BokkyPooBahs Red-Black Binary Search Tree Library

Status: Currently being tested and bug bounty open. Don't use in production without an audit, 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.

Check out the online Red-black tree visualization tool- type in some 4 digit numbers and press Insert or Delete to see the insertions and rebalances animated.

See also Hitchen's Order Statistics Tree, an extension built from this library.

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

Following is a chart with the minimum, average and maximum gas cost for insertions and deletions from a Red-Black Tree. The result have been generated by insering random data in the first case, and sequential data in the second case.

Data and chart - docs/GasStatistics.xlsx. These statistics have been generated using the test/02_test2.sh script with the results logged in test/test2results.txt - the parameters NUMBEROFITEMS, BATCHSIZE were varied for the results with different number of items, and the insertItems = shuffle(insertItems); and removeItems = shuffle(removeItems); commented out for sequential insertions.

Random Insertion Gas Cost

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	68,913	68,913	68,913	44,654	44,654	44,654
5	68,913	99,588	140,404	29,827	40,891	56,405
10	68,913	108,635	141,518	27,375	44,880	90,688
50	68,913	121,753	182,645	27,375	64,977	179,109
100	68,913	122,549	186,766	27,375	65,447	143,832
500	68,913	124,790	191,559	27,375	66,629	215,994
1,000	68,977	128,550	195,719	27,375	67,331	195,574
5,000	68,977	127,029	200,233	27,375	66,966	258,858
10,000	68,977	127,516	200,907	27,375	66,781	240,152

Note that the statistics above will change with each execution, as the data inserted is randomised.

Sequential Insertion Gas Cost

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	68,913	68,913	68,913	44,654	44,654	44,654
5	68,913	107,761	141,129	29,827	44,883	80,922
10	68,913	116,896	149,938	29,827	53,457	104,650
50	68,913	138,234	158,844	29,827	61,485	151,002
100	68,913	143,145	163,297	29,827	63,540	174,178
500	68,913	149,417	191,134	29,859	66,239	219,978
1,000	68,913	150,878	208,360	29,859	66,774	243,154
5,000	68,913	153,219	242,813	29,859	67,276	304,485
10,000	68,913	154,017	260,040	29,859	67,352	327,661

History

Version	Date	Notes
v0.90-pre-release	Feb 17 2019	Bug bounty added
v1.0 pre-release-a	Mar 8 2019	Incorporated suggestions from Rob Hitchens

Bug Bounty Scope And Donations

Details of the bug bounty program for this project can be found at BokkyPooBah's Hall Of Fame And Bug Bounties. Please consider donating to support the bug bounty, and the development and maintenance of decentralised applications.

The scope of the bug bounty for this project follows:

contracts/BokkyPooBahsRedBlackTreeLibrary.sol

Bounties awarded for this project:

Feb 25 2018 - Rob Hitchens submitted a set of performance and readability improvements to this library and has been awarded 5 ETH.

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

What would I use this library for?

Any smart contract where you need to maintain a sorted list of unsigned integers. One major use case is for this RBT library to maintain a decentralised exchange orderbook, sorted by price.

Why does this library only store unsigned integers and not any additional data?

This library was designed to be a simple component to be used within your smart contract project.

Store any additional data, e.g., key/value pairs, by adding the functionality into your smart contract. Sample code from contracts/TestBokkyPooBahsRedBlackTree.sol follows:

contract TestBokkyPooBahsRedBlackTree {
    using BokkyPooBahsRedBlackTreeLibrary for BokkyPooBahsRedBlackTreeLibrary.Tree;

    BokkyPooBahsRedBlackTreeLibrary.Tree tree;
    mapping(uint => uint) values;

    event Log(string where, uint key, uint value);

    ...

    function getNode(uint _key) public view returns (uint key, uint parent, uint left, uint right, bool red, uint value) {
        if (tree.exists(_key)) {
            BokkyPooBahsRedBlackTreeLibrary.Node memory node = tree.getNode(_key);
            (key, parent, left, right, red) = (_key, node.parent, node.left, node.right, node.red);
            value = values[_key];
        }
    }

    function insert(uint _key, uint _value) public {
        tree.insert(_key);
        values[_key] = _value;
        emit Log("insert", _key, _value);
    }
    function remove(uint _key) public {
        tree.remove(_key);
        emit Log("remove", _key, values[_key]);
        delete values[_key];
    }
}

Why can't I use 0 as a key?

This library has been configured with the EMPTY marker set to '0'. This can be set to non-0, but you will have to add some additional functionality - you can get an idea of the functionality you will need to add back from an older untested version of the library. Search for init(...) and the last line of getNode(...). Please test carefully!

Can duplicate keys be inserted?

No

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 EMPTY that is set to 0 in the library source code by default
The insert(...) and remove(...) functions have internal visibility so it is easy to deploy your contract, as these function calls will be inlined. There may be cases where you may want to specify a public visibility so the is not inlined and duplicated in the deployment EVM code.

root

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

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

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
`EMPTY`	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
`EMPTY`	Tree empty

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

isEmpty

function isEmpty(uint key) internal pure returns (bool);

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

getEmpty

function getEmpty() internal pure returns (uint);

Returns the value of the EMPTY variable

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 EMPTY if key does not exist in the tree.

insert

function insert(uint key) internal;

Insert the key key into the tree.

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

remove

function remove(uint key) internal;

Remove the key key from the tree.

Transaction	Condition
success	`key` has been successfully removed from the tree
failure	`key` does not 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

Red-black tree visualization
RBTree Animation
Red–black tree - Wikipedia
Algorithms for Red Black Tree Operations - from Introduction to Algorithms by Cormen, Leiserson, Rivest and Stein.
Iterative Algorithm for Red-Black Tree
Laboratory Module 6 - Red-Black Trees - Mihail Buricea. A copy has been saved to docs/lab8ASD.pdf
https://www.cs.dartmouth.edu/~thc/cs10/lectures/0519/0519.html
- https://www.cs.dartmouth.edu/~thc/cs10/lectures/0519/RBTree.java
https://www.csee.umbc.edu/courses/undergraduate/341/fall13/section3/lectures/10-Red-Black-Trees.pdf
http://cseweb.ucsd.edu/~kube/cls/100/Lectures/lec7/lec7.pdf
https://www.geeksforgeeks.org/red-black-tree-set-1-introduction-2/
https://www.scaler.com/topics/data-structures/binary-search-algorithm/

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

And thanks to Rob Hitchens for the suggestions.

Enjoy!

(c) BokkyPooBah / Bok Consulting Pty Ltd - Jan 18 2020. The MIT Licence.

Name		Name	Last commit message	Last commit date
Latest commit History 127 Commits
contracts		contracts
docs		docs
flattened		flattened
images		images
scripts		scripts
test		test
.gitattributes		.gitattributes
.gitignore		.gitignore
LICENSE		LICENSE
README.md		README.md

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

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

License

bokkypoobah/BokkyPooBahsRedBlackTreeLibrary

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BokkyPooBahs Red-Black Binary Search Tree Library

Table Of Contents

Overview

Binary Search Tree

Red-Black Binary Search Tree

Red-Black Tree With Random Insertion

Red-Black Tree With Sequential Insertion

Gas Cost

Random Insertion Gas Cost

Sequential Insertion Gas Cost

History

Bug Bounty Scope And Donations

Deployment

Questions And Answers

What would I use this library for?

Why does this library only store unsigned integers and not any additional data?

Why can't I use 0 as a key?

Can duplicate keys be inserted?

Functions

root

first

last

next

prev

exists

isEmpty

getEmpty

getNode

insert

remove

Algorithm

Testing

References

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