efficient local diagnosis algorithm for multiprocessor systems under the pmc model

Cheng-Kuan LinTzu-Liang KungJimmy J. M. Tan

Efficient Local Diagnosis Algorithm for Multiprocessor Systems under the PMC Model

• Abstract• The PMC Model• Local Diagnosability• Local Diagnosis Algorithm (Voting Method)• g-good-neighbor Conditional Diagnosability• Local-Diagnose-Under-Conditional-Faults

Algorithm (LDUCF)

3

Abstract

• In this proposal, we study some variant of diagnosis problems, such as local diagnosability, strongly local-diagnosable property and diagnosis algorithm.


Algorithm (LDUCF)

5

10

0/10/1

The PMC Model

Example： The definition for PMC model.

i jr(i,j)

( , )r i j

PMC model (Preparata, Metze and Chien. 1967)

1

2

3

4

G

6

Example： 1

2

3

4

1

2

3

4

The testing graph

The PMC Model

1 2 3 4

{1, 2}.syndrome: ( (1, 2), (1,3), (2,1), (2,4), (3,1), (3, 4), (4,2), (4,3))

{( , , , ,1,0,1,0) | {0,1},1 4}

16

F

F

i

Fr r r r r r r r

x x x x x i

G

1

0/1

0/1

0/1

0/1

1

0 0

7

The PMC Model

• In this model – ： be a syndrome– F： all syndromes if F is the set of faulty set– (in)distinguishable： F1 F2 ()=

8

• Lemma 1 (Dahbura, 1984)(F1, F2) is distinguishable-pair F1 F2

OR

The PMC Model

F1 F2

u

v

u

v

F1 F2

u v:

1:

0:

F1 F2

u v:

0:

1:

9

1

3

F1 F2

2

4

5

6 7

Example 1： A description of indistinguishable pair

( (1,2), (1,3), (2,1), (2,5), (3,1), (3, 4), (3, 7), (4,3), (4,5), (5, 2), (5,4), (6,7), (7,3))

r r r r r r r r r rr r r

1 1 2 3 4 5 6 7{( , , , , , , ,1, 0, 1, 0, 0,1) | {0,1},1 7}F ix x x x x x x x i 2 1 2 3 4 5 6 7{(0, 1, 0, 1, , , , , , , ,0,1) | {0,1},1 7}F ix x x x x x x x i

The PMC Model

10

1

3

F1 F2

2

4

5

6

Example 2： A description of distinguishable pair

( (1,2), (1,3), (2,1), (2,5), (2,6), (3,1), (3, 4), (4,3), (4,5), (5, 2), (6, (5,4 , )2))

r r r r r r r r r rrr

1 1 2 3 4 5 6 7{( , , , , , , , 1, 0, 1, 0, ) | {0,1},1 71 }F ix x x x x x x x i 2 1 2 3 4 5 6{(0, 1, 0, 1, 0, , , , , , , ) | {0,1},1 6}0F ix x x x x x x i

The PMC Model

PMC Model

• |F1| t and |F2| t

or

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• Lemma 2G is t-diagnosableF1, F2 V(G), F1 F2, |F1| t and |F2| t, then (F1, F2) is distinguishable-pair.

The PMC Model


Algorithm (LDUCF)

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Local Diagnosability

• Motivationusing connectivity as an example

u

v

(Q2) = 2

(Q3) = 3

(G) = 1

Menger’s Theorem: (G) = min{(u,v) | for all u,vV(G)}

(u, v) = 3

15

1

v 2 3

F1 = {v, 1, 2, 3}, |F1| 4

F2 = {1, 2, 3}, |F2| 4

(F1, F2) is indistinguishable-pair. ∴Q3 is not 4-diagnosable.

Q3v :F1 F2

In fact, t(Qn) = n


16

Qi

Qj


t(Qi) = i

i << j

t(Qj) = j

t(G) i

The strong diagnosabiliry of Qj is disregarded.

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• DefinitionLet G(V,E) be a graph and vV be a vertex. G is locally t-diagnosable at vertex v if, given a syndrome F produced by a set of faulty vertices FV containing vertex v with |F| t, every set of faulty vertices F’ compatible with F and |F’| t, must also contain vertex v.


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• DefinitionLet G(V,E) be a graph and vV be a vertex. The local diagnosability of vertex v, written as tl(v), is defined to be the maximum value of t such that G is locally t-diagnosable at vertex v.


