arbo/tree.go

/*
Package arbo implements a Merkle Tree compatible with the circomlib
implementation of the MerkleTree (when using the Poseidon hash function),
following the specification from
https://docs.iden3.io/publications/pdfs/Merkle-Tree.pdf and
https://eprint.iacr.org/2018/955.

Also allows to define which hash function to use. So for example, when working
with zkSnarks the Poseidon hash function can be used, but when not, it can be
used the Blake3 hash function, which improves the computation time.
*/
package arbo

import (
	"bytes"
	"encoding/binary"
	"encoding/hex"
	"fmt"
	"io"
	"math"
	"sync"
	"sync/atomic"
	"time"

	"github.com/iden3/go-merkletree/db"
)

const (
	// PrefixValueLen defines the bytes-prefix length used for the Value
	// bytes representation stored in the db
	PrefixValueLen = 2

	// PrefixValueEmpty is used for the first byte of a Value to indicate
	// that is an Empty value
	PrefixValueEmpty = 0
	// PrefixValueLeaf is used for the first byte of a Value to indicate
	// that is a Leaf value
	PrefixValueLeaf = 1
	// PrefixValueIntermediate is used for the first byte of a Value to
	// indicate that is a Intermediate value
	PrefixValueIntermediate = 2
)

var (
	dbKeyRoot   = []byte("root")
	dbKeyNLeafs = []byte("nleafs")
	emptyValue  = []byte{0}
)

// Tree defines the struct that implements the MerkleTree functionalities
type Tree struct {
	sync.RWMutex
	tx         db.Tx
	db         db.Storage
	lastAccess int64 // in unix time
	maxLevels  int
	root       []byte

	hashFunction HashFunction
	emptyHash    []byte
}

// NewTree returns a new Tree, if there is a Tree still in the given storage, it
// will load it.
func NewTree(storage db.Storage, maxLevels int, hash HashFunction) (*Tree, error) {
	t := Tree{db: storage, maxLevels: maxLevels, hashFunction: hash}
	t.updateAccessTime()

	t.emptyHash = make([]byte, t.hashFunction.Len()) // empty

	root, err := t.dbGet(dbKeyRoot)
	if err == db.ErrNotFound {
		// store new root 0
		t.tx, err = t.db.NewTx()
		if err != nil {
			return nil, err
		}
		t.root = t.emptyHash
		if err = t.tx.Put(dbKeyRoot, t.root); err != nil {
			return nil, err
		}
		if err = t.setNLeafs(0); err != nil {
			return nil, err
		}
		if err = t.tx.Commit(); err != nil {
			return nil, err
		}
		return &t, err
	} else if err != nil {
		return nil, err
	}
	t.root = root
	return &t, nil
}

func (t *Tree) updateAccessTime() {
	atomic.StoreInt64(&t.lastAccess, time.Now().Unix())
}

// LastAccess returns the last access timestamp in Unixtime
func (t *Tree) LastAccess() int64 {
	return atomic.LoadInt64(&t.lastAccess)
}

// Root returns the root of the Tree
func (t *Tree) Root() []byte {
	return t.root
}

// AddBatch adds a batch of key-values to the Tree. This method will be
// optimized to do some internal parallelization. Returns an array containing
// the indexes of the keys failed to add.
func (t *Tree) AddBatch(keys, values [][]byte) ([]int, error) {
	t.updateAccessTime()
	if len(keys) != len(values) {
		return nil, fmt.Errorf("len(keys)!=len(values) (%d!=%d)",
			len(keys), len(values))
	}

	t.Lock()
	defer t.Unlock()

	var err error
	t.tx, err = t.db.NewTx()
	if err != nil {
		return nil, err
	}

	var indexes []int
	for i := 0; i < len(keys); i++ {
		err = t.add(keys[i], values[i])
		if err != nil {
			indexes = append(indexes, i)
		}
	}
	// store root to db
	if err := t.tx.Put(dbKeyRoot, t.root); err != nil {
		return indexes, err
	}
	// update nLeafs
	if err = t.incNLeafs(uint64(len(keys) - len(indexes))); err != nil {
		return indexes, err
	}

	if err := t.tx.Commit(); err != nil {
		return nil, err
	}
	return indexes, nil
}

