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A Distributed Formation of Orthogonal Convex Polygons in Mesh-Connected

A Distributed Formation of Orthogonal Convex Polygons in Mesh-Connected
A Distributed Formation of Orthogonal Convex Polygons in Mesh-Connected

A Distributed Formation of Orthogonal Convex Polygons in Mesh-Connected

Multicomputers

Jie Wu

Department of Computer Science and Engineering

Florida Atlantic University

Boca Raton,FL33431

Abstract

The rectangular faulty block model is the most commonly

used fault model in designing a fault-tolerant and deadlock-

free routing algorithm in mesh-connected multicomputers.

The convexity of a rectangle facilitates simple and ef?cient

ways to route messages around fault regions using relatively

few virtual channels to avoid deadlock.However,such a

faulty block may include many nonfaulty nodes which are

disabled,i.e.,they are not involved in the routing process.

Therefore,it is important to de?ne a fault region that is con-

vex,and at the same time,to include a minimum number of

nonfaulty nodes.In this paper,we propose a simple and ef-

?cient distributed algorithm that can quickly construct a set

of special convex polygons,called orthogonal convex poly-

gons,from a given set of rectangular faulty blocks in a2-D

mesh(or2-D torus).The formation of orthogonal convex

polygons is done through a labeling scheme based on itera-

tive message exchanges among neighboring nodes.

1.Introduction

In a mesh-connected multicomputer,processors(also

called nodes)exchange data and coordinate their efforts

by sending and receiving messages through the underlying

mesh network.Thus,the performance of such a system

depends heavily on the end-to-end cost of communication

mechanisms.Routing is the process of transmitting data

from one node to another node in a given system.As the

number of nodes in a mesh-connected multicomputer in-

creases,the chance of failure also increases.At the same

time,applications that run on such a system are often criti-

cal and may have real-time constraints.Therefore,the abil-

ity to tolerate failure is becoming increasingly important,

especially in routing.

Most literatures on fault-tolerant routing use disjoint

rectangular blocks([1],[2],[6])to model node faults(link

number of nonfaulty nodes in the block subject to the condi-tion that the resultant region(s)is still an orthogonal convex

polygon(s).

A simple and ef?cient distributed algorithm is presented in this paper that determines a set of small orthogonal

convex polygons to cover all the faults in a given faulty block.Let denote the diameter of faulty block,

while represents the maximum diameter of all the faulty blocks in the faulty mesh.This algorithm is based on iterative message exchanges among neighboring

nodes.Speci?cally,this approach consists of two phases. In phase one,disjoint faulty blocks are constructed through rounds of message exchanges among neigh-bors in a given faulty mesh.In phase two,some non-faulty nodes in faulty blocks are activated,by removing

them from the associated faulty blocks,through up to an-other rounds of message exchanges between neighbors.

We show that a resultant region,called a disabled re-gion,generated after removing activated nodes from the given faulty block is the smallest orthogonal convex poly-gon that covers all the faults in the region.Note that there may have several disabled regions generated from a given faulty block.In addition,we show that the number of non-faulty nodes covered in these disabled regions(from a given faulty block)is no more than that in the smallest orthogo-nal convex polygon that includes all the faulty nodes in the original faulty block.We note that for certain cases,a dis-abled region can be further partitioned and more nonfaulty nodes in the region can be removed.This brings the fol-lowing open problem:For a given faulty block,?nd a set of orthogonal convex polygons that covers all the faults in the faulty block and contains a minimum number of nonfaulty nodes.This problem is conjectured to be NP-complete[3].

2.Preliminaries

We consider only node faults and assume that faulty nodes just cease to work.Also,each nonfaulty node knows the status of its neighbors only;that is,there is no a priori global information of fault distribution.A-D mesh with nodes has an interior node degree of and a net-work diameter of.Each node has an address ,where.Two nodes:

and:are connected if their addresses dif-fer in one and only one dimension,say dimension.More-over,.Similarly,if they differ in dimension ,then.

