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tape.md |
README.md
simdjson : Parsing gigabytes of JSON per second
A C++ library to see how fast we can parse JSON with complete validation.
JSON documents are everywhere on the Internet. Servers spend a lot of time parsing these documents. We want to accelerate the parsing of JSON per se using commonly available SIMD instructions as much as possible while doing full validation (including character encoding). This library is part of the Awesome Modern C++ list.
Real-world usage
Research article (VLDB Journal)
A description of the design and implementation of simdjson is in our research article:
- Geoff Langdale, Daniel Lemire, Parsing Gigabytes of JSON per Second, VLDB Journal (to appear)
We also have an informal blog post providing some background and context.
Some people enjoy reading our paper:
Performance results
simdjson uses three-quarters less instructions than state-of-the-art parser RapidJSON and fifty percent less than sajson. To our knowledge, simdjson is the first fully-validating JSON parser to run at gigabytes per second on commodity processors.
On a Skylake processor, the parsing speeds (in GB/s) of various processors on the twitter.json file are as follows.
parser | GB/s |
---|---|
simdjson | 2.2 |
RapidJSON encoding-validation | 0.51 |
RapidJSON encoding-validation, insitu | 0.71 |
sajson (insitu, dynamic) | 0.70 |
sajson (insitu, static) | 0.97 |
dropbox | 0.14 |
fastjson | 0.26 |
gason | 0.85 |
ultrajson | 0.42 |
jsmn | 0.28 |
cJSON | 0.34 |
JSON for Modern C++ (nlohmann/json) | 0.10 |
Requirements
- We support 64-bit platforms like Linux or macOS, as well as Windows through Visual Studio 2017 or later.
- A processor with
- AVX2 (i.e., Intel processors starting with the Haswell microarchitecture released 2013 and AMD processors starting with the Zen microarchitecture released 2017),
- or SSE 4.2 and CLMUL (i.e., Intel processors going back to Westmere released in 2010 or AMD processors starting with the Jaguar used in the PS4 and XBox One)
- or a 64-bit ARM processor (ARMv8-A): this covers a wide range of mobile processors, including all Apple processors currently available for sale, going as far back as the iPhone 5s (2013).
- A recent C++ compiler (e.g., GNU GCC or LLVM CLANG or Visual Studio 2017), we assume C++17. GNU GCC 7 or better or LLVM's clang 6 or better.
- Some benchmark scripts assume bash and other common utilities, but they are optional.
License
This code is made available under the Apache License 2.0.
Under Windows, we build some tools using the windows/dirent_portable.h file (which is outside our library code): it under the liberal (business-friendly) MIT license.
Code usage and example
The main API involves populating a ParsedJson
object which hosts a fully navigable document-object-model (DOM) view of the JSON document. The DOM can be accessed using JSON Pointer paths, for example. The main function is json_parse
which takes a string containing the JSON document as well as a reference to pre-allocated ParsedJson
object (which can be reused multiple time). Once you have populated the ParsedJson
object you can navigate through the DOM with an iterator (e.g., created by ParsedJson::Iterator pjh(pj)
, see 'Navigating the parsed document').
#include "simdjson/jsonparser.h"
using namespace simdjson;
/...
const char * filename = ... //
// use whatever means you want to get a string (UTF-8) of your JSON document
padded_string p = get_corpus(filename);
ParsedJson pj;
pj.allocate_capacity(p.size()); // allocate memory for parsing up to p.size() bytes
const int res = json_parse(p, pj); // do the parsing, return 0 on success
// parsing is done!
if (res != 0) {
// You can use the "simdjson/simdjson.h" header to access the error message
std::cout << "Error parsing:" << simdjson::error_message(res) << std::endl;
}
// the ParsedJson document can be used here
// pj can be reused with other json_parse calls.
It is also possible to use a simpler API if you do not mind having the overhead of memory allocation with each new JSON document:
#include "simdjson/jsonparser.h"
using namespace simdjson;
/...
const char * filename = ... //
padded_string p = get_corpus(filename);
ParsedJson pj = build_parsed_json(p); // do the parsing
if( ! pj.is_valid() ) {
// something went wrong
std::cout << pj.get_error_message() << std::endl;
}
Though the padded_string
class is recommended for best performance, you can call json_parse
and build_parsed_json
, passing a standard std::string
object.
