# BATON Overlay

BATON, BAlanced Tree Over-lay Network, is a distributed tree structure for peer-to-peer (P2P) systems. Different from other overlays that use a distributed hash table (DHT), such as in the Chord system, BATON organizes peers in a distributed tree to support range search. In addition, BATON tries to keep the tree in a balanced manner as the AVL tree. And hence, the search cost is bounded by ${\displaystyle O(\log N)}$.

## Architecture

BATON is a binary tree. Each node in BATON keeps four kinds of links:

1. link to its parent node
2. links to its child nodes
3. links to its adjacent nodes in in-order
4. links to the routing nodes in the same level

In each tree level, the node is named by its position in the tree. For example, node h is named 3:0, node i is named 3:1 and node p is named 4:6. For a node at position ${\displaystyle p}$, it will fill its left routing table by nodes at position ${\displaystyle p-2^{x}}$ for any valid ${\displaystyle x\geq 0}$ and fill its right routing table by nodes at position ${\displaystyle p+2^{y}}$ for any valid ${\displaystyle y\geq 0}$.

## Node Joining and Leaving

The new node's joining request will always be forwarded to the leaf node. The leaf node will check to see whether if routing table is full. If the routing table is full, this level is full of nodes and the leaf node can accept the new node as its child to create a new level node. Otherwise, it must forward the new node to take over one of the empty positions.

When a node wants to leave the network, it must update the routing tables of its parent node, child nodes, adjacent nodes and routing nodes. If this node is a leaf node, it can leave the network safely. Otherwise, it must find a leaf node to replace its position.

## Routing

In BATON, each node maintains a continuous key space. Once a new node joins as its child, it splits its space and assigns half of it to the child. In this partition way, if we travel the tree in in-order, we can search the whole space in ascendant order. That's why BATON supports range queries.

For a range query q, BATON first locats its left bound, q.low. And then the search process will travel the tree in in-order (by adjacent link), until reach the upper bound, q.up. For locating a single key, BATON performs the similar routing strategy as Chord. First, the request is routed to the farthest routing nodes which does not over hit the key. If no such routing nodes exist, the parent link, child link or adjacent link is used.

## Restructure

When a node x accepts a joining node y as its child and detects that the tree balance is violated, it initiates the restructuring process. Without loss of generality, suppose that this restructuring is towards the right. Assume that y joins as x's left child. To rebalance the system, x notifies y to replace its position, and notifies its right adjacent node z that x will replace z's position. z then checks its right adjacent node t to see if its left child is empty. If it is, and adding a child to t does not affect the tree balance, z takes the position of t's left child as its new position and the restructuring process stops. If t's left child is full or t cannot accept x as its left child without violating the balance property, z occupies t's position while t needs to find a new position for itself by continuing to its right adjacent node.