1 Introduction
Digg
, a news sharing-based social network [18]. The data include cascades with a depth of up to six links, i.e., the maximum path length from the initiator of the content to the users who eventually get the content. When we partitioned the graph by just minimizing the number of links straddling between 32 balanced partitions (using Metis [6]), the majority of the traffic remained between the servers, as opposed to staying local. This traffic goes over a relatively small fraction of the links. Only \(0.01\%\) of the links occur in \(20\%\) of the cascades, and these links carry \(80\%\) of the traffic observed in these cascades. It is important to identify the highly active edges and avoid placing them crossing the partitions.Variable | Description |
---|---|
\(\varPi = \left\{ V_1, \dots , V_k \right\} \)
| A K-way partition of a graph \(G=(V,E)\) |
\(V_k\)
| Part k of partition \(\varPi \) |
\(\chi (\varPi )\)
| Cut size under partition \(\varPi \) |
\(I_g(v)\)
| Random propagation tree |
\(\lambda ^{\varPi }_g(v)\)
| Communication operations induced by propagation tree \(I_g(v)\) under \(\varPi \) |
\(g \sim G\)
| Unweighted directed graph g drawn from the distribution induced by G |
\({\mathbb {E}}_{v,g \sim G}[\lambda ^{\varPi }_g(v)] \)
| Expected number of communication operations during a propagation process |
\(w_{ij}\)
| Propagation probability along edge \(e_{ij}\) |
\(p_{ij}\)
| Probability of edge \(e_{ij}\) being involved in a random propagation process |
I
| The set of random propagation trees generated by the estimation technique |
\(F_I(e_{ij})\)
| The number of trees in I that contains edge \(e_{ij}\) |
N
| The size of set I (i.e., \(N = |I|\)) |
2 Background
2.1 Graph partitioning
2.2 Related work
2.2.1 Graph partitioning and replication
2.2.2 Multi-query optimization
2.2.3 Influence propagation
3 Problem definition
3.1 Propagation trees
3.2 Cascade-aware graph partitioning
4 Solution
4.1 Computation of the \(p_{ij}\) values
4.2 Implementation of the estimation method
4.3 Algorithm
4.4 Determining the size of set I
4.5 Complexity analysis
4.6 Extension to the LT model
4.7 Processes starting from multiple users
5 Extensions and limitations
5.1 Non-cascading queries
5.2 Intra-propagation balancing among servers
5.3 Repartitioning
5.4 Replication
6 Experimental evaluation
Number of | In degree | Out degree | Descripton | ||||
---|---|---|---|---|---|---|---|
Vertices | Edges | max | avg | max | avg | ||
Facebook | 4039 | 176,468 | 1045 | 44 | 1045 | 44 | Social network from Facebook |
wiki-Vote | 7115 | 103,689 | 893 | 15 | 457 | 15 | Wikipedia who-votes-on-whom network |
HepPh | 34,546 | 421,534 | 411 | 12 | 846 | 12 | Arxiv High Energy Physics paper citation network |
email-Enron | 36,692 | 367,662 | 1383 | 10 | 1383 | 10 | Email communication network |
Epinions | 75,879 | 508,837 | 1801 | 7 | 3035 | 7 | Who-trusts-whom network of Epinions.com |
Twitter (small) | 81,306 | 1,768,135 | 1205 | 22 | 3383 | 22 | Social network from Twitter |
Slashdot | 82,168 | 870,161 | 2510 | 11 | 2552 | 11 | Slashdot social network from February 2009 |
