Peer-to-Peer (P2P) systems DHT, Chord, Pastry, ….

Peer-to-Peer (P2P) systems

DHT, Chord, Pastry, …

Unstructured vs Structured Unstructured P2P networks allow resources

to be placed at any node. The network topology is arbitrary, and the growth is spontaneous.

Structured P2P networks simplify resource location and load balancing by defining a topology and defining rules for resource placement.

Guarantee efficient search for rare objectsWhat are the rules???

Distributed Hash Table (DHT)

Hash Tables Store arbitrary keys

and satellite data (value)

put(key,value) value = get(key)

Lookup must be fast Calculate hash function

h() on key that returns a storage cell

Chained hash table: Store key (and optional value) there

What Is a DHT?

Single-node hash table:key = Hash(name)put(key, value)get(key) -> value

How do I do this across millions of hosts on the Internet? Distributed Hash Table

DHT

Distributed hash table

Distributed application

get (key) data

node node node….

put(key, data)

Lookup service

lookup(key) node IP address

• Application may be distributed over many nodes• DHT distributes data storage over many nodes

(DHash)

(Chord)

Why the put()/get() interface?

API supports a wide range of applications

DHT imposes no structure/meaning on keys

Distributed Hash Table Hash table functionality in a P2P network :

lookup of data indexed by keys

Key-hash node mapping Assign a unique live node to a key Find this node in the overlay network quickly and

cheaply

Maintenance, optimization Load balancing : maybe even change the key-hash

node mapping on the fly Replicate entries on more nodes to increase

robustness

Distributed Hash Table

The lookup problem

Internet

N1

N2 N3

N6N5

N4

Publisher

Key=“title”Value=MP3 data…

ClientLookup(“title”)

?

Centralized lookup (Napster)

Publisher@

Client

Lookup(“title”)

N6

N9 N7

DB

N8

N3

N2N1SetLoc(“title”, N4)

Simple, but O(N) state and a single point of failure


N4

Flooded queries (Gnutella)

N4Publisher@

Client

N6

N9

N7N8

N3

N2N1

Robust, but large number of messages per lookup


Lookup(“title”)

Routed queries (Chord, Pastry, etc.)

N4Publisher

Client

N6

N9

N7N8

N3

N2N1

Lookup(“title”)


Routing challenges

Define a useful key nearness metric Keep the hop count small Keep the tables small Stay robust despite rapid change

Freenet: emphasizes anonymity Chord: emphasizes efficiency and

simplicity

What is Chord? What does it do?

In short: a peer-to-peer lookup service

Solves problem of locating a data item in a collection of distributed nodes, considering frequent node arrivals and departures

Core operation in most p2p systems is efficient location of data items

Supports just one operation: given a key, it maps the key onto a node

Chord Characteristics

Simplicity, provable correctness, and provable performance

Each Chord node needs routing information about only a few other nodes

Resolves lookups via messages to other nodes (iteratively or recursively)

Maintains routing information as nodes join and leave the system

Mapping onto Nodes vs. Values

Traditional name and location services provide a direct mapping between keys and values

What are examples of values? A value can be an address, a document, or an arbitrary data item

Chord can easily implement a mapping onto values by storing each key/value pair at node to which that key maps

Napster, Gnutella etc. vs. Chord

Compared to Napster and its centralized servers, Chord avoids single points of control or failure by a decentralized technology

Compared to Gnutella and its widespread use of broadcasts, Chord avoids the lack of scalability through a small number of important information for routing

Addressed Difficult Problems (1)

Load balance: distributed hash function, spreading keys evenly over nodes

Decentralization: chord is fully distributed, no node more important than other, improves robustness

Scalability: logarithmic growth of lookup costs with number of nodes in network, even very large systems are feasible

Addressed Difficult Problems (2)

Availability: chord automatically adjusts its internal tables to ensure that the node responsible for a key can always be found

Flexible naming: no constraints on the structure of the keys – key-space is flat, flexibility in how to map names to Chord keys

