#+TITLE:     Graph 500 Benchmark 1 ("Search")
#+AUTHOR:    Graph 500 Steering Committee
#+EMAIL:     sc@graph500.org
#+LANGUAGE:  en
#+OPTIONS:   H:3 num:t toc:t \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
#+OPTIONS:   TeX:t LaTeX:t skip:nil d:nil todo:t pri:nil tags:not-in-toc
#+STYLE: <style>body {margin-left: 10%; margin-right: 10%;} table {margin-left:auto; margin-right:auto;}</style>

Contributors: David A. Bader (Georgia Institute of Technology),
Jonathan Berry (Sandia National Laboratories), Simon Kahan (Pacific
Northwest National Laboratory and University of Washington), Richard
Murphy (Sandia National Laboratories), E. Jason Riedy (Georgia
Institute of Technology), and Jeremiah Willcock (Indiana University).

Version History:
 - V0.1 :: Draft, created 28 July 2010
 - V0.2 :: Draft, created 29 September 2010
 - V0.3 :: Draft, created 30 September 2010
 - V1.0 :: Created 1 October 2010
 - V1.1 :: Created 3 October 2010

Version 0.1 of this document was part of the Graph 500 community
benchmark effort, led by Richard Murphy (Sandia National
Laboratories).  The intent is that there will be at least three
variants of implementations, on shared memory and threaded systems, on
distributed memory clusters, and on external memory map-reduce
clouds. This specification is for the first of potentially several
benchmark problems.

References: "Introducing the Graph 500," Richard C. Murphy, Kyle
B. Wheeler, Brian W. Barrett, James A. Ang, Cray User’s Group (CUG),
May 5, 2010.

"DFS: A Simple to Write Yet Difficult to Execute Benchmark," Richard
C. Murphy, Jonathan Berry, William McLendon, Bruce Hendrickson,
Douglas Gregor, Andrew Lumsdaine, IEEE International Symposium on
Workload Characterizations 2006 (IISWC06), San Jose, CA, 25-27 October
2006.

* Brief Description of the Graph 500 Benchmark

Data-intensive supercomputer applications are an increasingly
important workload, but are ill-suited for platforms designed for 3D
physics simulations. Application performance cannot be improved
without a meaningful benchmark. Graphs are a core part of most
analytics workloads. Backed by a steering committee of 30
international HPC experts from academia, industry, and national
laboratories, this specification establishes a large-scale benchmark
for these applications. It will offer a forum for the community and
provide a rallying point for data-intensive supercomputing
problems. This is the first serious approach to augment the Top 500
with data-intensive applications.

The intent of this benchmark problem ("Search") is to develop a
compact application that has multiple analysis techniques (multiple
kernels) accessing a single data structure representing a weighted,
undirected graph. In addition to a kernel to construct the graph from
the input tuple list, there is one additional computational
kernel to operate on the graph.

This benchmark includes a scalable data generator which produces edge
tuples containing the start vertex and end vertex for each
edge. The first kernel constructs an /undirected/ graph in a format
usable by all subsequent kernels. No subsequent modifications are
permitted to benefit specific kernels. The second kernel performs a
breadth-first search of the graph. Both kernels are timed.

There are five problem classes defined by their input size:
  - toy :: 17GB or around 10^10 bytes, which we also call level 10,
  - mini :: 140GB (10^11 bytes, level 11),
  - small :: 1TB (10^12 bytes, level 12),
  - medium :: 17TB (10^13 bytes, level 13),
  - large :: 140TB (10^14 bytes, level 14), and
  - huge :: 1.1PB (10^15 bytes, level 15).

Table \ref{tbl:classes} provides the parameters used by the graph
generator specified below.

** References

D.A. Bader, J. Feo, J. Gilbert, J. Kepner, D. Koester, E. Loh,
K. Madduri, W. Mann, Theresa Meuse, HPCS Scalable Synthetic Compact
Applications #2 Graph Analysis (SSCA#2 v2.2 Specification), 5
September 2007.

* Overall benchmark

The benchmark performs the following steps:

  1. Generate the edge list.
  2. Construct a graph from the edge list (*timed*, kernel 1).
  3. Randomly sample 64 unique search keys with degree at least one,
     not counting self-loops.
  4. For each search key:
     1. Compute the parent array (*timed*, kernel 2).
     2. Validate that the parent array is a correct BFS search tree
        for the given search tree.
  5. Compute and output performance information.

