kvs/btree/btree.go

package btree

import (
	"fmt"
)

// Tree is the tree itself
type Tree struct {
	root *Node // Pointer to the Node root
	t    int   // Minimum degree
}

// Node is a Node of a Btree
type Node struct {
	numberOfKeys int // The number of keys really stored
	degree       int // The value of degree dependes upon disk blok size
	isLeaf       bool
	keys         []int
	children     []*Node
}

// Constructors

// NewBtree creates a new btree
func NewBtree(t int) *Tree {

	return &Tree{
		root: nil,
		t:    t,
	}
}

func newNode(degree int, isLeaf bool) *Node {

	return &Node{
		numberOfKeys: 0,
		degree:       degree,
		isLeaf:       isLeaf,
		keys:         make([]int, 2*degree-1),
		children:     make([]*Node, 2*degree),
	}
}

// Tree methods

// Traverse the tree
func (t *Tree) Traverse() {

	if t.root != nil {
		t.root.traverse()
	}
}

// Search k in the tree
func (t *Tree) Search(k int) *Node {

	if t.root == nil {
		return nil
	}

	return t.root.search(k)
}

// Remove k in the tree
func (t *Tree) Remove(k int) error {
	if t.root == nil {
		return fmt.Errorf("The tree is empty")
	}

	err := t.root.remove(k)

	// If the root node has 0 keys, makes its first child as the new root
	// If it has no child, set root as nil
	if t.root.numberOfKeys == 0 {
		if t.root.isLeaf {
			t.root = nil
			return err
		}
		t.root = t.root.children[0]
	}

	return err
}

// Insert k in the tree
func (t *Tree) Insert(k int) {
	// If the tree is empty
	if t.root == nil {
		t.root = newNode(t.t, true)
		t.root.keys[0] = k
		t.root.numberOfKeys = 1
		return
	}

	// If the tree is not empty
	if !t.root.isFull() {
		// If root is not full, insert in non full root
		t.root.insertNonFull(k)
		return
	}

	// If the root is full, then the tree grows in height
	s := newNode(t.t, false)

	// Make the old root as a child of the new root
	s.children[0] = t.root

	// Split the old root and move 1 key to the new root
	s.splitChild(0, t.root)

	// The new root has two children now.
	// We decide which of the two children is going to have the new key
	i := 0
	if s.keys[0] < k {
		i++
	}
	s.children[i].insertNonFull(k)

	// Change root
	t.root = s
}

// Node methods

// traverse all nodes in a subtree rooted with this node
func (n *Node) traverse() {

	// There are n entries and n+1 children, treverse trough n keys and n first children
	for i := 0; i < n.numberOfKeys; i++ {
		// If this is not a leaf, then traverse the subtree before printing the key
		if !n.isLeaf {
			n.children[i].traverse()
		}
		fmt.Printf(" %d", n.keys[i])
	}

	// Print the subtree rooted with the last child
	if !n.isLeaf {
		n.children[n.numberOfKeys].traverse()
	}
}

// search k in the subtree rooted with this node
func (n *Node) search(k int) *Node {

	// Find the first entry greater than or equal to k
	i := 0
	for i < n.numberOfKeys && k > n.keys[i] {
		i++
	}

	// If theh found key is equal to  k, return this node
	if n.keys[i] == k {
		return n
	}

	// If the key is not found here and this is a leaf node
	if n.isLeaf {
		return nil
	}

	// Go to the approipriate child
	return n.children[i].search(k)
}

func (n *Node) isFull() bool {
	return n.numberOfKeys == 2*n.degree-1
}

func (n *Node) insertNonFull(k int) {

	// Initialize the index as the index of the rightmost element
	i := n.numberOfKeys - 1

	// If this is a leaf node
	if n.isLeaf {
		// Finds the location of the new key to be inserted
		// Moves all greater keys to one place ahead
		for i >= 0 && n.keys[i] > k {
			n.keys[i+1] = n.keys[i]
			i--
		}

		// Insert the new key at the found location
		n.keys[i+1] = k
		n.numberOfKeys++
		return
	}

	// If this is not a leaf
	// Finds the child wich is going to have the new key
	for i >= 0 && n.keys[i] > k {
		i--
	}

	// Check if the found chird is full
	if n.children[i+1].isFull() {
		// If the child is full, then split it
		n.splitChild(i+1, n.children[i+1])

