Routing algorithms Routing algorithms establish the path followed

Routing algorithms • Routing algorithms establish the path followed by each message or packet. • Routing algorithms for wormhole routing are also valid for other switching techniques. • Routing algorithms can be classified according several criteria, such as adaptivity, progressiveness and minimality. ＮＣ２ (No. 5) 1

Taxonomy of routing algorithms Routing algorithms Unicast Routing source Routing table lookup multicast Routing distributed (incremental) Routing Finite-state machine (algorithmic routing) ＮＣ２ (No. 5) 2

Adaptivity (1/2) • Deterministic routing algorithm always supply the same path between a given source / destination pair. • Adaptive routing algorithms use information about network traffic and/or channel status to avoid congested or faulty regions. • Adaptive routing can be classified as progressive or backtracking. ＮＣ２ (No. 5) 3

Adaptivity (2/2) • Minimal (profitable) or nonminimal: Forward packets closer to their destination or not • Fully adaptive or partially adaptive: All the packets can be adaptively routable, or A part of the set of packets (e. g. classified by their destination) can be adaptively routed, and the rest of them cannot. • In practical, adaptive routing can be beaten by deterministic algorithms on some metrics. ＮＣ２ (No. 5) 4

Source routing (1/2) • The source node specifies the routing path on the basis of a deadlock-free routing algorithm. • The computed path is stored in the packet header, being used at intermediate nodes to reserve channels. • It has been mainly used in computer networks with irregular topologies (e. g. Myrinet). ＮＣ２ (No. 5) 5

Source routing (2/2) • When the long header is transmitted through the network, it comsumes network bandwidth. • Because the shorter header is preferable, street-sign routing uses the header contains only the node address at which the turn will take place and the direction of the turn (e. g. i. Warp). ＮＣ２ (No. 5) 6

Source routing (examples) d e f c a Naive source routing b c d e f b Street-sign routing b(upper) d(right) ＮＣ２ (No. 5) Output port encoding E N N E E X 7

Distributed routing • A path of a packet is determined in a distributed manner while the packet travels across the network. • For efficiency reasons, most hardware routers use distributed routing. • The header requires the destination address and a few implementation-dependent control bits (very compact). ＮＣ２ (No. 5) 8

Table-lookup routing • A routing table, which contains the number of entries equal to the number of nodes in the network, is placed at each node. • In the case of source routing, a table entry indicates the whole path to reach the destination node. • In the distributed routing, each table entry indicates which outgoing channel should be used to forward the packet toward its destination. (e. g. interval routing in Inmos T 9000 transputer and its router C 104) ＮＣ２ (No. 5) 9

Node-Table routing • Table-based routing can be performed by storing the routing table in the routing nodes rather than the terminals. • Node-table routing is more appropriate for adaptive routings because it enables the use of perhop network state information to select among several possible next hops at each stage of the route. • Performing a lookup at every step of the route, rather than just at the source, significantly increases the latency for a packet to pass through the router. ＮＣ２ (No. 5) 10

Interval routing Y+ Y- 0 4 8 12 1 5 9 13 2 6 10 14 3 X- 7 11 Routing table for node 5 X+ 8 -15 X- 0 -3 Y+ 4 Y- 6 -7 15 X+ ＮＣ２ (No. 5) 11

Finite-state machine routing • Most parallel computers use topologies that can be decomposed into orthogonal dimensions (e. g. mesh, hypercube tree). • In these topologies, it is possible to use simple routing algorithms based on finite-state machines. • For example, dimension-order routing routes packets by crossing dimensions in increasing order, nullifying the offset in one dimension before routing in the next one. • Such algorithmic routing is often more efficient in terms of both area and speed than table-based routing. ＮＣ２ (No. 5) 12

Dimension-order routing (e-cube routing) Y+ 0 1 2 3 4 5 6 7 8 9 10 11 Y- 12 13 14 15 X- X+ ＮＣ２ (No. 5) 13

Deterministic ruoting • It always provides the same output channel for the same destination. • Routing decision is independent of (i. e. oblivious to) the state of the network. • It became popular when wormhole switching (compact and fast hardware router is desirable) was invented. ＮＣ２ (No. 5) 14

X-Y routing for 2 -D meshes Inputs: coordinates of current node (Xc, Yc) destination node (Xd, Yd) Xo=Xd-Xc; Yo=Yd-Yc; If Xo<0 then channel=X-; If Xo>0 then channel=X+; If Xo=0 and Yo<0 then channel=Y-; If Xo=0 and Yo>0 then channel=Y+; If Xo=0 and Yo=0 then channel= internal; ＮＣ２ (No. 5) 15

