Fault-tolerant wormhole routing in
two-dimensional mesh networks with convex faults
Pao-Hwa Sui, Sheng-De Wang
*Department of Electrical Engineering, EE building, National Taiwan University, Room 441, 1 Roosevelt Road, Sec. 4, Taipei 106, Taiwan, ROC
Received 1 June 1996; received in revised form 3 November 1998; accepted 4 May 1999
Abstract
Faulty blocks are expanded, by disabling nodes, to form rectangular faults in many existing works to facilitate the designing of deadlock-free routing schemes for meshes recently. In this paper, faulty blocks are diused to be rectangular faults, which are composed of faulty and fault-diused nodes. These rectangular faults are then shrunk, by recovering fault-diused nodes, to form convex faults for reducing the number of nodes disabled. Simulation results show that up to 70% of the disabled nodes, which are needed to form rectangular faults, can be recovered if the number of faulty nodes is less than 10% of the total network nodes.
Both non-adaptive and adaptive fault-tolerant routing algorithms are proposed to handle these resulted convex faults. The adaptive routing algorithm is enhanced from the non-adaptive counterpart by utilizing the virtual channels that are not used in the non-adaptive algorithm; hence, the number of virtual channels per physical channel used in the adaptive algorithm is the same to that used in the non-adaptive algo-rithm. Ó 1999 Published by Elsevier Science Inc. All rights reserved.
1. Introduction
Direct networks have become a popular means for interconnecting com-ponents of massively parallel computer systems. In direct networks, nodes
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E-mail address: sdwang@cc.ee.ntu.edu.tw (S.-D. Wang).
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(computers) are connected to only a few nodes, its neighbors, according to the topology of the network and communicate with each other by passing mes-sages. The n-dimensional mesh (n-D mesh) network is currently the most popular topology for massively parallel computer systems. Low-dimensional mesh networks, due to their low node degree, are more popular than the high-dimensional mesh networks. The two-high-dimensional mesh (2-D mesh) topology has been adopted by Symult 2010 [4], Intel Touchstone DELTA [5] and Intel paragon; the MIT J-machine adopts three-dimensional mesh topology.
The wormhole switching technique by Dally and Seitz [3] has been widely used in contemporary multiprocessors for exchanging messages. Virtual-cut-through [1] and store-and-forward [2] are the other two often used switching techniques. In the wormhole switching, a message is divided into packets and a packet is composed of ¯ow control digits or ¯its. The header ¯it governs the route. As the header advances along a speci®c route, the remaining ¯its follow in a pipeline fashion. If the header encounters a busy channel, it is blocked until the channel become available and all the ¯its in same packet remain in the ¯it buer along the speci®ed route. A survey of wormhole routing for direct net-works can be found in [8]. Routing schemes proposed in this paper are based on wormhole switching.
An interconnection network cannot avoid having failure components in real world. Simulating multiple virtual channels on each physical channel has been used to support fault-tolerance and adaptability [6,7,9±12,14]. Adaptive rout-ing can tolerate faults by selectrout-ing message paths but the fault-tolerance is not guaranteed.
Linder and Harden [6] extend the concept of virtual channel to multiple virtual interconnection networks that provide adaptability and fault-tolerance. Messages routing in a virtual network are constrained to travel the virtual networks in a pre-de®ned order. The main disadvantage of their method is that the number of virtual channels per physical channel needed is large.
Chien and Kim [7] present a partially adaptive algorithm for mesh networks. Only three virtual channels per physical channel are required in their methods. Faults are expanded to be rectangular, convex in [7], faults to ensure the cor-rectness of their algorithms. Extra nodes are disabled when faults are located on the boundaries of meshes. For instance, even only one node or link fault on a boundary row of a two-dimensional mesh, all nodes on that row must be disabled. Dally and Aoki [9] present two adaptive routing algorithms for direct net-works. Both algorithms, based on the concept of dimension reversal, are deadlock-free by eliminating cycles in the resource dependency graph. Faults are not expanded to be rectangular faults. However, the number of faults tolerated depends on the number of virtual channels used, in the static algo-rithm, and depends on the locations of the faults, in the dynamic algorithm.
Boppana and Chalasani [10±12] present techniques to enhance existing routing algorithms for fault-tolerance in meshes with multiple rectangular
faults. The concepts of f-rings and f-chains are introduced and are used for routing messages around rectangular faults. Two, three or four virtual chan-nels are required for dierent fault situations. Arbitrarily located rectangular faults are considered in [12] but not in [10,11]. Although Boppana and Chalasani claim that their methods in [12] provide deadlock-free message routing in two-dimensional meshes, deadlocks among column messages may occur under some fault situation [13].
