conditional diagnosability of optical multi mesh hypercube networks under the comparison diagnosis model abstract due to integrated positive features of both hypercubes and tori optical multi mesh hypercube ommh networks are regarded as a class of promising optical interconnection topologies the notion of conditional diagnosability helps enhance the self diagnosing capability of multicomputers this paper determines the conditional diagnosabilities of ommh networks under the maeng malek comparison model

the g good neighbor conditional diagnosability of star graphs under the pmc and mm model diagnosability plays an important role in measuring the fault tolerance and reliability of interconnection networks the g good neighbor conditional faulty set is a special faulty set that every fault free vertex should have at least g fault free neighbors the g good neighbor conditional diagnosability is a metric that can give the maximum cardinality of g good neighbor conditional faulty set that the system is guaranteed to identify this paper shows that the g good neighbor conditional diagnosabilities of star graph sn under the pmc and mm model are ng g 1 1 for n4 and 0gn2 our study shows the g good neighbor conditional diagnosability of star graph is several times larger than the classical diagnosability of star graph

an 0 n 2 5 fault identification algorithm for diagnosable systems consider a system composed of n independent processors each of which tests a subset of the others it is assumed that at most tp of these processors are permanently faulty and that the outcome of a test is reliable if and only if the processor which performed the test is fault free such a system is said to be tp diagnosable if given any complete collection of test results the set of faulty processors can be uniquely identified in this paper it is shown that tp diagnosable systems due to their robust interconnection structure possess heretofore unknown graph theoretic properties relative to vertex cover sets and maximum matchings an 0 n2 5 algorithm is given which exploits these properties to identify the set of faulty processors in a tp diagnosable system the algorithm is shown to be correct complete not based on any conjecture and superior to any other known fault identification algorithm for the general class of tp diagnosable systems

strong rabin numbers of folded hypercubes the strong rabin number of a network w of connectivity k is the minimum l so that for any k 1 nodes s d1 d2 dk of w there exist k node disjoint paths from s to d1 d2 dk respectively whose maximal length is not greater than l where s d1 d2 dk and d1 d2 dk are not necessarily distinct in this paper we show that the strong rabin number of a k dimensional folded hypercube is k 2 1 where k 2 is the diameter of the k dimensional folded hypercube each node disjoint path we obtain has length not greater than the distance between the two end nodes plus two this paper solves an open problem raised by liaw and chang

rg conditional diagnosability a novel generalized measure of system level diagnosis abstract system level diagnosis has become an important diagnosis method for multiprocessor systems among all system level diagnosis measures diagnosability is relatively small the conditional diagnosability constraint that each vertex has at least one good neighbor is relatively conservative when the dimension is far greater than 1 and g good neighbor conditional diagnosability does not consider this restriction on faulty vertices therefore a thorough study of diagnosability under the condition that each vertex has at least g good neighbors is an appealing subject motivated by r g vertex connectivity in this paper we introduce a novel generalized system level diagnosis measure named r g conditional diagnosability which assumes that every processor has at least g good neighbors the popular conditional diagnosability is a special case of r g conditional diagnosability when g 1 then we determine that the r g conditional diagnosability of n dimensional hypercube q n under the preparata metze chien pmc model is 2 2 g n 2 g 2 2 g 1 1

diagnosability of hypercubes and enhanced hypercubes under the comparison diagnosis model a sengupta and a dahbura 1992 discussed how to characterize a diagnosable system under the comparison diagnosis model proposed by j maeng and m malek 1981 and a polynomial algorithm was given to identify the faulty processors provided that the system s diagnosability is known however for a general system the determination of its diagnosability is not algorithmically easy this paper proves that for the important hypercube structured multiprocessor systems n cubes the diagnosability under the comparison model is n when n spl ges 5 the paper also studies the diagnosability of enhanced hypercube which is obtained by adding 2 sup n 1 more links to a regular hypercube of 2 sup n processors it is shown that the augmented communication ability among processors also increases the system s diagnosability under the comparison model we prove that the diagnosability is n 1 for an enhanced hypercube when n spl ges 6

