placement_sort

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Fast O(n) stable sorting algorithm. C++ implementation. It outperforms std::sort and std::stable_sort on N > 100 for both primitive types and complex objects.

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Placement sort
==============

A fast O(n) stable sorting algorithm.

Interface
---------

```cpp
template <typename T>
void placement_sort::sort(T* first, size_t count);

template <class RandomIt>
void placement_sort::sort(RandomIt first, RandomIt last);

template <typename T>
void placement_sort::reverse_sort(T* first, size_t count);

template <class RandomIt>
void placement_sort::reverse_sort(RandomIt first, RandomIt last); 
```
Above is similar to std::sort interface.
Requires that T implements `operator[](size_t)` which returns a value of a fundamental type (int, float, etc..). The value will be used to define order.
Also requires T or dereferenced RandomIt to support std::move.

Interface below allows to implement a custom accessor to a value to sort by:
```cpp
template <typename T, typename TValueAccessor>
void placement_sort::sort(T* first, size_t count, const TValueAccessor& valueAccessor);

template <class RandomIt, typename TValueAccessor>
void placement_sort::sort(RandomIt first, RandomIt last, const TValueAccessor& valueAccessor) 
```
where valueAccessor is
```cpp
auto valueAccessor = [](const T* element) -> fund_type {return element->getValueToSortBy()};
```
or
```cpp
template <class T>
struct ValueAccessor {
    fund_type operator() (const T* element) const { return element->getValueToSortBy();}
};

```
This to be used on structs or classes to select a field or a method to pick value from.

Example
-------

```cpp
placement_sort::sort(array.begin(), array.end(), [](const T* sportsman) -> float { return sportsman->height;});
```

Idea shortly
------------

To sort numbers 4,2,3,1 just place number i on i-th position.
If numbers range is larger or smaller than size, use shift and scale operations to fit a number in the size range. If there are collisions after shift and scale, recursively sort out all the elements which hit the same place.

Idea in details
---------------

Let's assume A is not sorted input array, and B is a sorted output.
Steps:
1. Find min and max values in the A.
2. Define placer(x) := (x - min) * size / (max - min)
3. Move A[i] to B[placer(A[i])] for i = 1..N.

Collissions resolution.
Except of cases of soring unque 1..N numbers there are collisions to appear. It means that place(A[i]) == place(A[j]) for some of i and j, or in other words that A[i] and A[j] compete for the same place with the current placer() .

To detect collisions the algorithm uses N counters which store a number of elements competing for corresponding place.
If each counter[i] == 1 for any i {1..N}, then there is no collisions. Just move A[i] to B[place[i]] and stop.
Otherwise there are following steps to perform:
1. Compute memory distribution. For example for array of 13 elements if counter[1] = 1, counter[2] = 10, counter[3] = 2, then it means that the element with place == 1 will be on the 1st position, elements with place == 2 will be on positions from 2nd to 11th, and elements with place == 3 will be on positions 12th and 13th. To save some memory just replace counter[i] with index from which the corresponding intervals start. Thus counter[1] = 1, counter[2] = 2, counter[3] = 12.
2. Move elements according to the memory distribution. From the example above the element in the 1st position will be smaller than any other element. Elements in positions from 2nd to 11th will be larger than the 1st, and smaller than 12th and 13th. This could be done be moving A[i] to B[counterr[place(A[i])]] and then incrementing counter[placer(A[i])] by one, so that it points to the next place in the interval to put there the following colliding element.
3. Sort out recursively intervals with more than one element. To keep the worst case operations count limit within O(N * log(N)) the following strategy is used. If one interval is larger than N / 2, such an interval is devided into two equal subintervals, and both sorted recursively and separately, than merge sort step is applied to join them.

Complexity
----------

O(n) in most cases.
Actually complexity < O(n * k), where  k = log(size, max - min) + 1 and is a number of possible recursion levels. k is typically 2 to 4 on random data.

Worst case: values like 1, N, N^2, N^3, ...,  N^N. In that case placer(A[N]) = N, and placer(A[i]) = 1 for i {1,N-1} on first step. Thus every element except one collide at one place on first recursion level. On the next recusrsion it'll repeat without last element. And there are potentially N such steps and O(N * N) operations. But due to split of intervals larger than N / 2 and merge sort fall back it'll come down only to O(N * log N).

There is 2 * N data moves one the first recursion level, and 2 * M for subintervals of size M.

Performance
-----------

Outperforms qsort on any size, std::sort(g++ 7.2.0) on N > 80, and std::stable_sort on N > 40. (as measured on Ubuntu 17.10 / Intel(R) Core(TM) i7-5820K)

[Place for chart GCC Ubuntu]

[Place for chart MSVC Windows]

Memory usage
------------

Uses `O((sizeof(T) + k * sizeof(index_t)) * size)` extra memory.

It is possible to implement a not stable version which can run on just `sizeof(index_t) * size` extra memory.

Applicability
-------------

Works great to sort numerical values, and even better to sort large objects by a numerical field because of low number of data moves.
Due to the nature of soring by numerical values it's tricky to sort out data which is not numbers by it's nature, eq strings.

Possible future features
------------------------

- SMP (openmp/pthreads) 
    - sort number of thread chunks and merge sort these into sorted result

- Find only i-th to k-th elements of sorted array

- DMP
    - MPI/Distributed
        prepare blocks to send to a node 
        
    - MAP-Reduce
        Sort local chunks and merge sort on reduce

LICENSE
-------
Copyright Alexandr Kobotov 2018-2019. Licensed under the Apache License, Version 2.0. See LICENSE file for more details.

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