A Graph Partitioning Approach to Sequential Diagnosis

• Authors: Sanjeev Khanna;W. Kent Fuchs
• Venue: IEEE Transactions on Computers
• Year: 1997

This paper describes a generalized sequential diagnosis algorithm whose analysis leads to strong diagnosability results for a variety of multiprocessor interconnection topologies. The overall complexity of this algorithm in terms of total testing and syndrome decoding time is linear in the number of edges in the interconnection graph and the total number of iterations of diagnosis and repair needed by the algorithm is bounded by the diameter of the interconnection graph. The degree of diagnosability of this algorithm for a given interconnection graph is shown to be directly related to a graph parameter which we refer to as the partition number. We approximate this graph parameter for several interconnection topologies and thereby obtain lower bounds on degree of diagnosability achieved by our algorithm on these topologies. If we let N denote total number of vertices in the interconnection graph and 驴 denote the maximum degree of any vertex in it, then our results may be summarized as follows. We show that a symmetric d-dimensional grid graph is sequentially $\Omega \left( {N^{{{d \over {d+1}}}}} \right)$-diagnosable for any fixed d. For hypercubes, symmeteric log N-dimensional grid graphs, it is shown that our algorithm leads to a surprising $\Omega \left( {{{{N\,{\rm log\,log}\,N} \over {log\,N}}}} \right)$ degree of diagnosability. Next we show that the degree of diagnosability of an arbitrary interconnection graph by our algorithm is $\Omega \left( {\sqrt {{{N \over \Delta }}}} \right).$ This bound translates to an $\Omega \left( {\sqrt N} \right)$ degree of diagnosability for cube-connected cycles and an $\Omega \left( {\sqrt {{{N \over k}}}} \right)$ degree of diagnosability for k-ary trees. Finally, we augment our algorithm with another algorithm to show that every topology is $\Omega \left( {N^{{{1 \over 3}}}} \right)$-diagnosable.