// Add inserts the key-value into the Tree.  If the inputs come from a *big.Int,
// is expected that are represented by a Little-Endian byte array (for circom
// compatibility).
func (t *Tree) Add(k, v []byte) error {
	t.updateAccessTime()

	t.Lock()
	defer t.Unlock()

	var err error
	t.tx, err = t.db.NewTx()
	if err != nil {
		return err
	}

	err = t.add(k, v)
	if err != nil {
		return err
	}
	// store root to db
	if err := t.tx.Put(dbKeyRoot, t.root); err != nil {
		return err
	}
	// update nLeafs
	if err = t.incNLeafs(1); err != nil {
		return err
	}
	return t.tx.Commit()
}

func (t *Tree) add(k, v []byte) error {
	// TODO check validity of key & value (for the Tree.HashFunction type)

	keyPath := make([]byte, t.hashFunction.Len())
	copy(keyPath[:], k)

	path := getPath(t.maxLevels, keyPath)
	// go down to the leaf
	var siblings [][]byte
	_, _, siblings, err := t.down(k, t.root, siblings, path, 0, false)
	if err != nil {
		return err
	}

	leafKey, leafValue, err := newLeafValue(t.hashFunction, k, v)
	if err != nil {
		return err
	}

	if err := t.tx.Put(leafKey, leafValue); err != nil {
		return err
	}

	// go up to the root
	if len(siblings) == 0 {
		t.root = leafKey
		return nil
	}
	root, err := t.up(leafKey, siblings, path, len(siblings)-1)
	if err != nil {
		return err
	}

	t.root = root
	return nil
}

// down goes down to the leaf recursively
func (t *Tree) down(newKey, currKey []byte, siblings [][]byte,
	path []bool, l int, getLeaf bool) (
	[]byte, []byte, [][]byte, error) {
	if l > t.maxLevels-1 {
		return nil, nil, nil, fmt.Errorf("max level")
	}
	var err error
	var currValue []byte
	if bytes.Equal(currKey, t.emptyHash) {
		// empty value
		return currKey, emptyValue, siblings, nil
	}
	currValue, err = t.dbGet(currKey)
	if err != nil {
		return nil, nil, nil, err
	}

	switch currValue[0] {
	case PrefixValueEmpty: // empty
		// TODO WIP WARNING should not be reached, as the 'if' above should avoid
		// reaching this point
		// return currKey, empty, siblings, nil
		panic("should not be reached, as the 'if' above should avoid reaching this point") // TMP
	case PrefixValueLeaf: // leaf
		if bytes.Equal(newKey, currKey) {
			return nil, nil, nil, fmt.Errorf("key already exists")
		}

		if !bytes.Equal(currValue, emptyValue) {
			if getLeaf {
				return currKey, currValue, siblings, nil
			}
			oldLeafKey, _ := readLeafValue(currValue)
			oldLeafKeyFull := make([]byte, t.hashFunction.Len())
			copy(oldLeafKeyFull[:], oldLeafKey)

			// if currKey is already used, go down until paths diverge
			oldPath := getPath(t.maxLevels, oldLeafKeyFull)
			siblings, err = t.downVirtually(siblings, currKey, newKey, oldPath, path, l)
			if err != nil {
				return nil, nil, nil, err
			}
		}
		return currKey, currValue, siblings, nil
	case PrefixValueIntermediate: // intermediate
		if len(currValue) != PrefixValueLen+t.hashFunction.Len()*2 {
			return nil, nil, nil,
				fmt.Errorf("intermediate value invalid length (expected: %d, actual: %d)",
					PrefixValueLen+t.hashFunction.Len()*2, len(currValue))
		}
		// collect siblings while going down
		if path[l] {
			// right
			lChild, rChild := readIntermediateChilds(currValue)
			siblings = append(siblings, lChild)
			return t.down(newKey, rChild, siblings, path, l+1, getLeaf)
		}
		// left
		lChild, rChild := readIntermediateChilds(currValue)
		siblings = append(siblings, rChild)
		return t.down(newKey, lChild, siblings, path, l+1, getLeaf)
	default:
		return nil, nil, nil, fmt.Errorf("invalid value")
	}
}