#include "simdjson/jsonparser.h"
using namespace simdjson;
/...
std::string mystring = ... //
ParsedJson pj;
pj.allocate_capacity(mystring.size()); // allocate memory for parsing up to p.size() bytes
// std::string may not overallocate so a copy will be needed
const int res = json_parse(mystring, pj); // do the parsing, return 0 on success
// parsing is done!
if (res != 0) {
// You can use the "simdjson/simdjson.h" header to access the error message
std::cout << "Error parsing:" << simdjson::error_message(res) << std::endl;
}
// pj can be reused with other json_parse calls.
or
#include "simdjson/jsonparser.h"
using namespace simdjson;
/...
std::string mystring = ... //
// std::string may not overallocate so a copy will be needed
ParsedJson pj = build_parsed_json(mystring); // do the parsing
if( ! pj.is_valid() ) {
// something went wrong
std::cout << pj.get_error_message() << std::endl;
}
As needed, the json_parse
and build_parsed_json
functions copy the input data to a temporary buffer readable up to SIMDJSON_PADDING bytes beyond the end of the data.
JSON streaming
API and detailed documentation found here.
Here is a simple example, using single header simdjson:
#include "simdjson.h"
#include "simdjson.cpp"
int parse_file(const char *filename) {
simdjson::padded_string p = simdjson::get_corpus(filename);
simdjson::ParsedJson pj;
simdjson::JsonStream js{p};
int parse_res = simdjson::SUCCESS_AND_HAS_MORE;
while (parse_res == simdjson::SUCCESS_AND_HAS_MORE) {
parse_res = js.json_parse(pj);
//Do something with pj...
}
}
Usage: easy single-header version
See the "singleheader" repository for a single header version. See the included file "amalgamation_demo.cpp" for usage. This requires no specific build system: just copy the files in your project in your include path. You can then include them quite simply:
#include <iostream>
#include "simdjson.h"
#include "simdjson.cpp"
using namespace simdjson;
int main(int argc, char *argv[]) {
const char * filename = argv[1];
padded_string p = get_corpus(filename);
ParsedJson pj = build_parsed_json(p); // do the parsing
if( ! pj.is_valid() ) {
std::cout << "not valid" << std::endl;
std::cout << pj.get_error_message() << std::endl;
} else {
std::cout << "valid" << std::endl;
}
return EXIT_SUCCESS;
}
Note: In some settings, it might be desirable to precompile simdjson.cpp
instead of including it.
Runtime dispatch
On Intel and AMD processors, we get best performance by using the hardware support for AVX2 instructions. However, simdjson also runs on older Intel and AMD processors. We require a minimum feature support of SSE 4.2 and CLMUL (2010 Intel Westmere or better). The code automatically detects the feature set of your processor and switches to the right function at runtime (a technique sometimes called runtime dispatch).
We also support 64-bit ARM. We assume NEON support, and if the cryptographic extension is available, we leverage it, at compile-time. There is no runtime dispatch on ARM.
Thread safety
The simdjson library is single-threaded. Thread safety is the responsability of the caller: it is unsafe to reuse a ParsedJson object between different threads.
If you are on an x64 processor, the runtime dispatching assigns the right code path the first time that parsing is attempted. The runtime dispatching is thread-safe.
Usage (old-school Makefile on platforms like Linux or macOS)
Requirements: recent clang or gcc, and make. We recommend at least GNU GCC/G++ 7 or LLVM clang 6. A 64-bit system like Linux or macOS is expected.
To test:
make
make test
To run benchmarks:
make parse
./parse jsonexamples/twitter.json
Under Linux, the parse
command gives a detailed analysis of the performance counters.
To run comparative benchmarks (with other parsers):
make benchmark
Usage (CMake on 64-bit platforms like Linux or macOS)
Requirements: We require a recent version of cmake. On macOS, the easiest way to install cmake might be to use brew and then type
brew install cmake
There is an equivalent brew on Linux which works the same way as well.
You need a recent compiler like clang or gcc. We recommend at least GNU GCC/G++ 7 or LLVM clang 6. For example, you can install a recent compiler with brew:
brew install gcc@8
Optional: You need to tell cmake which compiler you wish to use by setting the CC and CXX variables. Under bash, you can do so with commands such as export CC=gcc-7
and export CXX=g++-7
.
