1、Brief Announcement: Speedups for Parallel Graph Triconnectivity,James Edwards and Uzi VishkinUniversity of Maryland,1,Motivation Begin with a theory of parallel algorithms (PRAM) Develop an architecture (XMT) based on theory Validate theory using architecture Validate architecture using theory In or
2、der to validate XMT, we need to move beyond simple benchmark kernels This is in line with the history of benchmarking of performance (e.g. SPEC) Triconnectivity is the most complex algorithm that has been tested on XMT. Only one serial implementation is publically available, and no prior parallel im
3、plementation Prior work of similar complexity on XMT includes biconnectivity EV12-PMAM/PPoPP and maximum flow CV11-SPAA.,2,Introduction,advanced planarity testing,advanced triconnectivity,planarity testing,triconnectivity,st-numbering,k-edge/vertex connectivity,minimum spanning forest,Euler tours,ea
4、r decompo- sition search,bicon- nectivity,strong orientation,centroid decomposition,tree contraction,lowest common ancestors,graph connectivity,tree Euler tour,list ranking,2-ruling set,prefix-sums,deterministic coin tossing,List, Tree, and Graph Algorithms,4,Triconnected Components,Input graph G,Tr
5、iconnected components of G,High-level structure Key insight for serial and parallel algorithms: separation pairs lie on cycles in the input graph Serial HT73: use depth-first search. Parallel RV88, MR92: use an ear decomposition.,5,Triconnectivity Algorithm,Ear decomposition of G,E1,E2,E3,Low-level
6、structure The bulk of the algorithm lies in general subroutines such as graph connectivity. Implementation of the triconnectivity algorithm was greatly assisted by reuse of a library developed during earlier work on biconnectivity (PMAM 12). Using this library, a majority of students successfully co
7、mpleted a programming assignment on biconnectivity in 2-3 weeks in a grad course on parallel algorithms.,6,Triconnectivity Algorithm,The Explicit Multi-Threading (XMT) architecture was developed at the University of Maryland with the following goals in mind: Good performance on parallel algorithms o
8、f any granularity Support for regular or irregular memory access Efficient execution of code derived from PRAM algorithms A 64-processor FPGA hardware prototype and a software toolchain (compiler and simulator) exist; the latter is freely available for download.,7,The XMT Platform,Random graph: Edge
9、s are added at random between unique pairs of vertices Planar3 graph: Vertices are added in layers of three; each vertex in a layer is connected to the other vertices in the layer and two vertices of the preceding layer Ladder: Similar to Planar3, but with two vertices per layer,8,Graph Families,9,S
10、peedup,10,Analytic vs. Experimental Runtime,T(n, m, s) = (2.38n + 0.238m + 4.75s) log2 n,The speedups presented here (up to 129x) in conjunction with prior results for biconnectivity (up to 33x) and max-flow (up to 108x) demonstrates that the advantage of XMT is not limited to small kernels. Biconne
11、ctivity was an exceptional challenge due to the compactness of the serial algorithm. This work completes the capstone of the proof-of-concept of PRAM algorithms on XMT. With this work, we now have the foundation in place to advance to work on applications.,11,Conclusion,CV11-SPAA G. Caragea, U. Vish
12、kin. Better Speedups for Parallel Max-Flow. Brief Announcement, SPAA 2011. EV12-PMAM J. Edwards and U. Vishkin. Better Speedups Using Simpler Parallel Programming for Graph Connectivity and Biconnectivity. PMAM, 2012. EV12-SPAA J. Edwards and U. Vishkin. Brief Announcement: Speedups for Parallel Gra
13、ph Triconnectivity. SPAA, 2012. HT73 J. E. Hopcroft and R. E. Tarjan. Dividing a graph into triconnected components. SIAM J. Computing, 2(3):135158, 1973.,12,References,MR92 G. L. Miller and V. Ramachandran. A new graph triconnectivity algorithm and its parallelization. Combinatorica, 12(1):5376, 19
14、92. KTCBV11 F. Keceli, A. Tzannes, G. Caragea, R. Barua and U. Vishkin. Toolchain for programming, simulating and studying the XMT many-core architecture. Proc. 16th Int. Workshop on High-Level Parallel Programming Models and Supportive Environments (HIPS), in conjunction with IPDPS, Anchorage, Alas
15、ka, May 20, 2011.,13,References,RV88 V. Ramachandran and U. Vishkin. Efficient parallel triconnectivity in logarithmic time. In Proc. AWOC, pages 3342, 1988. TV85 R. E. Tarjan and U. Vishkin. An Efficient Parallel Biconnectivity Algorithm. SIAM J. Computing, 14(4):862874, 1985. WV08 X. Wen and U. Vi
16、shkin. FPGA-Based Prototype of a PRAM-on-Chip Processor. In Proceedings of the 5th Conference on Computing Frontiers, CF 08, pages 5566, New York, NY, USA, 2008. ACM.,14,References,Backup slides,15,PRAM algorithms are not a good match for current hardware: Fine-grained parallelism = overheads Requir
17、es managing many threads Synchronization and communication are expensive Clustering reduces granularity, but at the cost of load balancing Irregular memory accesses = poor locality Cache is not used efficiently Performance becomes sensitive to memory latency,16,The Problem with the PRAM,Main feature
18、 of XMT: Using similar hardware resources (e.g. silicon area, power consumption) as existing CPUs and GPUs, provide a platform that to a programmer looks as close to a PRAM as possible. Instead of 8 “heavy” processor cores, provide 1,024 “light” cores for parallel code and one “heavy” core for seria
19、l code. Devote on-chip bandwidth to a high-speed interconnection network rather than maintaining coherence between private caches.,17,The XMT Platform,For the PRAM algorithm presented, the number of HW threads is more important than the processing power per thread because they happen to perform more
20、 work than an equivalent serial algorithm. This cost is overridden by sufficient parallelism in hardware. Balance between the tight synchrony of the PRAM and hardware constraints (such as locality) is obtained through support for fine-grained multithreaded code, where a thread can advance at it own speed between (a form of) synchronization barriers.,18,The XMT Platform,Maximal planar graph Built layer by layer The first layer has three vertices and three edges. Each additional layer has three vertices and nine edges.,19,Evaluation: Graph Families,