conditional diagnosability of alternating group graphs let an be the alternating group of degree n with n 3 set s 1 2 i 1 i 2 3 i n the alternating group graph denoted by agn is defined as the cayley graph on an with respect to s jwo et al networks 23 1993 315 326 introduced alternating group graph agn as an interconnection network topology for computing systems conditional diagnosability a new measure of diagnosability introduced by lai et al ieee transactions on computers 54 2 2005 165 175 can better measure the diagnosability of regular interconnection networks this paper determines that under pmc model the conditional diagnosability of agn is 4 for n 4 and 6n 18 for each n 5

the g good neighbor conditional diagnosability of twisted hypercubes under the pmc and mm model abstract connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks the g good neighbor conditional faulty set is a special faulty set that every fault free vertex should have at least g fault free neighbors the rg vertex connectivity of a connected graph g is the minimum cardinality of a g good neighbor conditional faulty set x v g such that g x is disconnected the g good neighbor conditional diagnosability is a metric that can give the maximum cardinality of g good neighbor conditional faulty set that the system is guaranteed to identify the twisted hypercube is a new variant of hypercubes with asymptotically optimal diameter introduced by x d zhu in this paper we first determine the rg vertex connectivity of twisted hypercubes then establish the g good neighbor conditional diagnosability of twisted hypercubes under the pmc model and mm model respectively

the triangular pyramid routing and topological properties in this paper a new topology for multicomputer interconnection networks based on triangular mesh is proposed the new network referred to as the triangular pyramid or tripy for short has l levels of triangular mesh we study some basic important properties of the proposed network as well as introduce a routing algorithm for the tripy network based on the routing of triangular meshes we prove that this form of pyramidal network is hamiltonian hamiltonian connected and pancyclic we also prove that the proposed network is 6 colorable and conduct a brief comparison of the tripy and its traditional pyramid counterpart our results show that the proposed network has higher scalability connectivity and total network bandwidth while preserving the important properties of the traditional pyramid network

on conditional diagnosability of the folded hypercubes the n dimensional folded hypercube fq n a variation of the hypercube proposed by ahmed et al a el amawy s latifi properties and performance of folded hypercubes ieee transactions on parallel and distributed systems 2 3 1991 31 42 is an n 1 regular n 1 connected graph conditional diagnosability a new measure of diagnosability introduced by lai et al pao lien lai jimmy j m tan chien ping chuang lih hsing hsu conditional diagnosability measures for large multiprocessor systems ieee transactions on computers 54 2 2005 165 175 can better measure the diagnosability of regular interconnection networks this paper determines that under pmc model the conditional diagnosability of fq n t c fq n is 4n 3 when n 5 or n 8 t c fq 3 3 t c fq 4 7

conditional diagnosability of arrangement graphs under the pmc model abstract processor fault diagnosis has played an important role in measuring the reliability of a multiprocessor system and the diagnosabilities of many well known multiprocessor systems have been investigated the conditional diagnosability has been widely accepted as a measure of diagnosability by assuming an additional condition that any fault set can t contain all the neighbors of any node in a multiprocessor system this paper considers the conditional diagnosability of an n k arrangement graph a n k a flexible interconnection network model for multiprocessor systems under the classical pmc diagnostic model and determines that the conditional diagnosability of a n k k 2 n k 2 is 4 k 4 n k 3 which is about four times its traditional diagnosability

on fault identification in diagnosable systems this paper begins by giving a characterization of t1 t1 diagnosable systems then a class of t0 diagnosable systems denoted by d n t0 x is considered it is shown for any member of this class that 1 necessary and sufficient conditions for t1 t1 diagnosability are greatly simplified 2 optimal diagnosis algorithms of time complexity 0 nt0 exist and most importantly 3 given the test results any set f of faults with f t1 can be identified to within a set f with f f and f min t1 f 1

