/BokkyPooBahsRedBlackTreeLibrary

BokkyPooBah's Red-Black Binary Search Tree Library

Primary LanguageSolidityMIT LicenseMIT

BokkyPooBahs Red-Black Binary Search Tree Library

Status: Work in progress. Documenting before getting further testing and review - don't use yet

A gas-efficient Solidity library using the iterative (rather than recursive) Red-Black binary search tree algorithm to help you maintain a sorted uint key index for your data. Insertions, deletions and searches are in O(log n) time (and ~gas). Note that the key of 0 is prohibited. Use the sorted keys as indices to your mapping tables of data to access your data in sorted order.

Inserting a key into an empty tree costs 68,459 gas. Inserting a key into a tree with 9,999 keys costs 127,210 gas on average. Removing an element from a tree with a single key costs 44,835 gas. Removing a key from a tree with 10,000 keys cost 81,486 gas on average.

An important use-case for this library is to maintain a sorted on-chain order book in decentralised exchange smart contracts, providing a provably fair order matching algorithm.


Table Of Contents


Overview

Binary Search Tree

The Red-Black Tree binary search tree is a self-rebalancing binary search tree. Following is a diagram of a fairly well-balanced binary search tree.

The following unbalanced binary search tree was generated by inserting the numbers 1 to 32 in sequential order [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]:

Inserting the keys into the binary search tree in sequential order will result in this unbalanced tree resembling a linked-list.


Red-Black Binary Search Tree

The red-black algorithm maintains a red or black colouring for each node in the tree, and uses this information to maintain a reasonably balanced tree. From Wikipedia:

In addition to the requirements imposed on a binary search tree the following must be satisfied by a red–black tree:

  • Each node is either red or black.
  • The root is black. This rule is sometimes omitted. Since the root can always be changed from red to black, but not necessarily vice versa, this rule has little effect on analysis.
  • All leaves (NIL) are black.
  • If a node is red, then both its children are black.
  • Every path from a given node to any of its descendant NIL nodes contains the same number of black nodes.

When an element is inserted into or removed from a red-black tree, the binary search tree is rebalanced to satisfy the red-black rules.

From Wikipedia's Red-Black Tree page, the following Red-Black tree was created by inserting the items [13,8,17,11,15,22,25,27,1,6]:


Red-Black Tree With Random Insertion

The following fairly well-balanced red-black tree was generated by inserting the numbers 1 to 32 in random order [15,14,20,3,7,10,11,16,18,2,4,5,8,19,1,9,12,6,17,13]:

The root node of the tree is 18, k represents the key numbers, p the parent, l the left node, r the right node. Nodes with l0 r0 are the leaves of the tree.


Red-Black Tree With Sequential Insertion

The following red-black tree was generated by inserting the numbers 1 to 32 in sequential order [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]:

A property of the red-black tree is that the path from the root to the farthest leaf is no more than twice as long as the path from the root to the nearest leaf. The shortest path has all black nodes and the longest path alternate between red and black nodes.

The shortest path is 4 levels deep: (8 black), (4 black), (2 black), (1 black).

The longest path is 8 levels deep: (8 black), (16 red), (20 black), (24 red), (28 black), (30 red), (31 black), (32 red). This is no more than twice as long as the shortest path.


Gas Cost

Average Case - Random Insert

The following table shows the minimum, average and maximum gas cost for the insertion of items in a random order and removal of items from a red-black tree:

Items Ins Min Ins Avg Ins Max Rem Min Rem Avg Rem Max
1 68,459 68,459 68,459 44,835 44,835 44,835
5 68,459 99,166 140,002 30,521 48,024 74,259
10 68,459 118,401 167,610 30,671 71,973 119,593
50 68,459 124,688 182,637 30,067 82,894 213,514
100 68,459 123,377 190,137 30,521 79,761 191,060
500 68,459 124,567 191,240 30,521 80,108 261,153
1,000 68,459 125,335 188,278 29,950 81,032 219,352
5,000 68,523 127,270 196,563 30,521 81,972 283,499
10,000 68,523 127,210 208,293 30,099 81,486 307,296

Worst Case - Sequential Insert

The following table shows the minimum, average and maximum gas cost for the insertion of items in a sequential order and removal of items from a red-black tree:

Items Ins Min Ins Avg Ins Max Rem Min Rem Avg Rem Max
1 68,459 68,459 68,459 44,835 44,835 44,835
5 68,459 107,349 140,753 29,918 56,072 86,075
10 68,459 116,513 149,588 30,067 75,757 119,064
50 68,459 137,949 158,598 30,067 86,002 213,537
100 68,459 142,907 163,103 30,521 87,266 218,371
500 68,459 149,290 191,089 30,521 86,200 266,556
1,000 68,459 150,802 208,341 29,950 87,266 287,889
5,000 68,459 153,263 242,846 30,521 88,336 310,295
10,000 68,459 154,113 260,099 29,950 88,003 333,443

If you find this library useful for your project, especially commercial projects, please donate to 0xb6dAC2C5A0222f6794265249ACE15568B750c2d1. I hope to cover my cost of getting this library independently audited.