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• Lemmavertex v is locally t-diagnosable for all F1, F2V(G), F1≠F2, and |F1| t, |F2| t,

vF1ΔF2, (F1, F2) is distinguishable-pair


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• TheoremG is locally t-diagnosable at vertex u there exists a subgraph called extended star TG(u; t) for vertex u as following


u

2

1

3

t

::

2'

1'

3'

t'

::

21

• example

3

3

3

3

3

3

3

3

Q3 is locally 3-diagnosable at every vertex.



Algorithm (LDUCF)

23

A Diagnosis Algorithm (Voting Method)

• Local syndrome

x

...0

1

1

0

1

1

24

v

0

0

0

1

1

0

1

1

n0,0+1

n0,1-1

n1,00

n1,10

+1 : 正一票 -1 : 負一票


t = n0,0 + n0,1 + n1,0 + n1,1

n0,0 n1,0 v is fault-free

n0,0 < n1,0 v is fault

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• example 1

v

0

0

0

0

0

1

0

1

( ) = 4 and the local syndrome of vertex is shown as following is fault-free

lt v vv

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• example 2

v

0

0

0

0

0

1

0

1

0

1

( ) = 5 and the local syndrome of vertex v is shown as following is faulty

lt vx

Local-Diagnosis Algorithm (LD)

Input: An extended star of order t rooted at node u, TG(u;t), in graph G.Output: The value is 0 or 1 if u is fault-free of faulty, respectively.

BEGIN n0,0 | { i | ((u1,1, u), (u2,1, u1,1) ) = (0, 0) } |; n1,0 | { i | ((u1,1, u), (u2,1, u1,1) ) = (1, 0) } |; If n0,0 n1,0

Return 0; Else

Return 1;END

extended star of order k rooted at node u, TG(u;k)

An extended star of full order rooted at a node v in Qn.

State of u Fault –Free Fault

Test Result 0 0 1 1 0 0 1 1

Test Result 0 1 0 1 0 1 0 1

# faulty nodes (x and y) 0 1 1 2 2 1 1 0

u

y

x

Theorem 1

Let TG(u; t) be an extended star of order t rooted at node u in graph G. Then the algorithm LD(G; TG(u; t)) correctly identifies the fault/fault-free status of node u if the total fault nodes in G does not exceed t.

Proof of Theorem 11. u is fault and n0,0 ≥ n1,0. total number of faulty nodes ≥ 2n0,0 +n0,1+n1,1+1 ≥ n0,0+n0,1+n1,0+n1,1 +1 = t+1

2. u is fault-free and n0,0 < n1,0. We have n1,0 ≥ n0,0 + 1. total number of faulty nodes≥ n0,1 + 2n1,0 + n1,1 ≥ n0,0 + n0,1 + n1,0 + n1,1+1 = t+1

State of u Fault –Free Fault

Test Result 0 0 1 1 0 0 1 1

Test Result 0 1 0 1 0 1 0 1

# faulty nodes (x and y) 0 1 1 2 2 1 1 0

u

y

x


Algorithm (LDUCF)

g-good-neighbor conditional diagnosability

A faulty set F V is called a g-good-neighbor conditional faulty set if |N(v) (V - F)| g for every vertex v in V - F.

A system G is g-good-neighbor conditional t-diagnosable if F1 and F2 are distinguishable, for each distinct pair of g-good-neighbor conditional faulty subsets F1 and F2 of V with |F1| t and |F2| t.

The g-good-neighbor conditional diagnosability tg(G) of a graph G is the maximum value of t such that G is g-good-neighbor conditional t-diagnosable

Related Works

• tg(Qn) 2g(n-g)+2g - 1• if g n – 3

• 2(n-1)+2-1 = 2n-1

Conditionally t∗-diagnosable at node v

Let G be a graph and v denote any node in G. Then G is conditionally t∗-diagnosable at node v if, given a syndrome F produced by any conditionally faulty set of nodes F ⊆V(G) with v ∈ F and |F| ≤ t, every conditionally faulty set F′ of nodes with which F is consistent must also contain node v.

Conditionally t∗-diagnosable at node v

u up p

Let G be a graph and let v be a node in G. Then G is conditionally t∗-diagnosable at node v if and only if for any two distinct conditionally fault sets F1 and F2 of V such that |F1| ≤ t, |F2| ≤ t and v ∈ F1△F2, (F1; F2) is a distinguishable pair.


