`vignettes/articles/methods.Rmd`

`methods.Rmd`

Sorting and searching are fundamental operations in computer and data
science. The objective of the `kdtools`

is to expose a C++
header file implementing efficient sorting and searching methods for
multidimensional tuple-like data. The interface mirrors that of the C++ Standard Library sorting
and searching functions. It is a single header-only library with no
external dependencies other than `std`

.

A common approach to improving search performance is to place data into an ordered tree structure, the most common being a binary tree and its variants. Computer textbooks and introductory courses are littered with examples of tree structures, however it has been pointed out by practitioners that many tree-type data structures have considerable storage overhead and poor memory locality. This results from using linked-list-like pointers to connect nodes. In many cases, a fully or partially sorted sequence combined with binary search will outperform pointer-based trees. Indeed, the Standard Library contains a set of algorithms for sorting or partitioning a data sequence and searching within partitioned sequences.

A limitation of the Standard Library algorithms and data structures is that they can only be ordered in a single dimension. Sort keys can possess higher cardinality however comparison operators and ordering are unidimensional. This is suboptimal for data that are intrinsically multidimensional, such as geographical coordinates or other multivariate data. In multiple dimensions, it is preferred to order a sequence of keys such the average distance between neighboring pairs in the sequence is small. There exist a vast array of techniques to accomplish sequence localization; however I will only discuss one known as the kd-tree.

A kd-tree is a type of binary tree that cycles among the dimensions of its stored objects at each level of the tree hierarchy. In the case of spatial data, the root node will contain the median key in the x-direction and each of its daughter nodes will contain the median in the y-direction of their respective partitions. Subsequent daughter nodes will again contain medians in the x-direction, and so on until there are no additional keys to the left or right. Searching a kd-tree involves recursive comparisons, cycling among the dimensions, until a leaf-node is encountered. Most searching operations can be accomplished in logarithmic time complexity yielding an efficient method.

Precisely as an ordinary binary tree can be replaced by a sorted range, a sequence of multidimensional elements can be sorted, in-place if desired, via recursive partitioning following the kd-tree scheme. Divide-and-conquer recursive partitioning is a well-known sorting algorithm and is the basis of the quicksort algorithm. Partitioning a sequence places all elements less than (or more generally meeting some predicate) to the left of a selected key, while all elements greater-than-or-equal-to follow to the right. The kd-sort algorithm presented here simply modifies quicksort to cycle among the dimensions of the stored elements. Searching follows exactly as the kd-tree except that nodes correspond to pivot-elements in the linear array.

Implementation of `kd_sort`

enables a set of additional
algorithms for searching and nearest neighbor queries. The following
table gives the exposed functions and corresponding Standard Library
routines.

Function templates | SL analog | Outcome |
---|---|---|

kd_sort | sort | sorts range |

kd_is_sorted | is_sorted | returns `true` if sorted |

kd_lower_bound | lower_bound | finds first element not less than key for all dimensions |

kd_upper_bound | upper_bound | finds first element greater than key for all dimensions |

kd_binary_search | binary_search | returns true if key exists in range |

kd_equal_range | equal_range | returns an iterator pair spanning all occurrences of a key |

kd_range_query | lower_bound, upper_bound | finds all elements not less than lower and less than upper |

kd_nearest_neighbor | finds nearest neighbor of key | |

kd_nearest_neighbors, kd_nn_indices | finds k-nearest neighbors of key | |

lex_sort | ordinary SL sort using kd_less for comparison |

The kd-sort algorithm in the kdtools package is implemented as a C++ function template parameterized by the starting dimension (usually 0) and the iterator type, which will be auto-deduced by the compiler.

```
template <size_t I, typename Iter>
void kd_sort(Iter first, Iter last)
{
using TupleType = iter_value_t<Iter>;
constexpr auto J = next_dim<I, TupleType>;
if (distance(first, last) > 1)
{
auto pred = kd_less<I>();
auto pivot = median_part(first, last, pred);
<J>(next(pivot), last);
kd_sort<J>(first, pivot);
kd_sort}
}
```

The `median_part`

function is a thin wrapper around the `nth_element`

function from the standard library, which is used to find the median
value in the current dimension. Elements to the left of the nth element
are not greater than the nth element and elements to the right are not
less than the nth element. This allows searching left or right depending
on whether the sought value is not greater than the median or not less
than the median (ties must search both directions).

