humboldt-universitÄt zu berlin institut fÜr informatik
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HUMBOLDT-UNIVERSITÄT ZU BERLIN INSTITUT FÜR INFORMATIK. DEPENDABLE SYSTEMS Vorlesung 4 Topological Testing Wintersemester 2000/2001 Leitung: Prof. Dr. Miroslaw Malek www.informatik.hu-berlin.de/~rok/zs. WHAT IS TOPOLOGICAL TESTING?. APPLICATION OF FORMAL GRAPH THEORY METHODS TO - PowerPoint PPT PresentationTRANSCRIPT
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HUMBOLDT-UNIVERSITÄT ZU BERLININSTITUT FÜR INFORMATIK
DEPENDABLE SYSTEMS
Vorlesung 4
Topological Testing
Wintersemester 2000/2001
Leitung: Prof. Dr. Miroslaw Malek
www.informatik.hu-berlin.de/~rok/zs
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WHAT IS TOPOLOGICAL TESTING?
APPLICATION OF FORMAL
GRAPH THEORY METHODS TO
TEST SYSTEMS WHOSE
ORGANIZATION OR BEHAVIOR
CAN BE DESCRIBED BY A
GRAPH
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TOPOLOGY OF A SYSTEM
• OBJECTIVE:– Given the topology of a system, minimize the test time
• TOPOLOGY OF A SYSTEM:– A graph description of a system that reflects either its physical
organization or its behavior
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DOMAINS OF APPLICABILITY
• BEHAVIOR:– Testing the Finite State Machine representation of a system,
e.g., testing protocol conformance
• ORGANIZATION:– Use the organization of a system to be tested in an optimal
fashion, e.g., use of Hamiltonians and Eulerians
• HIERARCHY:– Partitioning for parallel testing
– System integration
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METHODS OF TOPOLOGICAL TESTING (1)• HAMILTONIAN: Testing the nodes
– Switch Model
– Graph Model
P 1 P2 P 3 P 4
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METHODS OF TOPOLOGICAL TESTING (2)
• EULERIAN: Testing the edges
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METHODS OF TOPOLOGICAL TESTING (3)
• TRAVELING SALESMAN PROBLEM (TSP):
• CHINESE POSTMAN PROBLEM:
A
4
B
1
C6D
3
E
1
4226
5
L3
L
L5
L1
L2
L6 4
Chinese Postman Tour: L1 L2 L3 L4L L6 L5 L4 5
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METHODS OF TOPOLOGICAL TESTING (4)
• PARTITIONING:
• COVERING:
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METHODS OF TOPOLOGICAL TESTING (5)
• PATH COVERING:
• DOMINATING SET:
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METHODS OF TOPOLOGICAL TESTING (6)
• BROADCAST AND COLLECTION SPANNING TREES:
• COLORING PROBLEM:
1
1
1
1
2
3
2 log N
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THE RUBIK'S CUBE OF TOPOLOGICAL TESTING
Partitioning + Path Covering
Hamiltonian + TSP
Eulerian + CPP
PMS
RTL
logic
behavior organization hierarchy
level
graph concept
type
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APPLICATION EXAMPLES
• TESTING MULTISTAGE INTERCONNECTION NETWORKS (BANYAN)
• TESTING HYPERCUBES
• PROTOCOL TESTING USING FINITE STATE MACHINES– Uyar and Dahbura
• MEMORY TESTING– Hayes
– Patel
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TESTING OF MULTISTAGE INTERCONNECTION NETWORKS (BANYANS)
• An (f, L) SW-Banyan is an L level Multistage Interconnection Network having N (=fL) inputs and outputs and using f x f switches
• This type of network has been used in ETL's Sigma I, Butterfly, IBM's RP3, TRAC, PASM and other computers
0 P1 P2 P3 P4 P5 P6 P 7
M M M M M M M M0 1 2 3 4 5 6 7
PP 0
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A 2 x 2 SWITCH
• FAULT MODEL:– Stuck-at and bridge faults on data and control lines
– Routing faults
– Conflict resolution
SW
8CTS DV DATA PARITY
11 11
11
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PROPERTIES OF BANYANS
• The system graph has a Hamiltonian and an Eulerian.– This property can be used to implement a serial (on-line) fault
detection on nodes and edges.
