Communication network problem. How the routers need to store and forward packets as in Internet/internets/intranets on a hop-by-hop basis.

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1 Graph Algorithms. A graph portrays the topology of a distributed system (both in 2-D and in 3-D). A special area of distributed systems algorithm is Graph algorithms. Typical examples: Communication network problem. How the routers need to store and forward packets as in Internet/internets/intranets on a hop-by-hop basis. On this: ๐ how to we compute shortest path from a router to other routers? ๐ how do we implement a scheme to generate, and update Routing tables (RT)? ๐ For a large domain, RT at a router is. This may be way too large. How do we design and implement a Compact RT system for a router node? ๐ How do we do broadcasting on a graph? One way of doing it would be via a spanning tree? How do we compute minimum spanning tree for a graph in a distributed manner? ๐ How do we color a graph? How do we obtain maximum flow between a pair of nodes in a connected network? 1. Computation of shortest path.

2 =, is our graph with nodes. is the set of edges,. Cost or weight associated with an edge, is = 1 if we are interested in number of hops =, message propagation time on the, link. Example. Dijkstra s centralized algorithm for shortest path in an acyclic network. (Dijkstra in 1972 receiving Turning award) From a single node 1 above, identify and compute the shortest paths to all the nodes in the acyclic graph. Our problem is to compute the shortest path from a designated source node 0 to all the nodes on the graph. Let = be this amount. Suppose, the shortest distance to, and the corresponding shortest path are as shown below.

3 0 i l p j n m Then the shortest paths,, etc. are all on it as shown. = Set of neighbors of node. Identify =, =, = if the shortest path to must traverse through,, etc. = + where is the shortest path to node. Initially for all neighbors, =, and for all other nodes = if. Consider the graph shown. We need find the shortest path from 0 to all nodes using a distributed algorithm Computation of shortest path in a distributed system framework.

4 Node 0 has two successors. Therefore, " =#1,2& and the cardinality of " =2. Similarly, " ( =#2&, " ) =#1,3,& ", =#2,5&, ". =#2,5& and " / =#3,& 0 sends out to both 1, and 2 its distance to 0, and its weight to the node 1 and 2, respectively. send(1, =0, ( = send(2, =0, ) =6 set deficit(0)=2 These deficits will be reduced by one only when 0 receives an a message from its successor showing the optimal distance. This message would be construed as an ACK sent to the parent of the node which knows its shortest path to the initiator node via the parent node. 1 first computes its distance to 0, which it next sends out to its successor {2}. Since, currently ( =, the revised version of ( is ( =min# ( =, ) + (), + ( &= with the tentative parent toward node 0 to be the one which is currently offering the shortest path to 0. So, parent(1, tentative) = 0 ( = msg to 2: send(2, ( =, () =5 set deficit(1)=1

5 Meanwhile, the node 2 upon receiving his message from 0 computes its shortest path to 0 from what he knows thus far ) =min# ) =, ( + (), + ) &=6 Accordingly, for node 2: parent(2, tentative) = 0 ) =6 msg to 1: send(1, ) =6,5h=5 set deficit(1)=1 Now both 1 and 2 got all the information from their successor set. They can finalize their distances now on the next set of exchanges. Node 1 computes: Best ( =min # ( 78, ) + () =11}= Since this cannot be improved any further as all messages from all successors are collected (deficit=0), it is time to acknowledge the parent formally by sending an ACK to its tentative parent. Every time a return from a neighbor is received deficit is reduced by 1 and an ACK is sent. Node 2 also repeats these steps. Finally, for every parent identified we attach an arrow from the node to its parent yielding

6 Thus the shortest path from 5 to 0 would be: The basic idea is once a node knows its best estimate to the target node by compiling all the successor s results it can identify its parent. And when it does so, it sends his ACK to his new parent, but not before. Therefore, the Chandy-Misra shortest path computation algorithm looks like the following: program shortest path (i) define D,S: distance; (S = distance value received so far via messages) parent: process; deficit: int; N(i): set of successors of i; (each message-format: (distance, sender) initially D=, parent = i, deficit =0 Process 0 { send (0,, i is 0 s neighbor deficit = ") }