// downVirtually is used when in a leaf already exists, and a new leaf which
// shares the path until the existing leaf is being added
func (t *Tree) downVirtually(siblings [][]byte, oldKey, newKey []byte, oldPath,
	newPath []bool, l int) ([][]byte, error) {
	var err error
	if l > t.maxLevels-1 {
		return nil, fmt.Errorf("max virtual level %d", l)
	}

	if oldPath[l] == newPath[l] {
		siblings = append(siblings, t.emptyHash)

		siblings, err = t.downVirtually(siblings, oldKey, newKey, oldPath, newPath, l+1)
		if err != nil {
			return nil, err
		}
		return siblings, nil
	}
	// reached the divergence
	siblings = append(siblings, oldKey)

	return siblings, nil
}

// up goes up recursively updating the intermediate nodes
func (t *Tree) up(key []byte, siblings [][]byte, path []bool, l int) ([]byte, error) {
	var k, v []byte
	var err error
	if path[l] {
		k, v, err = newIntermediate(t.hashFunction, siblings[l], key)
		if err != nil {
			return nil, err
		}
	} else {
		k, v, err = newIntermediate(t.hashFunction, key, siblings[l])
		if err != nil {
			return nil, err
		}
	}
	// store k-v to db
	if err = t.tx.Put(k, v); err != nil {
		return nil, err
	}

	if l == 0 {
		// reached the root
		return k, nil
	}

	return t.up(k, siblings, path, l-1)
}

func newLeafValue(hashFunc HashFunction, k, v []byte) ([]byte, []byte, error) {
	leafKey, err := hashFunc.Hash(k, v, []byte{1})
	if err != nil {
		return nil, nil, err
	}
	var leafValue []byte
	leafValue = append(leafValue, byte(1))
	leafValue = append(leafValue, byte(len(k)))
	leafValue = append(leafValue, k...)
	leafValue = append(leafValue, v...)
	return leafKey, leafValue, nil
}

func readLeafValue(b []byte) ([]byte, []byte) {
	if len(b) < PrefixValueLen {
		return []byte{}, []byte{}
	}

	kLen := b[1]
	if len(b) < PrefixValueLen+int(kLen) {
		return []byte{}, []byte{}
	}
	k := b[PrefixValueLen : PrefixValueLen+kLen]
	v := b[PrefixValueLen+kLen:]
	return k, v
}

func newIntermediate(hashFunc HashFunction, l, r []byte) ([]byte, []byte, error) {
	b := make([]byte, PrefixValueLen+hashFunc.Len()*2)
	b[0] = 2
	b[1] = byte(len(l))
	copy(b[PrefixValueLen:PrefixValueLen+hashFunc.Len()], l)
	copy(b[PrefixValueLen+hashFunc.Len():], r)

	key, err := hashFunc.Hash(l, r)
	if err != nil {
		return nil, nil, err
	}

	return key, b, nil
}

func readIntermediateChilds(b []byte) ([]byte, []byte) {
	if len(b) < PrefixValueLen {
		return []byte{}, []byte{}
	}

	lLen := b[1]
	if len(b) < PrefixValueLen+int(lLen) {
		return []byte{}, []byte{}
	}
	l := b[PrefixValueLen : PrefixValueLen+lLen]
	r := b[PrefixValueLen+lLen:]
	return l, r
}

func getPath(numLevels int, k []byte) []bool {
	path := make([]bool, numLevels)
	for n := 0; n < numLevels; n++ {
		path[n] = k[n/8]&(1<<(n%8)) != 0
	}
	return path
}

// Update updates the value for a given existing key. If the given key does not
// exist, returns an error.
func (t *Tree) Update(k, v []byte) error {
	t.updateAccessTime()

	t.Lock()
	defer t.Unlock()

	var err error
	t.tx, err = t.db.NewTx()
	if err != nil {
		return err
	}

	keyPath := make([]byte, t.hashFunction.Len())
	copy(keyPath[:], k)
	path := getPath(t.maxLevels, keyPath)

	var siblings [][]byte
	_, valueAtBottom, siblings, err := t.down(k, t.root, siblings, path, 0, true)
	if err != nil {
		return err
	}
	oldKey, _ := readLeafValue(valueAtBottom)
	if !bytes.Equal(oldKey, k) {
		return fmt.Errorf("key %s does not exist", hex.EncodeToString(k))
	}

	leafKey, leafValue, err := newLeafValue(t.hashFunction, k, v)
	if err != nil {
		return err
	}

	if err := t.tx.Put(leafKey, leafValue); err != nil {
		return err
	}

	// go up to the root
	if len(siblings) == 0 {
		t.root = leafKey
		return t.tx.Commit()
	}
	root, err := t.up(leafKey, siblings, path, len(siblings)-1)
	if err != nil {
		return err
	}

	t.root = root
	// store root to db
	if err := t.tx.Put(dbKeyRoot, t.root); err != nil {
		return err
	}
	return t.tx.Commit()
}