Only the sections marked as *timed* are included in the performance
information.  Note that all uses of "random" permit pseudorandom
number generation.

* Generating the edge list

** Brief Description

The scalable data generator will construct a list of edge tuples
containing vertex identifiers. Each edge is undirected with its
endpoints given in the tuple as StartVertex and EndVertex. 

The intent of the first kernel below is to convert a list with no
locality into a more optimized form.  The generated list of input
tuples must not exhibit any locality that can be exploited by the
computational kernels.  Thus, the vertex numbers must be randomized
and a random ordering of tuples must be presented to kernel 1.
The data generator may be parallelized, but the vertex names
must be globally consistent and care must be taken to minimize effects
of data locality at the processor level.

** Detailed Text Description

The edge tuples will have the form <StartVertex, EndVertex> where
StartVertex is one endpoint vertex label and EndVertex is the
other endpoint vertex label.  The space of labels is the set of integers
beginning with *zero* up to but not including the number of vertices N
(defined below).  The kernels are not provided the size N explicitly
but must discover it.

The input values required to describe the graph are:

 - SCALE :: The logarithm base two of the number of vertices.

 - edgefactor :: The ratio of the graph's edge count to its vertex count (i.e.,
                 half the average degree of a vertex in the graph).

These inputs determine the graph's size:

 - N :: the total number of vertices, 2^SCALE. An implementation may
   use any set of N distinct integers to number the vertices, but at
   least 48 bits must be allocated per vertex number. Other parameters
   may be assumed to fit within the natural word of the machine. N is
   derived from the problem’s scaling parameter.

 - M :: the number of edges. M = edgefactor * N.

The graph generator is a Kronecker generator similar to the Recursive
MATrix (R-MAT) scale-free graph generation algorithm [Chakrabarti, et
al., 2004]. For ease of discussion, the description of this R-MAT
generator uses an adjacency matrix data structure; however,
implementations may use any alternate approach that outputs the
equivalent list of edge tuples. This model recursively sub-divides the
adjacency matrix of the graph into four equal-sized partitions and
distributes edges within these partitions with unequal
probabilities. Initially, the adjacency matrix is empty, and edges are
added one at a time. Each edge chooses one of the four partitions with
probabilities A, B, C, and D, respectively.  These probabilities, the
initiator parameters, are provided in Table \ref{tbl:initiator}.

#+CAPTION: Initiator parameters for the Kronecker graph generator
#+LABEL:   tbl:initiator
| A = 0.57 | B = 0.19             |
| C = 0.19 | D = 1-(A+B+C) = 0.05 |

The next section details a high-level implementation for this
generator.  High-performance, parallel implementations are included in
the reference implementation.

The graph generator creates a small number of multiple edges between
two vertices as well as self-loops. Multiple edges, self-loops, and
isolated vertices may be ignored in the subsequent kernels but must
be included in the edge list provided to the first kernel. The
algorithm also generates the data tuples with high degrees of
locality. Thus, as a final step, vertex numbers must be randomly
permuted, and then the edge tuples randomly shuffled.

It is permissible to run the data generator in parallel. In this case,
it is necessary to ensure that the vertices are named globally, and
that the generated data does not possess any locality, either in local memory
or globally across processors.

The scalable data generator should be run before starting kernel 1, storing its
results to either RAM or disk.
If stored to disk, the data may be retrieved before
starting kernel 1. The data generator and retrieval operations need not be
timed.

** Sample high-level implementation of the Kronecker generator

The GNU Octave routine in Algorithm \ref{alg:generator} is an
attractive implementation in that it is embarrassingly parallel and
does not require the explicit formation of the adjacency matrix.

#+CAPTION: High-level generator code
#+LABEL: alg:generator
#+BEGIN_SRC Octave
function ij = kronecker_generator (SCALE, edgefactor)
%% Generate an edgelist according to the Graph500
%% parameters.  In this sample, the edge list is
%% returned in an array with two rows, where StartVertex
%% is first row and EndVertex is the second.  The vertex
%% labels start at zero.
%%
%% Example, creating a sparse matrix for viewing:
%%   ij = kronecker_generator (10, 16);
%%   G = sparse (ij(1,:)+1, ij(2,:)+1, ones (1, size (ij, 2)));
%%   spy (G);
%% The spy plot should appear fairly dense. Any locality
%% is removed by the final permutations.