Once the first dimension is partitioned, the ranges right and left of
the pivot are partitioned, and so on until first and last span a single
or no value. The dimension index `I`

is incremented at
compile-time using template metaprogramming. The `value`

member of the templated struct `next_dim`

holds the
incremented value of `I`

that has been wrapped to cycle
through successive dimensions of `TupleType`

. The result of
`kd_sort`

is a sequence ordered as if it were inserted into a
kd-tree data structure. The following figure illustrates the resulting
order in 2-dimensions.

Here the sequence order is shown with a continuous color gradient. The resulting patchiness indicates a hierarchical tree-like partitioning of the space providing the desired locality. Placing nearby points in proximity is what allows for efficient searches once the data are sorted. Sort times for vectors of tuple-like objects are relatively efficient, taking roughly twice the time of an ordinary lexicographical sort.

In the above results, `lex_sort`

is the Standard Library
built-in `std::sort`

algorithm using `kd_less`

as the predicate. The results
indicate that `kd_sort`

is reasonably fast, requiring roughly
twice the time as the Standard Library built-in sorting algorithm.
Because of the divide-and-conquer nature of `kd_sort`

, it is
trivially parallelizable. The threaded version is equal to or faster
than the regular sort. Performance comparable to the build-in Standard
Library `sort`

function indicates that this reference
implementation is reasonable.

To ensure the correct behavior in the presence of multiple ties, or
even when all keys in one dimension are equal, the `kd_less`

predicate implements circular lexicographic comparison. If elements in
one dimension are tied, the next dimension is interrogated to break the
tie. The comparisons cycle from trailing dimensions to leading
dimensions using modulo arithmetic. It is implemented as a recursive,
templated function object with the dimension index incremented at
compile-time. The recursion is terminated using constexpr
if.

```
template <size_t I, size_t K = 0>
struct kd_less
{
template <typename T>
bool operator()(const T& lhs, const T& rhs) const
{
if constexpr (is_last<K, T>) {
return less_nth<I>()(lhs, rhs);
} else {
constexpr auto J = next_dim<I, T>;
return equal_nth<I>()(lhs, rhs) ?
<J, K + 1>()(lhs, rhs) :
kd_less<I>()(lhs, rhs);
less_nth}
}
};
```

The predicates `equal_nth`

and `less_nth`

compare the \(I^{th}\) element of the
tuple. They have specializations for handling pointers to tuple-like
object, so all functions can operate on a range over pointers to tuples,
allowing arbitrary indexing into the original data.

A more general function object template `kd_compare`

that
takes an arbitrary binary predicate is also provided by
`kdtools`

. Some examples applying `kd_less`

,
including degenerate, all-ties cases, are shown here.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

1 | 4 | 3 | 2 | 5 | 8 | 9 | 6 | 7 |

2 | 1 | 4 | 5 | 9 | 3 | 6 | 7 | 8 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

2 | 5 | 4 | 1 | 9 | 7 | 8 | 3 | 6 |

1 | 4 | 3 | 2 | 5 | 8 | 9 | 6 | 7 |

2 | 1 | 4 | 5 | 9 | 3 | 6 | 7 | 8 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Notice that `kd_sort`

is a variant of the quicksort
algorithm in one dimension. It also is not effected by all-ties. The
order of subsequent dimensions can be changed when the leading dimension
contains all ties, however the sequence is still sorted and can be used
for searching.

Once data are ordered appropriately, searching can be achieved
efficiently. The `kdtools`

package provides analogs of the
Standard Library `lower_bound`

,
`upper_bound`

,
`equal_range`

and `binary_search`

functions for querying sorted ranges. An additional function,
`kd_range_query`

, does efficient searching within a boxed
region.