Chord properties

Efficient: O(log(N)) messages per lookup N is the total number of servers

Scalable: O(log(N)) state per node

Robust: survives massive failures

The Base Chord Protocol (1)

Specifies how to find the locations of keys

How new nodes join the system

How to recover from the failure or planned departure of existing nodes

Consistent Hashing

Hash function assigns each node and key an m-bit identifier using a base hash function such as SHA-1

ID(node) = hash(IP, Port) ID(key) = hash(key)

Properties of consistent hashing:

Function balances load: all nodes receive roughly the same number of keys

When an Nth node joins (or leaves) the network, only an O(1/N) fraction of the keys are moved to a different location

Chord IDs

Key identifier = SHA-1(key) Node identifier = SHA-1(IP, Port) Both are uniformly distributed

Both exist in the same ID space

How to map key IDs to node IDs?

Chord IDs

consistent hashing (SHA-1) assigns each node and object an m-bit ID

IDs are ordered in an ID circle ranging from 0 – (2m-1).

New nodes assume slots in ID circle according to their ID

Key k is assigned to first node whose ID ≥ k

successor(k)

Consistent hashing

N32

N90

N105

K80

K20

K5

Circular 7-bitID space

Key 5Node 105

A key is stored at its successor: node with next higher ID

6

1

2

6

0

4

26

5

1

3

7

2identifiercircle

identifier

node

X key

Successor Nodes

successor(1) = 1

successor(2) = 3successor(6) = 0

Node Joins and Departures

6

1

2

0

4

26

5

1

3

7successor(6) = 7

6

1

successor(1) = 3

Consistent Hashing – Join and Departure

When a node n joins the network, certain keys previously assigned to n’s successor now become assigned to n.

When node n leaves the network, all of its assigned keys are reassigned to n’s successor.

Consistent Hashing – Node Join

0

4

26

5

1

3

7

keys1

keys2

keys

keys

7

5

Consistent Hashing – Node Dep.

0

4

26

5

1

3

7

keys1

keys2

keys

keys6

7

Scalable Key Location

A very small amount of routing information suffices to implement consistent hashing in a distributed environment

Each node need only be aware of its successor node on the circle

Queries for a given identifier can be passed around the circle via these successor pointers

A Simple Key Lookup

Pseudo code for finding successor:// ask node n to find the successor of idn.find_successor(id)

if (id (n, successor])return successor;

else// forward the query around the

circlereturn

successor.find_successor(id);

Scalable Key Location

Resolution scheme correct, BUT inefficient:

it may require traversing all N nodes!

Acceleration of Lookups Lookups are accelerated by maintaining additional

routing information

Each node maintains a routing table with (at most) m entries (where N=2m) called the finger table

ith entry in the table at node n contains the identity of the first node, s, that succeeds n by at least 2i-1 on the identifier circle

s = successor(n + 2i-1) (all arithmetic mod 2m)

s is called the ith finger of node n, denoted by n.finger(i).node

Scalable Key Location – Finger Tables

0

4

26

5

1

3

7

124

130

finger tablestart succ.

keys1

235

330


keys2

457

000


keys6

0+20

0+21

0+22

For.

1+20

1+21

1+22

For.

3+20

3+21

3+22

For.

Finger Tables

0

4

26

5

1

3

7

124

[1,2)[2,4)[4,0)

130

finger tablestart int. succ.

keys1

235

[2,3)[3,5)[5,1)

330


keys2

457

[4,5)[5,7)[7,3)

000


keys6

Finger Tables - characteristics

Each node stores information about only a small number of other nodes, and knows more about nodes closely following it than about nodes farther away

A node’s finger table generally does not contain enough information to determine the successor of an arbitrary key k

Repetitive queries to nodes that immediately precede the given key will lead to the key’s successor eventually

Node Joins – with Finger Tables

0

4

26

5

1

3

7

124

[1,2)[2,4)[4,0)

130


keys1

235

[2,3)[3,5)[5,1)

330


keys2

457

[4,5)[5,7)[7,3)

000


keys


keys

702

[7,0)[0,2)[2,6)

003

6

6

66

6

Node Departures – with Finger Tables

0

4

26

5

1

3

7

124

[1,2)[2,4)[4,0)