email-EuAll | 265,214 | 418,956 | 929 | 2 | 7631 | 2 | Email communication network |
dblp | 317,080 | 2,099,732 | 343 | 7 | 343 | 7 | DBLP collaboration network |
youtube | 1,134,890 | 5,975,248 | 28,754 | 5 | 28,754 | 5 | Youtube online social network |
Pokec | 1,632,803 | 30,622,564 | 8763 | 19 | 13,733 | 19 | Pokec online social network |
wiki-Talk | 2,394,385 | 5,021,410 | 100,022 | 2 | 3311 | 2 | Wikipedia talk (communication) network |
LiveJournal | 4,847,571 | 68,475,391 | 20,292 | 14 | 13,905 | 14 | LiveJournal online social network |
Twitter (large) | 11,316,811 | 85,331,843 | 564,512 | 7 | 214,381 | 7 | Social network from Twitter |
uk-2002 | 18,484,123 | 292,243,668 | 194,942 | 16 | 2449 | 16 | Web graph crawled (2002) under .uk domain |
sinaweibo | 58,655,849 | 522,642,104 | 278,490 | 9 | 278,490 | 9 | Sina Weibo online social network |
webbase-2001 | 118,142,155 | 1,019,903,190 | 816,127 | 8 | 3841 | 8 | Web graph crawled (2001) by WebBase [44] |
random-social-network | 65,536 | 910,479 | 9613 | 19 | 3233 | 19 | Generated by graph500 power law graph generator |
6.1 Experimental setup
6.1.1 Datasets
Facebook
‐LiveJournal
) are collected from Stanford Large Network Dataset Collection1 [45], and they contain friendship, communication or citation relationships between users in various real-world social network applications. Twitter (large)
is collected from [46], uk-2002
and webbase-2001
are collected from Laboratory for Web Algorithmics2 [47], and sinaweibo
is collected from Network Repository3 [48]. Additionally, we also make use of a synthetic graph, named as random-social-network
, which we generate by using graph500 [49] power law random graph generator. The graph500 tool is initialized with two parameters, namely as edge-factor and scale, in order to produce graphs with \(2^\mathrm{scale}\) vertices and edge-factor\(\times 2^\mathrm{scale}\) directed edges. We set both scale and edge-factor to 16 to produce random-social-network
dataset.6.1.2 Baseline partitioning (BLP) algorithm
6.1.3 Other tested algorithms
6.1.4 Content propagations
6.1.5 Partitioning framework
6.2 Experimental results
\(K=32\)
|
\(K=64\)
| |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
RP | BLP | CAP | RP | BLP | CAP | |||||||
comm | comm | cut | comm | cut | %imp | comm | comm | cut | comm | cut | %imp | |
Facebook | 3787 | 1818 | 0.32 | 1647 | 0.38 | 9.40 | 3854 | 2344 | 0.52 | 2187 | 0.57 | 6.66 |
wiki-Vote | 2127 | 1681 | 0.74 | 1360 | 0.82 | 19.11 | 2160 | 1824 | 0.82 | 1497 | 0.87 | 17.91 |
HepPh | 12,048 | 4092 | 0.29 | 3155 | 0.44 | 22.91 | 12,580 | 5095 | 0.36 | 3858 | 0.52 | 24.28 |
email-Enron | 25,153 | 6539 | 0.39 | 5083 | 0.45 | 22.27 | 25,514 | 8073 | 0.45 | 6193 | 0.52 | 23.29 |
Epinions | 29,801 | 12,781 | 0.59 | 8290 | 0.66 | 35.14 | 30264 | 13,232 | 0.64 | 8699 | 0.70 | 34.26 |
Twitter (small) | 68,391 | 11,901 | 0.18 | 8827 | 0.23 | 25.83 | 69,470 | 14,348 | 0.22 | 10,379 | 0.27 | 27.66 |
email-EuAll | 27,543 | 3709 | 0.21 | 2592 | 0.31 | 30.11 | 28,050 | 24,421 | 0.75 | 2934 | 0.35 | 87.99 |
