for the new parent of $x$.

Since $H$ is a balanced binary tree, it has a maximum depth of $\lceil {\log }_{2}n\rceil +1$. Since the maximum number of times that the sift operation can occur is constrained by the depth of the tree, the worst case time complexity for heap insertion is $𝒪(\log n)$.This means that a heap can be built from scratch to hold a multiset of $n$ values in $𝒪(n\log n)$ time.

What follows is the pseudocode for implementing a heap insertion.For the given pseudocode, we presume that the heap is actually represented implicitly in an array (see the binary tree entry for details).

• $\mathrm{HeapInsert}(H,n,⪯,x)$Input: A heap $(H,⪯)$ (represented as an array) containing $n$ values and a new value $x$ to be inserted into $H$.Output: $H$ and $n$, with $x$ inserted and the heap property preserved.Procedure:$n\leftarrow n+1$$H[n]\leftarrow x$$child\leftarrow n$$parent\leftarrow n\text{ div }2$$\text{𝗐𝗁𝗂𝗅𝖾}$$parent\ge 1$$\text{𝖽𝗈}$$\text{𝗂𝖿}$$H[child]⪯H[parent]$$\text{𝗍𝗁𝖾𝗇}$$child\leftarrow parent$$parent\leftarrow parent\text{ div }2$$\text{𝖾𝗅𝗌𝖾}$$\text{𝖾𝗅𝗌𝖾}parent\leftarrow 0$$$fi$$od