Hypergraph Sparsifica/on and Its Applica/on to Par//oning

Transcription

1 Hypergraph Sparsifica/on and Its Applica/on to Par//oning Mehmet Deveci 1,3, Kamer Kaya 1, Ümit V. Çatalyürek 1,2 1 Dept. of Biomedical Informa/cs, The Ohio State University 2 Dept. of Electrical & Computer Engineering, The Ohio State University 3 Dept. of Computer Science & Engineering, The Ohio State University

2 Mo+va+on Problem: Sparsifica/on of large- scale data modeled as a hypergraph for a scalable computa/on and analysis Today data is big and its u/liza/on and analysis require complex algorithms and immense amount of compu/ng power. The techniques to make the data smaller are very important. We should avoid any redundancy in the data and we can even sacrifice some part of it to reduce the size. Applica/on (in this work): Hypergraph par//oning Used in many problems in parallel scien/fic compu/ng such as sparse matrix reordering, sta/c and dynamic load balancing, clustering, and recommenda/on. Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 2

3 Contribu+on Proposed hypergraph sparsifica/on techniques Iden/cal net removal Already exist in some par//oning tools but our implementa/on is faster Iden/cal vertex removal Similar net removal To our best knowledge, there is no work that analyzes the effec/veness of the sparsifica/on on hypergraphs. Implemented under UMPa [Catalyurek12], a mul/- objec/ve hypergraph par//oner Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 3

4 Hypergraph Par++oning Hypergraph H=(V,N) V: vertex set, N: net set c(n): cost of a net w(v): weight of a vertex Objec/ve: Par//on the hypergraph Balanced load distribu/on Wk < Wavg( 1+ ε ) for 1 k K Minimized communica/on between parts CV = n N c(n) ( λ n 1) n 1, n 2, n 3 and n 5, n 6 are identical nets v 2, v 4 are identical vertices. Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 4

5 Par++oning Example P 1 P 2! c(n i ) =1, w(v i ) =1 $ # & # λ n1 = λ n2 = λ n3 = λ n5 = λ n6 =1, & # & " λ n4 = 2, % # CV =1 & % ( $ W 1 = 2,W 2 = 3 imbal = 0.2' Par//oning criteria: Communica/on volume and par//oning /me. Be_er volume reduces the parallel execu/on /me. However, par//oning /me can dominate applica/on /me. We want to reduce the par//oning /me by sparsifica/on Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 5

6 Mul+- level Approach Three phases: Coarsening: obtain smaller and similar hypergraphs to the original Ini+al par++oning: find a solu/on for the smallest hypergraph. Uncoarsening: project the ini/al solu/on to the finer hypergraphs and refine it itera/vely un/l a solu/on for the original hypergraph obtained. Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 6

7 Iden+cal Net Removal (INR) Two nets are iden/cal if their pin sets are the same Pairwise comparison is very expensive Instead, we use hashing If two nets are iden/cal the sum of their pin id s must be iden/cal Calculate a hash value for each net, and compare only the ones with the same hash value Choose one representa/ve net for an iden/cal net set Coarsening sparsifies the vertices. INR is done after coarsening level. Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 7

8 INR Hash Func+on Hash func/ons: CSj(n) = i pins[n] Murmur Hash [Appleby12] i j The quality of the hash func/on depends on the number of collusions, e.g., CS(n 1 ) =CS(n 2 ) for two nets n 1 and n 2 which are not iden/cal nets False- posi/ve cost: Number of pairwise comparisons for non- iden/cal nets Checksum occupancy: The average number of representa/ves having the same checksum value Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 8

9 INR Variants INR- SRT: Calculates hash values for each net, then sorts it w.r.t. hash values. Reduces the false- posi/ve, and occupancy rate. However, sor/ng can be expensive. INR- MEM: Uses two arrays first and next to store the hash values in a linked list structure. Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 9

10 Hashing Example Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 10

11 Hashing Example CS1(n 1 ) = 1+3 = 4 Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 11

12 Hashing Example CS1(n 2 ) = 1+3 = 4 Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 12

13 Hashing Example CS1(n 3 ) = 1+3 = 4 Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 13

14 Hashing Example CS1(n 4 ) = = 9 9 mod 7 = 2 Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 14

15 Hashing Example CS1(n 5 ) = = mod 7 = 4 Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 15

16 Hashing Example CS1(n 6 ) = = mod 7 = 4 c(n 1 ) = c(n 1 )+c(n 2 )+c(n 3 ) c(n 5 ) = c(n 5 )+c(n 6 ) Occupancy = (2 + 1) / 2 = 1.5 Occupancy = ( ) / 3 = 1 Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 16

17 Iden+cal Vertex Removal (IVR) Two ver/ces are iden/cal if they are connected to the same nets. Same methods applied to INR Although INR does not affect the par//oning result, IVR affects the quality of the par//oning by taking early decisions on the part assignments. Coarsening sparsifies the identical vertices during coarsening. There is no need for IVR. But IVR performed at the beginning of the coarsening can reduce its execution time. Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 17