Slashdot | 57,426 | 29,724 | 0.71 | 21,497 | 0.77 | 27.68 | 58,313 | 29,416 | 0.74 | 22,433 | 0.79 | 23.74 |
dblp | 240,487 | 36,939 | 0.17 | 33,853 | 0.20 | 8.36 | 244,170 | 40,818 | 0.19 | 37,270 | 0.22 | 8.69 |
youtube | 648,399 | 147,215 | 0.33 | 122,535 | 0.40 | 16.76 | 659,107 | 163,117 | 0.38 | 134,613 | 0.46 | 17.47 |
Pokec | 429,046 | 129,345 | 0.26 | 117,943 | 0.31 | 8.82 | 435,335 | 154,124 | 0.33 | 141,410 | 0.39 | 8.25 |
wiki-Talk | 729,773 | 115,825 | 0.44 | 93,005 | 0.47 | 19.70 | 740,897 | 125,724 | 0.47 | 98,511 | 0.50 | 21.64 |
LiveJournal | 1,152,319 | 279,846 | 0.24 | 234,804 | 0.30 | 16.10 | 1,168,167 | 317,001 | 0.29 | 267,697 | 0.36 | 15.55 |
Twitter (large) | 4,183,322 | 1,950,300 | 0.57 | 1,194,675 | 0.75 | 38.74 | 4,250,880 | 1,942,093 | 0.68 | 1,268,234 | 0.79 | 34.70 |
uk-2002 | 6,933,807 | 107,593 | 0.01 | 60,539 | 0.03 | 43.73 | 7,046,263 | 114,284 | 0.01 | 64,713 | 0.03 | 43.38 |
sinaweibo | 40,403,232 | 12,192,759 | 0.46 | 9,541,550 | 0.56 | 21.74 | 40,404,365 | 12,356,095 | 0.50 | 10,169,184 | 0.61 | 17.70 |
webbase-2001 | 29,301,906 | 614,458 | 0.02 | 329,646 | 0.03 | 46.35 | 29,774,985 | 651,124 | 0.02 | 358,722 | 0.04 | 44.91 |
norm avgs wrto RP | 1.00 | 0.21 | 0.16 | 25.16 | 1.00 | 0.26 | 0.17 | 31.82 |
\(K=128\)
|
\(K=256\)
| |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Facebook | 3884 | 2774 | 0.70 | 2629 | 0.73 | 5.24 | 3893 | 3107 | 0.82 | 2980 | 0.85 | 4.09 |
wiki-Vote | 2174 | 1931 | 0.88 | 1837 | 0.96 | 4.89 | 2184 | 2125 | 0.96 | 2101 | 0.97 | 1.13 |
HepPh | 12,409 | 5954 | 0.44 | 4389 | 0.60 | 26.29 | 12,850 | 7034 | 0.53 | 5048 | 0.68 | 28.23 |
email-Enron | 25,714 | 9596 | 0.53 | 7636 | 0.59 | 20.42 | 25,820 | 11,505 | 0.59 | 10,208 | 0.65 | 11.27 |
Epinions | 30,506 | 13,285 | 0.68 | 9094 | 0.72 | 31.55 | 30,653 | 13,635 | 0.71 | 9574 | 0.74 | 29.78 |
Twitter (small) | 69,993 | 17,111 | 0.27 | 12,336 | 0.34 | 27.90 | 70,169 | 20,687 | 0.34 | 14,709 | 0.41 | 28.89 |
email-EuAll | 28,266 | 24,241 | 0.79 | 3231 | 0.41 | 86.67 | 28,348 | 24,172 | 0.81 | 4207 | 0.47 | 82.60 |
Slashdot | 58,716 | 29,501 | 0.76 | 23,149 | 0.81 | 21.53 | 58,946 | 30,265 | 0.79 | 23,667 | 0.81 | 21.80 |
dblp | 245,923 | 43,376 | 0.20 | 39,053 | 0.24 | 9.97 | 247,006 | 45,240 | 0.21 | 40,534 | 0.25 | 10.40 |
youtube | 664,146 | 177,097 | 0.43 | 146,781 | 0.51 | 17.12 | 666,821 | 195,400 | 0.49 | 158,057 | 0.54 | 19.11 |
Pokec | 438,930 | 182,158 | 0.40 | 168,153 | 0.49 | 7.69 | 440,504 | 210,461 | 0.48 | 201,080 | 0.61 | 4.46 |
wiki-Talk | 747,415 | 130,856 | 0.49 | 111,842 | 0.51 | 14.53 | 747,916 | 142,779 | 0.50 | 138,546 | 0.52 | 2.97 |
LiveJournal | 1,177,061 | 347,405 | 0.33 | 295,910 | 0.40 | 14.82 | 1,182,573 | 368,749 | 0.36 | 318,370 | 0.44 | 13.66 |
Twitter (large) | 4,284,834 | 3,449,396 | 0.86 | 1,308,157 | 0.81 | 62.08 | 4,301,563 | 3,610,392 | 0.93 | 1,343,795 | 0.83 | 62.78 |
uk-2002 | 7,101,990 | 117,968 | 0.01 | 68,482 | 0.03 | 41.95 | 7,129,786 | 126,795 | 0.01 | 71,213 | 0.03 | 43.84 |
sinaweibo | 40,726,570 | 12,836,470 | 0.53 | 10,679,002 | 0.63 | 16.81 | 40,886,216 | 13,232,762 | 0.57 | 11,027,580 | 0.65 | 16.66 |
