1. Parallel Routing Bruce, Chiu-Wing Sham

2. Overview Background Routing in parallel computers Routing in hypercube network –Bit-fixing routing algorithm –Randomized routing algorithm

3. Parallel Computer Architectures Parallel computers consist of multiple processing elements interconnected by a specific interconnection topology Example: –linear array –hypercube –mesh –fat tree

4. Interconnection Topologies

5. Routing in Parallel Computers Parallel computers are modeled by directed graphs All interconnections between processors (nodes) occur in synchronous steps Each link can carry at most one unit message (packet) in one step During a step, a node can send at most one packet to each of its neighbors Each node is uniquely identified by a number between 1 and N

6. Permutation Routing Problem A network of N nodes, {1, …, N} Each node i contains one packet v i that should be routed to the destination node Each destination node d(i) for each node i, for 1  i  N, should form a permutation of {1, …, N}, i.e., every node is the destination of exactly one packet

7. Oblivious Routing Algorithm Properties: –A route between each node i and each destination node d(i) is specified –The route between the node i and the node d(i) depends on i and d(i) only

8. Oblivious Routing Algorithm Theorem 1: –For any deterministic oblivious permutation routing algorithm on a network of N nodes each of degree d, there is an instance of permutation routing requiring  ( ) steps Proof: –Paper: C. Kaklamanis, D. Krizanc, T. Tsantilas, “Tight Bounds for Oblivious Routing in the Hypercube”, Pro. of ACM symp. on Parallel alg. & architectures, 1990

9. Hypercube Topology

10. Hypercube Network An n-dimensional hypercube network: –Number of nodes: N = 2 n –Degree: n –The node i with address (i 1, i 2, …, i n )  {0, 1} n and the node j with address (j 1, j 2, …, j n )  {0, 1} n are connected if the hamming distance between (i 1, i 2, …, i n ) and (j 1, j 2, …, j n ) is 1

11. Bit-Fixing Routing Algorithm Algorithm: –Given a destination address d(i) and an intermediate node  (i) –Compare the bits of d(i) with  (i) from left to right –Identify the first bit position at which these two addresses differ –Route this packet to its neighbor n(i) such that  (i) and n(i) differ only in this bit position

12. Bit-Fixing Routing Algorithm Example: –Source: (0, 0, 0, 0, 0, 0) –Destination: (1, 0, 1, 0, 1, 1) –(0, 0, 0, 0, 0, 0)  (1, 0, 0, 0, 0, 0)  (1, 0, 1, 0, 0, 0)  (1, 0, 1, 0, 1, 0)  (1, 0, 1, 0, 1, 1)

13. Bit-Fixing Routing Algorithm Corollary 1: –On an n-dimensional hypercube, there is an instance (e.g. transpose permutation) of permutation routing requiring  ( ) steps for the bit-fixing routing algorithm –It satisfies Theorem 1 where N = 2 n and d = n

14. Bit-Fixing Routing Algorithm Proof: –Let (i.j) be the address of a node, where i and j are two binary strings each of length n/2 and. is the string concatenation operation –Consider the packet stored on node (i.j) is routed to the destination node (j.i) (transpose permutation) and look at the sources where j = 0 only

15. Bit-Fixing Routing Algorithm Proof: –i.0  0.i –if i is odd, the packet must pass through node (1.0) –No. of nodes = 2 n/2 /2 –Only one packet can be routed on the same edge at a time –Lower bound = 2 n/2 /2

16. Randomized Routing Algorithm For i = 1 to N –Route a packet v i by executing the following two steps independently of all the other packets Choose a random intermediate destination t i from {1, …, N}, and route v i from i to t i using bit-fixing algorithm Route v i from t i to its final destination d(i) using bit- fixing algorithm Queuing: FIFO (delay occurs)

17. Randomized Routing Algorithm Lemma 1: –If the bit-fixing algorithm is used to route a packet v i from i to t i and v j from j to t j then their routes do not rejoin after they separate

18. Randomized Routing Algorithm Proof (lemma 1): –Assume k is the node at which the two paths separate and l is the node at which they rejoin –According to bit-fixing scheme, v i and v j from k to l depends only on the bit representations of k and l –v i and v j must follow the same route –Contradict to the assumption

19. Randomized Routing Algorithm Let the route of packet v i follow the sequence of edges p i = (e 1, e 2, …, e k ) Let S be the set of packets (other than v i ) whose routes pass through at least one of {e 1, e 2, …, e k } Lemma 2: –The delay incurred by v i is at most |S|

20. Randomized Routing Algorithm Proof (lemma 2): –Define lag l for any packet w, l=t – j (a packet is ready to follow edge e j at time t –If the lag of v i increase from l to l + 1, some packet should have lag l in front of v i

21. Randomized Routing Algorithm Proof (lemma 2): –Let t j be the last time step at which any packet in S has lag l –A packet w must follow the edge e j where l= t j – j and it must leave at t j +1.

22. Randomized Routing Algorithm Proof (lemma 2): –If the lag of v i reaches l + 1, some packet in S leaves p i with lag l –By lemma 1, the routes of different packets will not rejoin after separate –Each member of S whose route intersects p i is charged at most one delay for v i

23. Randomized Routing Algorithm Define a random variable H ij as: Let delay i be the total delay incurred by v i, then: From linearity of expectation:

24. Randomized Routing Algorithm For an edge e of the hypercube, let the random variable T(e) be the number of routes that pass through e. If p i = (e 1, …, e k ), then: We have:

25. Randomized Routing Algorithm All edges in the hypercube are symmetric –E[T(e l )] = E[T(e m )] for any two edges e l and e m –Total number of edges: Nn –The expected length of each route is n/2 –Expected length of total route is Nn/2 –E[T(e)] = 1/2 for all edges We have:

26. Randomized Routing Algorithm Theorem 2 (Chernoff bound): –Let X 1, X 2, …, X n be the independent Poisson trials such that, for 1  i  n, Pr[X i = 1] = p i, where 0  p i  1. –X = –  = E[X] =

27. Randomized Routing Algorithm

28. We have: By using: Put  = 11:

29. Randomized Routing Algorithm Theorem 3: –With probability at least 1-2 -5n, the packet v i reaches t i in 7n or fewer steps Proof: –Since the total number of packets is 2 n, the probability that any of them have a delay exceeding 6n is less than 2 n* 2 -6n = 2 -5n –The packet requires addition n steps to route from the source to the destination

30. Randomized Routing Algorithm Theorem 4: –A packet reaches its destination in 14n or fewer steps with a probability larger than (1-1/N) Proof: –Phase 2 of the Valiant’s scheme is identical to Phase 1 –Fail probability = 2*2 -5n < 2 -n = 1/N

31. Conclusion Oblivious routing algorithm may give very poor result at some specific cases Randomized routing algorithm can give satisfactory result for all cases with high probability