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Compact Routing in Theory and Practice Lenore J. Cowen Tufts University

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The Theory Problem Consider the following distributed view of shortest path routing on an n node network: u v e1 e2 e3 V e2 T(u)

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The Theory Problem Consider the following distributed view of shortest path routing on an n node network: u e1 e2 e3 V e2 v t T(u) T(t)

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The Theory Problem Consider the following distributed view of shortest path routing on an n node network: u e1 e2 e3 V e2 v t T(u) T(t) Space: O(n log n) bit tables

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The Compact Routing Problem Can we “compact” the local routing tables if we allow short rather than shortest paths? u v e1 e2 e3 V e2 h T(u) F(h,v,T(u))=e

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Compact Routing Scheme The stretch of a path p(u,v) from u to v is | p(u,v)|/d(u,v) where d(u,v) is the length of the shortest path from u to v. The stretch of a routing scheme is the maximum stretch of any path. Want: small stretch and small table size.

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Reality This graph algorithms problem has little to do with this:

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Reality Each router advertises some of its known routes to (some of the) destinations in the network it knows about, to its neighbors. Routers prefer short routes; if a router thinks a route is undesirable, it will fail to pass it on, or depreciate it by artificially inflating its length. Complicated by peering relationships Tables and routes can be arbitrarily long; no stretch bounds

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The Geni Initiative Clean slate redesign: what should the new routing protocols accomplish? - low stretch -diff. QoS - scalable table size -mobile nodes - traffic engineering -updatable - policy - security

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The Geni Initiative Clean slate redesign: what should the new routing protocols accomplish? - low stretch -diff. QoS - scalable table size -mobile nodes - traffic engineering -updatable - policy - security

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Back to Compact Routing Theory Early work: Santoro-Khatib (85) [rings, trees], van Leeuwen-Tan (86) [complete networks, grids], Fredrickson-Janardan (88) [outerplanar graphs, small separators] Compact Routing on Trees Universal compact routing schemes

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Compact Routing on Trees Interval routing: 1 2 3 4 5 6 8 7 9 10 11 12

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Compact Routing on Trees Interval routing: 1 2 3 4 5 6 8 7 9 10 11 12 1-2 3-11 Stretch=1 Space=O(d log n) Header= O(logn)

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Compact Routing on Trees Space hack (C-00): Big nodes > sqrt(n) only remember big children 1 3 4 5 6 8 7 9 10 11

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Compact Routing on Trees Space hack (C-01): Big nodes > sqrt(n) only remember big children 1 3 4 5 6 8 7 9 10 11 Packet header: destination; Name of the last big node on The path from the root to it edge from last big node

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The Compact Routing Problem Can we “compact” the local routing tables if we allow short rather than shortest paths? u v e1 e2 e3 V e2 h T(u) F(h,v,T(u))=e

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Compact Routing on Trees Space hack (C-01): Big nodes > sqrt(n) only remember big children 1 3 4 5 6 8 7 9 10 11 Packet header: destination; Name of the last big node on The path from the root to it edge from last big node Stretch=1 Space=O(sqrt n log n) Header= O(logn)

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Compact Routing on Trees Fraignaud/Gavoille [01] and Thorup/Zwick [01]: Stretch 1 Space O(log n) !!!! Header size O(log n) 2

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Compact Routing on Trees Single source shortest path routing: For each node, identify Its heavy child; the one with most descendents

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Compact Routing on Trees Single source shortest path routing: For each node, identify Its heavy child; the one with most descendents Packet header is the names of all the light edges from root to destination along with their parent names

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Compact Routing on Trees Single source shortest path routing: Packet header is the names of all the light edges from root to destination along with their parent names Can only be O(log n) light edges on any path!!!

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Compact Routing on Trees This works to route down; need to know when to route up: Add interval labeling to know when to go toward the root

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Universal Compact Routing Schemes Work on any graph: -- ABLP-90, AP-92 early schemes -- EGP-98, stretch 5 -- C-00, stretch 3 -- TZ-01, stretch 3 (improved table size) Gavoille and Gengler proved a lower bound of 3 on the stretch on any routing scheme with sublinear-sized routing tables.

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Universal Compact Routing Schemes Define a node’s local neighborhood to be its closest sqrt n nodes. From set cover: choose a set of sqrt n log n landmarks to hit every local neighborhood.

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Universal Compact Routing Schemes Within a local neighborhood, store exact shortest path routing information Outside store routing information to all landmarks; route through a destination’s closest landmark, and then use tree- routing from landmark to destination.

