[[abstract]]Composed of various topologies, the k-ary n-cube system is desirable for accepting and executing topologically different tasks. We propose a new allocation strategy to utilize the large amount of processor resources in the k-ary n-cubes. Our strategy is an extension of the TC strategy on hypercubes and is able to recognize all subcubes with different topologies. Simulation results show that with such full subcube recognition ability and no internal fragmentation, our strategy depicts constantly better performance than other strategies, such as the Free list strategy on k-ary n-cubes and the Sniffing strategy[[sponsorship]]台灣大學[[conferencetype]]國際[[conferencedate]]19971218~19971220[[conferencelocation]]臺北市, 臺

Chuang, Po-jen

Composed of various topologies, the k-ary n-cube system is desirable for accepting and executing topologically different tasks. We propose a new allocation strategy to utilize the large amount of processor resources in the k-ary n-cubes. Our strategy is an extension of the TC strategy on hypercubes and is able to recognize all subcubes with different topologies. Simulation results show that with such full subcube recognition ability and no internal fragmentation, our strategy depicts constantly better performance than other strategies, such as the Free list strategy on k-ary n-cubes and the Sniffing strategy台灣大學國際19971218~19971220臺北市, 臺

Tamkang University Institutional Repository

Processor Allocation in k-ary n-cube Multiprocessors Po-Jen Chuang and Chih-Ming Wu Department of Electrical Engineering Tam kang University Tamsui, Taipei Hsien, Taiwan 25137, R. 0. (2. Abstract Composed of various topologies, the k -ary  n-cvbe system is desirable f o r  accepting and executing topo- logically different tasks. I n  this paper, we propose a new allocation strategy to .utilite the large amount of processor resources in the k - a r y  n-cubes. Our  strategy is an extension of the TC' strategy on hypercubes an,d is able io recognize all subcubes with different topolo- gies. Simulatton results sh0.w that with such full sub- cube recognition ability and no  internal fragmentation, o'ur strategy depicts consiantly better performance than the other sirategies, such as the Free-list strategy on IC- a r y  n-cubes and the Srizfing strategy. 1 Introduction Composed of various topologies, such as the multi- dimensional toris, meshes and binary cubes, the k- ary n-cube is a generalized form in the multiprocessor system [1,2]. As in real practice the incoming tasks to a system can involve different topologies, the k- ary n-cube becomes very desirable for accepting and executing topologically different tasks. To utilize its large amount of processor resources, processor alloca- tion strategies play quite an important role. When a task arrives dynamically a t  a k-ary n-cube system re- questing processors connected in a specific topology, a good processor allocation strategy makes it possible to find a subsystem swift,ly with the right topology and size. As a n  incoming task has a proper r (2  5 r 5 k )  and m(0 5 m 5 n ) ,  an exact amount of processors with a particular topology must be allocated to it. A k-ary n-cube, denoted by H k , n ,  comprises k" nodes, each with 2n connection links serving as com- munication channels directly to 2n immediate neigh- bors. The parameter n is the dimension of the cube and k is the radix or the number of nodes along each dimension. A 4-ary 3-cube is shown in Fig. 1. A node in the k-ary n-cube can be identified by an n- digit radix-k address An-lAn-2.. .A1Ao, where Ai represents the node's position in the i th  dimension. Rings, meshes, tori, binary n-cubes (hypercubes), and Omega networks are topologically isomorphic to the family of k-ary n-cubes [2]. The computation pat- tern (topology) and the size of the corresponding sub- cube requested by each incoming task must be spec- ified first. The  processors in an H k , n  must be thus allocated to all such tasks so that processor utiliza- tion is maximized (or equivalently, a task is assigned an available subcube whenever it exists). Due to the 1087-4089/97 $10.00 0 1997 IEEE special structure of the k-ary n-cube, it is nontrivial to detect the availability of a subcube. To accomplish this, several previous strategies have been proposed. The  Free-list strategy on k-ary n-cubes [3] is an ex- tension of the Free-list strategy on hypercubes. It can recognize all the free subcubes, but its k is restricted to the power of two. The  Sniffing strategy [4] performs an exhaustive search under some subcube representa- tions and is unable to achieve full subcube recogni- tion. The  k-ary Partner strategy [5] is based on the concept of "partners" in the k-ary tree representation and can allocate only r-ary subcubes with r = k. To achieve full subcube recognition ability with no limita- tions, we propose a new processor allocation strategy which is an extension .from the Tree-Collapsing (TC) strategy on hypercubes [6].  