Portable Roaring bitmaps in C (and C++) with full support for your favorite compiler (GNU GCC, LLVM's clang, Visual Studio). Included in the Awesome C list of open source C software.
Bitsets, also called bitmaps, are commonly used as fast data structures. Unfortunately, they can use too much memory. To compensate, we often use compressed bitmaps.
Roaring bitmaps are compressed bitmaps which tend to outperform conventional compressed bitmaps such as WAH, EWAH or Concise. They are used by several major systems such as Apache Lucene and derivative systems such as Solr and Elasticsearch, Metamarkets' Druid, LinkedIn Pinot, Netflix Atlas, Apache Spark, OpenSearchServer, Cloud Torrent, Whoosh, InfluxDB, Pilosa, Bleve, Microsoft Visual Studio Team Services (VSTS), and eBay's Apache Kylin. The CRoaring library is used in several systems such as Apache Doris. The YouTube SQL Engine, Google Procella, uses Roaring bitmaps for indexing.
We published a peer-reviewed article on the design and evaluation of this library:
- Roaring Bitmaps: Implementation of an Optimized Software Library, Software: Practice and Experience 48 (4), 2018 arXiv:1709.07821
Roaring bitmaps are found to work well in many important applications:
Use Roaring for bitmap compression whenever possible. Do not use other bitmap compression methods (Wang et al., SIGMOD 2017)
There is a serialized format specification for interoperability between implementations. Hence, it is possible to serialize a Roaring Bitmap from C++, read it in Java, modify it, serialize it back and read it in Go and Python.
The primary goal of the CRoaring is to provide a high performance low-level implementation that fully take advantage of the latest hardware. Roaring bitmaps are already available on a variety of platform through Java, Go, Rust... implementations. CRoaring is a library that seeks to achieve superior performance by staying close to the latest hardware.
(c) 2016-... The CRoaring authors.
- Linux, macOS, FreeBSD, Windows (MSYS2 and Microsoft Visual studio).
- We test the library with ARM, x64/x86 and POWER processors. We only support little endian systems (big endian systems are vanishingly rare).
- Recent C compiler supporting the C11 standard (GCC 7 or better, LLVM 7.0 or better, Xcode 11 or better), there is also an optional C++ class that requires a C++ compiler supporting the C++11 standard.
- CMake (to contribute to the project, users can rely on amalgamation/unity builds if they do not wish to use CMake).
- Under x64 systems, the library provides runtime dispatch so that optimized functions are called based on the detected CPU features. It works with GCC, clang (version 9 and up) and Visual Studio (2017 and up). Other systems (e.g., ARM) do not need runtime dispatch.
Hardly anyone has access to an actual big-endian system. Nevertheless, We support big-endian systems such as IBM s390x through emulators---except for IO serialization which is only supported on little-endian systems (see issue 423).
If you like CMake, you can just a few lines in you CMakeLists.txt file to grab a CRoaring release. See our demonstration for further details.
If you installed the CRoaring library locally, you may use it with CMake's find_package function as in this example:
cmake_minimum_required(VERSION 3.15)
project(test_roaring_install VERSION 0.1.0 LANGUAGES CXX C)
set(CMAKE_CXX_STANDARD 11)
set(CMAKE_CXX_STANDARD_REQUIRED ON)
set(CMAKE_C_STANDARD 11)
set(CMAKE_C_STANDARD_REQUIRED ON)
find_package(roaring REQUIRED)
file(WRITE main.cpp "
#include <iostream>
#include \"roaring/roaring.hh\"
int main() {
roaring::Roaring r1;
for (uint32_t i = 100; i < 1000; i++) {
r1.add(i);
}
std::cout << \"cardinality = \" << r1.cardinality() << std::endl;
return 0;
}")
add_executable(repro main.cpp)
target_link_libraries(repro PUBLIC roaring::roaring)The CRoaring library can be amalgamated into a single source file that makes it easier for integration into other projects. Moreover, by making it possible to compile all the critical code into one compilation unit, it can improve the performance. For the rationale, please see the SQLite documentation, https://www.sqlite.org/amalgamation.html, or the corresponding Wikipedia entry (https://en.wikipedia.org/wiki/Single_Compilation_Unit). Users who choose this route, do not need to rely on CRoaring's build system (based on CMake).