		// After the split, the middle key of children[i] goes up and
		// children[i] is splitted into two
		// See which of those two is going to have the new key
		if n.keys[i+1] < k {
			i++
		}
	}

	n.children[i+1].insertNonFull(k)
}

func (n *Node) splitChild(i int, y *Node) {

	// Create a new node that will store (t-1) keys of y
	z := newNode(y.degree, y.isLeaf)
	z.numberOfKeys = n.degree - 1

	// Copy the last (t-1) keys of y to z
	for j := 0; j < n.degree-1; j++ {
		z.keys[j] = y.keys[j+n.degree]
		if !y.isLeaf {
			z.children[j] = y.children[j+n.degree]
		}
	}

	// Copy the last t children of y to z
	if !y.isLeaf {
		z.children[n.degree-1] = y.children[2*n.degree-1]
	}

	// Reduce the number of keys in y
	y.numberOfKeys = n.degree - 1

	// Since this node is going to have a new child, create space for it
	for j := n.numberOfKeys; j >= i+1; j-- {
		n.children[j+1] = n.children[j]
	}

	// Link the new child to this node
	n.children[i+1] = z

	// A key of y will move to this node
	// Find the location of the new key and move all greater keys ahead
	for j := n.numberOfKeys - 1; j >= i; j-- {
		n.keys[j+1] = n.keys[j]
	}

	// Copy the middle key of y to this node
	n.keys[i] = y.keys[n.degree-1]

	// Increment the count of keys in this node
	n.numberOfKeys++
}

// findKey returns the index of the first key that is greater than or equal to k
func (n *Node) findKey(k int) int {

	index := 0
	for index < n.numberOfKeys && n.keys[index] < k {
		index++
	}
	return index
}

// remove the key k from the sub-tree rooted with this node
func (n *Node) remove(k int) error {

	index := n.findKey(k)

	// The key to be removed is in this node
	if index < n.numberOfKeys && n.keys[index] == k {
		if n.isLeaf {
			return n.removeFromLeaf(index)
		}
		return n.removeFromNonLeaf(index)
	}

	// If this is a leaf, the key is not in the tree
	if n.isLeaf {
		return fmt.Errorf("The key %d does not exist in the tree", k)
	}

	isInLastChild := false
	if index == n.numberOfKeys {
		isInLastChild = true
	}

	// If the child where is the key has less than t keys, wi fill it
	if n.children[index].numberOfKeys < n.degree {
		n.fill(index)
	}

	// If the last child has been merged, it must be merged with the previous
	// child and so we recurse on the (index-1)th child.
	if isInLastChild && index > n.numberOfKeys {
		return n.children[index-1].remove(k)
	}

	// We recurse on the (index)th child which now has at least t keys
	return n.children[index].remove(k)
}

// removeFromLeaf the index-th key from this node which is a leaf node
func (n *Node) removeFromLeaf(index int) error {

	// Move all the keys after the index-th position one place backward
	for i := index + 1; i < n.numberOfKeys; i++ {
		n.keys[i-1] = n.keys[i]
	}

	n.numberOfKeys--
	return nil
}

// removeFromNonLeaf the index-th key from this node which is not a leaf node
func (n *Node) removeFromNonLeaf(index int) error {

	k := n.keys[index]

	// If the child that precedes k has at least t keys,
	// find the predecessor of k in the subtree and replace k with it
	// Recursively delete the predecessor in the child
	if n.children[index].numberOfKeys >= n.degree {
		pred := n.getPred(index)
		n.keys[index] = pred
		return n.children[index].remove(pred)
	}

	// If the child has less than t keys, examine children[index+1]
	// If it has at least t keys, find the successor of k in this subtree
	// Replace k by its successor and recursively delete the successor in the subtree
	if n.children[index+1].numberOfKeys >= n.degree {
		succ := n.getSucc(index)
		n.keys[index] = succ
		return n.children[index+1].remove(succ)
	}

	// Merge k and all of children[index+1] int children[index]
	// Free children[index+1] and recursively delete k from children[index]
	n.merge(index)
	return n.children[index].remove(k)
}

// getPred returns the predecessor of keys[index]
func (n *Node) getPred(index int) int {

	// Keep moving to the rightmost node until we reach a leaf
	current := n.children[index]
	for !current.isLeaf {
		current = current.children[current.numberOfKeys]
	}

	// Return the last key of the leaf
	return current.keys[current.numberOfKeys-1]
}