Su and Shin [14] present fault-tolerant routing algorithms, which decompose the network into two virtual interconnection networks, VIN1 and VIN2. VIN1 supports deadlock-free routing and VIN2 support fully adaptive routing. VIN2 trades its adaptability for fault-tolerance when the requested channels in VIN1 are not connected to safe nodes. For mesh networks, faulty blocks are ex-panded to disconnected rectangular blocks, which are composed of unsafe and faulty nodes. Two rectangular blocks, B1and B2, are disconnected, if for each
node x12 B1 and x22 B2, the distance between x1 and x2 in at least
one-di-mension is no smaller than three. Expanding faulty blocks to be disconnected rectangular faults may cause even more nodes disabled than just expanding faulty blocks to be rectangular faults. Although only two virtual channels per physical channel are needed in their method, deadlocks can also occur under some fault situation [17].
In all the routing methods mentioned above, either the number of faults tolerated is up to the number of virtual channels used [9] or the faults are restricted to or expanded to be rectangular faults [7,10±12,14]. Simulating multiple virtual channels on every physical channel is not a bargain without cost. Large number of virtual channels per physical channel causes the in-crease of hardware complexity, cost and delay in routing logic. Expanding faults to be rectangular faults causes good nodes unnecessarily disabled. The more the good nodes were disabled, the less the computing and transmission power was left. In [14], though not all nodes in the expanded rectangular fault are disabled, only corner nodes of each rectangular fault could be in unsafe state if they are not faulty. These unsafe nodes can transmit and re-ceive but not relay messages. In this paper, for the purpose of reducing the number of nodes disabled, each expanded rectangular fault is shrunk, by recovering nodes, to form one or many convex faults. A non-adaptive fault-tolerant routing algorithm is ®rst proposed. Only four virtual channels per physical channel are required in the algorithm. The non-adaptive algorithm is then enhanced to be an adaptive algorithm by utilizing the virtual channels that are not used in the non-adaptive algorithm. Since the adaptability is introduced by utilizing the virtual channels that are not used in the non-adaptive counterpart, the same number of virtual channels per physical channel are used in these two algorithms. The rest of this paper is organized as follows: Section 2 describes terminology. The methods for diusing faults and shrinking rectangular faults are also described in Section 2. Simulation
results of the fault shrink approach are also given in this section. A non-adaptive routing algorithm is proposed in Section 3. The non-non-adaptive al-gorithm is enhanced in Section 4 to be an adaptive alal-gorithm. This paper concludes with Section 5.
2. Preliminaries
An n-dimensional mesh has knÿ1; knÿ2; . . . ; k0 nodes, ki nodes along
dimen-sion i, 0 6 i 6 n ÿ 1, and kiP 2. Each node x is uniquely indexed by an n-tuple
(xnÿ1; xnÿ2; . . . ; x0), where 0 6 xi6 kiÿ 1. Nodes x xnÿ1; xnÿ2; . . . ; x0) is a
positive (resp. negative) neighbor to node y ynÿ1; ynÿ2; . . . ; y0 if and only if
xi yifor all i except one, j, where xj yj 1 (resp. yjÿ 1). Each node has
from n to 2n neighbors up to their location on the mesh. Neighboring nodes are connected by a direct link implemented by two unidirectional physical channels with opposite directions. When node x is a positive (resp. negative) neighbor of node y, the channel from node y to node x is a positive (resp. negative) channel. The four directions of a two-dimensional mesh are labeled, hereinafter, as North, East, South and West.
A faulty block is a set of connected faulty nodes and links. The fault boundary, f-boundary, of a faulty block is the set of good nodes and links immediately surrounding the faulty block. Faulty blocks in any shape can be diused to be rectangular faults, whose f-boundaries are in shapes of rect-angles. A faulty block is a convex fault if there are no good nodes located outside the block are surrounded by the fault in the two opposite directions of a dimension. A node is a corner node of an f-boundary if it is on the f-boundary and is located at distance two to the fault surrounded by the f-boundary.
2.1. Fault-diusion
Fault-diusion: Each good node that has two or more faulty or fault-diused links in dierent dimensions set itself to be a fault-diused node.