We maintain pre-generated amalgamated files at https://github.com/lemire/CRoaringUnityBuild for your convenience.
To generate the amalgamated files yourself, you can invoke a bash script...
./amalgamation.sh(Bash shells are standard under Linux and macOS. Bash shells are available under Windows as part of the GitHub Desktop under the name Git Shell. So if you have cloned the CRoaring GitHub repository from within the GitHub Desktop, you can right-click on CRoaring, select Git Shell and then enter the above commands.)
It is not necessary to invoke the script in the CRoaring directory. You can invoke it from any directory where you want the amalgamation files to be written.
It will generate three files for C users: roaring.h, roaring.c and amalgamation_demo.c... as well as some brief instructions. The amalgamation_demo.c file is a short example, whereas roaring.h and roaring.c are "amalgamated" files (including all source and header files for the project). This means that you can simply copy the files roaring.h and roaring.c into your project and be ready to go! No need to produce a library! See the amalgamation_demo.c file.
For example, you can use the C code as follows:
#include <stdio.h>
#include "roaring.c"
int main() {
roaring_bitmap_t *r1 = roaring_bitmap_create();
for (uint32_t i = 100; i < 1000; i++) roaring_bitmap_add(r1, i);
printf("cardinality = %d\n", (int) roaring_bitmap_get_cardinality(r1));
roaring_bitmap_free(r1);
return 0;
}The script will also generate C++ files for C++ users, including an example. You can use the C++ as follows.
#include <iostream>
#include "roaring.hh"
#include "roaring64map.hh"
using namespace roaring;
int main() {
Roaring r1;
for (uint32_t i = 100; i < 1000; i++) {
r1.add(i);
}
std::cout << "cardinality = " << r1.cardinality() << std::endl;
Roaring64Map r2;
for (uint64_t i = 18000000000000000100ull; i < 18000000000000001000ull;
i++) {
r2.add(i);
}
std::cout << "cardinality = " << r2.cardinality() << std::endl;
return EXIT_SUCCESS;
}If you prefer a silent output, you can use the following command to redirect stdout :
./amalgamation.sh > /dev/nullThe C interface is found in the file include/roaring/roaring.h. We have C++ interface at cpp/roaring.hh.
Some users have to deal with large volumes of data. It may be important for these users to be aware of the addMany (C++) roaring_bitmap_or_many (C) functions as it is much faster and economical to add values in batches when possible. Furthermore, calling periodically the runOptimize (C++) or roaring_bitmap_run_optimize (C) functions may help.
For general users, CRoaring would apply default allocator without extra codes. But global memory hook is also provided for those who want a custom memory allocator. Here is an example:
#include <roaring.h>
int main(){
// define with your own memory hook
roaring_memory_t my_hook{my_malloc, my_free ...};
// initialize global memory hook
roaring_init_memory_hook(my_hook);
// write you code here
...
}This example assumes that CRoaring has been build and that you are linking against the corresponding library. By default, CRoaring will install its header files in a roaring directory. If you are working from the amalgamation script, you may add the line #include "roaring.c" if you are not linking against a prebuilt CRoaring library and replace #include <roaring/roaring.h> by #include "roaring.h".
