btree: add basic in memory b-tree with integers

btree/btree.go

@@ -0,0 +1,498 @@
+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
+	t            int // The value of t 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(t int, isLeaf bool) *Node {
+
+	return &Node{
+		numberOfKeys: 0,
+		t:            t,
+		isLeaf:       isLeaf,
+		keys:         make([]int, 2*t-1),
+		children:     make([]*Node, 2*t),
+	}
+}
+
+// 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.t-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.t, y.isLeaf)
+	z.numberOfKeys = n.t - 1
+
+	// TODO: make optimisations
+	// Copy the last (t-1) keys of y to z
+	for j := 0; j < n.t-1; j++ {
+		z.keys[j] = y.keys[j+n.t]
+	}
+
+	// Copy the last t children of y to z
+	if !y.isLeaf {
+		for j := 0; j < n.t; j++ {
+			z.children[j] = y.children[j+n.t]
+		}
+	}
+
+	// Reduce the number of keys in y
+	y.numberOfKeys = n.t - 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.t-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.t {
+		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.t {
+		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.t {
+		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.t {
+		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.t {
+		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]
+
+	// TODO: optimize loops here
+	// Moving all keys in sibling one step behind
+	for i := 1; i < sibling.numberOfKeys; i++ {
+		sibling.keys[i-1] = sibling.keys[i]
+	}
+
+	// Moving the child pointers one step behind
+	if !sibling.isLeaf {
+		for i := 1; i <= sibling.numberOfKeys; i++ {
+			sibling.children[i-1] = sibling.children[i]
+		}
+	}
+
+	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.t-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.t] = sibling.keys[i]
+	}
+
+	// Copies the child pointers from C[index+1] to children[index]
+	if !child.isLeaf {
+		for i := 0; i <= sibling.numberOfKeys; i++ {
+			child.children[i+n.t] = sibling.children[i]
+		}
+	}
+
+	// Moves all keys after index in the current node one step before
+	// to fill the gap created by moving keys[index] to children[index]
+	for i := index + 1; i < n.numberOfKeys; i++ {
+		n.keys[i-1] = n.keys[i]
+	}
+
+	// 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 + 2; i <= n.numberOfKeys; i++ {
+		n.children[i-1] = n.children[i]
+	}
+
+	// Updates the key count of child and the current node
+	child.numberOfKeys += sibling.numberOfKeys + 1
+	n.numberOfKeys--
+}

btree/btree_test.go

@@ -0,0 +1,123 @@
+package btree
+
+import (
+	"fmt"
+	"testing"
+)
+
+func TestTree_Search(t *testing.T) {
+	tree := NewBtree(3)
+	tree.Insert(6)
+
+	k := 6
+	if tree.Search(k) == nil {
+		t.Errorf("Not present %d", k)
+	}
+
+	k = 15
+	if tree.Search(k) != nil {
+		t.Errorf("Present %d", k)
+	}
+}
+
+func TestTree_Remove(t *testing.T) {
+	tree := NewBtree(3)
+	toInsert := []int{10, 20, 5, 6, 12, 30, 7, 17}
+
+	for _, k := range toInsert {
+		tree.Insert(k)
+	}
+
+	tree.Remove(toInsert[0])
+	for _, k := range toInsert[1:] {
+		if tree.Search(k) == nil {
+			t.Errorf("Not present %d", k)
+		}
+	}
+
+	if tree.Search(toInsert[0]) != nil {
+		t.Errorf("Present %d", toInsert[0])
+	}
+
+}
+
+func TestTree_Insert(t *testing.T) {
+	tree := NewBtree(3)
+	toInsert := []int{10, 20, 5, 6, 12, 30, 7, 17}
+
+	// Test before insertion
+	for _, k := range toInsert {
+		if tree.Search(k) != nil {
+			t.Errorf("Present %d", k)
+		}
+	}
+
+	for _, k := range toInsert {
+		tree.Insert(k)
+	}
+
+	// Test after insertion
+	for _, k := range toInsert {
+		if tree.Search(k) == nil {
+			t.Errorf("Not present %d", k)
+		}
+	}
+}
+
+/*
+func TestTree_Traverse(t *testing.T) {
+	tree := NewBtree(3)
+
+	toInsert := []int{1, 3, 7, 10, 11, 13, 14, 15, 18, 16, 19, 24, 25, 26, 21, 4, 5, 20, 22, 2, 17, 12, 6}
+	for _, k := range toInsert {
+		tree.Insert(k)
+	}
+
+	tree.Traverse()
+	fmt.Println("\n 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26")
+}
+*/
+
+func ExampleTree_Traverse() {
+	tree := NewBtree(3)
+
+	toInsert := []int{1, 3, 7, 10, 11, 13, 14, 15, 18, 16, 19, 24, 25, 26, 21, 4, 5, 20, 22, 2, 17, 12, 6}
+	for _, k := range toInsert {
+		tree.Insert(k)
+	}
+
+	tree.Traverse()
+	fmt.Println("")
+
+	tree.Remove(6)
+	tree.Traverse()
+	fmt.Println("")
+
+	tree.Remove(13)
+	tree.Traverse()
+	fmt.Println("")
+
+	tree.Remove(7)
+	tree.Traverse()
+	fmt.Println("")
+
+	tree.Remove(4)
+	tree.Traverse()
+	fmt.Println("")
+
+	tree.Remove(2)
+	tree.Traverse()
+	fmt.Println("")
+
+	tree.Remove(16)
+	tree.Traverse()
+
+	// Output:
+	// 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26
+	//  1 2 3 4 5 7 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26
+	//  1 2 3 4 5 7 10 11 12 14 15 16 17 18 19 20 21 22 24 25 26
+	//  1 2 3 4 5 10 11 12 14 15 16 17 18 19 20 21 22 24 25 26
+	//  1 2 3 5 10 11 12 14 15 16 17 18 19 20 21 22 24 25 26
+	//  1 3 5 10 11 12 14 15 16 17 18 19 20 21 22 24 25 26
+	//  1 3 5 10 11 12 14 15 17 18 19 20 21 22 24 25 26
+}