Each node is in good, fault-diused or faulty state after fault-diusion. Links incident to fault-diused nodes are fault-diused links. All fault-diused nodes can be identi®ed recursively. Fig. 1 shows an 8 8 mesh with faulty nodes (shown as ®lled circles). After fault-diusion, nodes a, b, c, d, e, f, g, h, i, j, k, l and m are set to be fault-diused nodes and two rectangular faults are formed. Heavy lines in Fig. 1 indicate f-boundaries and dotted lines indicate fault-diused links. It is possible that more than one faulty blocks are con-tained in a rectangular fault after fault-diusion.
2.2. Fault-shrink
Fault-shrink: Each fault-diused node that has a good positive (resp. neg-ative) neighbor on dimension i, 0 6 i 6 1, generates and sends a ¯ag, f1, to its negative (resp. positive) neighbor on dimension i. Flag f1 is propagated with-out changing direction until a fault node is reached. Any fault-diused node that receives and/or generates two f1 ¯ags is recovered to be a good node. Each recovered node then generates and sends a ¯ag, f 2, to its fault-diused neighbors along the direction that is opposite to the direction on which f1 received. Flag f 2 is propagated without changing direction until a good node is reached. Any fault-diused node that accepts one or more f 2 ¯ags is also re-covered to be a good node. A fault-diused node that is not rere-covered sets itself to be a faulty node after the fault-shrink process.
Each rectangular fault may become several convex faults after fault-shrink. Fig. 2 is the ®nal result of Fig. 1 after fault-shrink. Nodes b, c, d, e, g, i, j and k are recovered due to the receiving of ¯ag f1 and nodes h, l and m are recovered due to the receiving of ¯ag f 2. Three convex faults are formed in Fig. 2.
To each f-boundary, the only set of connected nodes which has largest index value on dimension 0 (resp. dimension 1) with respect to all other nodes on the f-boundary is called the EMAX (resp. NMAX) of the f-boundary; the only set of connected nodes which has smallest index value on dimension 0 (resp. di-mension 1) with respest to all other nodes on the f-boundary is called the EMIN (resp. NMIN) of the f-boundary. We assume that the length of each
convex fault is smaller than its corresponding dimension length, for a whole row or column failure disconnects two-dimensional meshes.
Lemma 1. Only convex faults exist after fault-shrink.
Proof. Lemma 1 is proved by showing that no good node is surrounded by the same fault in the two opposite directions of a dimension. Good nodes are classi®ed into three disjointed classes. Class A contains nodes that are never contained in rectangular fault. Class B contains nodes that are recovered due to the receiving of ¯ag f1 and class C contains nodes that are recovered due to the receiving of ¯ag f 2.
Since all faulty blocks are rectangular faults after fault-diusion, each node in class A has at most one link connects to a rectangular fault. Therefore, no node in class A can be surrounded by same faulty block in the two opposite directions of a dimension. Each node in class B can reach f-boundaries, which are composed of good nodes and links, in two directions of dierent dimen-sions; hence, it cannot be surrounded by same fault in the two opposite di-rections of a dimension. A rectangular fault is decomposed, by nodes of classes B and C, into several convex faults if there are fault-diused nodes recovered due to the receiving of ¯ag f 2. Therefore, the two convex faults located at the two opposite sides of nodes in class C are disconnected and, hence, no node in class C is surrounded by a fault in the two opposite directions of a dimension. From above arguments, no good nodes are surrounded by a fault in the two opposite directions of a dimension, hence only convex faults exist after fault-shrink.
2.3. Simulation results
In order to evaluate how many percentages of the fault-diused nodes can be recovered by fault-shrink, we simulated a 16 16 mesh with dierent per-centages of the total network nodes being faulty. Symbols F and N denote the total number of faulty and network nodes. Fault-diusion and fault-shrink are performed each time after the required faulty nodes are generated randomly. Table 1 shows Nd, the total number of fault-diused nodes, Nr1, the number of
nodes recovered due to the receiving of ¯ag f1, and Nr2, the number of nodes
recovered due to the receiving of ¯ag f 2, of 1000 experiments in each of the dierent fault situations. The last column of Table 1 indicates how many percentages of the fault-diused nodes are recovered and shows that the smaller the percentage of the total network nodes being faulty, the larger the percentage of the diused nodes can be recovered. Upto 70% of the fault-diused nodes are recovered when the number of total faulty nodes is less than 10% of the total network nodes.
3. Non-adaptive fault-tolerant routing
The following notations are de®ned to facilitate our discussion: Destination node d d1; d0.
Current node c c1; c0.