#include <roaring/roaring.h>
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
bool roaring_iterator_sumall(uint32_t value, void *param) {
*(uint32_t *)param += value;
return true; // iterate till the end
}
int main() {
// create a new empty bitmap
roaring_bitmap_t *r1 = roaring_bitmap_create();
// then we can add values
for (uint32_t i = 100; i < 1000; i++) roaring_bitmap_add(r1, i);
// check whether a value is contained
assert(roaring_bitmap_contains(r1, 500));
// compute how many bits there are:
uint32_t cardinality = roaring_bitmap_get_cardinality(r1);
printf("Cardinality = %d \n", cardinality);
// if your bitmaps have long runs, you can compress them by calling
// run_optimize
uint32_t expectedsizebasic = roaring_bitmap_portable_size_in_bytes(r1);
roaring_bitmap_run_optimize(r1);
uint32_t expectedsizerun = roaring_bitmap_portable_size_in_bytes(r1);
printf("size before run optimize %d bytes, and after %d bytes\n",
expectedsizebasic, expectedsizerun);
// create a new bitmap containing the values {1,2,3,5,6}
roaring_bitmap_t *r2 = roaring_bitmap_of(5, 1, 2, 3, 5, 6);
roaring_bitmap_printf(r2); // print it
// we can also create a bitmap from a pointer to 32-bit integers
uint32_t somevalues[] = {2, 3, 4};
roaring_bitmap_t *r3 = roaring_bitmap_of_ptr(3, somevalues);
// we can also go in reverse and go from arrays to bitmaps
uint64_t card1 = roaring_bitmap_get_cardinality(r1);
uint32_t *arr1 = (uint32_t *)malloc(card1 * sizeof(uint32_t));
assert(arr1 != NULL);
roaring_bitmap_to_uint32_array(r1, arr1);
roaring_bitmap_t *r1f = roaring_bitmap_of_ptr(card1, arr1);
free(arr1);
assert(roaring_bitmap_equals(r1, r1f)); // what we recover is equal
roaring_bitmap_free(r1f);
// we can go from arrays to bitmaps from "offset" by "limit"
size_t offset = 100;
size_t limit = 1000;
uint32_t *arr3 = (uint32_t *)malloc(limit * sizeof(uint32_t));
assert(arr3 != NULL);
roaring_bitmap_range_uint32_array(r1, offset, limit, arr3);
free(arr3);
// we can copy and compare bitmaps
roaring_bitmap_t *z = roaring_bitmap_copy(r3);
assert(roaring_bitmap_equals(r3, z)); // what we recover is equal
roaring_bitmap_free(z);
// we can compute union two-by-two
roaring_bitmap_t *r1_2_3 = roaring_bitmap_or(r1, r2);
roaring_bitmap_or_inplace(r1_2_3, r3);
// we can compute a big union
const roaring_bitmap_t *allmybitmaps[] = {r1, r2, r3};
roaring_bitmap_t *bigunion = roaring_bitmap_or_many(3, allmybitmaps);
assert(
roaring_bitmap_equals(r1_2_3, bigunion)); // what we recover is equal
// can also do the big union with a heap
roaring_bitmap_t *bigunionheap =
roaring_bitmap_or_many_heap(3, allmybitmaps);
assert(roaring_bitmap_equals(r1_2_3, bigunionheap));
roaring_bitmap_free(r1_2_3);
roaring_bitmap_free(bigunion);
roaring_bitmap_free(bigunionheap);
// we can compute intersection two-by-two
roaring_bitmap_t *i1_2 = roaring_bitmap_and(r1, r2);
roaring_bitmap_free(i1_2);
// we can write a bitmap to a pointer and recover it later
uint32_t expectedsize = roaring_bitmap_portable_size_in_bytes(r1);
char *serializedbytes = malloc(expectedsize);
roaring_bitmap_portable_serialize(r1, serializedbytes);
// For additional safety, you may replace roaring_bitmap_portable_deserialize by
// roaring_bitmap_portable_deserialize_safe.
// Note: it is expected that the input follows the specification
// https://github.com/RoaringBitmap/RoaringFormatSpec
// otherwise the result may be unusable.
roaring_bitmap_t *t = roaring_bitmap_portable_deserialize(serializedbytes);
assert(roaring_bitmap_equals(r1, t)); // what we recover is equal
roaring_bitmap_free(t);
// we can also check whether there is a bitmap at a memory location without
// reading it
size_t sizeofbitmap =
roaring_bitmap_portable_deserialize_size(serializedbytes, expectedsize);
assert(sizeofbitmap ==
expectedsize); // sizeofbitmap would be zero if no bitmap were found
// we can also read the bitmap "safely" by specifying a byte size limit:
t = roaring_bitmap_portable_deserialize_safe(serializedbytes, expectedsize);
// It is still necessary for the content of seriallizedbytes to follow
// the standard: https://github.com/RoaringBitmap/RoaringFormatSpec
// This is guaranted when calling 'roaring_bitmap_portable_deserialize'.