• There exists f pair wise edge-disjoint test graphs, each with n disjoint paths between pairs of processors.– This property can be used to implement parallel (off-line) fault
detection.
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PARALLEL TESTING OF DATA PATHS
• Two tests are sufficient to detect any number of s-a- faults on the data part of the vertices
• 2 f tests are sufficient to detect any number of multiple s-a- faults on the data part of the edges
• Fault location:– Vertices s-a-: 2f tests can locate up to f-1 faults
– Edges s-a-: 2f + log L tests
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TESTING ROUTING
• Only f tests are needed to test the routing capabilities of the switches in the entire network
Use the f edge-disjoint test graphs (f = 2)
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TESTING THE CONTROL AND PRIORITY LOGIC (1)
• Objective: Test the correct behavior of the switches under any input pattern especially in case of contention on the outputs
• To test completely for control faults, every mapping of inputs to outputs has to be tested
• Number of tests for an f x f switch is:
T f i1
f
if fi 1 f f
1
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TESTING THE CONTROL AND PRIORITY LOGIC (2)
2 x 2 switch: T2 = 8
4 x 4 switch: T4 = 624
8x8 switch: T8 = 43,046,721
• In the case of round-robin priority, tests should be repeated for every priority state
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TESTING TECHNIQUE • Test separately the Arbitration Logic Blocks (ALB), the
Routing and Storage Blocks and the SELECT/DESELECT lines
Design of the control part of a 4 x 4 switch
• f2f-1 tests are sufficient to test the conflict resolution capabilities of a switch in the case of fixed priority and f2ff tests are sufficient in the case of round-robin priority
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Finite State Machine
• Finite State Machine of the ALB of a 4 x 4 switch using round-robin priority
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TESTING CONFLICT RESOLUTION IN THE ENTIRE NETWORK
• i-conflict i inputs requesting the same output in a switch• Two tests are sufficient to produce an i-conflict and an (f-i)-
conflict in every switch in the network independently of the number of levels
• 2-conflicts and 1-conflicts produced by the same test on all even numbered levels of the network
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ESTIMATION OF TESTING TIME FOR THE SIGMA-1 COMPUTER
• SIGMA-1 interconnection network:• L = 2 two levels• 10 x 10 switches configured as 8 x 8's• Round-robin priority• Time assumed for traversal of the network and memory
access: t = 120 ns• Estimated testing time using a pseudoexhaustive method:
~20 hours• Actual testing time: ~22.5 hours • Estimated testing time using our method: 26 seconds
OVER 3000 TIMES BETTER!