webbase-2001 | 30,011,343 | 682,865 | 0.02 | 383,526 | 0.04 | 43.84 | 30,129,511 | 717,763 | 0.02 | 405,536 | 0.04 | 43.50 |
norm avgs wrto RP | 1.00 | 0.28 | 0.19 | 32.04 | 1.00 | 0.31 | 0.21 | 29.97 |
\(K=512\)
|
\(K=1024\)
| |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Facebook | 3912 | 3625 | 0.95 | 3555 | 0.95 | 1.91 | 3914 | 3721 | 0.97 | 3687 | 0.97 | 0.91 |
wiki-Vote | 2186 | 2135 | 0.97 | 2131 | 0.97 | 0.21 | 2190 | 2138 | 0.96 | 2115 | 0.97 | 1.11 |
HepPh | 12,822 | 7956 | 0.61 | 5555 | 0.76 | 30.18 | 12,863 | 8910 | 0.70 | 6075 | 0.82 | 31.82 |
email-Enron | 25,835 | 13,560 | 0.66 | 12,733 | 0.71 | 6.10 | 25,897 | 15,679 | 0.72 | 14,621 | 0.75 | 6.75 |
Epinions | 30,631 | 14,396 | 0.74 | 10,320 | 0.75 | 28.32 | 30,674 | 15,302 | 0.77 | 11,424 | 0.77 | 25.34 |
Twitter (small) | 70,309 | 24,979 | 0.42 | 18,020 | 0.50 | 27.86 | 70,430 | 33,555 | 0.55 | 23,861 | 0.64 | 28.89 |
email-EuAll | 28,388 | 24,255 | 0.83 | 6109 | 0.58 | 74.81 | 28,390 | 25,421 | 0.86 | 10,569 | 0.70 | 58.42 |
Slashdot | 59,053 | 30,532 | 0.81 | 24,340 | 0.82 | 20.28 | 59,161 | 30,944 | 0.82 | 24,938 | 0.83 | 19.41 |
dblp | 247,367 | 47,106 | 0.22 | 41,875 | 0.26 | 11.11 | 247,658 | 49,605 | 0.24 | 43,372 | 0.27 | 12.57 |
youtube | 668,146 | 200,723 | 0.52 | 166,605 | 0.56 | 17.00 | 668,800 | 212,618 | 0.55 | 180,943 | 0.58 | 14.90 |
Pokec | 441,892 | 232,194 | 0.55 | 214,309 | 0.67 | 7.70 | 442,150 | 249,415 | 0.60 | 223,356 | 0.72 | 10.45 |
wiki-Talk | 752,402 | 635,183 | 0.90 | 622,024 | 0.92 | 2.07 | 752,048 | 672,420 | 0.93 | 663,852 | 0.92 | 1.27 |
LiveJournal | 1,185,101 | 389,357 | 0.39 | 340,986 | 0.48 | 12.42 | 1,185,655 | 412,393 | 0.42 | 362,391 | 0.52 | 12.12 |
Twitter (large) | 4,310,133 | 3,498,242 | 0.94 | 1,472,766 | 0.85 | 57.90 | 4,314,342 | 3,917,869 | 0.97 | 3,397,889 | 0.94 | 13.27 |
uk-2002 | 7,144,002 | 138,545 | 0.02 | 75,149 | 0.04 | 45.76 | 7,150,971 | 156,001 | 0.02 | 82,082 | 0.05 | 47.38 |
sinaweibo | 40,966,473 | 13,630,831 | 0.60 | 11,516,372 | 0.67 | 15.51 | 40,983,365 | 14,209,656 | 0.63 | 11,904,051 | 0.69 | 16.23 |
webbase-2001 | 30,188,276 | 735,969 | 0.02 | 427,812 | 0.04 | 41.87 | 30,218,102 | 760,751 | 0.02 | 441,437 | 0.04 | 41.97 |
norm avgs wrto RP | 1.00 | 0.36 | 0.26 | 27.36 | 1.00 | 0.38 | 0.30 | 22.14 |
Facebook
and wiki-Vote
datasets), improvements of Metis significantly degrades as can be seen from Table 3. However, for the case of web graphs (e.g., uk-2002
and webbase-2001
), Metis provides significantly better partitions, providing cut ratios below 0.1 (i.e., structures of graphs also have effect on quality of partitions produced by Metis). As a result, the number of inter-partition communication operations is significantly less in cases of these graphs as compared to other cases.email-EuAll
social network for \({K = 64}\), where the partitions obtained by CAP achieve \(88\%\) less communication operations than those by BLP. Also in this partitioning instance, CAP achieves a cut ratio of 0.35 which is significantly less than 0.75 of BLP. However, as the value of K increases, the improvement of CAP over BLP decreases for some social networks, especially for wiki-Talk