fault diagnosability of arrangement graphs the growing size of the multiprocessor system increases its vulnerability to component failures it is crucial to locate and to replace the faulty processors to maintain a system s high reliability the fault diagnosis is the process of identifying faulty processors in a system through testing in this paper we show that the largest connected component of the survival graph contains almost all of the remaining vertices in the n k arrangement graph a n k when the number of moved faulty vertices is up to twice or three times the traditional connectivity using this fault resiliency we establish the conditional diagnosability of a n k under the comparison model and prove that the conditional diagnosability of a n k is 3k 2 n k 3 for k 4 n k 2 and the conditional diagnosability of a n n 1 is 3n 7 for n 5

the g good neighbor diagnosability of n k star graphs many large scale multiprocessor or multi computer systems take interconnection networks as underlying topologies fault diagnosis is especially important to identify fault tolerability of such systems the g good neighbor conditional diagnosability such that every fault free node has at least g fault free neighbors is a novel measure of diagnosability in this paper we show that the g good neighbor diagnosability of the n k star graph s n k under the pmc model 2 k n 1 and 1 g n k and the comparison model 2 k n 1 and 2 g n k is n g k 1 1 respectively in addition we derive that 1 good neighbor diagnosability of s n k under the comparison model is n k 2 for 3 k n 1 and n 4 as a supplement we also derive that the g good neighbor diagnosability of the n 1 star graph s n 1 1 g n 2 1 and n 4 under the pmc model and the comparison model is n 2 1 respectively we explore fault diagnosability of multiprocessor systems based on combinatorial network theory we establish the g good neighbor diagnosability of the n k star graph s n k under the pmc model we establish the g good neighbor diagnosability of the n k star graph s n k under the comparison model

on g good neighbor conditional diagnosability of n k star networks abstract the g good neighbor conditional diagnosability is a new measure for fault diagnosis of systems xu et al 2017 27 determined the g good neighbor conditional diagnosability of n k star networks s n k i e t g s n k with 1 k n 1 for 1 g n k under the pmc model and the mm model in this paper we determine t g s n k for all the remaining cases with 1 k n 1 for 1 g n 1 under the two models from which we can obtain the g good neighbor conditional diagnosability of the star graph obtained by li et al 2017 16 for 1 g n 2

on reliability of the folded hypercubes in this paper we explore the 2 extra connectivity and 2 extra edge connectivity of the folded hypercube fq n we show that k 2 fq n 3n 2 for n 8 and l 2 fq n 3n 1 for n 5 that is for n 8 resp n 5 at least 3n 2 vertices resp 3n 1 edges of fq n are removed to get a disconnected graph that contains no isolated vertices resp edges when the folded hypercube is used to model the topological structure of a large scale parallel processing system these results can provide more accurate measurements for reliability and fault tolerance of the system

adaptive finite element solution of compressible turbulent flows an adaptive e nite element method for solving compressible turbulent e ows up to transonic regime is presented pressure based methods previously developed for laminar compressible e ows and turbulent incompressible e ows are combined to solve compressible turbulent e ows turbulence is incorporated via the k model the algorithm uses the logarithms of k and as computational variables to preserve positivity solutionsare obtained in primitive variables using quadratic e nite elements on unstructured grids the error is estimated by a local projection method the solution algorithm and error estimation are validated on problems with known analytical solutions the method is then applied to compressible subsonic and transonic e ows and predictions are compared with experimental measurements

the 1 2 good neighbor conditional diagnosabilities of regular graphs abstract fault diagnosis of systems is an important area of study in the design and maintenance of multiprocessor systems in 2012 peng et al proposed a new measure for the fault diagnosis of systems namely g good neighbor conditional diagnosability which requires that any fault free vertex has at least g fault free neighbors in the system the g good neighbor conditional diagnosabilities of a graph g under the pmc model and the mm model are denoted by t g pmc g and t g mm g respectively in this paper we first determine that t g pmc g t g mm g if g 2 second we establish a general result on the 1 2 good neighbor conditional diagnosabilities of some regular graphs as applications the 1 2 good neighbor conditional diagnosabilities of bc graphs folded hypercubes and four classes of cayley graphs namely unicyclic transposition graphs wheel transposition graphs complete transposition graphs and tree transposition graphs are determined under the pmc model and the mm model in addition we determine the r2 connectivities of bc graphs and folded hypercubes