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TESTING OF HYPERCUBES (1)
PROPERTIES OF HYPERCUBES:
• Distributed-memory, message-passing multiprocessor• N = 2N processors consecutively numbered by binary
integers from 0 through 2N - 1• Each processor connected to all other processors whose
binary tags differ from its own by exactly one bit• Degree of each vertex = n• Homogeneous
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TESTING OF HYPERCUBES (2)• Cube of each dimension is obtained by replicating the
one of next lower dimension, then connecting corresponding nodes
• Partitioning into smaller sub-cubes is easy
An example of a hypercube of dimension 4
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ROUTING ON A HYPERCUBE
• At each stage, the routing scheme is simply to send the message to the neighbor whose binary tag agrees with the tag of the ultimate destination in the next bit position that differs between the sender and final destination
• Alternatively,
Source SnSn-1 …… S2S1
Destination dndn-1 …… d2d1
Bit wise EXOR xnxn-1 …… x2x1
where xi = Si + di for i = 1, ……, n
• Those values of i for which xi = 1 indicate the dimension that must be traversed to transfer a message from source to destination
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COMMUNICATION
• No shared memory
• Message-passing communication system
• Store-and-forward function at each node, e.g., a receiving processor checks the address of the message and reroutes the message if not intended for it
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TESTING TECHNIQUE (1)
1. Partition circuit into node and edge disjoint Q2’s (or C4’s )
2. Perform a ring test, i.e., each node sends a packet to the diagonal node, all in the same direction, first clockwise, then anti-clockwise
3. Repeat for all ( ) partitionsn2
0000
0010
0110
0100 0101 11011100
11101111
100010010001
0111
0011 1011 1010
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TESTING TECHNIQUE (2)
• Number of node and edge disjoint Q2‘s in each partition = 2n-2 where n is the dimension of the cube
• There are nC2such partitions possible. All partitions are to be tested
• Time is of the order of
0(n2) or 0(log2N)
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PARTITIONING A 3-CUBE
• First Phase: [000, 001, 011, 010]
and [100, 101, 111, 110]
• Second Phase: [000, 001, 101, 100]
and [010, 011, 111, 110]
• Third Phase: [000, 010, 110, 100]
and [001, 011, 111, 100]
100 101
000 001
010 011
110 111
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WHAT IS TESTED BY THE ABOVE TEST?
• All nodes• All communication links• All paths between any pair of processors
TESTING CONTENTION:
• To test contention of two (a 2-conflict), partition Qn into Q2’s as before. Within each Q2, only one node can be tested at a time
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Testing Contention of 3 on a Q3 (1)
• Estimation of Test Time:• Time to test the hypercube with pseudoexhaustive test =
Nn(2n-1-1)t
0(N2logN)
000 001
101
111110
010
100
011
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Testing Contention of 3 on a Q3 (2)
• IN FACT, TESTING MAXIMAL CONTENTION CAN BE DEFINED AS GRAPH COLORING OF G2
FOR Qn Q2n
• T: 2 log2
(n+1) COLORS ARE SUFFICIENT TO COLOR Q2n
FOR Mn M2n
• Five colors are necessary to color M2n
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EXAMPLES (1)
1. A HYPERCUBE
Q3 = 2 log 2
3
4=
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EXAMPLES (2)
2. A MESH
• COMMERCIAL QUALITY TEST DEVELOPED FOR A MESH NETWORK IN SYMULT, STATE-OF-THE-ART MULTIPROCESSOR, REQUIRES CONSTANT NUMBER OF TESTS AND TEST TIME IS LESS THAN 2.5 ms, REGARDLESS OF THE SYSTEM SIZE
M4 5
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Time
• Time to test the hypercube with our tests =
• For hypercube of dimension 10 (Q10) time to test for contention decreases from the range of about 7 min - 8 hours 40 min to 5 sec - 6.5 min, which results in about 80 times improvement
log12nC2 +
n
i=3
niC 2
(i + 1) i ] t[
0 log 3 N
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EXAMPLES (3)
Q4 = 2 log 2
5 = 8
M7 = 5
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EXAMPLES (4)
Q4 = 2 log 2
5 = 8
1 2 3 4 5 1 2
3
5
2
4
1
3
4 5 2 3 41
5 1 2 3 4 5
3 4 5 1 2 3
2 3 4 5 1 2
4 5 1 2 3 4
1 2 3 4 5 1
M7 = 5
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CONCLUSIONS
• TOPOLOGICAL TESTING EXPLORES POWER OF GRAPH
THEORY TO TEST– BEHAVIOR
– ORGANIZATION
– HIERARCHY
– OF COMPUTER SYSTEMS AND NETWORKS
• EFFICIENT ALGORITHMS FOR TESTING MULTISTAGE
NETWORKS AND HYPER-CUBES ACHIEVE OVER THREE ORDERS OF MAGNITUDE SPEEDUP WITHOUT COMPRO-MISING TEST COVERAGE