This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Unicast in Hypercubes with Large Number of Faulty Nodes
October 1999 (vol. 10 no. 10)
pp. 964-975

Abstract—Unicast in computer/communication networks is a one-to-one communication between a source node $s$ and a destination node $t$. We propose three algorithms which find a nonfaulty routing path between $s$ and $t$ for unicast in the hypercube with a large number of faulty nodes. Given the $n$-dimensional hypercube $H_n$ and a set $F$ of faulty nodes, node $u\in H_n$ is called $k$-safe if $u$ has at least $k$ nonfaulty neighbors. The $H_n$ is called $k$-safe if every node of $H_n$ is $k$-safe. It has been known that for $0\leq k\leq n/2$, a $k$-safe $H_n$ is connected if $|F|\leq 2^k(n-k)-1$. Our first algorithm finds a nonfaulty path of length at most $d(s,t)+4$ in $O(n)$ time for unicast between 1-safe $s$ and $t$ in the $H_n$ with $|F|\leq 2n-3$, where $d(s,t)$ is the distance between $s$ and $t$. The second algorithm finds a nonfaulty path of length at most $d(s,t)+6$ in $O(n)$ time for unicast in the $2$-safe $H_n$ with $|F|\leq 4n-9$. The third algorithm finds a nonfaulty path of length at most $d(s,t)+O(k^2)$ in time $O(|F|+n)$ for unicast in the $k$-safe $H_n$ with $|F|\leq 2^k(n-k)-1$ ($0\leq k\leq n/2$). The time complexities of the algorithms are optimal. We show that in the worst case, the length of the nonfaulty path between $s$ and $t$ in a $k$-safe $H_n$ with $|F|\leq 2^k(n-k)-1$ is at least $d(s,t)+ 2(k+1)$ for $0\leq k\leq n/2$. This implies that the path lengths found by the algorithms for unicast in the 1-safe and 2-safe hypercubes are optimal.

[1] J.C. Bermond, “Interconnection Networks,” Discrete Applied Math, special issue, 1992.
[2] J.C. Bermond, N. Homobono, and C. Peyrat, “Large Fault-Tolerant Interconnection Networks,” Graphs and Combinatorics, vol. 5, 1989.
[3] J. Bruck, R. Cypher, and D. Soroker, "Tolerating Faults in Hypercubes Using Subcube Partitioning," IEEE Trans. Computers, vol. 41, no. 5, pp. 599-605, May 1992.
[4] M.S. Chen and K.G. Shin, "Depth-First Search Approach for Fault-Tolerant Routing in Hypercube Multicomputers," IEEE Trans. Parallel and Distributed Systems, vol. 1, no. 2, pp. 152-159, Apr. 1990.
[5] H. Chernoff, “A Measure of Asymptotic Efficiency for Tests of a Hypotheses Based on the Sum of Observations,” Ann. Math. Stat., vol. 23, pp. 493-509, 1952.
[6] P. Erdös and I. Spencer, Probabilistic Method in Combinatorics. New York: Academic Press, 1974.
[7] A.H. Esfahanian, “Generalized Measures of Fault Tolerance with Application to n-Cube Networks,” IEEE Trans. Computers, vol. 38, no. 11, pp. 1,586-1,591, 1989.
[8] J.M. Gordon and Q.F. Stout, “Hypercube Message Routing in the Presence of Faults,” Proc. Third Conf. Hypercube Concurrent Computers and Applications, pp. 318-327, Jan. 1988.
[9] Q. Gu and S. Peng, “Optimal Algorithms for Node-to-Node Fault Tolerant Routing in Hypercubes,” The Computer J., vol. 39, no. 7, pp. 626-629, 1996.
[10] Q.P. Gu and S. Peng, “k-Pairwise Cluster Fault Tolerant Routing in Hypercubes,” IEEE Trans. Computers, vol. 46, no. 9, pp. 1042-1049, Sept. 1997.
[11] Q. Gu and S. Peng, “Cluster Fault Tolerant Routing in Hypercubes,“ Proc. 1998 Int'l Conf. of Parallel Processing, pp. 252-255, 1998.
[12] D.F. Hsu, “Interconnection Networks and Algorithms,” Networks, vol. 23, no. 4, 1993.
[13] D.F. Hsu, “On Container with Width and Length in Graphs, Groups, and Networks,” IEICE Trans. Fundamental of Electronics, Information, and Computer Sciences, vol. E77-A, no. 4, pp. 668-680, 1994.
[14] Y. Lan, “A Fault-Tolerant Routing Algorithm in Hypercubes,“ In Proc. 1994 Int'l Conf. Parallel Processing, pp. III163-III166, 1994.
[15] S. Latifi, “Combinatorial Analysis of the Fault Diameter of the$n$-Cube,” IEEE Trans. Computers, vol. 42, no.1, pp. 27-33, 1993.
[16] S. Latifi, M. Hedge, and M. Naraghi-Pour, “Conditional Connectivity Measures for Large Multipprocessor Systems,” IEEE Trans. Computers, vol. 43, no. 2, pp. 218-222, 1994.
[17] T.C. Lee and J.P. Hayes,“A fault-tolerant communication scheme for hypercube computers,” IEEE Trans. Computers, vol. 41, no. 10, pp. 1,242-1,256, Oct. 1992.
[18] M. Peercy and P. Banerjee, “Optimal Distributed Deadlock-Free Algorithms for Routing and Broadcasting in Arbitrarily Faulty Hypercubes,“ Proc. 20th Int'l Symp. on Fault-Tolerant Computing, 1990.
[19] C.S. Raghavendra, P.J. Yang, and S.B. Tien, “Free Dimensions—An Efficient Approach to Archieving Fault Tolerance in Hypercube,“ IEEE Trans. Computers, vol. 44, pp. 1,152-1,157, 1995.
[20] Y. Saad and M. Schultz, "Topological Properties of Hypercubes," IEEE Trans. Computers, vol. 37, no. 7, pp. 867-872, July 1988.
[21] S.-B. Tien and C.S. Raghavendra,“Algorithms and bounds for shortest paths and diameter in faultyhypercubes,” IEEE Trans. Parallel and Distributed Systems, pp. 713-718, June 1993.
[22] J. Wu, “Safety Levels—An Efficient Mechanism for Achieving Reliable Broadcasting in Hypercubes,” IEEE Trans. Computers, vol. 44, no. 5, pp. 702-706, May 1995.
[23] J. Wu, "Unicasting in Faulty Hypercubes Using Safety Levels," IEEE Trans. Computers, vol. 46, no. 2, pp. 241-247, Feb. 1997.

Index Terms:
Fault tolerance, interconnection network, off-line routing algorithm, unicast, hypercubes.
Citation:
Qian-Ping Gu, Shietung Peng, "Unicast in Hypercubes with Large Number of Faulty Nodes," IEEE Transactions on Parallel and Distributed Systems, vol. 10, no. 10, pp. 964-975, Oct. 1999, doi:10.1109/71.808128
Usage of this product signifies your acceptance of the Terms of Use.