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Universal Compact Routing Schemes Stretch bound follows from the triangle inequality: Suppose d(u,v) >= d(L(v),v) Then d(u,L(v)) + d(L(v),v) <= d(u,v) + d(v,L(v)) + d(L(v),v) <= 3d(u,v) u v L(v)

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Universal Compact Routing Schemes The construction sketched above achieves stretch 3 with tables of average size O(sqrt n log^2 n) to store local information for each node’s closest sqrt n nodes, and routing information for every landmark. u v L(v)

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Universal Compact Routing Schemes Remark: needs undirected Graphs for triangle ineq bound Roundtrip routing (C-Wagner-04 ) measure d(u,v)+d(v,u) u v L(v)

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The Internet Graph The Internet graph is not an arbitrary graph; it is believed to have a power-law topology. Faloutsos^3 -99 (but see also CCGJSW-02)

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The Internet Graph

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Performance of universal schemes on both synthetic models of power-law topologies and “real” maps of the inter-AS graph is much better on average than worst-case guarantees!

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So: shouldn’t we be designing schemes for Internet graphs? Brady-C 06: Compact Routing with Additive Stretch on power-law graphs. Breaks graph into a dense core, of diameter d; the rest of the graph is called the fringe.

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BC Scheme Let e be the number of edges that need to be removed from the fringe to make it into a forest. Theorem. For any unweighted, undirected network, there is a routing scheme that uses O(e log^2 n)-bit headers and routing tables, and has additive stretch d.

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BC Scheme Constructed using a shortest path tree to route in the core, and e spanning trees in the fringe, and a distance labeling to choose between them. Estimates on the Inter-AS graph would give a worst-case additive stretch of 10 (and average stretch much smaller).

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Internet Graph Simulations: Our Results Definition. A power-law graph G=(V,E) is an unweighted, undirected graph whose degree distribution approximates a power- law, i.e it has c vertices of degree 1 c/x vertices of degree x

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Internet Graph Simulations Definition. A power-law graph G=(V,E) is an unweighted, undirected graph whose degree distribution approximates a power-law, i.e it has c vertices of degree 1 c/x vertices of degree x Note: how to generate them randomly is non-trivial; slight differences in the model

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Internet Graph Simulations: Our Results

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Internet Graph Simulations Results

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Internet Graph Simulations: Our Results

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Real Internet Data

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The Geni Initiative Clean slate redesign: what should the new routing protocols accomplish? - low stretch -diff. QoS -mobile nodes - scalable table size -mobile nodes - traffic engineering -updatable - policy - security

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So far: Compact Routing is Static Routing table setup is centralized Node names are topology dependent: -- in interval routing and BC scheme depended on place in the tree -- in Cowen or TZ schemes depended on nearest landmark

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So far: Compact Routing is Static Routing table setup is centralized Node names are topology dependent: -- in interval routing and BC scheme depended on place in the tree -- in Cowen or TZ schemes depended on nearest landmark Separate naming from packet forwarding layers!!!

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Name-Independent Compact Routing v is any unique identifier h is initially empty! u v e1 e2 e3 V e2 h T(u) F(h,v,T(u))=e

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Name dependent routing 000102 03 10 11 1213 20 22 21 23 30 31 32 33

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Name independent routing 2 10 5 15 6 138 1 3 1612 9 14 1147

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Name Independent Routing Discover information about network topology as you “wander around” Idea (Peleg): place a distributed dictionary on top that couples names to new topology-dependent names, and stores them in the packet header.

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Name independent routing 2 10 5 15 6 138 1 3 1612 9 14 1147 9??

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Name independent routing 2 10 5 15 6 138 1 3 1612 9 14 1147 9?? 9 =23!

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Name dependent routing 000102 03 10 11 1213 20 22 21 23 30 31 32 33 9 =23!

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Name-Independent Universal Schemes Table size (soft O) Header size Max stretch ABLP-89O(n^1/2)O(log n)2592 AP-90O(n^1/2)O(log^2 n)1088 ACLRT-03O(n^1/2)O(log^2 n)5 ACLRT-03O(n^2/3)O(log n)5 AGMNT-04O(n^1/2)O(log^2 n)3

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Name-independent simulations

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But is it practical yet? No. Still have to deal with updating routing tables in response to changing network topology. Brady-Cowen protocol has some attractive features that make this easier than for the universal schemes (work in progress)

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The Genie Initiative Clean slate redesign: what should the new routing protocols accomplish? - low stretch -diff. QoS - scalable table size -mobile nodes - traffic engineering -updatable - policy - security

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Back to theory A distance labeling is an assignment of short (polylog n)-bit strings to vertices so that given the labels of two vertices their exact (or approximate) distance can be inferred. Exact distance labeling schemes are known for graphs with small separators or separators with small diameter.

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Theory Not possible for planar graphs (Gavoille, Peleg, x) Theorem (Brady-C 06) Exact distance labeling yield additive stretch compact routing schemes. Other connections between spanners, distance labelings, compact routing??

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Theory: open problems Stretch 1 or additive stretch compact routing schemes for planar graphs: do they exist? More classes of exact additive spanners More special classes of graphs that have exact distance labeling schemes

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Collaborators Marta Arias … and thanks to Arthur Brady the National Science Kevin Fall Foundation for their Dima Krikourov support of this work. Kofi Laing Rajmohan Rajaraman Ori Taka

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