assert(roaring_bitmap_equals(r1, t)); // what we recover is equal
roaring_bitmap_free(t);
free(serializedbytes);
// we can iterate over all values using custom functions
uint32_t counter = 0;
roaring_iterate(r1, roaring_iterator_sumall, &counter);
// we can also create iterator structs
counter = 0;
roaring_uint32_iterator_t *i = roaring_create_iterator(r1);
while (i->has_value) {
counter++; // could use i->current_value
roaring_advance_uint32_iterator(i);
}
// you can skip over values and move the iterator with
// roaring_move_uint32_iterator_equalorlarger(i,someintvalue)
roaring_free_uint32_iterator(i);
// roaring_bitmap_get_cardinality(r1) == counter
// for greater speed, you can iterate over the data in bulk
i = roaring_create_iterator(r1);
uint32_t buffer[256];
while (1) {
uint32_t ret = roaring_read_uint32_iterator(i, buffer, 256);
for (uint32_t j = 0; j < ret; j++) {
counter += buffer[j];
}
if (ret < 256) {
break;
}
}
roaring_free_uint32_iterator(i);
roaring_bitmap_free(r1);
roaring_bitmap_free(r2);
roaring_bitmap_free(r3);
return EXIT_SUCCESS;
}This example assumes that CRoaring has been build and that you are linking against the corresponding library. By default, CRoaring will install its header files in a roaring directory so you may need to replace #include "roaring.hh" by #include <roaring/roaring.hh>. If you are working from the amalgamation script, you may add the line #include "roaring.c" if you are not linking against a CRoaring prebuilt library.
#include <iostream>
#include "roaring.hh"
using namespace roaring;
int main() {
Roaring r1;
for (uint32_t i = 100; i < 1000; i++) {
r1.add(i);
}
// check whether a value is contained
assert(r1.contains(500));
// compute how many bits there are:
uint32_t cardinality = r1.cardinality();
// if your bitmaps have long runs, you can compress them by calling
// run_optimize
uint32_t size = r1.getSizeInBytes();
r1.runOptimize();
// you can enable "copy-on-write" for fast and shallow copies
r1.setCopyOnWrite(true);
uint32_t compact_size = r1.getSizeInBytes();
std::cout << "size before run optimize " << size << " bytes, and after "
<< compact_size << " bytes." << std::endl;
// create a new bitmap with varargs
Roaring r2 = Roaring::bitmapOf(5, 1, 2, 3, 5, 6);
r2.printf();
printf("\n");
// create a new bitmap with initializer list
Roaring r2i = Roaring::bitmapOfList({1, 2, 3, 5, 6});
assert(r2i == r2);
// we can also create a bitmap from a pointer to 32-bit integers
const uint32_t values[] = {2, 3, 4};
Roaring r3(3, values);
// we can also go in reverse and go from arrays to bitmaps
uint64_t card1 = r1.cardinality();
uint32_t *arr1 = new uint32_t[card1];
r1.toUint32Array(arr1);
Roaring r1f(card1, arr1);
delete[] arr1;
// bitmaps shall be equal
assert(r1 == r1f);
// we can copy and compare bitmaps
Roaring z(r3);
assert(r3 == z);
// we can compute union two-by-two
Roaring r1_2_3 = r1 | r2;
r1_2_3 |= r3;
// we can compute a big union
const Roaring *allmybitmaps[] = {&r1, &r2, &r3};
Roaring bigunion = Roaring::fastunion(3, allmybitmaps);
assert(r1_2_3 == bigunion);
// we can compute intersection two-by-two
Roaring i1_2 = r1 & r2;
// we can write a bitmap to a pointer and recover it later
uint32_t expectedsize = r1.getSizeInBytes();
char *serializedbytes = new char[expectedsize];
r1.write(serializedbytes);
// readSafe will not overflow, but the resulting bitmap
// is only valid and usable if the input follows the
// Roaring specification: https://github.com/RoaringBitmap/RoaringFormatSpec/
Roaring t = Roaring::readSafe(serializedbytes, expectedsize);
assert(r1 == t);
delete[] serializedbytes;
// we can iterate over all values using custom functions
uint32_t counter = 0;
r1.iterate(
[](uint32_t value, void *param) {
*(uint32_t *)param += value;
return true;
},
&counter);
// we can also iterate the C++ way
counter = 0;
for (Roaring::const_iterator i = t.begin(); i != t.end(); i++) {
++counter;
}
// counter == t.cardinality()
// we can move iterators to skip values
const uint32_t manyvalues[] = {2, 3, 4, 7, 8};
Roaring rogue(5, manyvalues);
Roaring::const_iterator j = rogue.begin();
j.equalorlarger(4); // *j == 4
return EXIT_SUCCESS;
}
CRoaring follows the standard cmake workflow. Starting from the root directory of the project (CRoaring), you can do:
mkdir -p build
cd build
cmake ..
cmake --build .
