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Issue No.10 - October (1999 vol.10)
pp: 964-975
ABSTRACT
<p><b>Abstract</b>—Unicast in computer/communication networks is a one-to-one communication between a source node <tmath>$s$</tmath> and a destination node <tmath>$t$</tmath>. We propose three algorithms which find a nonfaulty routing path between <tmath>$s$</tmath> and <tmath>$t$</tmath> for unicast in the hypercube with a large number of faulty nodes. Given the <tmath>$n$</tmath>-dimensional hypercube <tmath>$H_n$</tmath> and a set <tmath>$F$</tmath> of faulty nodes, node <tmath>$u\in H_n$</tmath> is called <tmath>$k$</tmath>-safe if <tmath>$u$</tmath> has at least <tmath>$k$</tmath> nonfaulty neighbors. The <tmath>$H_n$</tmath> is called <tmath>$k$</tmath>-safe if every node of <tmath>$H_n$</tmath> is <tmath>$k$</tmath>-safe. It has been known that for <tmath>$0\leq k\leq n/2$</tmath>, a <tmath>$k$</tmath>-safe <tmath>$H_n$</tmath> is connected if <tmath>$|F|\leq 2^k(n-k)-1$</tmath>. Our first algorithm finds a nonfaulty path of length at most <tmath>$d(s,t)+4$</tmath> in <tmath>$O(n)$</tmath> time for unicast between 1-safe <tmath>$s$</tmath> and <tmath>$t$</tmath> in the <tmath>$H_n$</tmath> with <tmath>$|F|\leq 2n-3$</tmath>, where <tmath>$d(s,t)$</tmath> is the distance between <tmath>$s$</tmath> and <tmath>$t$</tmath>. The second algorithm finds a nonfaulty path of length at most <tmath>$d(s,t)+6$</tmath> in <tmath>$O(n)$</tmath> time for unicast in the <tmath>$2$</tmath>-safe <tmath>$H_n$</tmath> with <tmath>$|F|\leq 4n-9$</tmath>. The third algorithm finds a nonfaulty path of length at most <tmath>$d(s,t)+O(k^2)$</tmath> in time <tmath>$O(|F|+n)$</tmath> for unicast in the <tmath>$k$</tmath>-safe <tmath>$H_n$</tmath> with <tmath>$|F|\leq 2^k(n-k)-1$</tmath> (<tmath>$0\leq k\leq n/2$</tmath>). The time complexities of the algorithms are optimal. We show that in the worst case, the length of the nonfaulty path between <tmath>$s$</tmath> and <tmath>$t$</tmath> in a <tmath>$k$</tmath>-safe <tmath>$H_n$</tmath> with <tmath>$|F|\leq 2^k(n-k)-1$</tmath> is at least <tmath>$d(s,t)+ 2(k+1)$</tmath> for <tmath>$0\leq k\leq n/2$</tmath>. This implies that the path lengths found by the algorithms for unicast in the 1-safe and 2-safe hypercubes are optimal.</p>
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 & Distributed Systems, vol.10, no. 10, pp. 964-975, October 1999, doi:10.1109/71.808128