README.md

part-ll-avl-morphing-dict

Python module implementing a partitioned linked-list/AVL-tree morphing dictionary, i.e. a data structure that manages key-value pairs where the keys are integers partitioned into a fixed number of blocks, each one associated with a linked list or an AVL tree depending on its size. First essay paper of Algorithm Engineering course.

Requirements

Given the following parameters in the set of integers $\mathbb{Z}$:

• $min$ and $max$ such that $max > min$
• $r = 6$
• $b > r$ such that $\left( max - min \right)$ is a multiple of $b$

Let's define $d = \left| \frac{max - min}{b} \right|$ and the set $B_i = \left\{ n \in \mathbb{Z} : min + i \cdot b \leq n < min + \left( i + 1 \right) \cdot b \right\}$ with $i \in \left\{0, \ldots, d - 1 \right\}$, $B_d = \left\{ n \in \mathbb{Z} : n < min \right\}$ and $B_{d + 1} = \left\{ n \in \mathbb{Z} : n \geq max \right\}$.

From the previous definitions it can be seen that $\bigcup\nolimits_{i=0}^{d+1} B_i = \mathbb{Z}$ and $B_i \cap B_j = \emptyset \quad \forall i,j \in \left\{1, \ldots, d + 1\right\}$ with $i \ne j$, therefore $\left\{ B_i \right\}_{i = 0}^{d + 1}$ forms a partition of $\mathbb{Z}$.

Implement a data structure that manages pairs of type $\left(key, value\right)$ (for simplicity $key \in \mathbb{Z}$) with the following characteristics:

• an array $v$ of size $d + 2$ where each cell is indexed by $i \in \left\{ 0, \ldots, d + 1\right\}$ and points to a known data structure (linked list or AVL tree) with $n_i$ elements
  • $v\left[ i \right]$ points to a linked list $\Leftrightarrow n_i < r$
  • $v\left[ i \right]$ points to an AVL tree $\Leftrightarrow n_i \ge r$
• implements the insert(key, value), delete(key), search(key) methods of the Dictionary abstract class
  • operations are performed on the data structure pointed to by $v\left[ i \right] \Leftrightarrow key \in B_i$ with $i \in \left\{ 0, \ldots, d + 1 \right\}$

N.B.: the insert and delete operations modify the number $n_i$ of elements present in the data structure pointed by $v\left[ i \right]$. The data structure, where necessary, must be changed accordingly. For example, if in the linked list pointed by $v\left[ 1 \right]$ there are $5$ elements ($n_1 < r$) and one more is added, $n_1$ is then greater than or equal to $r$ and the list is transformed into an AVL tree with the same elements. Now $v\left[ 1 \right]$ will point to the tree and no longer to the linked list. If instead from the tree of $v\left[ 1 \right]$ we remove an element we arrive at $n_1 = 5$ and the tree is transformed into a linked list.

Finally, experimentally compare the average execution time of the methods search and delete of the previously described data structure and of the Python dictionary with respect to the number $n$ of elements.