# follow by 'ctest' if you want to test.
# you can also type 'make install' to install the library on your system
# C header files typically get installed to /usr/local/include/roaring
# whereas C++ header files get installed to /usr/local/include/roaring
(You can replace the build directory with any other directory name.)
By default all tests are built on all platforms, to skip building and running tests add -DENABLE_ROARING_TESTS=OFF to the command line.
As with all cmake projects, you can specify the compilers you wish to use by adding (for example) -DCMAKE_C_COMPILER=gcc -DCMAKE_CXX_COMPILER=g++ to the cmake command line.
If you are using clang or gcc and you know your target architecture, you can set the architecture by specifying -DROARING_ARCH=arch. For example, if you have many server but the oldest server is running the Intel haswell architecture, you can specify -DROARING_ARCH=haswell. In such cases, the produced binary will be optimized for processors having the characteristics of a haswell process and may not run on older architectures. You can find out the list of valid architecture values by typing man gcc.
mkdir -p build_haswell
cd build_haswell
cmake -DROARING_ARCH=haswell ..
cmake --build .
For a debug release, starting from the root directory of the project (CRoaring), try
mkdir -p debug
cd debug
cmake -DCMAKE_BUILD_TYPE=Debug -DROARING_SANITIZE=ON ..
ctest
To run real-data benchmark
./real_bitmaps_benchmark ../benchmarks/realdata/census1881
where you must adjust the path "../benchmarks/realdata/census1881" so that it points to one of the directories in the benchmarks/realdata directory.
To check that your code abides by the style convention (make sure that clang-format is installed):
./tools/clang-format-check.sh
To reformat your code according to the style convention (make sure that clang-format is installed):
./tools/clang-format.sh
We are assuming that you have a common Windows PC with at least Visual Studio 2015, and an x64 processor.
To build with at least Visual Studio 2015 from the command line:
- Grab the CRoaring code from GitHub, e.g., by cloning it using GitHub Desktop.
- Install CMake. When you install it, make sure to ask that
cmakebe made available from the command line. - Create a subdirectory within CRoaring, such as
VisualStudio. - Using a shell, go to this newly created directory. For example, within GitHub Desktop, you can right-click on
CRoaringin your GitHub repository list, and selectOpen in Git Shell, then typecd VisualStudioin the newly created shell. - Type
cmake -DCMAKE_GENERATOR_PLATFORM=x64 ..in the shell while in theVisualStudiorepository. (Alternatively, if you want to build a static library, you may use the command linecmake -DCMAKE_GENERATOR_PLATFORM=x64 -DROARING_BUILD_STATIC=ON ...) - This last command created a Visual Studio solution file in the newly created directory (e.g.,
RoaringBitmap.sln). Open this file in Visual Studio. You should now be able to build the project and run the tests. For example, in theSolution Explorerwindow (available from theViewmenu), right-clickALL_BUILDand selectBuild. To test the code, still in theSolution Explorerwindow, selectRUN_TESTSand selectBuild.
To build with at least Visual Studio 2017 directly in the IDE:
- Grab the CRoaring code from GitHub, e.g., by cloning it using GitHub Desktop.
- Select the
Visual C++ tools for CMakeoptional component when installing the C++ Development Workload within Visual Studio. - Within Visual Studio use
File > Open > Folder...to open the CRoaring folder. - Right click on
CMakeLists.txtin the parent directory withinSolution Explorerand selectBuildto build the project. - For testing, in the Standard toolbar, drop the
Select Startup Item...menu and choose one of the tests. Run the test by pressing the button to the left of the dropdown.
We have optimizations specific to AVX2 in the code, and they are turned dynamically based on the detected hardware at runtime.
You can install the library using the conan package manager:
$ echo -e "[requires]\nroaring/0.3.3" > conanfile.txt
$ conan install .
vcpkg users on Windows, Linux and macOS can download and install roaring with one single command from their favorite shell.
On Linux and macOS:
$ ./vcpkg install roaring
will build and install roaring as a static library.