The `kd_lower_bound`

, `kd_upper_bound`

and
`kd_equal_range`

algorithms behave differently than their
unidimensional analogs. This is because the only reasonable definition
of multidimensional less is to require less in all dimensions
simultaneously. As illustrated below, this leads to cases where some,
but not all, elements are less to be included in the interval spanned by
`kd_lower_bound`

, `kd_upper_bound`

and
`kd_equal_range`

. In other words, in a single dimension,
these routines are both necessary and sufficient, whereas in
multiple-dimensions, they are necessary, but not sufficient to uniquely
identify elements in a boxed region.

In other schemes, such as Morton codes,
elaborate subroutines are required to prune the extra elements falling
outside the region of interest (see lower-left panel below). This is a
result of using a fundamentally unidimensional search on
multidimensional data. Instead `kd_sort`

implements direct
binary search for points falling within a boxed region.

```
template <size_t I,
typename Iter,
typename TupleType,
typename OutIter>
void kd_range_query(Iter first, Iter last,
const TupleType& lower,
const TupleType& upper,
)
OutIter outp{
if (distance(first, last) > 32) {
auto pred = less_nth<I>();
auto pivot = find_pivot<I>(first, last);
constexpr auto J = next_dim<I, TupleType>;
if (within(*pivot, lower, upper)) *outp++ = *pivot;
if (!pred(*pivot, lower)) // search left
<J>(first, pivot, lower, upper, outp);
kd_range_queryif (pred(*pivot, upper)) // search right
<J>(next(pivot), last, lower, upper, outp);
kd_range_query} else {
(first, last, outp, [&](const TupleType& x){
copy_ifreturn within(x, lower, upper);
});
}
return;
}
```

Here, each pivot element is tested to see whether it falls within the
boxed region, in which case, it is assigned to the output iterator. The
template `find_pivot`

searches for the partition point whose
position may or may not be the middle of the sequence. Leaf tuples are
also tested and copied to the output iterator. We check in each
dimension whether there might be additional points in the region of
interest to the left or right in the array. These are searched
recursively until reaching the span falls below a threshold and is more
profitably searched via sequential scan. In the following plot, the blue
points indicate those not meeting the predicate and the aquamarine
points are those where the predicate returns true.

The Boost Geometry library is a highly sophisticated and performant framework for working with geometric features. It includes an R-Tree index that allows access to objects within a bounding region in logarithmic time. Here I compare the speed of building and querying a collection of points. The only purpose of this comparison is to look for large (i.e., 5 to 10 fold) differences, which would indicate a problem in the implementation. Small differences in measured times are probably not meaningful and will likely change on different systems. Reported results are the median of 21 trials using 10^{7} tuples.

Test | Time | Ratio |
---|---|---|

kd_sort_threaded | 0.421 | 1.0 |

kd_sort inplace | 1.890 | 4.5 |

kd_sort w/ copy | 1.900 | 4.5 |

rtree pack linear | 2.100 | 5.0 |

rtree pack quadratic | 2.090 | 5.0 |

rtree pack rstar | 2.090 | 5.0 |

Test | Time | Ratio |
---|---|---|

kd_range_query | 0.000294 | 1.0 |

rtree query linear | 0.001330 | 4.5 |

rtree query quadratic | 0.001160 | 3.9 |

rtree query rstar | 0.001170 | 4.0 |

The results indicate that sorting and searching with kdtools is at least as performant as using Boost Geometry. Generally, an R-Tree will have an advantage with many individual insertions and deletions, whereas a sorted range will be more compact and show equal, if not better, query performance.

The kdtools package also provides routines for nearest- and k-nearest- neighbor search.

Test | Time | Ratio |
---|---|---|

kd_nearest_neighbor | 1.10e-06 | 1.0 |

rtree nearest neighbor linear | 2.67e-05 | 25.0 |

rtree nearest neighbor quadratic | 7.10e-06 | 6.6 |

rtree nearest neighbor rstar | 4.10e-06 | 3.8 |

Test | Time | Ratio |
---|---|---|

kd_nearest_neighbors 100 | 1.71e-05 | 1.00 |

rtree nearest 100 linear | 8.38e-05 | 4.90 |

rtree nearest 100 quadratic | 4.05e-05 | 2.40 |

rtree nearest 100 rstar | 1.63e-05 | 0.95 |

As before, the results show kdtools to be as fast or faster than Boost Geometry search. This is not surprising as the R-Tree data structure is optimized for indexing boxed regions, not points. The exception is rstar search for 100 nearest neighbors, which is faster. This might have to do with differences in the priority queue implementation or it could be that the rtree buckets allow rapid retrieval of groups of nearby points all at once. The main conclusion here is that the current implementation of kdtools is reasonably efficient.