and wiki-Vote
, where \(19.11\%\) and \(19.70\%\) improvements of CAP over BLP for \(K=32\), respectively, decrease to \(1.11\%\) and \(1.27\%\) for \({K = 1024}\). This can be attributed to Eq. (20), since as the value of K increases, the number of cut edges is also expected to increase. As shown in Eq. (20), the number of cut edges directly affects the error made by CAP algorithm: the upper bound of the error made by CAP algorithm is shown to be proportional to the number of cut edges. Indeed, the performance improvement of CAP over BLP is observed to be the lowest on the partitioning instances for which CAP incurs the highest cut ratio. For instance, on datasets Facebook
, wiki-Talk
and wiki-Vote
for \({K=1024}\), partitions generated by CAP have cut ratios of 0.97, 0.97 and 0.92, respectively.random-social-network
. As seen in Table 4, partitions obtained by CAP algorithm cause \(43\%\) less communication operations for \({K = 32}\) even though the fraction of edges are \(6\%\) more than that of BLP. As noted in previous experimental results, the improvement of CAP over BLP decreases as the value of K and the cut ratio increases: the percent improvement of CAP over BLP decreases from \(43\%\) to \(30\%\) as the fraction of cut edges increases from 0.91 to 0.95 on \(K=32\) and \(K=1024\), respectively.random-social-network
. “cut” column denotes the fraction of edges remain in the cut after partitioning, “comm” column denotes the average number of communication operations and “%imp” column denotes the percent improvement of CAP over BLP
K
| RP | BLP | CAP | %imp | |||
---|---|---|---|---|---|---|---|
cut | comm | cut | comm | cut | comm | ||
32 | 0.97 | 24,690 | 0.85 | 20,981 | 0.91 | 11,890 | 43.33 |
64 | 0.98 | 25,119 | 0.92 | 18,638 | 0.93 | 12,317 | 33.91 |
128 | 0.99 | 25,328 | 0.93 | 18,484 | 0.94 | 12,635 | 31.64 |
256 | 0.99 | 25,378 | 0.94 | 18,546 | 0.94 | 12,961 | 30.11 |
512 | 0.99 | 25,400 | 0.95 | 19,911 | 0.95 | 13,515 | 32.12 |
1024 | 0.99 | 25,511 | 0.95 | 20,279 | 0.95 | 14,229 | 29.83 |
6.2.1 Sensitivity analysis
random-social-network
. In Fig. 3, we designate the size of set I as \(|I|=10,10^2,10^3,10^4\) and \(10^5\), respectively. Experiments are performed under K-way partitions for \(K=32,64,128\) and 256. We plot the percent increase in the performance of CAP over RP on y-axis. The accuracy values are computed for confidence \(\delta =0.05\) and displayed on the right side of the figure. As seen in the figure, with increasing size of set I, the value of \(\theta \) decreases exponentially and the improvement of CAP increases logarithmically. Additionally, as also observed earlier, the relative performance of CAP decreases with increasing K. The best performance improvement is obtained for \(K=32\) where CAP performs 2x better than RP. These results can be attributed to the results of Eq. 20, since for higher values of K, both the cut ratio and the error made by the CAP algorithm increase. However, as the accuracy increases, the error made by CAP decreases and the overall optimization quality improves.