relating diagnosability strong diagnosability and conditional diagnosability of strong networks an interconnection network s diagnosability is an important measure of its self diagnostic capability based on the classical notion of diagnosability strong diagnosability and conditional diagnosability were proposed later to better reflect the networks self diagnostic capability under more realistic assumptions in this paper we study a class of interconnection networks called strong networks which are n regular n 1 connected and with cn number no more than n 3 we build a relationship among the three diagnosability measures for strong networks under both pmc and rm mm ast models given a strong network g with diagnosability t we prove that g is strongly t diagnosable if and only if g s conditional diagnosability is greater than t a simple check can show that almost all well known regular interconnection networks are strong networks the significance of this paper s result is that it reveals an important relationship between strong and conditional diagnosabilities and the proof of strong diagnosability for many interconnection networks under rm mm ast or pmc model is not necessary if their conditional diagnosability can be shown to be strictly larger than their diagnosability

an approach to conditional diagnosability analysis under the pmc model and its application to torus networks a general technique is proposed for determining the conditional diagnosability of interconnection networks under the pmc model several graph invariants are involved in the approach such as the length of the shortest cycle the minimum number of neighbors p resp p over all p vertex subsets resp cycles and a variant of connectivity called the r super connectivity an n dimensional torus network is defined as a cartesian product of n cycles ck1 ckn where ckj is a cycle of length kj for 1 j n the proposed technique is applied to the two or higher dimensional torus networks and their conditional diagnosabilities are established completely the conditional diagnosability of every torus network g is equal to 4 g 1 excluding the three small ones c3 c3 c3 c4 and c4 c4 in addition p g as well as 4 g is derived for 2 p 4 and the r super connectivity is also derived for 1 r 3

diagnosable evaluation of dcc linear congruential graphs under the pmc diagnostic model a dcc disjoint consecutive cycles linear congruential graph g f n consists of n nodes and is generated by a set of linear functions f with special properties it was proved that g f n is a 2t regular graph and has connectivity 2t where t f and 1 3 and 2

on simplifying incremental remap based transport schemes the flux form incremental remapping transport scheme introduced by dukowicz and baumgardner 1 converts the transport problem into a remapping problem this involves identifying overlap areas between quadrilateral flux areas and regular square grid cells which is non trivial and leads to some algorithm complexity in the simpler swept area approach originally introduced by hirt et al 2 the search for overlap areas is eliminated even if the flux areas overlap several regular grid cells the resulting simplified scheme leads to a much simpler and robust algorithm we show that for sufficiently small courant numbers approximately cfl 1 2 the simplified or swept area scheme can be more accurate than the original incremental remapping scheme this is demonstrated through a von neumann stability analysis an error analysis and in idealized transport test cases on the sphere using the incremental remapping based scheme called ff cslam flux form version of the conservative semi lagrangian multi tracer scheme on the cubed sphere

edge fault tolerance analysis of a class of interconnection networks fault tolerant measures have played an important role in the reliability of an interconnection network edge connectivity restricted edge connectivity extra edge connectivity and super edge connectivity of many well known interconnection networks have been explored in this paper we study the 2 extra edge connectivity of a special class of graphs g g0 g1 m proposed by chen et al appl math comput 140 2003 245 254 then by showing that several well known interconnection networks such as hyper