History


Deployment

This library has been designed to be automatically compiled into your Ethereum Solidity contract or library, instead of having to deploy this library and then linking your contract or library to this library.


Questions And Answers


Functions

See contracts/TestBokkyPooBahsRedBlackTree.sol (or the flattened version) for an example contract that uses this library.

Notes:

  • The function parameter Tree storage self has been omitted in the documentation below, as Solidity automatically injects the library data structure in place of this first parameter
  • There is a constant SENTINEL that is set to 0 in the library source code by default

init

function init() internal

Call this function from your contract constructor to initialises some data. Note that this is really only necessary when SENTINEL is non-0.


root

function root() internal view returns (uint _key);

Returns the root of the tree, or SENTINEL is the tree is empty.


inserted

function inserted() internal view returns (uint _inserted);

Number of keys inserted into the tree.


removed

function removed() internal view returns (uint _removed);

Number of keys removed from the tree.


count

function count() internal view returns (uint _count);

Number of items in the tree.


first

function first() internal view returns (uint _key);

Returns the smallest key in the tree.

Return Value Condition
{first key} Tree has at least one key
SENTINEL Tree empty

last

function last() internal view returns (uint _key);

Returns the largest key in the tree.

Return Value Condition
{last key} Tree has at least one key
SENTINEL Tree empty

next

function next(uint x) internal view returns (uint y);

Returns the next key in the tree with a value larger than x.

Return Value Condition
{next key} There exists a key with a value larger than the x key
SENTINEL Tree empty
SENTINEL x is not an existing key in the tree
SENTINEL x is the only key in the tree
SENTINEL x is the last key in the tree

prev

function prev(uint x) internal view returns (uint y);

Returns the previous key in the tree with a value smaller than x.

Return Value Condition
{prev key} There exists a key with a value smaller than the x key
SENTINEL Tree empty
SENTINEL x is not an existing key in the tree
SENTINEL x is the only element in the tree
SENTINEL x is the last element in the tree

exists

function exists(uint key) internal view returns (bool _exists);

Returns true if the key exists in the tree, false otherwise.

Return Value Condition
true key is an existing key in the tree
false Tree empty
false key is not an existing key in the tree

getNode

function getNode(uint key) internal view returns (uint _returnKey, uint _parent, uint _left, uint _right, bool _red);

Returns the node information if key exists in the tree. All uint values will be set to SENTINEL if key does not exist in the tree.


parent

function parent(uint key) internal view returns (uint _parent);

Returns the parent of key.

Return Value Condition
{parent key} key has a parent, when there are more than one item in the tree
SENTINEL Tree is empty
SENTINEL key is not an existing key in the tree
SENTINEL key is the only element in the tree

grandparent

function grandparent(uint key) internal view returns (uint _grandparent);

Returns the grandparent of key.

Return Value Condition
{grandparent key} key has a parent that has a parent
SENTINEL Tree is empty
SENTINEL key is not an existing key in the tree
SENTINEL key does not have a parent, or the parent does not have a parent

sibling

function sibling(uint key) internal view returns (uint _sibling);

Returns the sibling of key.

Return Value Condition
{sibling key} key has a sibling
SENTINEL Tree is empty
SENTINEL key is not an existing key in the tree
SENTINEL key does not have a sibling

uncle

function uncle(uint key) internal view returns (uint _uncle);

Returns the uncle of key.

Return Value Condition
{uncle key} key has an uncle
SENTINEL Tree is empty
SENTINEL key is not an existing key in the tree
SENTINEL key does not have an uncle

insert

function insert(uint z) internal;

Insert the key z into the tree.

Transaction Condition
success The key z has been successfully inserted into the tree
failure The key z already exists in the tree

remove

function remove(uint z) internal;

Remove the key z from the tree.

Transaction Condition
success The key z has been successfully removed from the tree
failure The key z does exist in the tree

Algorithm

The main Red-Black binary search tree algorithm is listed in Algorithms for Red Black Tree Operations (from CLRS text).

Note that this algorithm is designed to work with memory pointers to the node data. The rebalancing process after the removal of an item from the tree may result in a swapping of data values between nodes.

As the nodes are stored as elements in a Solidity mapping data structure, Iterative Algorithm for Red-Black Tree provides an alternative algorithm to perform this swapping. In particular, the function RB-Delete in the main Red-Black algorithm will need the line then key[z] := key[y] replaced with the alternative swapping algorithm.


Testing

  • Test random insertions and deletions of 1, 10, 100, 1000 and 10000 keys
  • Test the insert function, including inserting a duplicate key
  • Test the remove function, including removing a non-existent key
  • Test the view functions, including what happens when a non-existent key is passed
  • Test sequential insertions
  • Test repeated random insertions and deletions

References



Thanks to James Zaki and Solidified for the 3 minor issues they picked up at the Web3 Summit.

Enjoy!

(c) BokkyPooBah / Bok Consulting Pty Ltd - Nov 03 2018. The MIT Licence.