18 Similar Net Removal (SNR) INR aims to remove the redundancy from hypergraph. only effec/ve when iden/cal nets, i.e., redundancy, exist SNR removes the similar nets even when there is no redundancy Lossy compression technique. Usually worsen the quality, but makes the par//oning faster. When the performance of the applica/on is not very sensi/ve against small changes in par//oning quality, this trade/off can be useful. Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 18

19 Similar Net Removal (SNR) The similarity between two nets n i and n j is defined with Jaccard Coefficient: J(n i, n j ) = pins[n i ] pins[n j ] pins[n i ] pins[n j ] Since the number of nets is large, it is infeasible to compute the similarity for each net pair. Instead, compute a footprint of each net using minhash Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 19

20 Similar Net Removal (SNR) σ is a random permuta/on of the integers from 1 to V, and min σ (n) is the first vertex id of a net n N under the permuta/on σ. We use t permuta/ons σ 1 to σ t to obtain a minwise footprint of each net. Two nets n i and n j are similar iff their minwise footprints are iden/cal, where mf(n) = (min σ1 (n),..., min σt (n)). We do the hashing and pairwise comparison only for this minwise footprint set, and choose one of the nets as the representa/ve of this set. Large (LRG): representa/ve is the net with the largest number of pins. Important (IMP): when calcula/ng the pin count, priori/zing the pins which are connected to heavy nets. Union (UNI): representa/ve is a virtual net that is connected to all pins of the nets in the set. Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 20

21 Experiments All the algorithms are implemented in UMPa. g++ version and O3 flag. Intel Xeon E5520 (quad- core clocked at 2.27 Ghz) 48 GBs of Memory 28 matrices from different matrix classes. K = 2, 8, 32, 128, 512, 1024 Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 21

22 Hash Func+on Comparison Normalized+w.r.t.+Cs12Sort+ 1.4" 1.2" 1" 0.8" 0.6" Elimina:on":me" False"posi:ve"cost" Checksum"occupancy" Quality is be_er with INRSRT, as no limit on hash- size. Except CS1, all other has an occupancy value close to 1 (op/mal occupancy). INRMEM equipped with CS2 has best performance. Checksum func/on is as good as CS3 and MurmurHash. 0.4" Computa/onally cheaper. 0.2" 0" InrSrt+Cs2" InrSrt+Cs3" InrSrt+Murmur" InrMem+Cs1" InrMem+Cs2" InrMem+Cs3" InrMem+Murmur" Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 22

23 Improvement on Time and CV BASE UMPa INR INR+IVR K Time CV Time- Speedup CV- improve Time- Speedup CV- improve , , , , , , to 3.30 speedups for INR+IVR 0.3% 2.4% quality improvement on average. The speedup values are increasing with K promising as the overhead of the partitioning problem is usually an issue for large K values. Most of the speedup is obtained with INR, as not all hypergraphs contain identical vertices. 14/28 of the matrices in the test set have less than 103 identical vertices Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 23

24 SNR improvement w.r.t INR+IVR 1.1$ 1$ SNR8LRG$ SNR8P48LRG$ SNR8IMP$ SNR8P48IMP$ SNR8UNI$ SNR8P48UNI$ 0.9$ 0.8$ 0.7$ Total$Volume$ Total$Time$ Total$Volume$ Total$Time$ Total$Volume$ Total$Time$ 4 permuta/on array (t=4). SNR- P4- X restricts the removal process to only the nets with 4 or more pins SNR- X and SNR- P4- X where X is a representa/ve selec/on method. 128$ 512$ 1024$ On 1024 processor SNR- LRG è 22% improvement on /me, 5% harm on CV 4.2 speedup w.r.t. Base 4% reduc/on on CV SNR- P4- LRG è 15% improvement on /me, 2% harm on CV 3.9 speedup w.r.t. Base 2% reduc/on on CV Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 24

25 Conclusion We proposed heuris/cs for lossless and lossy hypergraph sparsifica/on. We show that the effec/veness of the heuris/c increases with the number of part numbers. This is promising as par//oning overhead is an issue for today s architectures with large number of processors. Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 25

26 References Catalyurek et al. "UMPA: A Mul/- objec/ve, mul/- level par//oner for communica/on minimiza/on." Graph Par??oning and Graph Clustering (2012). A. Appleby, SMHasher & MurmurHash, 2012, h_p://code. google.com/p/smhasher/. Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 26

27 Thanks For more informa/on Visit h_p://bmi.osu.edu/~umit or h_p://bmi.osu.edu/hpc Acknowledgement of Support Deveci et al. "Hypergraph Sparsifica+on and its Applica+on to Par++oning" 27