Building: While in the project repository, do the following:
mkdir build
cd build
cmake ..
make
make test
CMake will build a library. By default, it builds a shared library (e.g., libsimdjson.so on Linux).
You can build a static library:
mkdir buildstatic
cd buildstatic
cmake -DSIMDJSON_BUILD_STATIC=ON ..
make
make test
In some cases, you may want to specify your compiler, especially if the default compiler on your system is too old. You may proceed as follows:
brew install gcc@8
mkdir build
cd build
export CXX=g++-8 CC=gcc-8
cmake ..
make
make test
Usage (CMake on 64-bit Windows using Visual Studio)
We assume you have a common 64-bit Windows PC with at least Visual Studio 2017 and an x64 processor with AVX2 support (2013 Intel Haswell or later) or SSE 4.2 + CLMUL (2010 Westmere or later).
- Grab the simdjson code from GitHub, e.g., by cloning it using GitHub Desktop.
- Install CMake. When you install it, make sure to ask that
cmake
be made available from the command line. Please choose a recent version of cmake. - Create a subdirectory within simdjson, such as
VisualStudio
. - Using a shell, go to this newly created directory.
- Type
cmake -DCMAKE_GENERATOR_PLATFORM=x64 ..
in the shell while in theVisualStudio
repository. (Alternatively, if you want to build a DLL, you may use the command linecmake -DCMAKE_GENERATOR_PLATFORM=x64 -DSIMDJSON_BUILD_STATIC=OFF ..
.) - This last command (
cmake ...
) created a Visual Studio solution file in the newly created directory (e.g.,simdjson.sln
). Open this file in Visual Studio. You should now be able to build the project and run the tests. For example, in theSolution Explorer
window (available from theView
menu), right-clickALL_BUILD
and selectBuild
. To test the code, still in theSolution Explorer
window, selectRUN_TESTS
and selectBuild
.
Usage (Using vcpkg
on 64-bit Windows, Linux and macOS)
vcpkg users on Windows, Linux and macOS can download and install simdjson
with one single command from their favorite shell.
On 64-bit Linux and macOS:
$ ./vcpkg install simdjson
will build and install simdjson
as a static library.
On Windows (64-bit):
.\vcpkg.exe install simdjson:x64-windows
will build and install simdjson
as a shared library.
.\vcpkg.exe install simdjson:x64-windows-static
will build and install simdjson
as a static library.
These commands will also print out instructions on how to use the library from MSBuild or CMake-based projects.
If you find the version of simdjson
shipped with vcpkg
is out-of-date, feel free to report it to vcpkg
community either by submiting an issue or by creating a PR.
Tools
json2json mydoc.json
parses the document, constructs a model and then dumps back the result to standard output.json2json -d mydoc.json
parses the document, constructs a model and then dumps model (as a tape) to standard output. The tape format is described in the accompanying filetape.md
.minify mydoc.json
minifies the JSON document, outputting the result to standard output. Minifying means to remove the unneeded white space characters.jsonpointer mydoc.json <jsonpath> <jsonpath> ... <jsonpath>
parses the document, constructs a model and then processes a series of JSON Pointer paths. The result is itself a JSON document.
Scope
We provide a fast parser, that fully validates an input according to various specifications. The parser builds a useful immutable (read-only) DOM (document-object model) which can be later accessed.
To simplify the engineering, we make some assumptions.
- We support UTF-8 (and thus ASCII), nothing else (no Latin, no UTF-16). We do not believe this is a genuine limitation, because we do not think there is any serious application that needs to process JSON data without an ASCII or UTF-8 encoding. If the UTF-8 contains a leading BOM, it should be omitted: the user is responsible for detecting and skipping the BOM; UTF-8 BOMs are discouraged.
- All strings in the JSON document may have up to 4294967295 bytes in UTF-8 (4GB). To enforce this constraint, we refuse to parse a document that contains more than 4294967295 bytes (4GB). This should accommodate most JSON documents.
- As allowed by the specification, we allow repeated keys within an object (other parsers like sajson do the same).
- Performance is optimized for JSON documents spanning at least a tens kilobytes up to many megabytes: the performance issues with having to parse many tiny JSON documents or one truly enormous JSON document are different.
We do not aim to provide a general-purpose JSON library. A library like RapidJSON offers much more than just parsing, it helps you generate JSON and offers various other convenient functions. We merely parse the document.