Our new strategy recog- nizes either cubic or non-cubic subcubes and allocates the exact amount of processors to an incoming task. By allocating the exact amount of processors, internal fragmentation can be substantially eliminated. 2 Our Proposed Strategy Our strategy is indeed the generalization of the TC Strategy for processor allocation in hypercubes [SI. The  TC Strategy uses an  n-bit nofa t ion  to represent exactly 2n-m distinct m-dimensional subcubes in an n-dimensional hypercube. For example, a 4-bit nota- tion 0 * U* represents 4 distinct 2-dimensional sub- cubes O*O*, O* 1*, 1 *O*, and 1* I* in a 4-dimensional hypercube. Algorithm G*, the basis of the T C  Strat- egy, is used to generate: the complete search space for an incoming request ( ix . ,  all the possible distinct sub- cubes of the requested subcube dimension, say m) by repeating the process 'of right rotating transform on the n-bit notation starting from the pr imary  notation O O . . . O  * * . . - * ,  with the leftmost n - m bits be- ing the select bits and the rightmost m bits being the don't care bits. The  Right Rotating Transform (R-transform),  as stated in [ 6 ] ,  operates on an n-bit notation, say A = aoal . . . u n - l .  Specifically, an  R- transform from bit i, (denoted by p i ,  performs right rotation on the rightmost n - i bits of A and produces a rotated notation A/ = a O / a l / .  . .an- l /  such that (i) a i /  = a i ,  for 0 5 j < i, (ii) a i /  = aj-1,  for i < j 5 n - 1, and (iii) ai/ = an- l .  To give an example, the rotated notation of per- forming p2 on a 6-bit notation U * 0 * U* is U * *U * U. 21 1 Algorithrn G* for the TC: S t r akgy  is given in the fol- lowing. Algorithm G*: Input: the primary notation (stored in primary-notation) and the dimension of a subcube to be allocated. m.  Procedure main( n,  m, prirnarynotation) begin if(m = 0 or m = n) then stop; for i := n - m - 1 down t80 0 call rotate(i, primarynotation, new-notation); call subsequent-rotate(i, 0 ,  newnota t ion) ;  end for end; Procedure subsequent-rot,at,e( pre-level, pre-step, begin p re -11 o t a t ion ) s t p p  := prest ,ep + 1: i f  s t e p  2 ~n then ret .urn else for i := pre-level to n - m - 1 call rotate( i+ st ep, pre-notation , new -notation) ; call subsequent-rotate( i, step, new-notation); end for end if end: It is clear that there are C ( n , m )  x 2"-" possible distinct. m-dimensional subcubes i n  a n  n-dimensional hypercube. As .Algorithm G* can generate a corn- plete search space for any request,ed m-dimensional subcubes in an n-dimensional hypercube with the pri- mary notation and the C(n. ,  m.)- 1 rotmated notations it produces, we believe the same notation can be general- ized and applied to an Hk," ( k  > 2) by using the n-bit notafion (with m don't care bits *'s  and n - m select bits 0 's)  to represent exactly k"-" distinct S k , m ' s .  For t.he hypercubes, the select bits ca.n be 0 or 1; for the H k , , z ,  they c a n  be 0,  1 , 2 >  . .  ' ,  k - 1. It is easy to derive that Algorithm G* can be used to generate the complete search space for a n y  requested S k , ,  in H k , "  .i.e., to generate the C ( n ,  m.) x L'"" possible distinct Our proposed strategy, to be called the ex- tended TC strategy, uses Algorithm G* to gener- a te  the complete search space and is able to rec- ognize any Sk,,,'s with radices being the same as the system's in the Hk,n. To recognize subcubes of the other topologies, such as meshes, hyper- cubes, and multi-dimensional meshes, our extended T C  strategy makes use of subnotations, each rep- resenting a subset of possible subcubes which may have different dimension radices. To give an  ex- ample, for notation 0 0 . .  . O  * * .  . . *, the subset of possible subcubes with different dimension radices ("-1, "-2,. . ., mo) can be represented by subno- tation m n - l  ...Om(*pm-l),.,,,-l where 0 E { 0 , 1 , .  . ., k - l } ;  pi (0 5 i 5 m-1) represents the start- ing node position in dimension i, thus 0 < pi  < k - 1; r, (0  5 j m. - 1) is the requested radices (number of nodes) in dimension j and 2 5 rj 5 k .  Assuming the incoming t a s k  is a two-dimensional 4 x 2 mesh and the system is a 4-ary 3-cube, a subset of t,he possible Sk ,m 'S. subcubes (noted by 0 * *) can be represented by sub- notation O ( * O ) ~ ( * O ) ~ .  The  starting node of dimension 0 is 0 and the dimension needs two neighboring nodes, say node 0 and node 1. Likewise, dimension 1 can take nodes 0 ,  1, 2,  and 3. A possible 4 x 2 mesh O(*o 4(*0)2  ( tha t  is, when 0 =O in the above subnotation I has 8 nodes (0,0,0), (0,0,1), (0,1,0), (0,1,1), (0,2,0), (Oq2,1),  (0,3,od) and (0,3,1). If all the 4 x 2 meshes repre- sente by the subnotation are busy, another subno- tation 0(*0)4 (*1) . .  