130


keys1

235

[2,3)[3,5)[5,1)

330


keys2

457

[4,5)[5,7)[7,3)

660


keys


keys

702

[7,0)[0,2)[2,6)

003

6

6

6

0

3

Simple Key Location SchemeN1

N8

N14

N21N32

N38

N42

N48

K45

Scalable Lookup SchemeN1

N8

N14

N21N32

N38

N42

N48

N51

N56N8+1 N14

N8+2 N14

N8+4 N14

N8+8 N21

N8+16 N32

N8+32 N42

Finger Table for N8

finger 1,2,3

finger 4

finger 6

finger [k] = first node that succeeds (n+2k-1)mod2m

finger 5

Chord key location Lookup in

finger table the furthest node that precedes key

-> O(log n) hops

Scalable Lookup Scheme

// ask node n to find the successor of idn.find_successor (id)

n0 = find_predecessor(id);return n0.successor;

// ask node n to find the predecessor of idn.find_predecessor (id)

n0 = n; while (id not in (n0, n0.successor] )

n0 = n0.closest_preceding_finger(id);return n0;

// return closest finger preceding idn.closest_preceding_finger (id)

for i = m downto 1if (finger[i].node belongs to (n, id))

return finger[i].node;return n;

Lookup Using Finger TableN1

N8

N14

N21N32

N38

N42

N51

N56

N48

lookup(54)

Scalable Lookup Scheme

Each node forwards query at least halfway along distance remaining to the target

Theorem: With high probability, the number of nodes that must be contacted to find a successor in a N-node network is O(log N)

“Finger table” allows log(N)-time lookups

N80

½¼

1/8

1/161/321/641/128

Dynamic Operations and Failures

Need to deal with: Node Joins and Stabilization Impact of Node Joins on Lookups Failure and Replication Voluntary Node Departures

Node Joins and Stabilization

Node’s successor pointer should be up to date For correctly executing lookups

Each node periodically runs a “Stabilization” Protocol Updates finger tables and successor

pointers

Node Joins and Stabilization

Contains 6 functions: create() join() stabilize() notify() fix_fingers() check_predecessor()

Create()

Creates a new Chord ring

n.create()predecessor = nil;successor = n;

Join()

Asks m to find the immediate successor of n.

Doesn’t make rest of the network aware of n.

n.join(m)predecessor = nil;successor = m.find_successor(n);

Stabilize()

Called periodically to learn about new nodes Asks n’s immediate successor about successor’s

predecessor p Checks whether p should be n’s successor instead Also notifies n’s successor about n’s existence, so that

successor may change its predecessor to n, if necessary

n.stabilize()x = successor.predecessor;if (x (n, successor))

successor = x;successor.notify(n);

Notify()

m thinks it might be n’s predecessor

n.notify(m)if (predecessor is nil or m (predecessor, n))

predecessor = m;

Fix_fingers()

Periodically called to make sure that finger table entries are correct

New nodes initialize their finger tables Existing nodes incorporate new nodes into their

finger tables

n.fix_fingers()next = next + 1 ;if (next > m)

next = 1 ;finger[next] = find_successor(n + 2next-1);

Check_predecessor()

Periodically called to check whether predecessor has failed If yes, it clears the predecessor pointer,

which can then be modified by notify()

n.check_predecessor()if (predecessor has failed)

predecessor = nil;

Theorem

If any sequence of join operations is executed interleaved with stabilizations, then at some time after the last join the successor pointers will form a cycle on all nodes in the network

Stabilization Protocol

Guarantees to add nodes in a fashion to preserve reachability

Impact of Node Joins on Lookups

If finger table entries are reasonably current

Lookup finds the correct successor in O(log N) steps

If successor pointers are correct but finger tables are incorrect

Correct lookup but slower If incorrect successor pointers

Lookup may fail

Impact of Node Joins on Lookups

Performance If stabilization is complete

Lookup can be done in O(log N) time If stabilization is not complete

Existing nodes finger tables may not reflect the new nodes

Doesn’t significantly affect lookup speed Newly joined nodes can affect the lookup speed, if

the new nodes ID’s are in between target and target’s predecessor

Lookup will have to be forwarded through the intervening nodes, one at a time

Theorem

If we take a stable network with N nodes with correct finger pointers, and another set of up to N nodes joins the network, and all successor pointers (but perhaps not all finger pointers) are correct, then lookups will still take O(log N) time with high probability