// GenProof generates a MerkleTree proof for the given key. If the key exists in
// the Tree, the proof will be of existence, if the key does not exist in the
// tree, the proof will be of non-existence.
func (t *Tree) GenProof(k []byte) ([]byte, error) {
	t.updateAccessTime()
	keyPath := make([]byte, t.hashFunction.Len())
	copy(keyPath[:], k)

	path := getPath(t.maxLevels, keyPath)
	// go down to the leaf
	var siblings [][]byte
	_, value, siblings, err := t.down(k, t.root, siblings, path, 0, true)
	if err != nil {
		return nil, err
	}

	leafK, leafV := readLeafValue(value)
	if !bytes.Equal(k, leafK) {
		fmt.Println("key not in Tree")
		fmt.Println(leafK)
		fmt.Println(leafV)
		// TODO proof of non-existence
		panic(fmt.Errorf("unimplemented"))
	}

	s := PackSiblings(t.hashFunction, siblings)
	return s, nil
}

// PackSiblings packs the siblings into a byte array.
// [      1 byte        | L bytes |       S * N bytes     ]
// [  bitmap length (L) |  bitmap |  N non-zero siblings  ]
// Where the bitmap indicates if the sibling is 0 or a value from the siblings
// array. And S is the size of the output of the hash function used for the
// Tree.
func PackSiblings(hashFunc HashFunction, siblings [][]byte) []byte {
	var b []byte
	var bitmap []bool
	emptySibling := make([]byte, hashFunc.Len())
	for i := 0; i < len(siblings); i++ {
		if bytes.Equal(siblings[i], emptySibling) {
			bitmap = append(bitmap, false)
		} else {
			bitmap = append(bitmap, true)
			b = append(b, siblings[i]...)
		}
	}

	bitmapBytes := bitmapToBytes(bitmap)
	l := len(bitmapBytes)

	res := make([]byte, l+1+len(b))
	res[0] = byte(l) // set the bitmapBytes length
	copy(res[1:1+l], bitmapBytes)
	copy(res[1+l:], b)
	return res
}

// UnpackSiblings unpacks the siblings from a byte array.
func UnpackSiblings(hashFunc HashFunction, b []byte) ([][]byte, error) {
	l := b[0]
	bitmapBytes := b[1 : 1+l]
	bitmap := bytesToBitmap(bitmapBytes)
	siblingsBytes := b[1+l:]
	iSibl := 0
	emptySibl := make([]byte, hashFunc.Len())
	var siblings [][]byte
	for i := 0; i < len(bitmap); i++ {
		if iSibl >= len(siblingsBytes) {
			break
		}
		if bitmap[i] {
			siblings = append(siblings, siblingsBytes[iSibl:iSibl+hashFunc.Len()])
			iSibl += hashFunc.Len()
		} else {
			siblings = append(siblings, emptySibl)
		}
	}
	return siblings, nil
}

func bitmapToBytes(bitmap []bool) []byte {
	bitmapBytesLen := int(math.Ceil(float64(len(bitmap)) / 8)) //nolint:gomnd
	b := make([]byte, bitmapBytesLen)
	for i := 0; i < len(bitmap); i++ {
		if bitmap[i] {
			b[i/8] |= 1 << (i % 8)
		}
	}
	return b
}

func bytesToBitmap(b []byte) []bool {
	var bitmap []bool
	for i := 0; i < len(b); i++ {
		for j := 0; j < 8; j++ {
			bitmap = append(bitmap, b[i]&(1<<j) > 0)
		}
	}
	return bitmap
}

