Compact Routing with Minimum Stretch

1. Compact Routingwith Minimum Stretch Kei Takahashi

2. Introduction • In distributed networks, message relaying is inevitable • All-to-all connections are physically impossible • Nodes can dynamically appear, move, and disappear • Some routing tactics are possible • Broadcast • Random relaying • Routing table V1 V3 V0 V2 Message To V3

3. Routing tables • A routing table knows the next hop to each node • The information need not be optimal • Stray messages can be rerouted at relay nodes • There are trade-off of the table size and efficiency • A complete routing table occupies large memory • Efficient routing is not always possible with partial routing tables V1 V3 Port 1 Port 0 Port 2 Port 1 V0 V2 V1 : Port 0 V2 : Port 1 V3 : Port 1 V0 : Port 0 V1 : Port 1 V3 : Port 2 Port 0 Message To V3

4. Naïve (complete) routing • Each node holds the next hop to all nodes • Always assures optimal routing • Table size is O(n log(n)) at each node • n : number of nodes • size(table) = size(column) * (n-1) • size(column) = size(nodename) + size(portname)≤ O(log(n)) log(n) V1 : Port 0 V2 : Port 1 V3 : Port 1 V4 : Port 0 V5 : Port 1 V6 : Port 1 … Port 0 Port 1 V0 n-1

5. The proposed method • Table size ≤ O(n2/3 log4/3 (n)) • Better than O(n log(n)) in the naïve method • In most cases, table is smaller than this bound • Maximum stretch ≤ 3 • Transfer cost is 3 times more than optimal case • In most cases, the cost is smaller V1 Stretch = 2 Cost = 5 5 6 Cost = 10 V0 V2 4

6. Agenda • Introduction • Problem definition • Basic Idea • Landmarks • Re-labeling • Routing • Details • Selecting landmarks • Constructing Routing tables • Proofs • Table size • Max stretch < 3 • Conclusion

7. Port V1 Port V2 Problem definition • Nodes are connected with undirected edges having weight • Node names can be relabeled as long as length ≤ O(log(n)) • Hostname, IP, processor number • It is freely changed in programs • Edges are identified with “port name” • Port names differ in each node, and cannot be relabeled • “Port” is something like a UNIX socket A302 A301 V1 NG V3 OK Port 1 Port 0 Port 2 A303 A300 Port 1 V0 V2 Port 0

8. label port V3 : V10 : -------------- V1 : Landmarks Near nodes Basic idea (1) : landmarks • Select global “Landmarks” • They are near to many neighbor nodes • Each node sets its own landmark as nearest one • Routing tables have columns for… • On landmarks : other landmarks (no entry for neighbors) • On the other nodes : landmarks and neighbors whose shortest path to their landmark goes through that node • In case of v0, v1 applies to this criteria Landmark v0 v1 v7 v8 v2 v3 v4 v9 v10 v11 Landmark v5 v6 v12 v13

9. Basic idea (2) : re-labeling • Re-label node names • v → (v, lv, elv(v)) • lv : Ladmark of v • elv(v) : Next-hop from lv to v • In a natural way, elv(v) is written in the routing table of lv, but then routing table of lv gets larger label port v0,v3  v0 v1,v3  v7,v10  v8,v10  v1 v7 v8 (V3, V3, -): (V10,V10,-): -------------- (V1,V3,) : V3 : V10 : -------------- V1 : Landmarks v2,v3  v4,v3  v2 v3 v4 v9 v10 v11 v3,v3 - v9,v10  v11,v10  v10,v10 - Near nodes v5,v3  v6,v3  v6 v5 v12 v13 v12,v10  v13,v10 

10. Basic idea (3) : routing algorism • The node u routes a message to (v, lv, elv(v)) as follows: • if u == lv : send to elv(v) • if v is in u’s table: send according to the table • otherwise: send to its landmark (lv)’ s next hop • Let’s see the node (v0, v3, ↑) in the following figure • (v1,v3,↓) : check the table and send to → • (v12, v10, ↓) : try to send to (v10,v10,-) v0,v3  v1,v3  v7,v10  v8,v10  (v3, v3, -): (v10,v10,-): -------------- (v1,v3,) : v2,v3  v3,v3 - v6,v3  v9,v10  v10,v10 - v11,v10  v5,v3  v6,v3  v12,v10  v13,v10 