Source of Inconsistencies:Concurrent Operations and Failures

Basic “stabilization” protocol is used to keep nodes’ successor pointers up to date, which is sufficient to guarantee correctness of lookups

Those successor pointers can then be used to verify the finger table entries

Every node runs stabilize periodically to find newly joined nodes

Stabilization after Join

np

su

cc(n

p)

= n

s

ns

n

pre

d(n

s)

= n

p

n joins predecessor = nil n acquires ns as successor via some n’

n notifies ns being the new predecessor

ns acquires n as its predecessor

np runs stabilize

np asks ns for its predecessor (now n)

np acquires n as its successor

np notifies n

n will acquire np as its predecessor

all predecessor and successor pointers are now correct

fingers still need to be fixed, but old fingers will still work

nil

pre

d(n

s)

= n

su

cc(n

p)

= n

Node joins and stabilization


• N26 joins the system

• N26 aquires N32 as its successor

• N26 notifies N32

• N32 aquires N26 as its predecessor


• N26 copies keys

• N21 runs stabilize() and asks its successor N32 for its predecessor which is N26.


• N21 aquires N26 as its successor

• N21 notifies N26 of its existence

• N26 aquires N21 as predecessor

Failure Recovery

Key step in failure recovery is maintaining correct successor pointers

To help achieve this, each node maintains a successor-list of its r nearest successors on the ring

If node n notices that its successor has failed, it replaces it with the first live entry in the list

stabilize will correct finger table entries and successor-list entries pointing to failed node

Performance is sensitive to the frequency of node joins and leaves versus the frequency at which the stabilization protocol is invoked

68

Impact of node joins on lookups

All finger table entries are correct => O(log N) lookups

Successor pointers correct, but fingers inaccurate => correct but slower lookups

68

69

Impact of node joins on lookups

Stabilization completed => no influence on performence

Only for the negligible case that a large number of nodes joins between the target‘s predecessor and the target, the lookup is slightly slower

No influence on performance as long as fingers are adjusted faster than the network doubles in size

69

70

Failure of nodes

Correctness relies on correct successor pointers

What happens, if N14, N21, N32 fail simultaneously?

How can N8 aquire N38 as successor?

70

71

Failure of nodes

Correctness relies on correct successor pointers

What happens, if N14, N21, N32 fail simultaneously?

How can N8 aquire N38 as successor?

71

Voluntary Node Departures

Can be treated as node failures Two possible enhancements

Leaving node may transfers all its keys to its successor

Leaving node may notify its predecessor and successor about each other so that they can update their links

Chord – facts

Every node is responsible for about K/N keys (N nodes, K keys)

When a node joins or leaves an N-node network, only O(K/N) keys change hands (and only to and from joining or leaving node)

Lookups need O(log N) messages

To reestablish routing invariants and finger tables after node joining or leaving, only O(log2N) messages are required

Pastry Self-organizing overlay network of nodes

With high probability, nodes with adjacent nodeId are diverse in geography, ownership, jurisdiction, network attachment, etc…

Pastry takes into account network locality (e.g. IP routing hops).