  %% Set number of vertices.
  N = 2^SCALE;

  %% Set number of edges.
  M = edgefactor * N;

  %% Set initiator probabilities.
  [A, B, C] = deal (0.57, 0.19, 0.19);

  %% Create index arrays.
  ij = ones (2, M);
  %% Loop over each order of bit.
  ab = A + B;
  c_norm = C/(1 - (A + B));
  a_norm = A/(A + B);

  for ib = 1:SCALE,
    %% Compare with probabilities and set bits of indices.
    ii_bit = rand (1, M) > ab;
    jj_bit = rand (1, M) > ( c_norm * ii_bit + a_norm * not (ii_bit) );
    ij = ij + 2^(ib-1) * [ii_bit; jj_bit];
  end

  %% Permute vertex labels
  p = randperm (N);
  ij = p(ij);

  %% Permute the edge list
  p = randperm (M);
  ij = ij(:, p);

  %% Adjust to zero-based labels.
  ij = ij - 1;
#+END_SRC

** Parameter settings

The input parameter settings for each class are given in Table \ref{tbl:classes}.

#+CAPTION: Problem class definitions and required storage for the edge list assuming 64-bit integers.
#+LABEL: tbl:classes
| Problem class     | SCALE | edge factor | Approx. storage size in TB |
|-------------------+-------+-------------+----------------------------|
| Toy (level 10)    |    26 |          16 |                     0.0172 |
| Mini (level 11)   |    29 |          16 |                     0.1374 |
| Small (level 12)  |    32 |          16 |                     1.0995 |
| Medium (level 13) |    36 |          16 |                    17.5922 |
| Large (level 14)  |    39 |          16 |                   140.7375 |
| Huge (level 15)   |    42 |          16 |                  1125.8999 |
#+TBLFM: $4=2^$2*$3*2*8/1e12;%.4f

** References

D. Chakrabarti, Y. Zhan, and C. Faloutsos, R-MAT: A recursive model
for graph mining, SIAM Data Mining 2004.

Section 17.6, Algorithms in C (third edition). Part 5 Graph
Algorithms, Robert Sedgewick (Programs 17.7 and 17.8)

P. Sanders, Random Permutations on Distributed, External and
Hierarchical Memory, Information Processing Letters 67 (1988) pp
305-309.

* Kernel 1 – Graph Construction

** Description

The first kernel may transform the edge list to any data structures
(held in internal or external memory) that are used for the remaining
kernels. For instance, kernel 1 may construct a (sparse) graph from a
list of tuples; each tuple contains endpoint vertex identifiers for an
edge, and a weight that represents data assigned to the edge.

#+ XXX: Unsure about this part.  I'd rather permit structural
#+ optimizations here so long as they're used for *all* kernels.
#+   - However, any layout or arrangement computations intended to
#+    increase locality, improve load-balance, or reduce hot-spotting
#+    must be counted in the benchmark execution time.  Therefore, /no
#+    comparison of or binning by vertex degrees is permitted during this
#+    kernel 1 (graph construction)/.

The graph may be represented in any manner, but it may not be modified
by or between subsequent kernels. Space may be reserved in the data
structure for marking or locking.
Only one copy of a kernel will be
run at a time; that kernel has exclusive access to any such marking or
locking space and is permitted to modify that space (only).

There are various internal memory representations for sparse graphs,
including (but not limited to) sparse matrices and (multi-level)
linked lists. For the purposes of this application, the kernel is
provided only the edge list and the edge list's size.  Further
information such as the number of vertices must be computed within this
kernel.  Algorithm \ref{alg:kernel1} provides a high-level sample
implementation of kernel 1.

The process of constructing the graph data structure (in internal or
external memory) from the set of tuples must be timed.

#+CAPTION: High-level implementation of kernel 1
#+LABEL: alg:kernel1
#+BEGIN_SRC Octave
function G = kernel_1 (ij)
%% Compute a sparse adjacency matrix representation
%% of the graph with edges from ij.