7 Process i { do (message = :,;)) :<)) if ((deficit > 0) (parent )) send(ack) parent; fi; parent = k; D = S; send(,,) "); deficit = deficit +1; [] (message =:,;)) : ) send(ack) sender; [] ack deficit --; [] (deficit=0) (parent ) send(ack) parent; od } Distance Vector Routing: This algorithm is used in packet routing networks. Idea is: arriving packets at a router would be stored (buffered), and then forwarded to the next router based on the local routing table RT. The next router is always one of the neighbors of the current router. The RT comprises a set of tuples for the current router as {, ;, }, where j: Destination node for the packet k: Next router which is a neighbor of the current router i : The distance to destination from the current router The component Ds are called distance-vector. If the number of routers is, the size of RT is 1. Intially, =1 if "), neighbor of i = 0, if = =, otherwise

8 Periodic updates interval depends on protocol. For RIP in IP, it is 30 secs. Updates are sent to , which would be treated as broadcasts. Full table updates. Split horizon. To ensure no routing loops, updates are not sent on router interface on which the current router s source update has been received. Count to infinity is addressed by limiting hopcount to 15. An alternative scheme. Link-state Routing. DVT reports stale info on distant nodes on their distances based on hops. Linkstate Routing (LSR) reports all { } for node where "). These are link states from node. The recipient of LSR are the neighboring nodes. Node would broadcast its LSR to all its neighbors. Nodes and their links crash and reappear dynamically. So most current LSR has to be sent out. Each LSR report has to have a sequence number and a time stamp to determine its currency. LSR allows creation of a topology of the system graph. LSR does not impose any limit to number of hops a packet can take to reach destination. TTL field of a packet limits it. Core strategy. type LSP: record {Link State Packet}{ sender: process; state: record { neighbor's id; weight(link); seq: int;

9 } time: local time;} define L: LSP((j,w(j,k) for each edge(j,k),0,t); s: int (initially 0) local: array [0..N-1] of LSP local[k]=lsp of node k do toplogy change detected local[i]:=(i, state, s); send(local[i]) s=s+1; time=current_time; [] recv(l) if L.sender = i discard L [] (L.sender!= i)^(l[sender].seq)> local[sender].seq local[sender]:=l; send(l); [] (L.sender i)^(l[sender].time)>local[sender].time local[sender]:=l; send(l); [] else skip; fi od Total number of LSP s moving at any time must be less than E. Each LSP gets stale after a while; thus, effectively each LSP has a TTL field after which it becomes useless Every time a node sends an LSP out, it reduces its TTL field by 1. A node may crash. In that case, its local array of LSPs must be relearnt. Interval Routing For a general graph G = (V,E) the routing table at a node could be ), where =. This could be enormously large.

10 One may be interested to reduce the routing table size at a node. Interval routing is a way to cope with this challenge. Example. A linear interval labelling scheme. We assume each node to have two ports: 0, and 1, or L, and R. A typical layout would look like as follows: 2,5 3,5, ,2 1,3 1,5 In this case every node has at most two entries. Interval routing scheme for tree topologies. Santoro and Khatib [85]. Each node has two ports. Port 0 is connected to higher id nodes, port 1 to lower id nodes. Basic scheme: Label the root node as 0. Traverse the tree in PREORDER mode and label the node reached with successive higher integer. For each node, label the port towards a child by the node label of the child. For each node, label the port towards the parent by C)+ D)+1) EF ", where C): label of the node D): Number of nodes in the subtree below but not Including An example (from Ghosh)

11 For a ring type network, we could use the following model. Let be the number of nodes on the ring. Nodes are labelled starting at some node as 0 with the successive nodes labelling on the ring incremented by 1. For each node, the two ports are labelled with +1) EF and +G FH 2I) EF This is shown on the following ring (Ghosh)