Pastry

Instead of organizing the id-space as a Chord-like ring, the routing is based on numeric closeness of identifiers

The focus is not only on the no. of routing hops, but also on network locality – as factors in routing efficiency

Pastry Identifier space:

Nodes and data items are uniquely associated with m-bit ids – integers in the range (0 – 2m -1) – m is typically 128

Pastry views ids as strings of digits to the base 2b where b is typically chosen to be 4

A key is located on the node to whose node id it is numerically closest

Routing Goal

Pastry routes messages to the node whose nodeId is numerically closest to the given key in less than log2b (N) steps:

“A heuristic ensures that among a set of nodes with the k closest nodeIds to the key, the message is likely to first reach a node near the node from which the message originates, in term of the proximity metric”

Routing Information Pastry’s node state is divided into 3 main

elements

The routing table – similar to Chord’s finger table – stores links to id-space

The leaf set contains nodes which are close in the id-space

Nodes that are closed together in terms of network locality are listed in the neighbourhood set

Routing Table A Pastry node’s routing table is made up of m/b

(log2b N) rows with 2b -1 entries per row On node n, entries in row i hold the identities of Pastry

nodes whose node-id share an i-digit prefix with n but differ in digit n itself

For ex, the first row is populated with nodes that have no prefix in common with n

When there is no node with an appropriate prefix, the corresponding entry is left empty

Single digit entry in each row shows the corresponding digit of the present node’s id – i.e. prefix matches the current id up to the given value of p – the next row down or leaf set should be examined to find a route.

Routing Table

Routing tables (RT) thus built achieve an effect similar to Chord finger table

The detail of the routing information increases with the proximity of other nodes in the id-space

Without a large no. of nearby nodes, the last rows of the RT are only sparsely populated – intuitively, the id-space would need to be fully exhausted with node-ids for complete RTs on all nodes

In populating the RT, there is a choice from the set of nodes with the appropriate id-prefix

During the routing process, network locality can be exploited by selecting nodes which are close in terms of proximity ntk. metric

Leaf Set

The Routing tables sort node ids by prefix. To increase lookup efficiency, the leaf set L of nodes holds the |L| nodes numerically closest to n (|L|/2 smaller and |L|/2 larger, L = 2 or 2 × 2b, normally)

The RT and the leaf set are the two sources of information relevant for routing

The leaf set also plays a role similar to Chord’s successor list in recovering from failures of adjacent nodes

Neighbourhood Set

Instead of numeric closeness, the neighbourhood set M is concerned with nodes that are close to the current node with regard to the network proximity metric

Thus, it is not involved in routing itself but in maintaining network locality in the routing information

Pastry Node State (Base 4)

LNodes that are numerically

closer to the present Node (2b or 2x2b entry)

RCommon prefix with

10233102-next digit-rest of NodeId (log2b (N) rows, 2b-1 columns)

MNodes that are closest

according to the proximity metric (2b or 2x2b entry)

Routing

Key D arrives at nodeId A

Ril enetry in routing table

at column i and row l

Li i-th closest nodeId in leaf set

Dl value of the l’s digit in the key D

shl(A,B) length of the prefix shared in digits

Routing

Routing is divided into two main steps:

First, a node checks whether the key K is within the range of its leaf set

If it is the case, it implies that K is located in one of the nearby nodes of the leaf set. Thus, the node forwards the query to the leaf set node numerically closest to K. In case this is the node itself, the routing process is finished.

Routing

If K does not fall within the range of the leaf set, the query needs to be forwarded over a large distance using the routing table

In this case, a node n tries to pass the query on to a node which shares a longer common prefix with K than n itself

If there is no such entry in the RT, the query is forwarded to a node which shares a prefix with K of the same length as n but which is numerically close to K than n

Routing

This scheme ensures that routing loop do not occur because the query is routed strictly to a node with a longer common identifier prefix than the current node, or to a numerically closer node with the same prefix

Routing performance

Routing procedure converges, each step takes the message to node that either:

Shares a longer prefix with the key than the local node

Shares as long a prefix with, but is numerically closer to the key than the local node.

Routing performance

Assumption: Routing tables are accurate and no recent node failures

There are 3 cases in the Pastry routing scheme:

Case 1: Forward the query (according to the RT) to a node with a longer prefix match than the current node.

Thus, the no. of nodes with longer prefix matches is reduced by at least a factor of 2b in each step, so the destination is reached in log2b N steps.

Routing performance

There are 3 cases:

Case 2: The query is routed via leaf set (one step). This increases the no. of hop by one

Routing performance

There are 3 cases: Case 3: The key is neither covered by the leaf set

nor does the RT contains an entry with a longer matching prefix than the current node

Consequently, the query is forwarded to a node with the same prefix length, adding an additional routing hop.