  %% Remove self-edges.
  ij(:, ij(1,:) == ij(2,:)) = [];
  %% Adjust away from zero labels.
  ij = ij + 1;
  %% Find the maximum label for sizing.
  N = max (max (ij));
  %% Create the matrix, ensuring it is square.
  G = sparse (ij(1,:), ij(2,:), ones (1, size (ij, 2)), N, N);
  %% Symmetrize to model an undirected graph.
  G = spones (G + G.');
#+END_SRC

** References

Section 17.6 Algorithms in C third edition Part 5 Graph Algorithms,
Robert Sedgewick (Program 17.9)

* Sampling 64 search keys

The search keys must be randomly sampled from the vertices in the
graph.  To avoid trivial searches, sample only from vertices that are
connected to some other vertex.  Their degrees, not counting self-loops,
must be at least one.  If there are fewer than 64 such vertices, run
fewer than 64 searches.  This should never occur with the graph sizes
in this benchmark, but there is a non-zero probability of producing a
trivial or nearly trivial graph.  The number of search keys used is
included in the output, but this step is untimed.

* Kernel 2 – Breadth-First Search

** Description

A Breadth-First Search (BFS) of a graph starts with a single source
vertex, then, in phases, finds and labels its neighbors, then the
neighbors of its neighbors, etc.  This is a fundamental method on
which many graph algorithms are based. A formal description of BFS can
be found in Cormen, Leiserson, and Rivest.  Below, we specify the
input and output for a BFS benchmark, and we impose some constraints
on the computation.  However, we do not constrain the choice of BFS
algorithm itself, as long as it produces a correct BFS tree as output.

This benchmark's memory access pattern (internal or external) is data-dependent
with small average prefetch depth.  As in a simple
concurrent linked-list traversal benchmark, performance reflects an
architecture's throughput when executing concurrent threads, each of
low memory concurrency and high memory reference density.  Unlike such
a benchmark, this one also measures resilience to hot-spotting when
many of the memory references are to the same location; efficiency
when every thread's execution path depends on the asynchronous
side-effects of others; and the ability to dynamically load balance
unpredictably sized work units.  Measuring synchronization performance
is not a primary goal here.

You may not search from multiple search keys concurrently.

*ALGORITHM NOTE* We allow a benign race condition when vertices at BFS
level k are discovering vertices at level k+1.  Specifically, we do
not require synchronization to ensure that the first visitor must
become the parent while locking out subsequent visitors.  As long as
the discovered BFS tree is correct at the end, the algorithm is
considered to be correct.

** Kernel 2 Output

For each search key, the routine must return an array containing valid
breadth-first search parent information (per vertex).  The parent of
the search_key is itself, and the parent of any vertex not included in
the tree is -1.  Algorithm \ref{alg:kernel2} provides a sample (and
inefficient) high-level implementation of kernel two.

#+CAPTION: High-level implementation of kernel 2
#+LABEL: alg:kernel2
#+BEGIN_SRC Octave
function parent = kernel_2 (G, root)
%% Compute a sparse adjacency matrix representation
%% of the graph with edges from ij.

  N = size (G, 1);
  %% Adjust from zero labels.
  root = root + 1;
  parent = zeros (N, 1);
  parent (root) = root;

  vlist = zeros (N, 1);
  vlist(1) = root;
  lastk = 1;
  for k = 1:N,
    v = vlist(k);
    if v == 0, break; end
    [I,J,V] = find (G(:, v));
    nxt = I(parent(I) == 0);
    parent(nxt) = v;
    vlist(lastk + (1:length (nxt))) = nxt;
    lastk = lastk + length (nxt);
  end

  %% Adjust to zero labels.
  parent = parent - 1;

#+END_SRC

* Validation

It is not intended that the results of full-scale runs of this
benchmark can be validated by exact comparison to a standard reference
result. At full scale, the data set is enormous, and its exact details
depend on the pseudo-random number generator and BFS algorithm used. Therefore,
the
validation of an implementation of the benchmark uses soft checking of
the results.

We emphasize that the intent of this benchmark is to exercise these
algorithms on the largest data sets that will fit on machines being
evaluated. However, for debugging purposes it may be desirable to run
on small data sets, and it may be desirable to verify parallel results
against serial results, or even against results from the executable
specification.

The executable specification verifies its results by comparing them
with results computed directly from the tuple list.

Kernel 2 validation: after each search, run (but do not time) a
function that ensures that the discovered breadth-first tree is
correct by ensuring that:

  1) the BFS tree is a tree and does not contain cycles,
  2) each tree edge connects vertices whose BFS levels differ by
     exactly one,
  3) every edge in the input list has vertices with levels that differ
     by at most one or that both are not in the BFS tree,
  4) the BFS tree spans an entire connected component's vertices, and
  5) a node and its parent are joined by an edge of the original graph.

