Interval Routing Presented by: Marc Segal

Motivation(1) In a computer network a routing method is required so that nodes can communicate with each other. Normally, an O(N) routing table is used. This method does not scale well for large networks:  Routing is inefficient  Storage space cost for nodes

Motivation(2) One possible solution is prefix-routing:  Node identities are words over an alphabet Σ  Routing is done using the entry in the routing table that has the longest common prefix with the destination address In this presentation we present an alternative method that requires O(d) size tables, where d is the degree of the node

Model A network is an N-node connected graph G(V,E), N=|V| Edges may have a non-negative cost value associated with them. The edges incident at a node are called its links. A node communicates with its neighbours via its links. Links are bidirectional.

Interval Labeling Scheme Nodes and links are labeled with labels from a linearly ordered set {i 0,i 1,…,i N-1 }  The label of node v is marked α( v )  For simplicity, we label nodes 0,1,…,N-1 An Interval Labeling scheme (ILS) is a scheme in which:  All nodes get different labels  At every node each link receives a distinct label Link labels are stored in a sorted cyclic table at each node

The routing routine Routing is performed to the first link α s such that the interval [α s, α s+1 ) contains the destination node. SEND(i,j,m) { if i==j process m else { i=first link α s such that α s ≤j<α s+1 ; SEND(i,j,m) }

Interval Routing – An example SEND(0,3,m) SEND(2,3,m) SEND(1,3,m) But… SEND(0,5,m) SEND(4,5,m) SEND(1,5,m) This ILS is invalid!

Valid ILS An ILS is valid if all messages sent from any source node reach their destination. Theorem: For every network G there exists a valid ILS. We show a DFS based algorithm for finding a valid ILS.

The DFS scheme Start with an arbitrary node, label it 0 Proceed in DFS order, labeling nodes with consecutive numbers. Label links with the label of the node they connect to If a node that admits no forward is reached, and i is the largest node number assigned until now, backtrack and label the link over which we backtrack (i+1) mod N. If backtracking to node u from node v that has a frond at node 0 and i=N-1, label the backtrack link by α( u ) (instead of 0).

The DFS scheme - example

DFS Scheme - correctness Since the graph is connected DFS will find a spanning tree A node labeled u whose maximum descendent is i has : u v w v w-1 i+1 All nodes in the interval [ v,w-1) will be under this subtree All nodes in the interval [ w,i) will be under this subtree

Optimum schemes The DFS labeling scheme is valid but is not necessarily optimal.  For example, A labeling scheme is called optimal if all paths between the nodes in the scheme are the shortest possible paths.

Optimum Schemes(2) The DFS scheme is optimal for Trees and for complete graphs. For rings the following scheme is optimal:  Orient the ring in one direction an label the nodes consecutively from 0 to N-1  For each node i Label the right link by label the left link by

Optimum Schemes(3) A k-labeled ILS is an ILS where:  Each link may receive up to k distinct labels  At every node all the link labels must be distinct Proposition: For any graph with N nodes there exists an (N-1)-labeled ILS that is optimum  For every node i, for every node j label the first link on the shortest path from i to j by j.  This is actually the traditional routing table

Neighbourly schemes An ILS is a neighbourly scheme if it is valid and all messages for a neighbour are delivered directly in one hop. Lemma: The only nodes in a DFS scheme that do not necessarily deliver messages to neighbours in one hop are those nodes k that have fronds to nodes i with i<k  Proof: j k Let j, k be negihbours Case 1: (j,k) is a frond k j Case 2: j is k ’s father Case 2a:If k has no fronds from k to i, i < j b Case 2b:If k has a frond from k to i, i <j i i

Neighbourly schemes(2) Theorem: There exists a two-labeled neighbourly scheme for any arbitrary graph.  Start with a DFS scheme  We only need to concern ourselves with the case in lemma 1  For such a link ( k, j ) double-label it with the label j  We must make sure that messages to to any node t in [ j, k+1 ) are routed correctly.  Since j≤t<k, the message is sent to j and does not return to k

Orderly DFS schemes A DFS scheme is orderly if whenever there is a ‘backward’ frond from node k to node i and x > k, either x must belong to the subtree of the DFS tree with k as a root or x does not belong to the subtree with i as a root. Lemma: in an orderly DFS scheme, if there is a backward frond from node k to node i and the backtrack link at k is labeled b, the backtrack link at i is also labeled b. k i b

Orderly DFS schemes(2) Theorem: There exists a neighbourly ILS for every graph that has an orderly DFS scheme.  Relabel the orderly DFS scheme: for each node k that has backward fronds the smallest of which is i and whose father is j: k j i b i k j i j b  The modified scheme is neighbourly : By the lemma, we only have to consider node k. Messages to j are now delivered in one hop. But so are messages to i since