For a moderate leaf set size ( |L| = 2 × 2b), the probability of this case is less than 0.6%. So, it is very unlikely that more than one additional hop is incurred.

Routing performance

As a result, the complexity of routing remains at O(log2b N) on average

Higher values of b leads to fast routing but also increases the amount of state that needs to managed at each node

Thus, b is typically 4 but Pastry implementation can choose an appropriate trade-off for specific application

Join and Failure

Join Use routing to find numerically closest node already in

network Ask state from all nodes on the route and initialize

own state

Error correction Failed leaf node: contact a leaf node on the side of the

failed node and add appropriate new neighbor

Failed table entry: contact a live entry with same prefix as failed entry until new live entry found, if none found, keep trying with longer prefix table entries

Self Organization: Node Arrival

The new node n is assumed to know a nearby Pastry node k based on the network proximity metric

Now n needs to initialize its RT, leaf set and neighbourhood set. Since K is assumed to be close to n, the nodes

in K’s neghbourhood set are reasonably good choices for n, too.

Thus, n copies the neighbourhood set from K.


To build its RT and leaf set, n routes a special join message via k to a key equal to n According to the standard routing rules, the

query is forwarded to the node c with the numerically closest id and hence the leaf set of c is suitable for n, so it retrieves c’s leaf set for itself.

The join request triggers all nodes, which forwarded the query towards c, to provide n with their routing information.


Node n’s RT is constructed from the routing information of these nodes starting at row 0. As this row is independent of the local node id,

n can use these entries at row zero of k’s routing table

In particular, it is assumed that n and k are close in terms of network proximity metric

Since k stores nearby nodes in its RT, these entries are also close to n.

In the general case of n and k not sharing a common prefix, n cannot reuse entries from any other row in K’s RT.


The route of the join message from n to c leads via nodes v1, v2, … vn with increasingly longer common prefixes of n and vi

Thus, row 1 from the RT of v1 is also a good choice for the same row of the RT of n

The same is true for row 2 on node v2 and so on

Based on this information, the RT of n can be constructed.


Finally, the new node sends its node state to all nodes in its routing data so that these nodes can update their own routing information accordingly

In contrast to lazy updates in Chord, this mechanism actively updates the state in all affected nodes when a new node joins the system

At this stage, the new node is fully present and reachable in the Pastry network

Node Failure Node failure is detected when a communication

attempt with another node fails. Routing requires contacting nodes from RT and leaf set, resulting in lazy detection of failures

During routing, the failure of a single node in the RT does not significantly delay the routing process. The local node can chose to forward the query to a different node from the same row in the RT. (Alternatively, a node could store backup nodes with each entry in the RT)

Node Failure To replace the failed node at entry i in row j of its RT

(Rji), a node contacts another node referenced in row

j Entries in the same row j of the remote node are valid

for the local node and hence it can copy entry Rji from

the remote node to its own RT

In case it failed as well, it can probe another node in row j for entry Rj

i

If no live node with appropriate nodeID prefix can be obtained in this way, the local node queries nodes from the preceding row Rj-1

Node Failure Repairing a failed entry in the leaf set of a node is

straightforward – utilizing the leaf set of other nodes referenced in the local leaf set.

Contacts the leaf set of the largest index on the side of the failed node

If this node is unavailable, the local node can revert to leaf set with smaller indices

Node Departure

Neighborhood node: asks other members for their M, checks the distance of each of the newly discovered nodes, and updates its own neighborhood set accordingly.

Locality “Route chosen for a message is likely to

be good with respect to the proximity metric”

Discussion: Locality in the routing table Route locality Locating the nearest among k nodes

Locality in the routing table

Node A is near X A’s R0 entries are close to A, A is close to X, and

triangulation inequality holds entries in X are relatively near A.

Likewise, obtaining X’s neighborhood set from A is appropriate.

