In computer science, a 2–3 heap is a data structure, a variation on the heap, designed by Tadao Takaoka in 1999. The structure is similar to the Fibonacci heap, and borrows from the 2–3 tree.

Time costs for some common heap operations are:

• Delete-min takes ${\displaystyle O(\log(n))}$ amortized time.
• Decrease-key takes constant amortized time.
• Insertion takes constant amortized time.

Polynomial of trees

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A linear tree of size ${\displaystyle r}$ is a sequential path of ${\displaystyle r}$ nodes with the first node as a root of the tree and it is represented by a bold ${\displaystyle \mathbf {r} }$ (e.g. ${\displaystyle \mathbf {1} }$ is a linear tree of a single node). Product ${\displaystyle P=ST}$ of two trees ${\displaystyle S}$ and ${\displaystyle T}$, is a new tree with every node of ${\displaystyle S}$ is replaced by a copy of ${\displaystyle T}$ and for each edge of ${\displaystyle S}$ we connect the roots of the trees corresponding to the endpoints of the edge. Note that this definition of product is associative but not commutative. Sum ${\displaystyle S+T}$ of two trees ${\displaystyle S}$ and ${\displaystyle T}$ is the collection of two trees ${\displaystyle S}$ and ${\displaystyle T}$.

An r-ary polynomial of trees is defined as ${\displaystyle P=\mathbf {a} _{k-1}\mathbf {r} ^{k-1}+\dots +\mathbf {a} _{1}\mathbf {r} +\mathbf {a} _{0}}$ where ${\displaystyle 0\leq a_{i}\leq r-1}$. This polynomial notation for trees of ${\displaystyle n}$ nodes is unique. The tree ${\displaystyle \mathbf {a} _{i}\mathbf {r} ^{i}}$ is actually ${\displaystyle a_{i}}$ copy of ${\displaystyle \mathbf {r} ^{i}}$ that their roots are connected with ${\displaystyle a_{i}-1}$ edges sequentially and the path of these ${\displaystyle a_{i}-1}$ edge is called the main trunk of the tree ${\displaystyle \mathbf {a} _{i}\mathbf {r} ^{i}}$. Furthermore, an r-ary polynomial of trees is called an r-nomial queue if nodes of the polynomial of trees are associated with keys in heap property.

(2,3)-heap

Source:^[1]

An ${\displaystyle (l,r)-}$ tree ${\displaystyle T(i)}$ is defined recursively by ${\displaystyle T(i)=T_{1}(i-1)\triangleleft \dots \triangleleft T_{s}(i-1)}$ for ${\displaystyle i\geq 1}$ (${\displaystyle s}$ is between ${\displaystyle l}$ and ${\displaystyle r}$ and ${\displaystyle (s-1)}$ ${\displaystyle \triangleleft }$ operations are evaluated from right to left) where for two trees, ${\displaystyle A}$ and ${\displaystyle B}$, the result of the operation${\displaystyle A\triangleleft B}$ is connecting the root of ${\displaystyle B}$ as a rightmost child to the root of ${\displaystyle A}$ and ${\displaystyle T(0)}$ is a single node tree. Note that the root of the tree ${\displaystyle T(i)}$ has degree ${\displaystyle i}$.

An extended polynomial of trees, ${\displaystyle P}$, is defined by ${\displaystyle P=a_{k-1}T(k-1)+\dots +a_{1}T(1)+a_{0}}$. If we assign keys into the nodes of an extended polynomial of trees in heap order it is called ${\displaystyle (l,r)-heap}$, and the special case of ${\displaystyle l=2}$ and ${\displaystyle r=3}$ is called ${\displaystyle (2,3)-heap}$.

References