Features
- The input string is unmodified. (Parsers like sajson and RapidJSON use the input string as a buffer.)
- We parse integers and floating-point numbers as separate types which allows us to support large signed 64-bit integers in [-9223372036854775808,9223372036854775808), like a Java
long
or a C/C++long long
and large unsigned integers up to the value 18446744073709551615. Among the parsers that differentiate between integers and floating-point numbers, not all support 64-bit integers. (For example, sajson rejects JSON files with integers larger than or equal to 2147483648. RapidJSON will parse a file containing an overly long integer like 18446744073709551616 as a floating-point number.) When we cannot represent exactly an integer as a signed or unsigned 64-bit value, we reject the JSON document. - We support the full range of 64-bit floating-point numbers (binary64). The values range from
std::numeric_limits<double>::lowest()
tostd::numeric_limits<double>::max()
, so from -1.7976e308 all the way to 1.7975e308. Extreme values (less or equal to -1e308, greater or equal to 1e308) are rejected: we refuse to parse the input document. - We test for accurate float parsing with a bound on the unit of least precision (ULP) of one. Practically speaking, this implies 15 digits of accuracy or better.
- We do full UTF-8 validation as part of the parsing. (Parsers like fastjson, gason and dropbox json11 do not do UTF-8 validation. The sajson parser does incomplete UTF-8 validation, accepting code point sequences like 0xb1 0x87.)
- We fully validate the numbers. (Parsers like gason and ultranjson will accept
[0e+]
as valid JSON.) - We validate string content for unescaped characters. (Parsers like fastjson and ultrajson accept unescaped line breaks and tabs in strings.)
- We fully validate the white-space characters outside of the strings. Parsers like RapidJSON will accept JSON documents with null characters outside of strings.
Architecture
The parser works in two stages:
- Stage 1. (Find marks) Identifies quickly structure elements, strings, and so forth. We validate UTF-8 encoding at that stage.
- Stage 2. (Structure building) Involves constructing a "tree" of sort (materialized as a tape) to navigate through the data. Strings and numbers are parsed at this stage.
JSON Pointer
We can navigate the parsed JSON using JSON Pointers as per the RFC6901 standard.
You can build a tool (jsonpointer) to parse a JSON document and then issue an array of JSON Pointer queries:
make jsonpointer
./jsonpointer jsonexamples/small/demo.json /Image/Width /Image/Height /Image/IDs/2
./jsonpointer jsonexamples/twitter.json /statuses/0/id /statuses/1/id /statuses/2/id /statuses/3/id /statuses/4/id /statuses/5/id
In C++, given a ParsedJson
, we can move to a node with the move_to
method, passing a std::string
representing the JSON Pointer query.
Navigating the parsed document
From a simdjson::ParsedJson
instance, you can create an iterator (of type simdjson::ParsedJson::Iterator
which is in fact simdjson::ParsedJson::BasicIterator<DEFAULT_MAX_DEPTH>
) via a constructor:
ParsedJson::Iterator pjh(pj); // pj is a ParsedJSON
You then have access to the following methods on the resulting simdjson::ParsedJson::Iterator
instance:
bool is_ok() const
: whether you have a valid iterator, will be false if your parent parsed ParsedJson is not a valid JSON.size_t get_depth() const
: returns the current depth (start at 1 with 0 reserved for the fictitious root node)int8_t get_scope_type() const
: a scope is a series of nodes at the same depth, typically it is either an object ({
) or an array ([
). The root node has type 'r'.bool move_forward()
: move forward in document orderuint8_t get_type() const
: retrieve the character code of what we're looking at:[{"slutfn
are the possibilitiesint64_t get_integer() const
: get the int64_t value at this node; valid only if get_type() is "l"uint64_t get_unsigned_integer() const
: get the value as uint64; valid only if get_type() is "u"const char *get_string() const
: get the string value at this node (NULL ended); valid only if get_type() is ", note that tabs, and line endings are escaped in the returned value, return value is valid UTF-8, it may contain NULL chars, get_string_length() determines the true string length.uint32_t get_string_length() const
: return the length of the string in bytesdouble get_double() const
: get the double value at this node; valid only if gettype() is "d"bool is_object_or_array() const
: self-explanatorybool is_object() const
: self-explanatorybool is_array() const
: self-explanatorybool is_string() const
: self-explanatorybool is_integer() const
: self-explanatorybool is_unsigned_integer() const
: Returns true if the current type of node is an unsigned integer. You can get its value withget_unsigned_integer()