E-cube routing for hypercube (1/2) Inputs: addresses of current node current and destination Dest Offset = current ＾ Dest; If Offset = 0 then channel = internal; Else channel = first. One(Offset); where first. One(Offset) returns the position of the first bit set to one ＮＣ２ (No. 5) 16

E-cube routing for hypercube (2/2) 0010 0101 0001 1110 1010 0011 0100 0000 1111 0110 1101 1001 0000 →　0001 →　0011 → 0111 →　1111 ＮＣ２ (No. 5) 17

Dimension-order routing for unidirectionial 2 -D tori (1/2) C 00 00 10 20 01 11 21 02 12 22 C 01 C 10 ＮＣ２ (No. 5) 18

Dimension-order routing for unidirectionial 2 -D tori (2/2) Inputs: coordinates of current node (Xc, Yc) destination node (Xd, Yd) Xo=Xd-Xc; Yo=Yd-Yc; If Xo<0 then channel=C 00; If Xo>0 then channel=C 01; If Xo=0 and Yo<0 then channel=C 10; If Xo=0 and Yo>0 then channel=C 11; If Xo=0 and Yo=0 then channel= internal; ＮＣ２ (No. 5) 19

Partially adaptive routing • Partially adaptive routing represent a trade-off between flexibility and cost. • Try to approach the flexibility of fully adaptive routing at the expense of a moderate increase in complexity with respect to deterministic routing. • Most of them rely upon the absence of cyclic dependencies between channels to avoid deadlock. ＮＣ２ (No. 5) 20

Planar-adaptive routing (1/2) • One of the partially adaptive routing for ndimensional meshes and hyperecubes[Chien and Kim]. • It provides adaptivity in only two dimension at a time. • A packet is routed adaptively in a series of 2 -D planes. ＮＣ２ (No. 5) 21

Planer-adaptive routing (2/2) destination A 0 Fully adaptive A 1 A 0 A 1 A 2 source Planar-adaptive ＮＣ２ (No. 5) 22

Turn model (1/3) • It is the minimal or nonminimal partially adaptive routing[Glass and Ni, 1992]. • It prohibits the turns that form a cycle. • Deadlock cab be avoided by prohibiting just enough turns to break all the cycles. ＮＣ２ (No. 5) 23

Turn model (2/3) North-last west-fast Negative-first X-Y dimension-order ＮＣ２ (No. 5) 24

Turn model (3/3) X X X X ＮＣ２ (No. 5) 25

Fully adaptive routing • A fully adaptive routing algorithms can use all the physical paths from source to destination. • It may restrict the use of virtual channels to avoid deadlock. • Some deadlock recovery algorithms have no restriction on the use of virtual channels (true fully adaptive). ＮＣ２ (No. 5) 26

Hop algorithms • Gopal proposed several fully adaptive minimal routing algorithms which are known as hop algorithms[Gopal, 1985]. • They use structured buffer pools to avoid deadlock. • If the network diameter is D, a minimum of D+1 buffers per node are required. • A packet in a buffer class i will be stored in a buffer class i+1 in the next node (positive-hop algorithm). ＮＣ２ (No. 5) 27

Structured buffer pool (1/2) node 0 node 1 node 2 node 3 Positive-hop Label 0 Label 1 negative-hop ＮＣ２ (No. 5) 28

Structured buffer pool (2/2) • Structured buffer pool was designed for store-andforwarding networks using central buffers. • Replacing the central buffers by virtual channels, the hop algorithms can suit for wormhole switching. • Resulting algorithms can be fully adaptive minimal routing for any topology at the expense of high number of virtual channels. ＮＣ２ (No. 5) 29

Virtual channel balance • A hop algorithm produces an unbalanced use of virtual channels because all the packets start from virtual channel 0, however few packets take the maximum number of hops. • It can be improved by giving each packet a number of bonus cards (MAX hops – actual hops). • At each node, a packet has some flexibility, the range of virtual channels equal to the bonus cards+1, in the selection of virtual channels. ＮＣ２ (No. 5) 30

