Spanning%20trees

1
Spanning trees

• Suppose you have a connected undirected graph
• Connected every node is reachable from every
  other node
• Undirected edges do not have an associated
  direction
• ...then a spanning tree of the graph is a
  connected subgraph in which there are no cycles

2
Finding a spanning tree

• To find a spanning tree of a graph,
• pick an initial node and call it part of the
  spanning tree
• do a search from the initial node
• each time you find a node that is not in the
  spanning tree, add to the spanning tree both the
  new node and the edge you followed to get to it

3
Minimizing costs

• Suppose you want to supply a set of houses (say,
  in a new subdivision) with
• electric power
• water
• sewage lines
• telephone lines
• To keep costs down, you could connect these
  houses with a spanning tree (of, for example,
  power lines)
• However, the houses are not all equal distances
  apart
• To reduce costs even further, you could connect
  the houses with a minimum-cost spanning tree

4
Minimum-cost spanning trees

• Suppose you have a connected undirected graph
  with a weight (or cost) associated with each edge
• The cost of a spanning tree would be the sum of
  the costs of its edges
• A minimum-cost spanning tree is a spanning tree
  that has the lowest cost

5
Small Example
6
Why Multicast

• When sending same data to multiple receivers
• better bandwidth utilization
• less host/router processing
• quicker participation
• Application
• Video/Audio broadcast (One sender)
• Video conferencing (Many senders)
• Real time news distribution
• Interactive gaming

7
Unicast/Multicast
128.146.116.0/24
128.146.199.0/24
128.146.222.0/24
128.146.226.0/24
8
Unicast
128.146.116.0/24
128.146.199.0/24
Receiver
Sender
128.146.222.0/24
128.146.226.0/24
Receivers
Receiver
9
Multicast
128.146.116.0/24
128.146.199.0/24
Receiver
Sender
128.146.222.0/24
128.146.226.0/24
Receivers
Receiver
10
Two Major Issues

• Who are the multicast members
• How to send the packets to the members

11
IGMP
224.0.0.1
224.2.127.254
Designated router queries LAN for group membership
Host informs router with IGMP report
12
IGMP Joining a group

• Example R joins to Group 224.2.0.1
• R sends IGMP Membership-Reportto 224.2.0.1
• DR receives it. DR will start forwarding packets
  for 224.2.0.1 to Network A
• DR periodically sends IGMP Membership-Query to
  224.0.0.1 (ALL-SYSTEMS.MCAST.NET)
• R answers IGMP Membership-Report to 224.2.0.1

IGMP Membership-Report
R
Network A
DR
Data to 224.2.0.1
Network B
R ReceiverDR Designated Router
13
IGMP Leaving a group

• Example R leaves from a Group 224.2.0.1
• R sends IGMP Leave-Group to 224.0.0.2
  (ALL-ROUTERS.MCAST.NET)
• DR receives it.
• DR stops forwarding packets for 224.2.0.1 to
  Network A if no more 224.2.0.1 group members on
  Network A.

IGMP Leave-Group
R
Network A
DR
Data to 224.2.0.1
Network B
R ReceiverDR Designated Router
14
Multicast Routing

• Goal find a tree (or trees) connecting routers
  having local mcast group members
• tree not all paths between routers used
• source-based different tree from each sender to
  rcvrs
• shared-tree same tree used by all group members

Shared tree
15
Approaches for building mcast trees

• Approaches
• source-based tree one tree per source
• shortest path trees
• reverse path forwarding
• group-shared tree group uses one tree
• minimal spanning (Steiner)
• center-based trees

we first look at basic approaches, then specific
protocols adopting these approaches
16
Shortest Path Tree

• mcast forwarding tree tree of shortest path
  routes from source to all receivers
• Dijkstras algorithm

S source
LEGEND
R1
R4
router with attached group member
R2
router with no attached group member
R5
link used for forwarding, i indicates order
link added by algorithm
R3
R7
R6
17
Reverse Path Forwarding

• rely on routers knowledge of unicast shortest
  path from it to sender
• each router has simple forwarding behavior

• if (mcast datagram received on incoming link on
  shortest path back to center)
• then flood datagram onto all outgoing links
• else ignore datagram

18
Reverse Path Forwarding example
S source
LEGEND
R1
R4
router with attached group member
R2
router with no attached group member
R5
datagram will be forwarded
R3
R7
R6
datagram will not be forwarded

• result is a source-specific reverse SPT
• may be a bad choice with asymmetric links

19
Reverse Path Forwarding pruning

• forwarding tree contains subtrees with no mcast
  group members
• no need to forward datagrams down subtree
• prune msgs sent upstream by router with no
  downstream group members

LEGEND
S source
R1
router with attached group member
R4
router with no attached group member
R2
P
P
R5
prune message
links with multicast forwarding
P
R3
R7
R6
20
Shared-Tree Steiner Tree

• Steiner Tree minimum cost tree connecting all
  routers with attached group members
• problem is NP-complete
• excellent heuristics exists
• not used in practice
• computational complexity
• information about entire network needed
• monolithic rerun whenever a router needs to
  join/leave

21
Center-based trees

• single delivery tree shared by all
• one router identified as center of tree
• to join
• edge router sends unicast join-msg addressed to
  center router
• join-msg processed by intermediate routers and
  forwarded towards center
• join-msg either hits existing tree branch for
  this center, or arrives at center
• path taken by join-msg becomes new branch of tree
  for this router

22
Center-based trees an example
Suppose R6 chosen as center
LEGEND
R1
router with attached group member
R4
3
router with no attached group member
R2
2
1
R5
path order in which join messages generated
R3
1
R7
R6
23
Overlay Multicast

• Constructs Overlay Multicast Data Delivery Tree
  among Group Members
• Intermediate Receiver can act as a Multicast
  Forwarder
• Data is delivered by Unicast Tunneling
  Mechanisms, hop-by-hop basis