// Get returns the value for a given key
func (t *Tree) Get(k []byte) ([]byte, []byte, error) {
	keyPath := make([]byte, t.hashFunction.Len())
	copy(keyPath[:], k)

	path := getPath(t.maxLevels, keyPath)
	// go down to the leaf
	var siblings [][]byte
	_, value, _, err := t.down(k, t.root, siblings, path, 0, true)
	if err != nil {
		return nil, nil, err
	}
	leafK, leafV := readLeafValue(value)
	if !bytes.Equal(k, leafK) {
		panic(fmt.Errorf("%s != %s", BytesToBigInt(k), BytesToBigInt(leafK)))
	}

	return leafK, leafV, nil
}

// CheckProof verifies the given proof. The proof verification depends on the
// HashFunction passed as parameter.
func CheckProof(hashFunc HashFunction, k, v, root, packedSiblings []byte) (bool, error) {
	siblings, err := UnpackSiblings(hashFunc, packedSiblings)
	if err != nil {
		return false, err
	}

	keyPath := make([]byte, hashFunc.Len())
	copy(keyPath[:], k)

	key, _, err := newLeafValue(hashFunc, k, v)
	if err != nil {
		return false, err
	}

	path := getPath(len(siblings), keyPath)
	for i := len(siblings) - 1; i >= 0; i-- {
		if path[i] {
			key, _, err = newIntermediate(hashFunc, siblings[i], key)
			if err != nil {
				return false, err
			}
		} else {
			key, _, err = newIntermediate(hashFunc, key, siblings[i])
			if err != nil {
				return false, err
			}
		}
	}
	if bytes.Equal(key[:], root) {
		return true, nil
	}
	return false, nil
}

func (t *Tree) dbGet(k []byte) ([]byte, error) {
	// if key is empty, return empty as value
	if bytes.Equal(k, t.emptyHash) {
		return t.emptyHash, nil
	}

	v, err := t.db.Get(k)
	if err == nil {
		return v, nil
	}
	if t.tx != nil {
		return t.tx.Get(k)
	}
	return nil, db.ErrNotFound
}

// Warning: should be called with a Tree.tx created, and with a Tree.tx.Commit
// after the setNLeafs call.
func (t *Tree) incNLeafs(nLeafs uint64) error {
	oldNLeafs, err := t.GetNLeafs()
	if err != nil {
		return err
	}
	newNLeafs := oldNLeafs + nLeafs
	return t.setNLeafs(newNLeafs)
}

// Warning: should be called with a Tree.tx created, and with a Tree.tx.Commit
// after the setNLeafs call.
func (t *Tree) setNLeafs(nLeafs uint64) error {
	b := make([]byte, 8)
	binary.LittleEndian.PutUint64(b, nLeafs)
	if err := t.tx.Put(dbKeyNLeafs, b); err != nil {
		return err
	}
	return nil
}

// GetNLeafs returns the number of Leafs of the Tree.
func (t *Tree) GetNLeafs() (uint64, error) {
	b, err := t.dbGet(dbKeyNLeafs)
	if err != nil {
		return 0, err
	}
	nLeafs := binary.LittleEndian.Uint64(b)
	return nLeafs, nil
}

// Iterate iterates through the full Tree, executing the given function on each
// node of the Tree.
func (t *Tree) Iterate(f func([]byte, []byte)) error {
	t.updateAccessTime()
	return t.iter(t.root, f)
}

func (t *Tree) iter(k []byte, f func([]byte, []byte)) error {
	v, err := t.dbGet(k)
	if err != nil {
		return err
	}

	switch v[0] {
	case PrefixValueEmpty:
		f(k, v)
	case PrefixValueLeaf:
		f(k, v)
	case PrefixValueIntermediate:
		f(k, v)
		l, r := readIntermediateChilds(v)
		if err = t.iter(l, f); err != nil {
			return err
		}
		if err = t.iter(r, f); err != nil {
			return err
		}
	default:
		return fmt.Errorf("invalid value")
	}
	return nil
}

// Dump exports all the Tree leafs in a byte array of length:
// [ N * (2+len(k+v)) ]. Where N is the number of key-values, and for each k+v:
// [ 1 byte | 1 byte | S bytes | len(v) bytes ]
// [ len(k) | len(v) |   key   |     value    ]
// Where S is the size of the output of the hash function used for the Tree.
func (t *Tree) Dump() ([]byte, error) {
	t.updateAccessTime()
	// WARNING current encoding only supports key & values of 255 bytes each
	// (due using only 1 byte for the length headers).
	var b []byte
	err := t.Iterate(func(k, v []byte) {
		if v[0] != PrefixValueLeaf {
			return
		}
		leafK, leafV := readLeafValue(v)
		kv := make([]byte, 2+len(leafK)+len(leafV))
		kv[0] = byte(len(leafK))
		kv[1] = byte(len(leafV))
		copy(kv[2:2+len(leafK)], leafK)
		copy(kv[2+len(leafK):], leafV)
		b = append(b, kv...)
	})
	return b, err
}