O ( * O ) 4 ( * 2 ) 2 ,  . . I ,  or 0 ( * 3 ) 4 ( * 3 h  is considered subsequently. (Note that the combinations 0 0 , 0 1 , 0 2 , 0 3 , 1 0 , 1 1 , ~  . . ,32,33 of all the possible start- ing node positions in the dimensions with don'tcure bits are considered, and beginning from the starting node position, the requested numbers of nodes in those dimensions are t.aken in a "wraparound" way.) All the possible subcubes (4 x 2 meshes) represented by eac,h of the above subnotations are to be said in a subset of subcubes dicfafed by the primary notation 0 * *. If neit.her of the subcubes i n  a n y  subcube subsets (rlic- tat.ed by the primary n o t d o n )  satisfies t,he incoming t a s k ,  start  performing Algorithm G* to get t,he rotated notations. one by one, from 0 * *. Repeat the above process for each rotated notation until a free requested subc,ube is found. The  operation of processor allocation using our ex- tended T C  strategy is provided below. Processor Allocation step 1: Set, m := the dimension and ri := t.he ra.clis for dimension i of the subcube required to accornmo- date task I . .  step 2: {et notation := priniarynot,ation. step 3: Search all the subsets (represented by sub- notations) of the possible subcubes (wit.11 the required dimension and radix in each dimension) dictated by notation until a n  available subcube is found. When found, go to step 5. st.ep 4: Perform Algorithm G* and do the same search process as Step 3 for one rotated notation (if any) at a time. If an available subcube is found, go to the next step; otherwise, go to Step 6. step 5: Set t.he corresponding allocation bits for each node of the available subc.ube to 1's and allocate the subcube to task ij. Stop. Step 6: Attach Ij to the task queue and wait unt,il a subcube is released. Processor Relinquishment: Reset those allocation bits for each node of the re- leased subcube to 0's .  3 Performance Evaluation Extended simulation runs have been conducted to collect performance results for the extended TC strat- egy, the Free-list strategy on k-ary n-cubes and the Sniffing strategy. (The /&Partner strategy is excluded due to its inability to allocate topologically different tasks.) The  simulation is carried out in a 4-ary 3- cube system and task allocation is handled by a task dispatcher independent of the system. All processors are initialized to be free and the 100 tasks are gen- erated and queued at. the dispatcher (no new tasks are generated during t.he course of simulation). T w o  residence time distribut,ions of uniform(5.0, 10.0) and 212 uiiiform( 10.0, 1.5.0) as well as an FCFS scheduling dis- cipline are considered. The  dimensions and radices of subcubes requested by the 100 tasks are respec- tively governed by two distributions, Uniform ( U )  and Normal (N).  Thus there are four combinations of the dimension-radix distribution: U*U,  U*N,  N*U and N*N, and they are able to form subcubes of various topologies. Performance measures such as the completion time, processor utilization, external fragmentation and in- ternal fragmentation are collected for analysis and comparison. All the results are the average values of 10 random seeds; given 95% of confidence, they will be correct. with the error range falling between 3 4 . 4 % .  Simulation results of the four performance measures (under the residence time distribut,ion of uni- form(5.0, 10.0) and the four dimension-radix distribu- tions with tasks requesting different topologies) are respectively depicted in Figs. 2 (  a)-(d).  A s  observed, for the U * U  distribution, the extended TC strategy depicts higher processor utilizat,ion t8han the Sniff- ing strategy by 20.56?6. This is mainly because our strategy is implemented wit.h ful l  subcube recognition ab i  I i t  y and wit, liou t, i n  t.er nal fragment at, ion , while the Sniffing strategy repeatedly allocates excessive proces- sors to tasks due to its singular internal fragmentation (44.34%’. the result of its special subcube represen- t.at,ions). Processor utilization for the extended TC strategy is also higher than that of the Free-list strat- egy (by 7.28%) because the Free-list strategy has to allocate more than needed free processors to a task when t,he requested radix is not the power of two, re- sult ing i n  reniarkable int~ernal fragnient.ation (28.66%). N0t.e that, both internal and external fragmentations can affect. processor utilization i n  the system but the former is more dominat,ing. I n  fact,, substantial in- ternal fragmentation has enact.ecl obvious impact on processor utilizat,ion and completion t.ime for both the Sniffing and Free-list strategies as theL.simulation re- sults dem0nstrat.e. For