On Windows (64-bit):
.\vcpkg.exe install roaring:x64-windows
will build and install roaring as a shared library.
.\vcpkg.exe install roaring:x64-windows-static
will build and install roaring 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 roaring 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.
Our AVX2 code does not use floating-point numbers or multiplications, so it is not subject to turbo frequency throttling on many-core Intel processors.
Like, for example, STL containers or Java's default data structures, the CRoaring library has no built-in thread support. Thus whenever you modify a bitmap in one thread, it is unsafe to query it in others. It is safe however to query bitmaps (without modifying them) from several distinct threads, as long as you do not use the copy-on-write attribute. For example, you can safely copy a bitmap and use both copies in concurrently. One should probably avoid the use of the copy-on-write attribute in a threaded environment.
Suppose you want to compute the union (OR) of many bitmaps. How do you proceed? There are many different strategies.
You can use roaring_bitmap_or_many(bitmapcount, bitmaps) or roaring_bitmap_or_many_heap(bitmapcount, bitmaps) or you may
even roll your own aggregation:
roaring_bitmap_t *answer = roaring_bitmap_copy(bitmaps[0]);
for (size_t i = 1; i < bitmapcount; i++) {
roaring_bitmap_or_inplace(answer, bitmaps[i]);
}
All of them will work but they have different performance characteristics. The roaring_bitmap_or_many_heap should
probably only be used if, after benchmarking, you find that it is faster by a good margin: it uses more memory.
The roaring_bitmap_or_many is meant as a good default. It works by trying to delay work as much as possible.
However, because it delays computations, it also does not optimize the format as the computation runs. It might
thus fail to see some useful pattern in the data such as long consecutive values.
The approach based on repeated calls to roaring_bitmap_or_inplace
is also fine, and might even be faster in some cases. You can expect it to be faster if, after
a few calls, you get long sequences of consecutive values in the answer. That is, if the
final answer is all integers in the range [0,1000000), and this is apparent quickly, then the
later roaring_bitmap_or_inplace will be very fast.
You should benchmark these alternatives on your own data to decide what is best.
Tom Cornebize wrote a Python wrapper available at https://github.com/Ezibenroc/PyRoaringBitMap Installing it is as easy as typing...
pip install pyroaring
Salvatore Previti wrote a Node/JavaScript wrapper available at https://github.com/SalvatorePreviti/roaring-node Installing it is as easy as typing...
npm install roaring
Jérémie Piotte wrote a Swift wrapper.
Brandon Smith wrote a C# wrapper available at https://github.com/RogueException/CRoaring.Net (works for Windows and Linux under x64 processors)
There is a Go (golang) wrapper available at https://github.com/RoaringBitmap/gocroaring
Saulius Grigaliunas wrote a Rust wrapper available at https://github.com/saulius/croaring-rs
Yuce Tekol wrote a D wrapper available at https://github.com/yuce/droaring
Antonio Guilherme Ferreira Viggiano wrote a Redis Module available at https://github.com/aviggiano/redis-roaring
Justin Whear wrote a Zig wrapper available at https://github.com/jwhear/roaring-zig
https://groups.google.com/forum/#!forum/roaring-bitmaps
- Daniel Lemire, Owen Kaser, Nathan Kurz, Luca Deri, Chris O'Hara, François Saint-Jacques, Gregory Ssi-Yan-Kai, Roaring Bitmaps: Implementation of an Optimized Software Library, Software: Practice and Experience Volume 48, Issue 4 April 2018 Pages 867-895 arXiv:1709.07821
- Samy Chambi, Daniel Lemire, Owen Kaser, Robert Godin, Better bitmap performance with Roaring bitmaps, Software: Practice and Experience Volume 46, Issue 5, pages 709–719, May 2016 arXiv:1402.6407
- Daniel Lemire, Gregory Ssi-Yan-Kai, Owen Kaser, Consistently faster and smaller compressed bitmaps with Roaring, Software: Practice and Experience Volume 46, Issue 11, pages 1547-1569, November 2016 arXiv:1603.06549
- Samy Chambi, Daniel Lemire, Robert Godin, Kamel Boukhalfa, Charles Allen, Fangjin Yang, Optimizing Druid with Roaring bitmaps, IDEAS 2016, 2016. http://r-libre.teluq.ca/950/