Notice that the only requirement of the algorithms is that the
`get<I>`

function return a type meeting the EqualityComparable
and LessThanComparable
C++ concepts. This means that different dimensions can contain different
data types, including strings, for example. The current implementation
uses `std::get`

and so is restricted to
`std::pair`

, `std::tuple`

and
`std::array`

, unless one chooses to specializes in the
`std`

namespace, which is generally not recommended. A
suitable mechanism for providing type-specific specializations is
to-be-implemented. The following example demonstrates sorting and
searching a container holding keys of type
`std::tuple<double, std::string>`

.

```
// [[Rcpp::depends(kdtools)]]
// [[Rcpp::plugins(cpp17)]]
#include "mixed_query.h"
// [[Rcpp::export]]
void mixed_query(NumericVector c1, CharacterVector c2)
{
auto n = c1.size();
range_type data;
.reserve(n);
data
(begin(c1), end(c1), begin(c2), back_inserter(data), make_key());
transform
auto t = time_it([&]{
(begin(data), end(data));
kd_sort
});
<< "\nSort time: " << t.count() << " seconds" << std::endl;
Rcout
<< "\nFirst and last " << N << " elements of sorted data:\n" << std::endl;
Rcout (data);
print_range
range_type result;
auto u = time_it([&]{
(begin(data), end(data),
kd_range_querykey_type(0.4, "w"), key_type(0.6, "z"),
(result));
back_inserter
});
<< "\nQuery time: " << u.count() << " seconds" << std::endl;
Rcout
<< "\nFirst and last " << N << " elements of query return:\n" << std::endl;
Rcout (begin(result), end(result)); print_range(result);
kd_sort
.clear();
result
auto v = time_it([&]{
(begin(data), end(data),
kd_nearest_neighborskey_type(0.5, "kdtools"), 100,
(result));
back_inserter
});
<< "\nNeighbors query time: " << v.count() << " seconds" << std::endl;
Rcout
<< "\nFirst and last " << N << " elements of neighbors query return:\n" << std::endl;
Rcout (result);
print_range}
```

First the data are sorted and then a range query is conducted. The
range query selects tuples where the first element ranges from
`0.4`

to `0.6`

and the second element begins with
letters `w`

through `y`

. For the test, we borrow
the words from the `parts_of_speech`

dataset from the
`tidytext`

package. It contains more than \(10^5\) words.

```
data("parts_of_speech")
numbers = signif(runif(nrow(parts_of_speech)), 4)
strings = sample(tolower(parts_of_speech[[1]]))
mixed_query(numbers, strings)
#>
#> Sort time: 0.0675884 seconds
#>
#> First and last 3 elements of sorted data:
#>
#> 0 0.0001333 abdul
#> 1 0.0008761 abominator
#> 2 0.001739 abranchiate
#> ...
#> 208256 0.9975 year-round
#> 208257 0.9989 zedekiah
#> 208258 0.9995 zibet
#>
#> Query time: 3.6292e-05 seconds
#>
#> First and last 3 elements of query return:
#>
#> 0 0.4091 waffle
#> 1 0.4084 walls
#> 2 0.4102 wallydrag
#> ...
#> 1069 0.5879 yumuk
#> 1070 0.5928 yordan
#> 1071 0.5993 you'd
#>
#> Neighbors query time: 0.000375708 seconds
#>
#> First and last 3 elements of neighbors query return:
#>
#> 0 0.4432 koslo
#> 1 0.4882 kolnos
#> 2 0.4816 kobold
#> ...
#> 97 0.3718 laparotomies
#> 98 0.3328 kimonoed
#> 99 0.324 kingston
```

The timing results indicate that sorting \(2 \times 10 ^ 5\) tuples is on the order of 1/10th second, whereas the range query returned in less than a millisecond.