t.he \I*N distribution, processor utilization for the extended T C  strategy is better than t.he Sniffing strategy (by 13.66%) and the Free-list strate y t,ribution). The  markdown for the Free-list Strategy is apparently the result of its uprising internal and exter- nal fragmentations because when the dimension-radix distribution is U*N, the most frequently requested subcubes will be subcubes whose radices are three. In this case, the internal fragmentation for the  Free-list strategy worsens and as a result external fragmenta- tion also takes a turn for the worse. For the N*U distribution, processor utilization for the extended TC strategy is still higher than  the other two strategies (19% higher than the Sniffing strategy and 7.72% higher than the Free-list strat.egy). As can be seen. the value for the Free-list strategy rises to t.he same level as in the U*U distribut,ion because the radix dist,ribut.ion becomes uniform again, cutting down t,he probability of prompting internal fragmen- tation for the strategy. However, when the  dimension- radix distribution turns N*N,  processor utilization for the Free-list strategy plunges again due to the same 16.15%) - a significant lift from 7.28% for the  U $ U jb! is reason noted before for the U * N  distribut,ion. With the same (climension-radix) distributions, simulation results under the residence time distribu- tion of uniform (10.0, 15.0) are also collected to see the effect of different residence times. As it turns out,  the results follow almost the same trend as what is collected for the residence time distribution of uni- form (5.0, 10.0). The  only difference is: The  longer the residence time (1C1.0, 15.0), the larger the exter- nal fragmentation for all the strategies under all the dimension-radix distributions. This is because extend- ing the residence time of each task tends to prompt external fragment a t  ion. 4 Conclusion In this paper we propose a new processor alloca- tion strategy for the le-ary n-cube with no limitation on the requested dimensions and radices. Our pro- posed strategy c a n  recognize either cubic or non-cubic subcubes and allocate the exact amount of processors to an incoming task. Tha t  is, our strategy is able to achieve ful l  subcube recognition for tasks request- ing different. topologies. Under such a strategy, in- ternal fragmentation can be substantially eliminated. As simulation results demonstrate, in spite of hav- ing larger external fragmentation, our strategy depicts constantly better proclssor utilization than the other investigated strat,egies due to its ability to achieve full subcube recognition under different topologies and also due to the absence of internal fragmentation. It. is also noted that internal fragmentation plays quite a n  important role in dominating processor utilization in  a syst.em. Hence, besides pursuing full subcube recog- nition and preventing external fragmentation, a good processor allocation strategy should first concentrate on the elimination of internal fragmentation. Acknowledgments This work was supported in part by the National Science Council, Taiwan, R. 0. C., under Grant No. NSC85-22 13-E-032-006. References LV. J .  Dally, “Performance Analysis of k-ary n.- cube Interconnection Networks,” IEEE Trans. on Computers ,  Vol. 39, No. 6,  pp. 775-785, June 1990. Ii. Hwang, Advanced Computer  Architecture: Par- allelism, Scalability, Programmability, McGraw-. Hill, 1993, pp. 86-87. H.-L. Chen and C.-T. King, “An Efficient Dy- namic Processor Allocation Scheme for K-ary n- cube Multicomputers,” Proc. 1993 Int’l Conf. on Parallel and Distributed Systems, Dec. 1993, pp. V. Gautam and V.  Chaudhary, “Subcube Allo- cation Strategies in a K-ary N-cube,” Proc. 1993 ISCA Int’l Conf.  o n  Parallel and Distributed Com- puting Sys tems ,  Oct. 1993, pp. 141-146. E;. Windisch, Virginia Lo, and B. Bose, “Con- tiguous and Non-Contiguous Processor Allocation AIgorit,hms for K-ary N-cubes,” Proc. 1995 In2 ’I 217-22 1. 213 [6] P.-J. Chuang arid N.-F. Tzeng, "A Fast Recognition-Complete Processor Allocation Strat- egy for Hypercube Computers," IEEE Trans. on. Compufprs. Yol. 41,  No. 4 ,  pp .  467-479, Apr. 1992. Fig. 1 .  A 4-ary 3-cube. (Hidden nodes and connection links are not shown.) 70 60 .o 50 5 40 30 C 3 g 20 0. IO 0 U'U U'N N'U N'N TC m Sniffing m Free-list U+U U" N'U N'N U'U U'N N*U N*N U'U U'N N'U N'N Fig. 2 .  (a) Processor utilization (b) Completion time (c) External fragmentation (d) Internal fragmentation for different strategies under the residence time distribution of uniform(5.0, 10.0). 214 

Processor allocation in k-ary n-cube multiprocessors

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Processor allocation in k-ary n-cube multiprocessors

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