A Graph Partitioning Approach to Sequential Diagnosis

1
A Graph Partitioning Approach to Sequential
Diagnosis

• Fabrizio Silvestri
• Computer Science Dept. Pisa
• CNUCE (CNR Institute) Pisa

2
Introduction

• System-level diagnosis identify faulty
  processors in a multiprocessor system
• PMC model Preparata Metz Chien model has
  been widely studied
• A system is composed by several units or
  processors
• Each unit can test each others along the
  available communication channel
• A unit u can be even faulty or fault-free and the
  outcome of the test is considered reliable iff
  its the testing unit is fault-free

3
Outline

• Basic concepts
• Graph partitioning and sequential diagnosis
• PARTITION algorithm
• Investigated topologies
• Symmetric grid
• Hypercube
• Cube-connected cycles and k-ary trees
• Arbitrary topologies

4
Basic Concepts (I)

• The PMC model
• Syndrome Decoding interpretation of the test
  results
• One-Step vs. Sequential Diagnosis
• Degree of Diagnosability of a diagnosis algorithm
  A is the maximum number of faults that A can
  correctly detect

5
Basic Concepts (II)

• For a given graph G(V,E)
• V(G) and E(G) denote respectively its set of
  vertices and its set of edges
• D(G) its the diameter
• ?(G) is the maximum degree of any vertex

6
The Syndrome Graph

• Let G be the interconnection graph for each edge
  (x,y) in E, x tests y and vice versa
• The Syndrome Graph is a graph GS with V(Gs)
  V(G) and E(Gs) contains the edges of G labelled
  with the outcome of the tests an edge (x,y) is
  labelled pass if x declare y fault-free and
  vice-versa is labelled fail if x and y declare
  each other to be faulty any other edges are
  labelled as conflict.

7
Example of a Syndrome Graph

• Derivation of the syndrome graph.

8
Graph Partitioning and Diagnosability (I)

• Lemma 1. Let Gp be the subgraph of the syndrome
  graph Gs induced by the edges labeled as pass.
  Then, in each connected component of Gp, either
  all vertices are fault-free or all of them are
  faulty.

1
Pass
7
Pass
6
2
Pass
5
Pass
4
3
9
Graph Partitioning and Diagnosability (II)

• Corollary 1. Let t be an upper bound on the total
  number of faulty processors in the system. If the
  graph Gp contains a connected component of size
  t1 or larger, then it must be the case that all
  these vertices correspond to fault-free
  processors.

1
Pass
7
Pass
6
2
Pass
5
Pass
4
3
10
k-partition Number

• Definition 1. Given a connected graph G, the
  k-partition number of G is the largest integer p
  s.t. for all p-element subsets S ? V(G), the
  subgraph of G induced by the vertices in V(G)
  S has a connected component of size k or larger.
• The k-partition number of G is denoted by ?G(k)
  and in general for a given G, it will be a
  function of k and V(G). This function is
  undefined for kgtV(G).

11
k-partition Number (example)

• k 3

• ?G(3) 2

12
Use of ?G(k)

• We choose t s.t. ?G(t1)? t.
• Gp must have a component of size t1 or larger.
• By Corollary 1, any such component must consist
  of vertices corresponding to fault-free
  processors.

13
The Generalized Sequential Diagnosis Algorithm

• Let G be the interconnection graph for a given
  system S
• Let t be a nonnegative integer such that ?G(t1)?
  t
• The we can use the following algorithm to
  correctly diagnose all the faults in S provided
  there are no more than t faulty processors in S.
• The PARTITION algorithm is composed of two
  phases.

14
PARTITION (phase 1) Fault-Free Subset
Identification

• Each processors test each one of its neighbors.
• Gp the graph induced by the pass labeled node
  of the sindrome graph Gs is built.
• We do a depth first search upon Gp to find the
  connected component of size t1.

15
PARTITION (phase 2) Iterative Diagnosis and
Repair

• Select an arbitrary fault-free processor, say u,
  from the component identified in Phase 1 and
  construct a breadth-first search tree rooted at
  u.
• Starting from u we do a testing iteration to
  detect the faulty processors.

16
Performances of PARTITION

• The degree of diagnosability is greater than t
  (?G(t1)? t).
• The total testing and syndrome decoding time is
  O(E(G)).
• The number of diagnosis and repair iteration
  needed by the algorithm is at most D(G).

17
?G Approximation

• ?G(t) is hard to determine exactly, we approx it
  using a two-step approach.
• First we compute ?G(t) an approximation of ?G(t)
  ?G(t) 1.
• Next we compute ?G(t) ?G(t) 1.
• ?G(t) ? ?G(t) for any t.

18
PARTITION on Symmetric Grids

• n-dimensional path of length d. Nnd vertices.
• The degree of diagnosability is at least the
  largest t satisfying
  The solution is ?(N(d/d1))
• The total number of edges is O(N).
• The diameter is d(N(1/d)-1).

19
PARTITION on Hypercubes

• Its a d-dimensional symmetric grid with each
  dimension of length 2. N2d .
• The degree of diagnosability is at least the
  largest t satisfying
  The solution is ?(N loglogN / logN)
• The total number of edges is O(NlogN).
• The diameter is logN.

20
PARTITION on Cube Connected Cycles

• Its a d-dimensional hypercube where each node is
  replaced by a d-length cycle. Nd2d.
• The degree of diagnosability is at least
  (1/2)(?(2N 2) 2).
• The total number of edges is O(N).
• The diameter is ?(5d / 2) - 1?.

21
PARTITION on k-ary Trees

• Its a tree where each vertex has no more than k
  children. Complete k-ary trees.
• The degree of diagnosability is at
  least(1/2k)(?(4kN k2 4) (k 2)).
• The total number of edges is O(N).
• The diameter is ?(2logk(Nk1N)2)?.

22
Arbitrary Topologies

• The MAX Algorithm. Degree of diagnosability is at
  least ??/2?.
• Ifthen use the MAX algorithm for diagnosis
  else use the PARTITION algorithm (MAX_PARTITION).

23
Generalization

• For any given interconnection graph G, the
  MAX_PARTITION algorithm has
• Degree of diagnosability equal to ?(V(G)1/3)
• Time complexity O(E(G)).
• Number of iterations D(G) 1.

24
Conclusions

• A generalized sequential diagnosis algorithm has
  been shown.
• Very high degree of diagnosability even when the
  vertex degrees are small in the interconnection
  graph.
• No upper bound on the degree of diagnosability.