. Only a large value, which is out of range of a 64-bit signed integer, is represented internally as an unsigned node. On the other hand, a typical positive integer, such as 1, 42, or 1000000, is as a signed node. Be aware this function returns false for a signed node.bool is_double() const
: self-explanatorybool is_number() const
: self-explanatorybool is_true() const
: self-explanatorybool is_false() const
: self-explanatorybool is_null() const
: self-explanatorybool is_number() const
: self-explanatorybool move_to_key(const char *key)
: when at {, go one level deep, looking for a given key, if successful, we are left pointing at the value, if not, we are still pointing at the object ({) (in case of repeated keys, this only finds the first one). We seek the key using C's strcmp so if your JSON strings contain NULL chars, this would trigger a false positive: if you expect that to be the case, take extra precautions. Furthermore, we do the comparison character-by-character without taking into account Unicode equivalence.bool move_to_key_insensitive(const char *key)
: as above, but case insensitive lookupbool move_to_key(const char *key, uint32_t length)
: as above except that the target can contain NULL charactersvoid move_to_value()
: when at a key location within an object, this moves to the accompanying, value (located next to it). This is equivalent but much faster than callingnext()
.bool move_to_index(uint32_t index)
: when at[
, go one level deep, and advance to the given index, if successful, we are left pointing at the value,i f not, we are still pointing at the arraybool move_to(const char *pointer, uint32_t length)
: Moves the iterator to the value correspoding to the json pointer. Always search from the root of the document. If successful, we are left pointing at the value, if not, we are still pointing the same value we were pointing before the call. The json pointer follows the rfc6901 standard's syntax: https://tools.ietf.org/html/rfc6901bool move_to(const std::string &pointer)
: same as above but with a std::string parameterbool next()
: Withing a given scope (series of nodes at the same depth within either an array or an object), we move forward. Thus, given [true, null, {"a":1}, [1,2]], we would visit true, null, { and [. At the object ({) or at the array ([), you can issue a "down" to visit their content. valid if we're not at the end of a scope (returns true).bool prev()
: Within a given scope (series of nodes at the same depth within either an array or an object), we move backward.bool up()
: moves back to either the containing array or object (type { or [) from within a contained scope.bool down()
: moves us to start of that deeper scope if it not empty. Thus, given [true, null, {"a":1}, [1,2]], if we are at the { node, we would move to the "a" node.void to_start_scope()
: move us to the start of our current scope, a scope is a series of nodes at the same levelvoid rewind()
: repeatedly calls up until we are at the root of the documentbool print(std::ostream &os, bool escape_strings = true) const
: print the node we are currently pointing at
Here is a code sample to dump back the parsed JSON to a string:
ParsedJson::Iterator pjh(pj);
if (!pjh.is_ok()) {
std::cerr << " Could not iterate parsed result. " << std::endl;
return EXIT_FAILURE;
}
compute_dump(pj);
//
// where compute_dump is :
void compute_dump(ParsedJson::Iterator &pjh) {
if (pjh.is_object()) {
std::cout << "{";
if (pjh.down()) {
pjh.print(std::cout); // must be a string
std::cout << ":";
pjh.next();
compute_dump(pjh); // let us recurse
while (pjh.next()) {
std::cout << ",";
pjh.print(std::cout);
std::cout << ":";
pjh.next();
compute_dump(pjh); // let us recurse
}
pjh.up();
}
std::cout << "}";
} else if (pjh.is_array()) {
std::cout << "[";
if (pjh.down()) {
compute_dump(pjh); // let us recurse
while (pjh.next()) {
std::cout << ",";
compute_dump(pjh); // let us recurse
}
pjh.up();
}
std::cout << "]";
} else {
pjh.print(std::cout); // just print the lone value
}
}
The following function will find all user.id integers:
void simdjson_scan(std::vector<int64_t> &answer, ParsedJson::Iterator &i) {
while(i.move_forward()) {
if(i.get_scope_type() == '{') {
bool found_user = (i.get_string_length() == 4) && (memcmp(i.get_string(), "user", 4) == 0);
i.move_to_value();
if(found_user) {
if(i.is_object() && i.move_to_key("id",2)) {
if (i.is_integer()) {
answer.push_back(i.get_integer());
}
i.up();
}
}
}
}
}
In-depth comparisons
If you want to see how a wide range of parsers validate a given JSON file:
make allparserscheckfile
./allparserscheckfile myfile.json
For performance comparisons:
make parsingcompetition
./parsingcompetition myfile.json
For broader comparisons:
make allparsingcompetition
./allparsingcompetition myfile.json
Both the parsingcompetition
and allparsingcompetition
tools take a -t
flag which produces
a table-oriented output that can be conventiently parsed by other tools.
