zig/lib/std /
sort/block.zig
Stable in-place sort. O(n) best case, O(n*log(n)) worst case and average case.
O(1) memory (no allocator required).
Sorts in ascending order with respect to the given lessThan function.
NOTE: The algorithm only works when the comparison is less-than or greater-than.
(See https://github.com/ziglang/zig/issues/8289)
const builtin = @import("builtin");
const std = @import("../std.zig");
const sort = std.sort;
const math = std.math;
const mem = std.mem;
block()
const Range = struct {
start: usize,
end: usize,
fn init(start: usize, end: usize) Range {
return Range{
.start = start,
.end = end,
};
}
fn length(self: Range) usize {
return self.end - self.start;
}
};
const Iterator = struct {
size: usize,
power_of_two: usize,
numerator: usize,
decimal: usize,
denominator: usize,
decimal_step: usize,
numerator_step: usize,
fn init(size2: usize, min_level: usize) Iterator {
const power_of_two = math.floorPowerOfTwo(usize, size2);
const denominator = power_of_two / min_level;
return Iterator{
.numerator = 0,
.decimal = 0,
.size = size2,
.power_of_two = power_of_two,
.denominator = denominator,
.decimal_step = size2 / denominator,
.numerator_step = size2 % denominator,
};
}
fn begin(self: *Iterator) void {
self.numerator = 0;
self.decimal = 0;
}
fn nextRange(self: *Iterator) Range {
const start = self.decimal;
self.decimal += self.decimal_step;
self.numerator += self.numerator_step;
if (self.numerator >= self.denominator) {
self.numerator -= self.denominator;
self.decimal += 1;
}
return Range{
.start = start,
.end = self.decimal,
};
}
fn finished(self: *Iterator) bool {
return self.decimal >= self.size;
}
fn nextLevel(self: *Iterator) bool {
self.decimal_step += self.decimal_step;
self.numerator_step += self.numerator_step;
if (self.numerator_step >= self.denominator) {
self.numerator_step -= self.denominator;
self.decimal_step += 1;
}
return (self.decimal_step < self.size);
}
fn length(self: *Iterator) usize {
return self.decimal_step;
}
};
const Pull = struct {
from: usize,
to: usize,
count: usize,
range: Range,
};
/// Stable in-place sort. O(n) best case, O(n*log(n)) worst case and average case.
/// O(1) memory (no allocator required).
/// Sorts in ascending order with respect to the given `lessThan` function.
///
/// NOTE: The algorithm only works when the comparison is less-than or greater-than.
/// (See https://github.com/ziglang/zig/issues/8289)
pub fn block(
comptime T: type,
items: []T,
context: anytype,
comptime lessThanFn: fn (@TypeOf(context), lhs: T, rhs: T) bool,
) void {
const lessThan = if (builtin.mode == .Debug) struct {
fn lessThan(ctx: @TypeOf(context), lhs: T, rhs: T) bool {
const lt = lessThanFn(ctx, lhs, rhs);
const gt = lessThanFn(ctx, rhs, lhs);
std.debug.assert(!(lt and gt));
return lt;
}
}.lessThan else lessThanFn;
// Implementation ported from https://github.com/BonzaiThePenguin/WikiSort/blob/master/WikiSort.c
var cache: [512]T = undefined;
if (items.len < 4) {
if (items.len == 3) {
// hard coded insertion sort
if (lessThan(context, items[1], items[0])) mem.swap(T, &items[0], &items[1]);
if (lessThan(context, items[2], items[1])) {
mem.swap(T, &items[1], &items[2]);
if (lessThan(context, items[1], items[0])) mem.swap(T, &items[0], &items[1]);
}
} else if (items.len == 2) {
if (lessThan(context, items[1], items[0])) mem.swap(T, &items[0], &items[1]);
}
return;
}
// sort groups of 4-8 items at a time using an unstable sorting network,
// but keep track of the original item orders to force it to be stable
// http://pages.ripco.net/~jgamble/nw.html
var iterator = Iterator.init(items.len, 4);
while (!iterator.finished()) {
var order = [_]u8{ 0, 1, 2, 3, 4, 5, 6, 7 };
const range = iterator.nextRange();
const sliced_items = items[range.start..];
switch (range.length()) {
8 => {
swap(T, sliced_items, &order, 0, 1, context, lessThan);
swap(T, sliced_items, &order, 2, 3, context, lessThan);