Routing tag R r1; r0 d1ÿ c1; d0ÿ c0).
f(R): number of non-zero elements in R.
v(R, i): the value of the ith element in R, 0 6 i 6 1.
VCi;j: the virtual channel of class j, 0 6 j 6 3, of dimension i, 0 6 i 6 1;
VCi;j (resp. VCi;ÿj) represents positive (resp. negative) virtual channel.
3.1. Non-adaptive fault-tolerant routing algorithm
The non-adaptive fault-tolerant routing algorithm proposed is based on the e-cube algorithm [3]. Messages are routed ®rstly to the columns on which their
Table 1 Simulation results F/N (%) Nd Nr1 Nr2 Nr1 Nr2=Nd 1 75 69 0 0.92 5 2474 1968 36 0.81 10 14 623 9588 894 0.72 15 58 092 17 573 4712 0.38 20 139 734 10 635 3587 0.10 25 175 073 2892 1205 0.02
destination nodes reside and then routed to their destinations. A message that has not yet reached the column on which its destination resides is a row message; a message that is not a row message is a column message. Row messages traveling from West to East (resp. East to West) are WE (resp. EW) messages. Similarly, column messages are classi®ed into NS and SN messages. Since each message is routed ®rstly to the column on which its destination resides and then routed to its destination, a row message may become a column message but a column message never becomes a row message.
Algorithm 3.1. non-adaptive fault-tolerant routing algorithm for two-dimen-sional meshes with convex faults.
1. Calculate R;
2. If R 0, route the message to the local processor and exit; 3. If the message is a WE (resp. EW) message
then if the requested VC0;0 (resp. VC0;ÿ0) is not connected to a fault
then request VC0;0 (resp. VC0;ÿ0)
else request VC1;1(resp. VC1;2), if v R; 1 P 0, or VC1;ÿ1 (resp. VC1;ÿ2),
if v R; 1 6 0, until the corner node is reached. /* If the mesh boundary is reached or the requested virtual channel of dimension 1 is faulty, then re-verse the direction */
4. If message is an NS (resp. SN) message
then if the requested VC1;ÿ0 (resp. VC1;0) is not connected to a fault
then request VC1;ÿ0 (resp. VC1;0)
else route message on the f-boundary in direction of clockwise until the corresponding node on dimension 1 is reached. /* If the mesh boundary is reached or the requested VC0;1(resp. VC0;ÿ2) is faulty, then reverse the
direction to be counter-clockwise */
WE (resp. EW) messages request VC0;0 (resp. VC0;ÿ0) if they are not
blocked by faults. If the requested virtual channel is connected to a fault, the WE (resp. EW) message requests VC1;1(resp. VC1;2), if v R; 1 P 0, or VC1;ÿ1
(resp. VC1;ÿ2), if v R; 1 6 0, until the corner node is reached. The message
performs a U-turn and reverses its direction if the mesh boundary is reached or the requested virtual channel of dimension 1 is connected to a fault before the corner node is reached. WE (resp. EW) messages request VC0;0(resp. VC0;ÿ0)
again at the corner nodes.
NS (resp. SN) messages requests VC1;ÿ0 (resp. VC1;0) if they are not
blocked by faults. If the requested virtual channel is connected to a fault at node x, the column message is then routed in the direction of clockwise along the f-boundary until the corresponding node on dimension 1 of x is reached. The corresponding node on dimension 1 of x x1; x0, which has a positive
(negative) faulty channel on dimension 1, is the node y y1; y0, which has a
the f-boundary of the same fault, where y16 x1 and y0 x1. NS (resp. SN)
messages use VC0;1, VC1;ÿ0, VC0;ÿ1 and VC1;3 (resp. VC0;ÿ2, VC1;0, VC0;2
and VC1;ÿ3) as they traveling in the corresponding direction along the
f-boundary in direction of clockwise. If the mesh f-boundary is reached or the requested VC0;1(resp. VC0;ÿ2) is connected to a fault, the message performs a
U-turn and reverses its direction to counter-clockwise. NS (resp. SN) messages use VC0;ÿ3, VC1;ÿ0, VC0;1, and VC1;3 (resp. VC0;3, VC1;0, VC0;ÿ2 and
VC1;ÿ3) as they traveling in the corresponding direction along the f-boundary
in direction of counter-clockwise. NS (resp. SN) messages request VC1;ÿ0(resp.
VC1;1) again at the corresponding node on dimension 1 of node x.