B’s R1 entries are reasonable choice for R1of X Entries in each successive row are chosen from an

exponentially decreasing set size. The expected distance from B to any of its R1 entry is

much larger than the expected distance traveled from node A to B.

Second stage: X requests the state from each of the nodes in its routing table and neighborhood set to update its entries to closer nodes.

Routing locality Each routing step moves the message closer to the

destination in the nodeId space, while traveling the least possible distance in the proximity space.

Given that: A message routed from A to B at distance d cannot

subsequently be routed to a node with a distance of less than d from A

The expected distance traveled by a message during each successive routing step is exponentially increasing

Since a message tends to make larger and larger strides with no possibility of returning to a node within di of any node i encountered on the route, the message has nowhere to go but towards its destination

Locating the nearest among k nodes

Goal: among the k numerically closest nodes to a

key, a message tends to first reach a node near the client.

Problem: Since Pastry routes primarily based on nodeId

prefixes, it may miss nearby nodes with a different prefix than the key.

Solution (using a heuristic): Based on estimating the density of nodeIds, it

detects when a message approaches the set of k and then switches to numerically nearest address based routing to locate the nearest replica.

Arbitrary node failures and network partitions Node continues to be responsive, but

behaves incorrectly or even maliciously.

Repeated queries fail each time since they normally take the same route.

Solution: Routing can be randomized The choice among multiple nodes that

satisfy the routing criteria should be made randomly

CAN : Content Addressable Network

Hash value is viewed as a point in a D-dimensional Cartesian space

Hash value points <n1, n2, …, nD>. Each node responsible for a D-dimensional “cube” in

the space Nodes are neighbors if their cubes “touch” at more

than just a point

• Example: D=2• 1’s neighbors: 2,3,4,6• 6’s neighbors: 1,2,4,5• Squares “wrap around”, e.g., 7 and 8 are neighbors• Expected # neighbors: O(D)

CAN : Routing To get to <n1, n2, …, nD> from <m1, m2, …, mD>

choose a neighbor with smallest Cartesian distance from <m1, m2, …, mD> (e.g., measured from neighbor’s center)

• e.g., region 1 needs to send to node covering X• Checks all neighbors, node 2 is closest• Forwards message to node 2• Cartesian distance monotonically decreases with each transmission• Expected # overlay hops: (DN1/D)/4

CAN : Join

To join the CAN: find some node in the CAN

(via bootstrap process) choose a point in the space

uniformly at random using CAN, inform the node

that currently covers the space that node splits its space in half

1st split along 1st dimension if last split along dimension i

< D, next split along i+1st dimension

e.g., for 2-d case, split on x-axis, then y-axis

keeps half the space and gives other half to joining node

The likelihood of a rectangle being selected is proportional to it’s size,

i.e., big rectangles chosen more frequently

CAN: Join

Bootstrap

node

new node

CAN: construction

I

Bootstrap

node

new node 1) Discover some node “I” already in CAN

CAN: Join

2) Pick random point in space

I

(x,y)

new node

CAN: Join

(x,y)

3) I routes to (x,y), discovers node J

I

J

new node

CAN: Join

newJ

4) split J’s zone in half… new owns one half

CAN Failure recovery

View partitioning as a binary tree Leaves represent regions covered by overlay nodes Intermediate nodes represents “split” regions that

could be “reformed” Siblings are regions that can be merged together

(forming the region that is covered by their parent)

CAN Failure Recovery

Failure recovery when leaf S is removed

Find a leaf node T that is either

S’s sibling Descendant of S’s sibling

where T’s sibling is also a leaf node

T takes over S’s region (move to S’s position on the tree)

T’s sibling takes over T’s previous region

Maintenance

Use zone takeover in case of failure or leaving of a node

Send your neighbor table to neighbors to inform that you are alive at discrete time interval t

If your neighbor does not send alive in time t, takeover its zone

Zone reassignment is needed

Zone reassignment

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Zoning

Partition tree

Zone reassignment

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Zoning

Partition tree

Zone reassignment

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Partition tree

Zone reassignment

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Zoning

Partition tree

Peer-to-Peer (P2P) systems DHT, Chord, Pastry, ….

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