Unicast in Hypercubes with Large Number of Faulty Nodes

• Authors: Qian-Ping Gu;Shietung Peng
• Venue: IEEE Transactions on Parallel and Distributed Systems
• Year: 1999

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.