11. An example of routing • A message from (v0, v3, ↑) to (v12, v10, ↓) • (v0, v3, ↑) Checks the entry of landmark (v10,v10,-) : ↓ • (v3, v3, -) Checks the entry of landmark (v10,v10,-) : → • (v6, v3, →) Checks the entry of landmark (v10,v10,-) : → • (v9, v10, ←) Checks the entry of landmark (v10,v10,-) : → • (v10, v10, -) Checks the label of the destination :↓ • (v13, v10, ↓) Checks the entry of (v12, v10, ↓) : ← • (v12, v10, ↓) Finally receives the message • Longer than optimal, but stretch < 3 v0,v3  v1,v3  v7,v10  v8,v10  v2,v3  v3,v3 - v6,v3  v9,v10  v10,v10 - v11,v10  Optimal route v5,v3  v6,v3  v12,v10  v13,v10  Obtained route

12. 高速道路のアナロジー • Naïve routing : 全ての目的地が全ての案内板に書かれている • 現実的に不可能 • インターチェンジをランドマークにする • あらゆるインターチェンジまでの経路は全ての案内板に書く • インターチェンジの案内板には何も書かれていない • 柏キャンパスを「柏インターを北」と覚える • (柏キャンパス, 柏インター, 北)とre-labeling • 遠くからのルーティング • 「柏インター」を目指して、案内板に沿って進む • 柏インターを北に進むと、柏キャンパスが載った案内板が出現する • 柏の近くからは、柏インターを経由せず到達できる

13. Agenda • Introduction • Problem definition • Basic Idea • Landmarks • Re-labeling • Routing • Details • Selecting landmarks • Constructing Routing tables • Proofs • Table size • Max stretch < 3 • Conclusion

14. Dijkstra algorithm • Gives shortest ways from one node to every nodein an undirected graph • For each reachable node, calculate min-cost • Take one node having min-cost • The adding can be stopped, then the n-nearest subset is obtained → truncated-Dijkstra method From v2 Costs : {v1 => 1, v3 => 3} Take v1 Costs : {v0 => 2, v3 => 3} Take v0 Costs : {v3 => 3} Take v3 V0 4 1 V1 V2 V3 1 3

15. Neighbors set • Neighbors set (Bv) : reachable from v • Sort the nodes according to the cost from v(if some costs are the same, sort by their name label) • Take nα nodes from the nearest one → Bv (v’s ball) • Reversed-neighbors set (Rv) : reachable to v • A set of nodes which have v as their neighborRv ← {y | y ∈ Bv} V0 nα = 3 2 1 3 V1 V2 V3 1

16. Requirements for landmarks • Only a limited number of nodes can be landmarks • They should be close to many other nodes • Adding landmarks makes every routing table bigger • If v is on the way from many nodes to their landmark, v should be their landmark instead • Otherwise the table on v is too big Not suitable for a landmark V3 : V10 : -------------- V0 : V1 : V2 : V4 : V5 : V6 : v1 v0 v7 v8 v2 v3 v4 v9 v10 v11 v5 v6 v12 v13

17. A) Landmarks should be famous • Choose “famous” nodes from set of Bvs → D • Easily reached from any v • ∃D ⊂ V such that • |D| = O(n1-α log n) • ∀v ∈V , D ∩ Bv ≠ φ • An algorithm is known to obtain D(called Extending dominating set”) nα = 3 V = {v0, v1, v2, v3} D = {v2} |D| = 1 Bv0 ∩ D = v2 Bv1 ∩ D = v2 Bv2 ∩ D = v2 Bv3 ∩ D = v2 V0 2 1 3 V1 V2 V3 1