De?nition1:A region is orthogonal convex if and only if the following condition holds:For any horizontal or vertical line,if two nodes on the line are inside the region,all the nodes on the line that are between these two nodes are also inside the region.

The difference between a standard convex region and an orthogonal convex region is that the line in the latter is re-stricted to only horizontal and vertical,whereas the line in the former can be along any direction in a standard convex region.Clearly,T-shape,L-shape,and+-shape fault regions are orthogonal convex polygons,whereas U-shape and H-shape fault regions are non-orthogonal convex polygons.

An application of orthogonal convex regions in achiev-ing fault-tolerant routing in2-D meshes has been discussed in Chalasani and Boppana’s extended-cube routing[2].

3.Node Status

Although the faulty block model for2-D meshes has been studied by many researchers,there is still no agree-ment about terminology.Here we introduce a set of con-cepts and terminology and express some existing models and concepts in terms of the ones in the set.Three orthog-onal classi?cations of nodes in2-D meshes are given:(1) faulty vs.nonfaulty,(2)safe vs.unsafe,and(3)enabled vs. disabled.

All nodes are either faulty or nonfaulty(healthy).To construct disjoint faulty blocks in2-D meshes,nonfaulty nodes are further classi?ed into safe and unsafe nodes.Un-safe nodes will be included in faulty blocks and implicitly disabled(i.e.,they are treated as faulty).Normally,a faulty block is constructed by?rst identifying unsafe nodes de-?ned as follows:

De?nition2a:All faulty nodes are unsafe.A nonfaulty node is unsafe if it has two or more unsafe neighbors;otherwise, it is safe.

A faulty block consists of connected unsafe nodes.Let :and:be two nodes in a2-D mesh,

denotes the distance between and.The distance between two faulty blocks and is de?ned as.It has been shown that faulty blocks in2-D meshes are disjoint rectangles and the distance between any two faulty blocks is at least3.Figure1(a)shows an example of a faulty block where black nodes represent faulty nodes and gray nodes represent nonfaulty but unsafe nodes.The faulty block is bounded by adjacent safe nodes(which are not shown in the?gure).Note that under this de?nition of boundary lines, faulty nodes()and()(without any other faulty nodes)are contained in one single region.

Clearly,a faulty block may include many nonfaulty nodes,which is an undesirable feature.To reduce the num-ber of nonfaulty nodes in a faulty block,the following en-hanced de?nition of safe/unsafe nodes is used.

De?nition2b:All faulty nodes are unsafe.A nonfaulty node is unsafe if it has an unsafe neighbor in both dimen-sions;otherwise,it is safe.

(a)

(b)

(g)(h)

Figure1.Faulty blocks and disabled regions.

Note that the difference between De?nitions2a and2b is the following:When a node has exactly two unsafe neigh-bors and both of them are along the same dimension,this node is an unsafe node based on De?nition2a and a safe node based on De?nition2b.

Figure1(b)shows the result of applying De?nition2b to the same example of Figure1(a).In this case,there are two disjoint faulty blocks.It can be proved that the distance between two faulty blocks is at least2.Besides,the total number of nonfaulty nodes included in faulty blocks is less than the one under De?nition2a.

To further reduce the number of nonfaulty nodes in a faulty block,the concept of enabled/disabled nodes can be introduced.Basically,a nonfaulty but unsafe node can be made enabled in a faulty block,i.e.,this node can be excluded from the faulty block.The enabled status of a node is de?ned based on the enabled/disabled status of its neighbors,rather than depending on safe/unsafe status of its neighbors as in the de?nition either by Boura and Das [1]or by Su and Shin[6].

De?nition3(Wu):All faulty nodes are marked disabled. All safe nodes are marked enabled.An unsafe node is ini-tially marked disabled,but it is changed to the enabled sta-tus if it has two or more enabled neighbors.

A disabled region consists of connected disabled nodes. Clearly,disabled regions are disjoint and the distance be-tween any two disabled regions is at least2.