6.2.2 Relationship with minimizing cut edges
HepPh
dataset on \({K=32}\) parts/servers (i.e., the \(\alpha \) value is multiplied by \(F_I(e_{ij})\) value of each edge). As seen in the figure, with increasing \(\alpha \), the average number of communication operations decreases, whereas the ratio of the number of cut edges increases. On the other hand, the increase in the cut size slows down after \(\alpha =10^{-2}\) and cut ratio becomes at most 0.44, since the cut size also has effect on the average number of communication operations. Note that for the smallest value of \(\alpha \), CAP becomes almost equivalent to CUT, providing equally well partitions in terms of number of cut edges (black-dashed curve denotes the cut value obtained by CUT algorithm). If the query workload is dominated by non-cascading queries and there is comparably small number of cascades, then \(\alpha \) value can be set to smaller values, or vice versa.6.2.3 Running times
sinaweibo
, consists of approximately 58 M vertices and 522 M edges. For this social network, the estimation of \(p_{ij}\) values took approximately 5.9 h and required 59 GB main memory. For the estimation phase, we employed shared memory parallelism and generated the set I via using 72 threads. The partitioning phase of undirected graph \(G'\) into \({K=1024}\) parts via Metis took approximately 1, 25 hours and required 71 GB main memory. It is worth to mention that partitioning time of sinaweibo
via Pulp [51], which is a label propagation-based partitioning tool, took approximately 1.1 h in case of BLP algorithm. Here, both Metis and pulp were run in serial mode for a fair comparison. Similarly, the biggest graph we used in our experiments, webbase-2001
, has 118 M vertices and 1B edges. For this dataset, the estimation phase took approximately 2.2 h and required 130 GB main memory. However, even though webbase-2001
has more vertices and edges than sinaweibo
, partitioning this graph into \({K=1024}\) parts took approximately 0.5 h and required 51 GB main memory. Note that the estimation phase of CAP takes only 4.7x and 4.4x more time than the partitioning phase on the two large datasets sinaweibo
and webbase-2001
, respectively. These relative runtime comparisons suggest that cascade-aware graph partitioning, considered as an offline process that will be used in relatively long time intervals, runs in reasonable time limits for large-scale social networks.
6.3 Experiments on digg social network with real propagation traces
Digg
5 [18] social news portal. Digg
is an OSN where users can share and vote for news stories, and designate others as friends to inform about their activities. Friendships can be designated as one-way relationships in such a way that if a user \(v_i\) declares another user \(v_j\) as a friend, then \(v_i\) is informed of activities of \(v_j\) but not vice versa. Users follow activities of their friends in their news feeds where the stories their friends shared or voted for are displayed. With these properties of Digg
social network, a story can propagate through users once it is shared or voted for. Propagation of news stories can be considered as propagation of contents in our problem definition.Digg
dataset contains a directed graph \(G=(V,E)\) representing the underlying social network which consists of 71, 367 users and 1, 731, 658 friendship links. As friendships are formed as one-way relationships, they are represented by directed edges. Each directed edge \(e_{ij} \in E\) means that user \(v_j\) is following the activities of user \(v_i\); therefore, the content propagation can occur in the direction of \(v_i\) to \(v_j\). Additionally, the dataset contains log \({\mathcal {L}}\) of past activities of users over a set \({\mathcal {N}}\) of news stories. Each entry \((v_i, n_k, t_i) \in {\mathcal {L}}\) means that user \(v_i \in V\) has voted for news story \(n_k \in {\mathcal {N}}\) at time \(t_i\). The dataset contains 3, 018, 197 votes made on 3, 553 news stories (i.e., \(|{\mathcal {L}}| = 3{,}018{,}197\) and \(|{\mathcal {N}}|=3{,}553\)).Digg
social network and produce partitions that minimize the number of friendship links crossing different parts. In this way, BLP and CUT algorithms become equivalent.Digg
social network. In addition to CAP and the modified version of BLP, we also include the results for partitions generated by RP. For each of these partitions, we compute the average number of communication operations induced on the propagation trees that are generated and sampled from 10 percent of log \({\mathcal {L}}\).K | RP | MO+ | 2Hop | BLP | CAP | %imp |
---|---|---|---|---|---|---|
32 | 192 | 189 | 151 | 101 | 40 | 60.80 |
64 | 195 | 193 | 160 | 86 | 44 | 48.11 |
128 | 196 | 196 | 167 | 119 | 62 | 47.97 |
256 | 197 | 197 | 172 | 128 | 77 | 39.65 |
512 | 198 | 198 | 174 | 133 | 102 | 23.06 |
1024 | 198 | 198 | 177 | 152 | 131 | 13.53 |