conditional diagnosability and strong diagnosability of split star networks under the pmc model an interconnection network s diagnosability is an important measure of its self diagnostic capability the classical problems of fault diagnosis are explored widely the conditional diagnosability is proposed by lai et al as a new measure of diagnosability which can better measure the diagnosability of regular interconnection networks the conditional diagnosability is an important indicator of the robustness of a multiprocessor system in presence of failed processors furthermore a multiprocessor system is strongly t diagnos able if it is t diagnosable and can achieve diagnosability t 1 except for the case where a node s neighbors are all faulty the conditional diagnosability and strong diagnosability were proposed later to better reflect the networks self diagnostic capability under more realistic assumptions in this paper we determine the conditional diagnosability of an n dimensional split star network denoted as s n 2 a well known interconnection network model for multiprocessor systems under the pmc preparata metze and chien model we show that the conditional diagnosability of s n 2 n 4 is 8 n 23 which is about four times of its traditional diagnosability as a byproduct the strong diagnosability of s n 2 is also obtained we investigate the combinatorial properties and fault tolerant properties of the split star network we show that as long as a fault set f has 6 n 17 or fewer nodes s n 2 f excluding a subset of at most two nodes contains one large connected component we show that as long as a fault set f has 8 n 25 or fewer nodes s n 2 f excluding a subset of at most three nodes contains one large connected component we show that the classic diagnosability of s n 2 n 2 under the pmc model is 2 n 3 we also establish that the conditional diagnosability and the strong diagnosability of s n 2 are 8 n 23 and 2 n 3 under the pmc model

on conditional fault tolerance and diagnosability of hierarchical cubic networks fault tolerance is especially important for interconnection networks since the growing size of networks increases their vulnerability to component failures a classical measure for the fault tolerance of a network in the case of vertex failures is its connectivity given a network based on a graph g and a positive integer h the r h connectivity of g is the minimum cardinality of a set of vertices in g if any whose deletion disconnects g and the minimum degree of every connected component is at least h this paper investigates the r h connectivity h 1 2 of the hierarchical cubic network hcn n n 2 and shows that 1 hcn n 2 n 2 hcn n 4 n 4 respectively furthermore the paper establishes the conditional diagnosability of hcn n under the pmc diagnostic model

a general technique to establish the asymptotic conditional diagnosability of interconnection networks we develop a general and demonstrably widely applicable technique for determining the asymptotic conditional diagnosability of interconnection networks prevalent within parallel computing under the comparison diagnosis model we apply our technique to replicate yet extend existing results for hypercubes and k ary n cubes before going on to obtain new results as regards folded hypercubes pancake graphs and augmented cubes in particular we show that the asymptotic conditional diagnosability of folded hypercubes fq n is 3n 2 pancake graphs p n is 3n 7 and augmented cubes aq n is 6n 17 we demonstrate how our technique is independent of structural properties of the interconnection network g in question and essentially only dependent upon the minimal size of the neighbourhood of a path of length 2 in g the number of neighbours any two distinct vertices of g have in common and the minimal degree of any vertex in g

conditional diagnosability of matching composition networks under the mm model diagnosability is a critical metric for determining the reliability of a multiprocessor system in 2005 lai et al proposed a new measure for the fault diagnosis of a system i e conditional diagnosability in which it is assumed that at least one of the neighbors of an arbitrary node in the system is not faulty in this paper we obtain a sufficient condition for a class of networks called matching composition networks mcns which are conditionally t diagnosable under the mm model then we apply the sufficient condition to show the conditional diagnosability of bijective connection bc networks finally we show that the sufficient condition can be applied to networks other than bc networks

constructing one to many disjoint paths in folded hypercubes routing functions have been shown effective in deriving disjoint paths in the hypercube in this paper using a minimal routing function k 1 disjoint paths from one node to another k 1 distinct nodes are constructed in a k dimensional folded hypercube whose maximal length is not greater than the diameter plus one which is minimum in the worst case for the general case the maximal length is nearly optimal spl les the maximal distance between the two end nodes of these k 1 paths plus two as a by product the rabin number of the folded hypercube is obtained which is an open problem raised by s c liaw and g j chang 1999

conditional matching preclusion sets abstract the matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings for many interconnection networks the optimal sets are precisely those induced by a single vertex in this paper we look for obstruction sets beyond these sets we introduce the conditional matching preclusion number of a graph it is the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost perfect matchings we find this number and classify all optimal sets for several basic classes of graphs