Based on the two sets of de?nitions for node status:one for safe/unsafe nodes based on either De?nition2a or Def-inition2b and the other for enabled/disabled nodes based on De?nition3,a faulty node must be unsafe and disabled. For a nonfaulty node,there are three possible cases:(1)safe and enabled,(2)unsafe and enabled,and(3)unsafe and dis-abled.Figures1(c)and(d)show the results of applying the enabled/disabled rule to the examples of Figures1(a)and (b),respectively.

To ensure that all boundary nodes in a mesh are treated the same as interior nodes,four additional lines are added which are adjacent to the boundaries lines of the mesh1. These additional lines become the new boundaries of the mesh.Nodes along these additional lines are called ghost nodes which are safe but they do not participate in any ac-tivities,such as in a routing process.Among nodes in the given mesh,only enabled nodes will participate in routing activities.

The enabled/disabled node de?nition in De?nition3 involves one subtle issue and deserves more discussion. Unlike the“recursive”de?nition of safe/unsafe status,all nodes are initially marked as disabled or enabled.Node sta-tus can be changed later from disabled to enabled following the rule in De?nition3.This is to ensure that the concept of disabled/enabled node is well-de?ned,i.e.,each node has one and only one possible assignment of enabled/disabled status.Suppose the enabled/disabled rule is de?ned recur-sively as follows:All faulty nodes are marked disabled.All safe nodes are marked enabled.An unsafe node is enabled if it has two or more enabled neighbors;otherwise,it is marked disabled.For a given system con?guration,unsafe nodes may have“double status”,i.e.,two or more different enabled/disabled assignments are possible that both satisfy this de?nition.

Figure2(a)shows an example of a faulty block with its upper right block containing only nonfaulty nodes and with the remaining nodes in the faulty block being faulty.Based on the recursive de?nition of enabled/disabled status,the upper right corner node should be marked enabled.Itera-tively we enable all the nonfaulty nodes in the upper right block.Figure2(b)shows a similar example,however,the block that contains only nonfaulty nodes is located at the upper center of the faulty block.In this case,we can either enable all the nonfaulty nodes in the block or disable them. Therefore,these nonfaulty nodes have double status.Based on De?nition,all the nodes in the faulty block of Figure 2(b)have the disabled status.Similar problems exist in all safe/unsafe de?nitions(De?nition2a and De?nition2b).In fact,each nonfaulty node should be assigned the safe status initially.

Two distributed algorithms are given:one for deciding safe/unsafe status and another one for enabled/disabled sta-tus.The?rst algorithm generates a set of faulty blocks and the second one produces a set of disabled regions that are orthogonal convex polygons.Both algorithms follow the same structure where each node exchanges its status with

(a)(b)

Figure2.Two sample faulty blocks.

Safe/unsafe status:

all faulty nodes are initialized to unsafe;

all nonfaulty nodes are initialized to safe;

repeat

doall

(1)nonfaulty node exchanges its status with its neighbors;

(2)change’s status to unsafe if it has an unsafe neighbor

in both dimensions

odall

until there is no status change

Enabled/disabled status:

all unsafe nodes are initialized to disabled;

all safe nodes are initialized to enabled;

repeat

doall

(1)nonfaulty but unsafe node exchanges its status

with its neighbors;

(2)change’s status to enabled if it has two or more

enabled neighbors.

odall

until there is no status change

its neighbors and changes its status based on the collected neighbors’status.To simplify our discussion,each itera-tive algorithm is assumed to be synchronous and each round of exchange and update is done in a lock-step mode.The algorithm for safe/unsafe status is based on De?nition2b. The enabled/disabled status is based on the proposed en-abled/disabled rule(De?nition).

Consider an example of a2-D mesh with three faulty nodes:(1,3),(2,1),and(3,2).Using the safe/unsafe rule, one faulty block is https://www.wendangku.net/doc/1910998200.html,-ing the enabled/disabled rule,the faulty block is split into two disabled regions:and.All the nonfaulty nodes in the faulty block are enabled.

4.Properties

In this section,we?rst show that after removing all the enabled nodes(based on De?nition)in a faulty block, a disabled region containing adjacent faulty and disabled nodes is an orthogonal convex

polygon.