Nearest neighbor searching requires a suitable distance function be
defined for all types. The template `scalar_dist`

can be
specialized for different types. It returns the distance along a single
dimension. Currently, nearest neighbor distances are hardwired as the L2
norm of the returned scalar differences computed for each dimension. The
computation above uses the following definitions.

```
template<class InputIt1, class InputIt2>
double set_similarity(InputIt1 first1, InputIt1 last1,
, InputIt2 last2)
InputIt2 first2{
double num = 0, denom = 0;
while (first1 != last1 && first2 != last2) {
if (*first1 < *first2) {
++first1; ++denom;
} else {
if (!(*first2 < *first1)) {
++first1; ++num;
}
++first2; ++denom;
}
}
+= std::distance(first1, last1) + std::distance(first2, last2);
denom return num / denom;
}
namespace kdtools {
template <>
double scalar_dist(const std::string& lhs, const std::string& rhs)
{
std::string a(lhs), b(rhs);
std::sort(begin(a), end(a));
std::sort(begin(b), end(b));
return 1 - set_similarity(begin(a), end(a),
(b), end(b));
begin}
} // namespace kdtools
```

The `set_similarity`

template is adapted from the Standard
Library `set_intersection`

template and produces a number
between 0 and 1. I have not attempted to prove this distance is metric,
so the results may not be entirely correct. Also, there is no facility
currently available to weight distances in different dimensions or
override the default root-sum-of-squares algorithm for the multivariate
distance.

Since version 0.4.0, `kd_tools`

all predicates and
distance functions have specializations or overloads to handle pointers
to tuple-like objects. Hence, any of the routines can sort or search a
vector of pointers to tuples, which act as indices into the original
data.

```
// [[Rcpp::depends(kdtools)]]
// [[Rcpp::plugins(cpp17)]]
#include "mixed_query.h"
// [[Rcpp::export]]
void sort_pointers(NumericVector c1, CharacterVector c2)
{
auto n = c1.size();
range_type data;
.reserve(n);
data
(begin(c1), end(c1), begin(c2), back_inserter(data), make_key());
transform
pointers_type idx1, idx2;
.reserve(n); idx2.reserve(n);
idx1
for (auto& x : data) {
.push_back(&x);
idx1.push_back(&x);
idx2}
(begin(idx1), end(idx1));
kd_sort(begin(idx2), end(idx2), greater_key());
kd_sort
<< "Original order\n\n"; print_range(data);
Rcout << "\nOrder by less\n\n"; print_range(idx1);
Rcout << "\nOrder by greater\n\n"; print_range(idx2);
Rcout }
```

Here, `greater_key`

has overloads for both string and
double types.

```
sort_pointers(numbers, strings)
#> Original order
#>
#> 0 0.006704 unself-centered
#> 1 0.4771 pectinatella
#> 2 0.3967 dominga
#> ...
#> 208256 0.8289 educate
#> 208257 0.7698 cnida
#> 208258 0.7565 coprophobia
#>
#> Order by less
#>
#> 0 0.0001333 abdul
#> 1 0.0008761 abominator
#> 2 0.001739 abranchiate
#> ...
#> 208256 0.9975 year-round
#> 208257 0.9989 zedekiah
#> 208258 0.9995 zibet
#>
#> Order by greater
#>
#> 0 0.9995 zibet
#> 1 0.9989 zedekiah
#> 2 0.9973 zooplankton
#> ...
#> 208256 0.001739 abranchiate
#> 208257 0.0008761 abominator
#> 208258 0.0001333 abdul
```

Note that the forward and reverse sorting do not give exactly the same result (in reverse) owing to ties.

The kdtools package implements efficient sorting and searching of arbitrary tuple-like object in C++. One of the major design decisions was to make the number of dimensions fixed at compile time. This is a reasonable tradeoff as these methods break down in high dimensions where search times will be no better than that of brute-force random searching. Nonetheless, a runtime-dimensioned extension would be easy to add and could allow working with runtime-determined types as is common in R. The current implementation is a demonstration, however it does illustrate the value of the Standard Library as the algorithms are constructed from Standard Library components. The generic nature of the kdtools implementation is an attractive feature as it permits flexibility and can be adapted to arbitrary tuple-like data as needed. Modern development of C++ has emphasized tuple-style data and algorithms and kdtools extends these developments.