Orderly DFS schemes(3)  The modified scheme is valid: The only intervals that were changed are [j,k) and [b,i) Messages for nodes in [j,k) were routed through i and then made their way down to j. Now they are routed directly to j. Messages for nodes in [b,i) were routed throuh j and made their way up the tree to i. Now they are routed directly to i Corollary: There exists a neighbourly scheme for any Hamiltonian graph

Scheme indices The index of a node in an ILS is the number of hops it takes a message sent by the node to itself to return to its sender The index of an ILS is the maximum of indices of all the nodes Proposition: a DFS scheme is of index 2  A message from a node i to itself can be routed only to a frond or to the node’s father. In either case, the message returns immediately.

Insertion & connection of schemes Definition: An ILS is sequential if each node i has a link labeled (i+1)mod N Proposition: A valid ILS remains valid after cyclically shifting the labels of all nodes and links by a constant  Because Proposition: There exists an algorithm for updating a valid ILS after unit-link insertion of a node for every network G with a valid sequential ILS

Insertion of nodes If node x is to be inserted to node i, cyclically shift every node and link label until node i becomes node N-1. Label the new node by N. Label the link (N-1,N) by N and link (N,N-1) by

Insertion of nodes (2)  Routing within the old network is unchanged except for node N-1  Because the scheme is sequential node N-1 has a link to 0. If β is the maximum link label at node N-1 before the addition of the new node, the only interval that is affected at node N-1 is [N-1,N).  If β is the maximum link label at node i, then all messages to N will be routed through this link. But so will all messages to N-1. So the messages to N will follow the same route as messages to N-1. From N-1 they will reach N in one hop. Corollary: A DFS scheme remains valid after a unit- link insertion of a node. β N N-1 α

Insertion of nodes(3) The previous algorithm does not work if the new node is connected through multiple links. Definition: A zero-biased ILS is an ILS in which every node has a zero link with the possible exception of the zero node. Unlike sequentiality, the zero-biasedness is not preserved under arbitrary cyclic shifts. Proposition: There exists an algorithm for updating a zero-biased ILS after multiple links insertion of a node if that node has a link to node N-1.  Label the new node N and connect all links, labeling linke (i,N) by N and linke (N,i) by i  The zero link guarantees that the only messages routed to N are messages intended for N  Messages to the new node N will either get routed throug N-1 or directly through one of the new fronds

Connecting two networks Theorem: There exists an algorithm for constructing a valid ILS for the unit-link connection of two arbitrary sequential ILS.  Cyclically shift G 1 so that the node that connects to the second graph is labeled N-1.  Cyclically shift G 2 so that the node that connects to G 1 is labeled 0 and then add N to all node and link labels  Relabel all links with value N in G 2 to 0, except at node N  Label the link between G 1 and G 2 (N-1,N) by N and link (N,N-1) by 0.

Leader election using an ILS We can utilisze the information embedded in the ILS it is possible to solve the leader election problem The approach : nodes send a probing message via the maximum link Lemma: In a ring network with a valid ILS of index 2 if every node sends a probing message via the maximum link, either the maximum node or one of its neighbours must eventually receive two probing messages  If any other node is connected by maximum links to both its neighbours the scheme is not valid  If all nodes’ maximum links are in the same direction the scheme is not of index 2

Leader election using an ILS - a ring Theorem: There is an algorithm for locating the maximum node in a ring network of N nodes with a valid ILS of index 2 that uses at most 2N+1 messages  Every node sends a probing message containing its identification number to one of its neighbours and a regular awake message to the other neighbour.  The node that receives two probing messages compares the identification number of the three nodes and decides if it is the leader or one of its neighbours  Since every node sends 2 messages and there may be one extra message to inform the leader the bound is 2N+1

Leader election using ILS – general graph Theorem: There is a distributed algorithm for locating the maximum node in a general network of N nodes given a valid ILS of index 2. This is achieved in at most 2E+N exchanges of message.  Let every woken node send a probing message through its maximum link and a regular awake message on all other links.  The node then awaits messages from all its neighbours.  Every node computes the maximum in its 1-neighbourhood and sends the computed result to the neighbour via the maximum link  The node that receives its own identification back again as a processed maximum declares itself the leader  There can be at most one such node because the ILS is valid  There is at least one such node because the ILS is of index 2

References “Interval Routing”, J. Van Leeuwen and R.B. Tan, The computer Journal Vo. 30 No.4, p , 1987 “Linear Interval Routing”, Erwin M.Bakker, Jan Van Leeuwen and Richard B. Tan