3.2. Usage of virtual channels
The usage of virtual channels by dierent types of messages is depicted in Fig. 3. It is noticed that the sets of virtual channels used by dierent types of messages are disjointed. Heavy lines in Fig. 3 indicate the mesh boundary or f-boundaries.
WE messages use VC0;0, VC1;1and VC1;ÿ1 and EW messages use VC0;ÿ0,
VC1;2 and VC1;ÿ2as they traveling in the corresponding directions. NS
mes-sages use VC1;ÿ0, VC0;1, VC1;3 and VC0;ÿ1 as they traveling in the
corre-sponding directions with the exception that an NS message whose routing direction on the f-boundary has been set to counter-clockwise uses VC0;ÿ3while
traveling from East to West. SN messages use VC1;0, VC0;2, VC1;ÿ3 and
VC0;ÿ2as they traveling in the corresponding directions with the exception that
an SN message whose direction has been set to counter-clockwise uses VC0;3
while traveling from West to East. 3.3. Example
Let us consider the example of routing a message from node a 1; 1 to node k 6; 4 in Fig. 4. The message is routed as a WE message from node a to node b using VC0;0. Since the requested VC0;0is faulty and v R; 1 5 at
node b, the message then requests VC1;1until the corner node c is reached. At
node c, the message requests VC0;0again until node d, at which the message
becomes an SN message, is reached. The message then requests VC1;0at node
d. Since the requested VC1;0is faulty and VC0;ÿ2 is also faulty at node e, the
message is routed on the f-boundary in counter-clockwise until node j, the corresponding node of e, is reached. VC0;3 is used from node e to node f;
VC1;0 is used from node f to node g; VC0;ÿ2 is used from node g to node h;
VC1;ÿ3is used from node h to node i and VC0;ÿ2is used from node i to node j.
At node j, the message requests VC1;0again until node k is reached.
3.4. Deadlock-freeness of Algorithm 3.1
In Algorithm 3.1, we have the following facts:
1. EW, WE, SN and NS messages use disjointed sets of virtual channels, 2. row messages (EW and WE) can become column messages (NS and SN),
but no column messages can become row messages and
3. EW and WE messages cannot change into each other and NS and SN can-not change into each other.
Thus, Algorithm 3.1 is a deadlock-free routing method if we can prove no deadlock will occur in each of the four message types.
Theorem 1. Algorithm 3.1 is a deadlock-free routing method for two-dimensional meshes with convex faults.
Proof. For a deadlock to occur among some messages, there must be a circular-wait on these messages. The circular-circular-wait may occur within one column, one row or on dierent columns and rows in two-dimensional meshes. In the fol-lowing, we show that no circular-wait can occur among row or column mes-sages in these three situations.
Since no WE (resp. EW) row messages can travel from East to West (resp. West to East), no circular wait can occur among WE (resp. EW) row messages within one row or on dierent columns and rows. For a circular-wait to occur among row messages within one column, there must be two row messages such that each of the row messages has performed a U-turn and is waiting for the virtual channel occupied by the other. Since the length of each convex fault is smaller than its corresponding dimension length and no concave faults exist after fault-shrink, the condition for a circular-wait to occur among row mes-sages within one column can never be satis®ed. From the above statement, it can be concluded that no deadlock will occur among row messages.
NS (resp. SN) messages never perform a U-turn while heading to South (resp. North), no circular-wait can occur among column messages within one column. Although NS column messages, while routing on a row along a f-boundary, may perform a U-turn and reverses its routing direction, no circu-lar-wait can occur among NS messages on a row because of the following channel requesting relationships:
1. an NS message routing on VC0;ÿ1may depend on an NS message routing on
VC0;ÿ1or VC0;1;
2. an NS message routing on VC0;1may depend on an NS message routing on
VC0;1or VC0;ÿ3;
3. an NS message routing on VC0;1never depends on an NS message routing
on VC0;ÿ1;
4. an NS message routing on VC0;ÿ3never depends on an NS message routing
on VC0;1;
By a symmetric argument, no circular-wait can occur among SN messages on a row only. Since (1) NS (resp. SN) messages always heading to South (resp. North) when they are not on f-boundaries and (2) no NS (resp SN) message can ever be routed on EMAX or EMIN nodes of any f-boundary while they are heading to North (resp. South), no circular-wait can occur among NS (resp. SN) column messages on dierent rows and columns.
From above statement, no circular-wait can occur among messages within one column, one row or on dierent columns and rows; therefore, Algorithm 3.1 is deadlock-free.