Algorithm \ref{alg:validate} shows a sample validation routine.

#+CAPTION: High-level implementation of kernel 2 validation
#+LABEL: alg:validate
#+BEGIN_SRC Octave
function out = validate (parent, ij, search_key)
  out = 1;
  parent = parent + 1;
  search_key = search_key + 1;

  if parent (search_key) != search_key,
    out = 0;
    return;
  end

  ij = ij + 1;
  N = max (max (ij));
  slice = find (parent > 0);

  level = zeros (size (parent));
  level (slice) = 1;
  P = parent (slice);
  mask = P != search_key;
  k = 0;
  while any (mask),
    level(slice(mask)) = level(slice(mask)) + 1;
    P = parent (P);
    mask = P != search_key;
    k = k + 1;
    if k > N,
      %% There must be a cycle in the tree.
      out = -3;
      return;
    end
  end

  lij = level (ij);
  neither_in = lij(1,:) == 0 & lij(2,:) == 0;
  both_in = lij(1,:) > 0 & lij(2,:) > 0;
  if any (not (neither_in | both_in)),
    out = -4;
    return
  end
  respects_tree_level = abs (lij(1,:) - lij(2,:)) <= 1;
  if any (not (neither_in | respects_tree_level)),
    out = -5;
    return
  end
#+END_SRC

* Computing and outputting performance information

** Timing

Start the time for a search immediately prior to visiting the search
root.  Stop the time for that search when the output has been written
to memory.  Do not time any I/O outside of the search routine.  If
your algorithm relies on problem-specific data structures (by our
definition, these are informed by vertex degree), you must include the
setup time for such structures in /each search/. The spirit of the
benchmark is to gauge the performance of a single search.  We run many
searches in order to compute means and variances, not to amortize data
structure setup time.

** Performance Metric (TEPS)

In order to compare the performance of Graph 500 "Search"
implementations across a variety of architectures, programming models,
and productivity languages and frameworks, we adopt a new performance
metric described in this section. In the spirit of well-known
computing rates floating-point operations per second (flops) measured
by the LINPACK benchmark and global updates per second (GUPs) measured
by the HPCC RandomAccess benchmark, we define a new rate called traversed
edges per second (TEPS). We measure TEPS through the benchmarking of
kernel 2 as follows. Let time_K2(n) be the measured execution time for
kernel 2. Let m be the number of input edge tuples within the
component traversed by the search, counting any multiple edges and
self-loops. We define the normalized performance rate (number of edge
traversals per second) as:

#+BEGIN_CENTER
		       TEPS(n) = m / time_K2(n)
#+END_CENTER

** Output

The output must contain the following information:
  - SCALE :: Graph generation parameter
  - edgefactor :: Graph generation parameter
  - NBFS :: Number of BFS searches run, 64 for non-trivial graphs
  - construction\_time :: The single kernel 1 time
  - min\_time, firstquartile\_time, median\_time, thirdquartile\_time, max\_time :: Quartiles for the kernel 2 times
  - mean\_time, stddev\_time :: Mean and standard deviation of the kernel 2 times
  - min\_nedge, firstquartile\_nedge, median\_nedge, thirdquartile\_nedge, max\_nedge :: Quartiles for the number of
       input edges visited by kernel 2, see TEPS section above.
  - mean\_nedge, stddev\_nedge :: Mean and standard deviation of the number of
       input edges visited by kernel 2, see TEPS section above.
  - min\_TEPS, firstquartile\_TEPS, median\_TEPS, thirdquartile\_TEPS, max\_TEPS :: Quartiles for the kernel 2 TEPS
  - harmonic\_mean\_TEPS, harmonic\_stddev\_TEPS :: Mean and standard
       deviation of the kernel 2 TEPS.  *Note*: Because TEPS is a
       rate, the rates are compared using *harmonic* means.

Additional fields are permitted.  Algorithm \ref{alg:output} provides
a high-level sample.