(b)

opening

(c)

(a)

Figure3.Three cases of disabled regions.

Theorem1:A disabled region is an orthogonal convex polygon.

Proof:Assume that the disabled region is concave.We can?nd a horizontal(or vertical)line[]with two nodes and on the line both being inside the region,but some node on the line within these two nodes is outside the region.Consider an enabled region that includes connected enabled nodes(including node )in the original faulty block.Line partitions into two disjoint enabled regions and.An enabled region is said to have an opening if it includes a node that has a neighbor outside the original faulty block and node is called an opening point.

The following two cases are considered:(1)If either

or does not have an opening,say,based on the en-abled/disabled rule,nodes in,as well as nodes in that are on line,should all be marked disabled.This brings a contradiction to the assumption that is marked enabled.In Fig-ure3(a)both and do not have an opening and in Figure 3(b)does not have an opening and has an opening.(2) If both and have an opening(see Figure3(c)),assume that and are two opening points in and,respec-tively;then a path in connecting from to and from to disconnects the disabled region into at least two components, with one containing node and the other containing node. This contradicts the assumption that the disabled region is con-nected.

is outside the disabled region.

Lemma1:In a disabled region,each corner node is a faulty node.

Proof:If a corner node of a disabled region is a nonfaulty node, based on De?nition4,it has two enabled neighbors,one along each dimension,that are outside the disabled region.Based on the enabled/disabled rule of De?nition3,this corner node is marked enabled and should be excluded from the fault region.This brings a contradiction.

Note that four quadrants overlap with each other either along the adjacent(or)axis or at the origin or both. One special case occurs when each quadrant contains only one faulty node which is the origin.In this case,all four quadrants are the same and contain the origin node only. Lemma3:Let be a node and let denote a disabled region.If is not contained in then at least one quadrant, as de?ned in Lemma2,does not contain any nodes in. Proof:When a quadrant contains nodes in,at least one node in is on the or axis;otherwise,we prove Lemma3,since all the other three quadrants do not contain any nodes in.(Because is connected and covers two quadrants,it must include one node on either the or axis.)Without loss of generality,assume that there is a node in on the axis and it is on the positive side of it,i.e.,it is in both quadrants()and().If there is another node in on the negative side of the axis,i.e.,it is in both()and(),then the line segment that contains both nodes violates the de?nition of the orthogonal convex polygon, because the origin is also on this segment but it is outside. Therefore,there is at least one node in along the positive side

of

Figure4.Two convex polygons that cover all

the faults within the region.

the axis and one node in along the negative side of the axis to ensure that both quadrants()and()contain at least one node in and that nodes in that are in()and() are connected without including any nodes on the negative side of the axis.Then the axis as a line segment violates the de?nition of the orthogonal convex polygon(origin is outside). Corollary:Given a faulty block,the number of nonfaulty nodes covered in disabled regions,generated by applying the proposed enabled/disabled rule on the faulty block,is no more than the one in the smallest orthogonal convex poly-gon that contains all the faulty nodes in the faulty block.

Note that our result is optimal under the assumption that each disabled region cannot be further partitioned.For cer-tain cases,such as the ones in Figures1(c)and(d),a dis-abled region can still be partitioned into several disjoint or-thogonal convex polygons.This brings the following open problem:For a given faulty block,?nd a set of orthogonal

1

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4.5

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100150200250300350N u m b e r o f r o u n d s

Number of faulty nodes(f)

faulty block disabled region

20

40608010012014040050060070080090010001100

N u m b e r o f r o u n d s

Number of faulty nodes(f)

faulty block disabled region

(a)

(b)

0.9

0.910.920.930.940.950.960.970.980.99150

100150200250300350e n a b l e d n o d e /u n s a f e b u t n o n f a u l t y n o d e

Number of faulty nodes ratio

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e n a b l e d n o d e /u n s a

f e b u t n o n f a u l t y n o d e

Number of faulty nodes

ratio

(c)

(d)

Figure 5.Simulation results.

convex polygons that cover all the faults in the faulty block with a minimum number of nonfaulty nodes inside these polygons.This problem is conjectured to be NP-complete [3].