Since (1) each row message can always get to the column on which its destination node sits, (2) each column message never was blocked by the same fault twice and (3) the number of convex faults is ®nite, Algorithm 3.1 is also livelock-free.
4. Adaptive fault-tolerant routing
4.1. Adaptive fault-tolerant routing algorithm
In Algorithm 3.1, messages are routed on virtual channels of class 0 if they are not blocked by faults. Virtual channels of classes 1, 2 and 3 of dimensions 0 and 1 on f-boundaries are intended and used for avoiding faults. All other virtual channels of classes 1, 2 and 3 are not used elsewhere. In order to im-prove network performance, these unused virtual channels are utilized to en-hance Algorithm 3.1 to be an adaptive algorithm. The adaptability is introduced by allowing messages to use adaptively these unused virtual chan-nels if the messages are not on the f-boundaries. Each adaptive routing step brings the message one step closer to its destination. Two more notations are de®ned.
VCi;a: the virtual channel of classes 0, 1, 2 and 3 of dimension i, 0 6 i 6 1:
VCi;b: the virtual channel of classes 1, 2 and 3 of dimension i, 0 6 i 6 1:
Algorithm 4.1. Adaptive fault-tolerant routing algorithm for two-dimensional meshes with convex faults.
1. Calculate R;
2. If R 0, route the message to local processor and exit; 3. If the message is not on f-boundary
then if the message is a row message then if v R; 1 6 0
then parallel request VC0;a and VC1;b and route the message on
the ®rst available virtual channel
/* each routing step using VC1;b brings the message one step
closer to its destination */
else parallel request VC0;aand route the message on the ®rst
avail-able virtual channel
else parallel request VC1;a and route the message on the ®rst available
virtual channel
When messages are not on f-boundaries, row messages parallel request VC0;a
and VC1;b, if v R; 1 6 0, or otherwise request VC0;a only; column messages
request VC1;a. Each routing step on VC1;b brings the row message one step
closer to its destination. Messages are routed according to Algorithm 3.1 when they are on f-boundaries. The sets of virtual channels used by dierent message types are not disjointed in Algorithm 4.1.
4.2. Deadlock-freeness of Algorithm 4.1
Mesh networks are decomposed into two virtual interconnection networks, VIN1 and VIN2, to facilitate our discussion. VIN1 contains all the virtual channels of class 0 and all the virtual channels on f-boundaries of classes 1, 2 and 3; VIN2 contains all the other virtual channels. That is, messages can be routed in VIN1 or VIN2 when they are not on f-boundaries and can only be routed in VIN1 when they are on f-boundaries.
Theorem 2. Algorithm 4.1 is a deadlock-free routing algorithm.
Proof. Since (1) messages of dierent types use disjointed sets of virtual channels when they are routed in VIN1 and (2) there is always one virtual channel of class 0, which is in VIN1, can be requested at each routing step by messages routing in VIN2, messages of each particular type can only be blocked by messages of the same type in VIN1.
Since (1) no deadlock can occur among each of the message types in VIN1, for Algorithm 3.1 is deadlock-free and (2) routing in VIN2 never changes a column message into a row message and never bring a message to a node visited before, no deadlock can occur in each of the message type. From above statement, Algorithm 4.1 is deadlock-free.
5. Conclusion
Designing a deadlock-free routing algorithm that can tolerate unlimited number of faults of arbitrary shape with constant number of virtual channels is not a easy job. Faulty blocks are expanded, by disabling good nodes, to be rectangular faults in existing literature to facilitate the designing of deadlock-free routing algorithms for mesh networks recently. The more the good nodes were disabled, the less the computation and transmission power was left. In this paper, for the purpose of reducing the number of good nodes disabled, we proposed a method to shrink, by recovering fault-diused nodes, these rect-angular faults. Only convex faults exist after fault-shrink. The simulation re-sults show that up to 70% of the disabled nodes, which are needed to form
rectangular faults, can be recovered if the number of original faulty nodes is less than 10% of the total network nodes.
We also proposed a non-adaptive and an adaptive routing algorithm to handle these convex faults. The adaptive algorithm is enhanced from the non-adaptive counterpart by utilizing the virtual channels that are not used in the non-adaptive algorithm. The method we used to enhance the non-adaptive algorithm is simple, easy and its principle is similar to the DuatoÕs theorems [15,16]. There is no restriction on the number of faults tolerated and only four virtual channels per physical channel are needed in the proposed algorithms.
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