Docker
One can run tests and benchmarks using docker. It especially makes sense under Linux. A privileged access may be needed to get performance counters.
git clone https://github.com/lemire/simdjson.git
cd simdjson
docker build -t simdjson .
docker run --privileged -t simdjson
Other programming languages
We distinguish between "bindings" (which just wrap the C++ code) and a port to another programming language (which reimplements everything).
- ZippyJSON: Swift bindings for the simdjson project.
- pysimdjson: Python bindings for the simdjson project.
- simdjson-rs: Rust port.
- simdjson-rust: Rust wrapper (bindings).
- SimdJsonSharp: C# version for .NET Core (bindings and full port).
- simdjson_nodejs: Node.js bindings for the simdjson project.
- simdjson_php: PHP bindings for the simdjson project.
- simdjson_ruby: Ruby bindings for the simdjson project.
- simdjson-go: Go port.
Various References
- Google double-conv
- How to implement atoi using SIMD?
- Parsing JSON is a Minefield 💣
- https://tools.ietf.org/html/rfc7159
- The Mison implementation in rust https://github.com/pikkr/pikkr
- http://rapidjson.org/md_doc_sax.html
- https://github.com/Geal/parser_benchmarks/tree/master/json
- Gron: A command line tool that makes JSON greppable https://news.ycombinator.com/item?id=16727665
- GoogleGson https://github.com/google/gson
- Jackson https://github.com/FasterXML/jackson
- https://www.yelp.com/dataset_challenge
- RapidJSON. http://rapidjson.org/
Inspiring links:
- https://auth0.com/blog/beating-json-performance-with-protobuf/
- https://gist.github.com/shijuvar/25ad7de9505232c87034b8359543404a
- https://github.com/frankmcsherry/blog/blob/master/posts/2018-02-11.md
Validating UTF-8 takes no more than 0.7 cycles per byte:
- https://github.com/lemire/fastvalidate-utf-8 https://lemire.me/blog/2018/05/16/validating-utf-8-strings-using-as-little-as-0-7-cycles-per-byte/
Remarks on JSON parsing
- The JSON spec defines what a JSON parser is:
A JSON parser transforms a JSON text into another representation. A JSON parser MUST accept all texts that conform to the JSON grammar. A JSON parser MAY accept non-JSON forms or extensions. An implementation may set limits on the size of texts that it accepts. An implementation may set limits on the maximum depth of nesting. An implementation may set limits on the range and precision of numbers. An implementation may set limits on the length and character contents of strings.
-
JSON is not JavaScript:
All JSON is Javascript but NOT all Javascript is JSON. So {property:1} is invalid because property does not have double quotes around it. {'property':1} is also invalid, because it's single quoted while the only thing that can placate the JSON specification is double quoting. JSON is even fussy enough that {"property":.1} is invalid too, because you should have of course written {"property":0.1}. Also, don't even think about having comments or semicolons, you guessed it: they're invalid. (credit:https://github.com/elzr/vim-json)
-
The structural characters are:
begin-array = [ left square bracket begin-object = { left curly bracket end-array = ] right square bracket end-object = } right curly bracket name-separator = : colon value-separator = , comma
Pseudo-structural elements
A character is pseudo-structural if and only if:
- Not enclosed in quotes, AND
- Is a non-whitespace character, AND
- Its preceding character is either: (a) a structural character, OR (b) whitespace.
This helps as we redefine some new characters as pseudo-structural such as the characters 1, G, n in the following:
{ "foo" : 1.5, "bar" : 1.5 GEOFF_IS_A_DUMMY bla bla , "baz", null }
Academic References
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Funding
The work is supported by the Natural Sciences and Engineering Research Council of Canada under grant number RGPIN-2017-03910.