18. B) Famous nodes should be landmarks • ∃C ⊂ V such that • ∀c ∈ C , |Rc| ≥ n(1+α)/2 • C is easy to be computed when Rv is given • C always satisfies |C| ≤O(n(1+α)/2) • Σ(∀v ∈ V) Rv = n*nα ≥ Σ(∀c ∈ C) Rc nα = 3 V = {v0, v1, v2, v3} C = {v1, v2} Rv1 = 4 > 3.6 = n(1+α)/2 Rv2 = 4 > 3.6 = n(1+α)/2 V0 2 1 3 V1 V2 V3 1

19. Selecting landmarks • Two sets are obtained • ∃D ⊂ V such that • |D| = O(n1-α log n) • ∀v ∈V , D ∩ Bv ≠ φ • ∃C ⊂ V such that • ∀c ∈ C , |Rc| ≥ n(1+α)/2 • C always satisfies |C| ≤O(n(1+α)/2) • Landmark set is derived by simply joining them • L = C ∪ D • L = O(n1-α log n) + O(n(1+α)/2) V0 2 1 3 V1 V2 V3 Landmarks 1

20. Constructing routing tables // Calculate paths to nα-shortest nodes from v For each v ∈ V, perform truncated-Dijkstra(nα) // Here, less than nα nodes are nearer than u from v For each u reached from v:// If ↓ is true, the best route is given by using that landmark If no landmark is on the path from v to u: store(v, eu(v)) at u // Calculate shortest paths from landmarks to every node For each l ∈ L, perform full-Dijkstra(nα) // v appearedFor each v ∈ V Store (l, eu(l)) at u

21. A set of landmarks (L) is O(n1-α log n + n(1+α)/2) Table size = size(label of node) * size(columns) Size(label of node) = O(log(n)) Size(columns) ≤ |L| + nα = O(n1-α log n + n(1+α)/2 + nα) ∴ Table size = O((n1-α log n + n(1+α)/2 + nα) log n) Search α which minimizes table size, and get α = 1/3 + (2 log log n) / (3 log n)Table size = O(n2/3 log4/3 (n)) Proof : table size = O(n2/3 log4/3 (n))

22. Proof : max stretch ≤ 3 • Theorem : route(u, v) ≤ 3 * d(u, v) • route(u,v) : the cost of the route from u to v given by this algorism • d(u,v) : the smallest cost from u to v • L : set of landmarks • X : points on the shortest path among u and v • If d(u,v) < d(lv, v): • d(x, v) < d(u, v) < d(lv, v) • lv ∈ Bv, so (u ∈ Bv) and (v ∈ Ru) and (∀x ∈ X, x ∈ Bv, x ∈ Ru) • X ∩ L = φ • Otherwise v should take that node as lv (⊥) • Then, u should have (v, eu(v)) on its routing table, so as ∀x • The message will not pass lv (⊥) (As a result, route(u, v) is always optimal in this case) u lv x v

23. Proof : max stretch ≤ 3 (Cont.) • Otherwise : d(lv,v) ≤ d(u,v) • Column (v, eu(v)) is not on u’s table • Otherwise (v ∈ Ru) so any x’s table has a column(v, ex(v)) • The message will not pass lv (⊥) • Y : nodes on the shortest path among u and lv • If any y does not have a column (v, ey(v)) on its table : • route (u, v) = d(u, lv) + d(lv, v) • Every y refers the column for lv, which gives the shortest path to lv • d(u,lv) ≤ d(u, v) + d(lv, v) (triangle equation) • d(u,lv) ≤ 2 * d(u, v), so route(u,v) ≤ 2 * d(u,v) + d(u,v) ≤ 3 * d(u,v) • Otherwise : (skip) • (It can be proved that route(u,v) ≤ d(u, lv) + d(lv, v) ) u y lv x v

24. Conclusion • A routing strategy was given which assures • Small table size : O(n2/3 log4/3 (n)) • Max stretch is less than 3 • It consists of 4 parts: • Selecting landmarks • Re-labeling nodes • Create routing tables • Routing algorism • For practical situations, alternative methods are proposed. They offer smaller table size and better routing on average