Algorithm (LDUCF)

Two distinct types

Let u be any node of graph G and let t be any positive integer with t 2. We set S1 = { u }, S2 = { ui | 1 i t }, S3 = { ui

j,1 | 1 i t, 1 j t-1, and 1 k 3 }, and S3,l = { ul

j,1 | 1 j t-1 } for every 1 l t.Let BG(u;t) = (V(u;t), E(u;t)) be a subgraph of G with V(u; t) = S1 S2 S3 and E(u;t) = { {u, ui} | 1 i t } { {ui, ui

j,1} | 1 i t and 1 j t – 1 } { {ui

j,k, uij,k+1} | 1 i t, 1 j t - 1, and 1 k 2 }.

We say BG(u;t) is a branch-of-tree of order t rooted at node u if 1. |Si Sj| = 0 for every i j,2. |S2| = t,3. |S3,i| = 3(t – 1) for every 1 i t,4. |S3,i S3,j| 1 for i j with 1 i t and 1 j t, 5. |S3,i S3,j S3,k | = 0 for any three distinct elements i, j, and k

Local Diagnosisunder PMC model

Local Diagnosis under PMC model

• d(x) = t• |F| t• A – B 0 x is good• A – B < 0 x is fault

Local Diagnosis under PMC model

• S = { ui | ui determine good by local diagnosis under PMC model }

S = { ui | ui determine good by Local diagnosis under PMC model }

|S| 3 VOTE|S| = 2, S = { up, uq }

up is delegate if np0,0 - np

1,0 nq0,0 - nq

1,0 uq is delegate if nq

0,0 - nq1,0 > np

0,0 - np1,0

|S| = 1, S = { x } x is delegate |S| = 0

if ni1,0 - ni

1,0 2 for every 1 i t u is a faulty node if ni

1,0 - ni1,0 = 1 for some 1 i t

Let k be an integer such that nk1,0 - nk

1,0 = 1 r |{ 1 j t-1 | ((uk

j,1,uk), (ukj,2, uk

j,1), (ukj,3, uk

j,2)) = (1,0,1)}| if r 1 uk is delegate if r = 0 u is a fault-free node

Local-Diagnose-Under-Conditional-Faults Algorithm (LDUCF)

branch-of-tree of Qn

y1 = 00000001

y2 = 00000010

y3 = 00000100

y4 = 00001000

y5 = 00010000

y6 = 00100000

y7 = 01000000

y8 = 10000000

yij,k i = 1 i = 2 i = 3 i = 4 i = 5 i = 6 i = 7 i = 8j = 1 & k = 1 00000011 00000011 00000101 00001001 00010001 00100001 01000001 10000001j = 1 & k = 2 00000111 00001011 00010101 00101001 01010001 00100011 01000011 10000101j = 1 & k = 3 00001111 00011011 00110101 01101001 01010011 01100011 01000111 10001101j = 2 & k = 1 00000101 00000110 00000110 00001010 00010010 00100010 01000010 10000010j = 2 & k = 2 00001101 00001110 00010110 00101010 00010011 10100010 01000110 10010010j = 2 & k = 3 00011101 00011110 00110110 01101010 00110011 10100110 11000110 10010011j = 3 & k = 1 00001001 00001010 00001100 00001100 00010100 00100100 01000100 10000100j = 3 & k = 2 00011001 00011010 00011100 00101100 01010100 00100110 01001100 10001100j = 3 & k = 3 00111001 00111010 00111100 00101110 01010101 01100110 01001101 10011100j = 4 & k = 1 00010001 00010010 00010100 00011000 00011000 00101000 01001000 10001000j = 4 & k = 2 00110001 00110010 00110100 00111000 01011000 10101000 01001010 10011000j = 4 & k = 3 01110001 10110010 01110100 01111000 01011100 10101001 01011010 10011010j = 5 & k = 1 00100001 00100010 00100100 00101000 00110000 00110000 01010000 10010000j = 5 & k = 2 01100001 01100010 00100101 01101000 01110000 10110000 01010010 10010100j = 5 & k = 3 11100001 11100010 01100101 11101000 01110010 10110100 11010010 10010101j = 6 & k = 1 01000001 01000010 01000100 01001000 01010000 01100000 01100000 10100000j = 6 & k = 2 01000101 11000010 11000100 01001001 11010000 11100000 01100100 10100100j = 6 & k = 3 11000101 11000011 11001100 01011001 11110000 11100100 01101100 10101100j = 7 & k = 1 10000001 10000010 10000100 10001000 10010000 10100000 11000000 11000000j = 7 & k = 2 10001001 10000011 10000110 10001010 10010001 10100001 11001000 11000001j = 7 & k = 3 10011001 10000111 10001110 10101010 11010001 10100011 11001010 11001001

Table of yi and yij,k of e8 = 00000000 on Q8

efficient local diagnosis algorithm for multiprocessor systems under the pmc model

Documents

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