Virtual networks • A network is split into several virtual networks using virtual channels (VCs). • They are disjoint channels sets. • Each virtual network can be defined in such a way that the corresponding routing function has no cyclic dependencies between channels. • Packets transfers between virtual networks are restricted in such a way that deadlock is avoided. ＮＣ２ (No. 5) 31

Linder and Harden’s adaptive algorithm • Split n-dimensional meshes into 2 n-1 virtual networks. • Each virtual network has channels in both directions in the 0 th dimension and only one direction in the remaining n-1 dimensions. • The number of VCs across each physical channel is 2 n-1, except for dimension 0 which has 2 n. ＮＣ２ (No. 5) 32

Virtual networks for a 2 -D mesh 0 1 2 3 4 5 6 7 8 9 a b c d e f Virtual network 0 Virtual network 1 ＮＣ２ (No. 5) 33

Turns allowed on each virtual network Virtual network 0 Virtual network 1 It is called double-x routing. ＮＣ２ (No. 5) 34

Duato’s protocol (1/4) • Duato proposed design methodologies for fully adaptive routing [Duato, 1991]. • They usually supply near-optimal minimal or nonminimal routing algorithms. • When restricted to minimal paths, the resulting algorithms are referred as Duato’s protocol (DP). ＮＣ２ (No. 5) 35

Duato’s protocol (2/4) 1. For an interconnection network I, select one of the existing routing functions R 1. It must be deadlock-free and connected. It can be deterministic or adaptive. Let C 1 be the set of channels for R 1. 2. Split each physical channel into a set of additional VCs. Let C be the set of all the VCs. Define the new routing function as; R(x, y) = R 1(x, y)∪（Cxy∩（C-C 1), where Cxy is the set of output channels from node x to node y ＮＣ２ (No. 5) 36

Duato’s protocol (3/4) • R(x, y) is the new routing function which can use any of the new channels or, alteratively, the channels supplied by R 1. • For wormhole switching, verify that the extended channel dependency graph for R 1 is acyclic. If it is, the routing function is valid. Otherwise, it must be discarded, returning to step 1. ＮＣ２ (No. 5) 37

Duato’s protocol (4/4) • For step 1, deterministic or turn model should be preferred for R 1. • Step 2 indicates how to add more VCs to the network and how to define a new fully adaptive routing function. • The last verification step is not required for VCT and SAF, but it is only required for wormhole switching. Many minimal functions pass this step, but few nonminimal functions pass it. ＮＣ２ (No. 5) 38

An example of DP (1/2) • For 2 -D mesh using SAF, select XY dimensionorder routing as R 1. • Split each physical channel ci into ai and bi. • Let C 1 be the set of bi channels (C-C 1=ai). • R supplies all the outgoing ai channels from the current node, also supplying one b channel according to dimension-order routing. • When restricted to minimal paths, R is also deadlock-free for wormhole switching. ＮＣ２ (No. 5) 39

An example of DP (2/2) 0 b 1 a a b 2 3 4 5 6 7 8 9 a b c d e f ＮＣ２ (No. 5) 40

History (routing) • Early networks performed routing in software. • For example, processors explicitly relay communications in the Solomon machine (1962). • By the mid-1980 s, router chips, such as for tori, were developed. • Those routers forward messages throw networks without processor intervention. • Today, the software switching that requires terminal’s resources is rarely used. ＮＣ２ (No. 5) 41

History (adaptive routing) • Many adaptive routing algorithms give poor worst -case performance because they use only local network state resulting in global imbalance. • Duato’s protocol enabled a family of adaptive routing algorithms including the one used in Cray T 3 E and IBM Bluegene/L. • Minimal adaptive routing on a fat tree was used in the CM-5 and Quadrics Q’s. Net. ＮＣ２ (No. 5) 42

History ( deadlock-prevention) • Early work on deadlock-free interconnection networks identified the technique of enumerating network resources and traversing these resources an increasing order. • Linder and Harden developed a method that makes arbitrary adaptive routing deadlock-free but at the cost of a number of virtual channels (1991). • Duato’s protocol has been used in several networks such as Cray T 3 E and the Alpha 21364. ＮＣ２ (No. 5) 43

Bibliographic notes • Table-based source routing is popular in implementation due to its flexibility (ex. IBM SP 1 and SP 2). • Larger networks, such as SGI Spider, Infiniband, and the Internet, adopt node-tables to reduce the table size. • Hierarchical routing and interval routing are also the approaches for reducing table size. • CAMs are common in the implementation of routing tables. ＮＣ２ (No. 5) 44