// ImportDump imports the leafs (that have been exported with the ExportLeafs
// method) in the Tree.
func (t *Tree) ImportDump(b []byte) error {
	t.updateAccessTime()
	r := bytes.NewReader(b)
	count := 0
	// TODO instead of adding one by one, use AddBatch (once AddBatch is
	// optimized)
	var err error
	for {
		l := make([]byte, 2)
		_, err = io.ReadFull(r, l)
		if err == io.EOF {
			break
		} else if err != nil {
			return err
		}
		k := make([]byte, l[0])
		_, err = io.ReadFull(r, k)
		if err != nil {
			return err
		}
		v := make([]byte, l[1])
		_, err = io.ReadFull(r, v)
		if err != nil {
			return err
		}
		err = t.Add(k, v)
		if err != nil {
			return err
		}
		count++
	}
	// update nLeafs (once ImportDump uses AddBatch method, this will not be
	// needed)
	t.tx, err = t.db.NewTx()
	if err != nil {
		return err
	}
	if err := t.incNLeafs(uint64(count)); err != nil {
		return err
	}
	if err = t.tx.Commit(); err != nil {
		return err
	}
	return nil
}

// Graphviz iterates across the full tree to generate a string Graphviz
// representation of the tree and writes it to w
func (t *Tree) Graphviz(w io.Writer, rootKey []byte) error {
	fmt.Fprintf(w, `digraph hierarchy {
node [fontname=Monospace,fontsize=10,shape=box]
`)
	nChars := 4
	nEmpties := 0
	err := t.Iterate(func(k, v []byte) {
		switch v[0] {
		case PrefixValueEmpty:
		case PrefixValueLeaf:
			fmt.Fprintf(w, "\"%v\" [style=filled];\n", hex.EncodeToString(k[:nChars]))
			// key & value from the leaf
			kB, vB := readLeafValue(v)
			fmt.Fprintf(w, "\"%v\" -> {\"k:%v\\nv:%v\"}\n",
				hex.EncodeToString(k[:nChars]), hex.EncodeToString(kB[:nChars]),
				hex.EncodeToString(vB[:nChars]))
			fmt.Fprintf(w, "\"k:%v\\nv:%v\" [style=dashed]\n",
				hex.EncodeToString(kB[:nChars]), hex.EncodeToString(vB[:nChars]))
		case PrefixValueIntermediate:
			l, r := readIntermediateChilds(v)
			lStr := hex.EncodeToString(l[:nChars])
			rStr := hex.EncodeToString(r[:nChars])
			eStr := ""
			if bytes.Equal(l, t.emptyHash) {
				lStr = fmt.Sprintf("empty%v", nEmpties)
				eStr += fmt.Sprintf("\"%v\" [style=dashed,label=0];\n",
					lStr)
				nEmpties++
			}
			if bytes.Equal(r, t.emptyHash) {
				rStr = fmt.Sprintf("empty%v", nEmpties)
				eStr += fmt.Sprintf("\"%v\" [style=dashed,label=0];\n",
					rStr)
				nEmpties++
			}
			fmt.Fprintf(w, "\"%v\" -> {\"%v\" \"%v\"}\n", hex.EncodeToString(k[:nChars]),
				lStr, rStr)
			fmt.Fprint(w, eStr)
		default:
		}
	})
	fmt.Fprintf(w, "}\n")
	return err
}

// PrintGraphviz prints the output of Tree.Graphviz
func (t *Tree) PrintGraphviz(rootKey []byte) error {
	if rootKey == nil {
		rootKey = t.Root()
	}
	w := bytes.NewBufferString("")
	fmt.Fprintf(w,
		"--------\nGraphviz of the Tree with Root "+hex.EncodeToString(rootKey)+":\n")
	err := t.Graphviz(w, nil)
	if err != nil {
		fmt.Println(w)
		return err
	}
	fmt.Fprintf(w,
		"End of Graphviz of the Tree with Root "+hex.EncodeToString(rootKey)+"\n--------\n")

	fmt.Println(w)
	return nil
}

// Purge WIP: unimplemented
func (t *Tree) Purge(keys [][]byte) error {
	return nil
}