#+CAPTION: High-level implementation of the output routine
#+LABEL: alg:output
#+BEGIN_SRC Octave
function output (SCALE, edgefactor, NBFS, kernel_1_time, kernel_2_time, kernel_2_nedge)
  printf ("SCALE: %d\n", SCALE);
  printf ("edgefactor: %d\n", edgefactor);
  printf ("NBFS: %d\n", NBFS);
  printf ("construction_time: %20.17e\n", kernel_1_time);

  S = statistics (kernel_2_time);
  printf ("min_time: %20.17e\n", S(1));
  printf ("firstquartile_time: %20.17e\n", S(2));
  printf ("median_time: %20.17e\n", S(3));
  printf ("thirdquartile_time: %20.17e\n", S(4));
  printf ("max_time: %20.17e\n", S(5));
  printf ("mean_time: %20.17e\n", S(6));
  printf ("stddev_time: %20.17e\n", S(7));

  S = statistics (kernel_2_nedge);
  printf ("min_nedge: %20.17e\n", S(1));
  printf ("firstquartile_nedge: %20.17e\n", S(2));
  printf ("median_nedge: %20.17e\n", S(3));
  printf ("thirdquartile_nedge: %20.17e\n", S(4));
  printf ("max_nedge: %20.17e\n", S(5));
  printf ("mean_nedge: %20.17e\n", S(6));
  printf ("stddev_nedge: %20.17e\n", S(7));

  TEPS = kernel_2_nedge ./ kernel_2_time;
  N = length (TEPS);
  S = statistics (TEPS);
  S(6) = mean (TEPS, 'h');
  %% Harmonic standard deviation from:
  %% Nilan Norris, The Standard Errors of the Geometric and Harmonic
  %% Means and Their Application to Index Numbers, 1940.
  %% http://www.jstor.org/stable/2235723
  tmp = zeros (N, 1);
  tmp(TEPS > 0) = 1./TEPS(TEPS > 0);
  tmp = tmp - 1/S(6);
  S(7) = (sqrt (sum (tmp.^2)) / (N-1)) * S(6)^2;
  
  printf ("min_TEPS: %20.17e\n", S(1));
  printf ("firstquartile_TEPS: %20.17e\n", S(2));
  printf ("median_TEPS: %20.17e\n", S(3));
  printf ("thirdquartile_TEPS: %20.17e\n", S(4));
  printf ("max_TEPS: %20.17e\n", S(5));
  printf ("harmonic_mean_TEPS: %20.17e\n", S(6));
  printf ("harmonic_stddev_TEPS: %20.17e\n", S(7));
#+END_SRC

** References

Nilan Norris, The Standard Errors of the Geometric and Harmonic Means
and Their Application to Index Numbers, The Annals of Mathematical
Statistics, vol. 11, num. 4, 1940.
http://www.jstor.org/stable/2235723

* Sample driver

A high-level sample driver for the above routines is given in
Algorithm \ref{alg:driver}.

#+CAPTION: High-level sample driver
#+LABEL: alg:driver
#+BEGIN_SRC Octave
SCALE = 10;
edgefactor = 16;
NBFS = 64;

rand ("seed", 103);

ij = kronecker_generator (SCALE, edgefactor);

tic;
G = kernel_1 (ij);
kernel_1_time = toc;

N = size (G, 1);
coldeg = full (spstats (G));
search_key = randperm (N);
search_key(coldeg(search_key) == 0) = [];
if length (search_key) > NBFS,
  search_key = search_key(1:NBFS);
else
  NBFS = length (search_key);
end
search_key = search_key - 1;  

kernel_2_time = Inf * ones (NBFS, 1);
kernel_2_nedge = zeros (NBFS, 1);

indeg = histc (ij(:), 1:N); % For computing the number of edges

for k = 1:NBFS,
  tic;
  parent = kernel_2 (G, search_key(k));
  kernel_2_time(k) = toc;
  err = validate (parent, ij, search_key (k));
  if err <= 0,
    error (sprintf ("BFS %d from search key %d failed to validate: %d",
		    k, search_key(k), err));
  end
  kernel_2_nedge(k) = sum (indeg(parent >= 0))/2; % Volume/2
end

output (SCALE, edgefactor, NBFS, kernel_1_time, kernel_2_time, kernel_2_nedge);
#+END_SRC

* Evaluation Criteria

In approximate order of importance, the goals of this benchmark are:
 - Fair adherence to the intent of the benchmark specification
 - Maximum problem size for a given machine
 - Minimum execution time for a given problem size

Less important goals:
 - Minimum code size (not including validation code)
 - Minimal development time
 - Maximal maintainability
 - Maximal extensibility