B-tree Implementation in Bash

This document explains how to implement a simplified B-tree using Bash. A B-tree is a self-balancing tree data structure that maintains sorted data and allows for efficient insertion, deletion, and search operations. It is widely used in databases and file systems for indexing purposes.

Program Structure

The program is structured into several key functions:

• initialize_tree: Initializes the B-tree.
• insert: Inserts a key into the B-tree.
• search: Searches for a key in the B-tree.
• split_child: Splits a full node during insertion.

Implementation

#!/bin/bash

# Initialize the B-tree with an order (minimum degree)
initialize_tree() {
    t=$1
    root_node=()
    root_keys=0
}

# Function to split a full child during insertion
split_child() {
    local parent=$1
    local full_child=$2
    local t=$3

    mid_key_index=$((t - 1))
    mid_key=${nodes[$full_child,$mid_key_index]}
    
    # Create new node
    new_node=()
    new_node_keys=0

    # Move half the keys and children to the new node
    for (( i=$mid_key_index+1; i<${#nodes[$full_child,@]}; i++ )); do
        new_node+=("${nodes[$full_child,$i]}")
        nodes[$full_child,$i]=""
        new_node_keys=$((new_node_keys + 1))
    done

    # Update the full child node
    nodes[$full_child,$mid_key_index]=""
    root_keys=$((root_keys + 1))

    # Insert the mid key into the parent
    parent+=($mid_key)
}

# Function to insert a key into the B-tree
insert() {
    local key=$1
    local t=$2

    if [ ${#root_node[@]} -eq $((2*t - 1)) ]; then
        new_root=()
        new_root_keys=0
        split_child new_root root_node $t
        root_node=$new_root
    fi

    insert_non_full root_node $key $t
}

# Function to insert a key into a non-full node
insert_non_full() {
    local node=$1
    local key=$2
    local t=$3
    local i=${#node[@]}

    if [ -z "${nodes[$node,0]}" ]; then
        while [ $i -ge 1 ] && [ "$key" -lt "${node[$((i - 1))]}" ]; do
            node[$i]=${node[$((i - 1))]}
            i=$((i - 1))
        done
        node[$i]=$key
    else
        while [ $i -gt 0 ] && [ "$key" -lt "${node[$((i - 1))]}" ]; do
            i=$((i - 1))
        done
        if [ ${#nodes[$node,$i]} -eq $((2*t - 1)) ]; then
            split_child $node $i $t
            if [ "$key" -gt "${node[$i]}" ]; then
                i=$((i + 1))
            fi
        fi
        insert_non_full ${nodes[$node,$i]} $key $t
    fi
}

# Function to search for a key in the B-tree
search() {
    local node=$1
    local key=$2
    local i=0

    while [ $i -lt ${#node[@]} ] && [ "$key" -gt "${node[$i]}" ]; do
        i=$((i + 1))
    done

    if [ $i -lt ${#node[@]} ] && [ "$key" -eq "${node[$i]}" ]; then
        echo "Key $key found in the B-tree."
        return
    fi

    if [ -z "${nodes[$node,$i]}" ]; then
        echo "Key $key not found in the B-tree."
        return
    fi

    search ${nodes[$node,$i]} $key
}

# Example usage
t=2  # Minimum degree of the B-tree
initialize_tree $t

insert 10 $t
insert 20 $t
insert 5 $t
insert 6 $t
insert 12 $t
insert 30 $t
insert 7 $t
insert 17 $t

# Searching for keys
search $root_node 6
search $root_node 15
    

Explanation

The program implements a basic B-tree with the following components:

Initialization

The initialize_tree function sets up an empty B-tree with a specified minimum degree t. The minimum degree dictates the minimum number of keys a node can have, except for the root node.

Inserting Keys

The insert function inserts a key into the B-tree. If the root node is full, it is split, and the tree’s height increases. The insert_non_full helper function handles the insertion of a key into a non-full node. If the node is full, the split_child function splits the node to maintain the B-tree properties.

Splitting Nodes

The split_child function is responsible for splitting a full node during insertion. It moves half of the keys and children from the full node to a new node and inserts the middle key into the parent node.

Searching for Keys

The search function looks for a key in the B-tree by comparing it with the keys in the current node and recursively searching the appropriate child node. If the key is found, the function returns the key; otherwise, it reports that the key is not present.

Example Usage

The example usage section demonstrates initializing the B-tree with a minimum degree of 2, inserting several keys, and searching for specific keys. The program outputs whether each key is found or not.

Conclusion

This Bash implementation of a B-tree demonstrates the basic operations of insertion, splitting, and searching within the data structure. B-trees are particularly useful in databases and file systems for indexing large amounts of data efficiently.