Mehdi Kargar Department of Computer Science and Engineering 1

 R-tree is useful for indexing and representing multidimensional objects.  An R-tree is a depth balanced tree with a dynamic index structure ◦ Leaf nodes point to actual keys ◦ The number of entries in a node is between m and N (1 < m ≤ N) ◦ Root might have between 1 and N entries. ◦ All leaf nodes are at the same level ◦ The key for each internal node is the minimum bounding rectangle of its child nodes ◦ keys at all levels might have overlap with each other 2

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 During the search for a key, it might be necessary to descend multiple sub-trees  Insertion is more complex than search ◦ After inserting a new key, the new bounding rectangle should be propagated up to the tree. ◦ If a node overflows, it should be split. The split should also be propagated up to the tree.  During the Insertion, only one sub-tree is traversed at each level. We should not descend multiple sub-trees. 4

1. The Monitor Solution 2. The Readers-Writers Solution 3. Locking Nodes (RLink-Tree) 5

public class MonitorRTree { public synchronized void add(Rectangle rect) {. } public synchronized boolean search(Rectangle rect) {. } 6

import java.util.concurrent.locks.ReentrantReadWriteLock; public class ReadWriteLockRTree { private ReentrantReadWriteLock lock;.... public void add(Rectangle rect) { this.lock.writeLock().lock();. this.lock.writeLock().unlock(); } public boolean search(Rectangle rect) { this.lock.readLock().lock();. this.lock.readLock().unlock(); } 7

 Basic idea : ◦ Logical Serial Number (LSN) is used to capture unfinished splits. ◦ Right Links is used to follow the split nodes. 8

 RLink-Tree uses the lock coupling strategy for inserting new entries.  The normal implementation of lock coupling leads to deadlocks. x ZY Process 1 Finding the best node for inserting new entries Read-Lock the nodes in a top-down approach Process 2 After Inserting new entry, the split and new bounding rectangle should be propagated Write-Lock the nodes in a bottom-up approach Write-Lock 2 1 Read-Lock 1 2 9

 The previous situation is called phantom problem.  It can be solved by predicate locks.  What are predicate locks ??  You can find more about it here : ◦ K. Eswaren, J. Gray, R. Lorie and I. Traiger, On the Notions of Consistency and Predicate Locks in a Database System, Comm. ACM, November 1976, Vol. 19, No. 11, pp. 624–

 Here, the root node is kept write-lock for the insertion of new nodes. Thus, only one process can insert at any time and we do not have multiple insertions.  The LSN and right links are the same as the original RLink-Tree.  Search algorithm and the read-lock mechanism is also the same as RLink-Tree. 11

12 public synchronized void readLock() { while (this.state == WRITE) { try { this.wait(); } catch (InterruptedException e) {} } this.increment(); } public synchronized void readUnlock() { this.decrement(); this.notifyAll(); } public synchronized void writeLock() { while (this.readers != 0 || state == WRITE) { try { this.wait(); } catch (InterruptedException e) {} } this.state = WRITE; } public synchronized void writeUnlock() { this.state = READ; this.notifyAll(); }

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 Having multiple insertion process without locking the root.  Finding better solution for phantom problem. ... 16

 Verifying the implementations using different applications and tools such as Java PathFinder 17

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