3/12/2013 Computer Engg, IIT(BHU) 1

HYPERCUBE ALGORITHMS-3

●Model: All-port, full-duplex, WH ● Lower bound: ● Intermediate research results: –Recursive tiling approach–Dilated-diagonal-based approach – Both, applicable only on two-dimensional tori.

●Assume torus

● Look at the torus as on mosaic of constant patterns

(5 ,5 )k kT

Recursive tiling approach

5 B-Patterns (k=1)

A-Pattern (k=1)

Recursive tiling approach

●Observation: Torus for any k can be represented as a single A-Pattern. (every A-Pattern consists of 5 B-Patterns).

●Define for nodes that transmit at any odd round i, for nodes that transmit at any even round i. (the broadcaster has A-sons(B-sons) for every odd(even) i. His A-sons has B-sons for every even i, and A-sons for i=3,5,…k(or k-1). And so on…)

(5 ,5 )k kT

iA son

Recursive tiling approach●The Algorithm (for any node t ) : –toSend = true; Msg contains value - for broadcasting node,

– toSend = false; Msg is null - for all other nodes

–At round i ( ) do:●If toSend = true and i is odd then–send Msg to all (5) ●Else If toSend = true and i is even then–send Msg to all (5)●Else if received message M` then–Msg M`–toSend true–End of round

1 2i k

iA sons

iB sons

Recursive tiling approach

●Complexity analysis:●# of rounds: 2k, (reminder: )●# of transmissions: every node receives message only once, therefore it is

●The algorithm is optimal!

●Note: The algorithm is applicable only to tori

22 2 1log 5 2k k

25 1k

(5 ,5 )k kT

Dilated-diagonal-based approach●Assume torus.●The algorithm consists out of 3 Phases.●Phase I: Broadcast the packet into all rows so that each row contains exactly one packet. (by recursive splitting the torus into 5 horizontal strips, sending the packet using XY routing).●Phase I takes rounds

(5 ,5 )k kT

5log N k

Dilated-diagonal-based approach

Dilated-diagonal-based approach

Dilated-diagonal-based approach

Dilated-diagonal-based approach●Phase II: Align the packets to the main diagonal in all rows in parallel.

●Phase II takes single round.

Dilated-diagonal-based approach●Phase III: Decompose the torus into 5 diagonal bands (of width ).–The main diagonal is in the middle diagonal of the first one.–Every node on middle diagonal sends 4 packets to the counterpart nodes on the middle diagonals of other 4 bands.–All packets are send on link-disjoint paths.–Recursively divide each band into 5 bands and continue with the algorithm.

●Phase III takes rounds.

15k

5log N k

Dilated-diagonal-based approach●Complexity analysis:●# of rounds: 2k+1● # of transmissions: every node receives message only once, therefore it is ●The algorithm is almost optimal!●Note: The algorithm can be generalized and applied to arbitrary 2-D torus, in which case it requires at most 5 rounds more than the lower bound.

25 1k