// getSucc returns the successor of keys[index]
func (n *Node) getSucc(index int) int {

	// Keep moving to the leftmost node starting from children[index+1] until we reach a leaf
	current := n.children[index+1]
	for !current.isLeaf {
		current = current.children[0]
	}

	// Return the first key of the leaf
	return current.keys[0]
}

// fill child children[index] which has less than t-1 keys
func (n *Node) fill(index int) {

	// If the previous child has more than t-1 keys, borrow a key from that child
	if index != 0 && n.children[index-1].numberOfKeys >= n.degree {
		n.borrowFromPrev(index)
		return
	}

	// If the next child has more than t-1 keys, borrow a key from that child
	if index != n.numberOfKeys && n.children[index+1].numberOfKeys >= n.degree {
		n.borrowFromNext(index)
		return
	}

	// Merge children[index] with its sibling
	if index != n.numberOfKeys {
		n.merge(index)
		return
	}

	// If this is the last child, merge with the previous sibling
	n.merge(index - 1)
}

// borrowFromPrev takes a key from children[index+1] and insert it in children[index]
func (n *Node) borrowFromPrev(index int) {

	child := n.children[index]
	sibling := n.children[index-1]

	// Moves all keys in children[index] one step ahead
	for i := child.numberOfKeys - 1; i >= 0; i-- {
		child.keys[i+1] = child.keys[i]
	}

	// If the child is not a leaf, move all its child pointers one step ahead
	if !child.isLeaf {
		for i := child.numberOfKeys; i >= 0; i-- {
			child.children[i+1] = child.children[i]
		}
	}

	// Sets child's first key equal to keys[index-1] from the current node
	child.keys[0] = n.keys[index-1]

	// Moves sibling's last child as children[index]'s first child
	if !child.isLeaf {
		child.children[0] = sibling.children[sibling.numberOfKeys]
	}

	// Moves the key from the sibling to the parent
	n.keys[index-1] = sibling.keys[sibling.numberOfKeys-1]

	child.numberOfKeys++
	sibling.numberOfKeys--
}

// borrowFromNext takes a key from children[index+1] and insert it in children[index]
func (n *Node) borrowFromNext(index int) {

	child := n.children[index]
	sibling := n.children[index+1]

	// keys[index] is inserted as the last key in children[index]
	child.keys[child.numberOfKeys] = n.keys[index]

	// Sibling's first child is inserted as the last child into children[index]
	if !child.isLeaf {
		child.children[child.numberOfKeys+1] = sibling.children[0]
	}

	// The first key from sibling is inserted into keys[index]
	n.keys[index] = sibling.keys[0]

	// Moving all keys in sibling one step behind
	for i := 1; i < sibling.numberOfKeys; i++ {
		sibling.keys[i-1] = sibling.keys[i]
		if !sibling.isLeaf {
			sibling.children[i-1] = sibling.children[i]
		}
	}

	// Moving the child pointers one step behind
	if !sibling.isLeaf {
		sibling.children[n.numberOfKeys-1] = sibling.children[n.numberOfKeys]
	}

	child.numberOfKeys++
	sibling.numberOfKeys--
}

// merge children[index] with children[index+1]
func (n *Node) merge(index int) {

	child := n.children[index]
	sibling := n.children[index+1]

	// Pulls a key from the current node ande insert it into the (t-1)th position
	child.keys[n.degree-1] = n.keys[index]

	// Copies the keys from children[index+1] to children[index] at the end
	for i := 0; i < sibling.numberOfKeys; i++ {
		child.keys[i+n.degree] = sibling.keys[i]
		if !child.isLeaf {
			child.children[i+n.degree] = sibling.children[i]
		}
	}

	// Copies the child pointers from C[index+1] to children[index]
	if !child.isLeaf {
		child.children[sibling.numberOfKeys+n.degree] = sibling.children[sibling.numberOfKeys]
	}

	// Moves all keys after index in the current node one step before
	// to fill the gap created by moving keys[index] to children[index]
	// Moves the child pointer after (index+1) in the current node one step before
	// This action marks sibling for deletion by the GC
	for i := index + 1; i < n.numberOfKeys; i++ {
		n.keys[i-1] = n.keys[i]
		n.children[i] = n.children[i+1]
	}

	// Updates the key count of child and the current node
	child.numberOfKeys += sibling.numberOfKeys + 1
	n.numberOfKeys--
}