5.Simulation

The following simulation study has been conducted:(1)For a given mesh (with in the simulation)and a given number of faults (with in the simulation),the averages of the maximum numbers of rounds needed to determine faulty blocks and disabled re-gions (after the formation of faulty blocks)are calculated (see Figures 5(a)and (b)).(2)For each faulty block that can be reduced to a set of orthogonal convex polygons,the average percentage of enabled nodes among unsafe but non-faulty nodes in the faulty block is calculated (see Figures 5(c)and (d)).

In our simulation study,faults (in all)are randomly selected among nodes in the mesh.The simu-lation results show that the averages of the maximum num-bers of rounds needed to construct faulty blocks and then disabled regions are both relatively row,much lower than the diameter of the mesh.The average number for disabled regions (after the formation of faulty blocks)is lower than the number for faulty blocks,because disabled regions are generated out of faulty blocks.The average percentage of enabled nodes among unsafe but nonfaulty nodes in faulty blocks stays very high,especially when the number of faults is relatively low.This high percentage is in part due to the fact that a random distribution tends to generate a set of small faulty blocks and the fact that nonfaulty nodes in small faulty blocks are easy to be enabled.In summary,the

simulation results con?rm the effectiveness (high percent-age of enabled nodes)of our approach with a relatively low cost (small number of rounds).

6Conclusions

In this paper,we have proposed a simple and ef?cient distributed algorithm that can quickly construct a set of or-thogonal convex polygons containing all the faulty nodes in a given rectangular faulty block.We have shown that the number of nonfaulty nodes covered in these orthogonal convex polygons is no more than the one in the smallest or-thogonal convex polygon that includes all the faulty nodes in the faulty block.Moreover,each orthogonal convex poly-gon is the smallest one that contains all the faults it covers.The simulation results con?rm the cost-effectiveness of the approach,i.e.,orthogonal convex polygons can be gener-ated quickly from a given set of faulty blocks.The con-vexity of a fault region facilitates ef?cient fault-tolerant and deadlock-free routing.Based on the results of this paper,we can provide a re?ned fault model to ef?ciently support sev-eral routing objectives,including optimality and freedom of deadlock.

References

[1]Y .M.Boura and C.R.Das.Fault-tolerant routing in mesh

networks.Proc.of 1995International Conference on Parallel Processing .August 1995,I 106-I 109.[2]S.Chalasani and R.V https://www.wendangku.net/doc/1910998200.html,munication in mul-ticomputers with nonconvex faults.IEEE Transactions on Computers .46,(5),May 1997,616-622.[3] D.Z.Chen.private communication.

[4] F.P.Preparata and https://www.wendangku.net/doc/1910998200.html,putational Geometry:

An Introduction .Springer-Verlag,1985.[5]J.D.Shih.Adaptive fault-tolerant wormhole routing al-gorithms for hypercube and mesh interconnection networks.Proc.of the 11th International Parallel Processing Sympo-sium .April 1997,333-340.[6] C.C.Su and K.G.Shin.Adpative fault-tolerant deadlock-free routing in meshes and hypercubes.IEEE Transactions on Computers .45,(6),June 1996,672-683.[7]P.H.Sui and S.D.Wang.An improved algorithm for fault-tolerant wormhole routing in meshes.IEEE Transactions on Computers .46,(9),September 1997,1040-1042.[8]Y .C.Tseng,M.H.Yang,and T.Y .Juang.An Euler-path-based multicasting model for wormhole-routed networks with multi-destination capability.Proc.of the 1998International Conference on Parallel Processing .August 1998,366-373.[9]J.Wu.Fault-tolerant adaptive and minimal routing in mesh-connected multicomputers using extended safety levels.IEEE Trans.on Parallel and Distributed Systems .11,(2),Feb.2000,149-159.

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新视野大学英语读写教程第一册课文翻译及课后答案

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新大学日语简明教程课文翻译

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成语解释及例句

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新视野大学英语第一册Unit 1课文翻译

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