swap(T, sliced_items, &order, 4, 5, context, lessThan);
swap(T, sliced_items, &order, 6, 7, context, lessThan);
swap(T, sliced_items, &order, 0, 2, context, lessThan);
swap(T, sliced_items, &order, 1, 3, context, lessThan);
swap(T, sliced_items, &order, 4, 6, context, lessThan);
swap(T, sliced_items, &order, 5, 7, context, lessThan);
swap(T, sliced_items, &order, 1, 2, context, lessThan);
swap(T, sliced_items, &order, 5, 6, context, lessThan);
swap(T, sliced_items, &order, 0, 4, context, lessThan);
swap(T, sliced_items, &order, 3, 7, context, lessThan);
swap(T, sliced_items, &order, 1, 5, context, lessThan);
swap(T, sliced_items, &order, 2, 6, context, lessThan);
swap(T, sliced_items, &order, 1, 4, context, lessThan);
swap(T, sliced_items, &order, 3, 6, context, lessThan);
swap(T, sliced_items, &order, 2, 4, context, lessThan);
swap(T, sliced_items, &order, 3, 5, context, lessThan);
swap(T, sliced_items, &order, 3, 4, context, lessThan);
},
7 => {
swap(T, sliced_items, &order, 1, 2, context, lessThan);
swap(T, sliced_items, &order, 3, 4, context, lessThan);
swap(T, sliced_items, &order, 5, 6, context, lessThan);
swap(T, sliced_items, &order, 0, 2, context, lessThan);
swap(T, sliced_items, &order, 3, 5, context, lessThan);
swap(T, sliced_items, &order, 4, 6, context, lessThan);
swap(T, sliced_items, &order, 0, 1, context, lessThan);
swap(T, sliced_items, &order, 4, 5, context, lessThan);
swap(T, sliced_items, &order, 2, 6, context, lessThan);
swap(T, sliced_items, &order, 0, 4, context, lessThan);
swap(T, sliced_items, &order, 1, 5, context, lessThan);
swap(T, sliced_items, &order, 0, 3, context, lessThan);
swap(T, sliced_items, &order, 2, 5, context, lessThan);
swap(T, sliced_items, &order, 1, 3, context, lessThan);
swap(T, sliced_items, &order, 2, 4, context, lessThan);
swap(T, sliced_items, &order, 2, 3, context, lessThan);
},
6 => {
swap(T, sliced_items, &order, 1, 2, context, lessThan);
swap(T, sliced_items, &order, 4, 5, context, lessThan);
swap(T, sliced_items, &order, 0, 2, context, lessThan);
swap(T, sliced_items, &order, 3, 5, context, lessThan);
swap(T, sliced_items, &order, 0, 1, context, lessThan);
swap(T, sliced_items, &order, 3, 4, context, lessThan);
swap(T, sliced_items, &order, 2, 5, context, lessThan);
swap(T, sliced_items, &order, 0, 3, context, lessThan);
swap(T, sliced_items, &order, 1, 4, context, lessThan);
swap(T, sliced_items, &order, 2, 4, context, lessThan);
swap(T, sliced_items, &order, 1, 3, context, lessThan);
swap(T, sliced_items, &order, 2, 3, context, lessThan);
},
5 => {
swap(T, sliced_items, &order, 0, 1, context, lessThan);
swap(T, sliced_items, &order, 3, 4, context, lessThan);
swap(T, sliced_items, &order, 2, 4, context, lessThan);
swap(T, sliced_items, &order, 2, 3, context, lessThan);
swap(T, sliced_items, &order, 1, 4, context, lessThan);
swap(T, sliced_items, &order, 0, 3, context, lessThan);
swap(T, sliced_items, &order, 0, 2, context, lessThan);
swap(T, sliced_items, &order, 1, 3, context, lessThan);
swap(T, sliced_items, &order, 1, 2, context, lessThan);
},
4 => {
swap(T, sliced_items, &order, 0, 1, context, lessThan);
swap(T, sliced_items, &order, 2, 3, context, lessThan);
swap(T, sliced_items, &order, 0, 2, context, lessThan);
swap(T, sliced_items, &order, 1, 3, context, lessThan);
swap(T, sliced_items, &order, 1, 2, context, lessThan);
},
else => {},
}
}
if (items.len < 8) return;
// then merge sort the higher levels, which can be 8-15, 16-31, 32-63, 64-127, etc.
while (true) {
// if every A and B block will fit into the cache, use a special branch
// specifically for merging with the cache
// (we use < rather than <= since the block size might be one more than
// iterator.length())
if (iterator.length() < cache.len) {
// if four subarrays fit into the cache, it's faster to merge both
// pairs of subarrays into the cache,
// then merge the two merged subarrays from the cache back into the original array
if ((iterator.length() + 1) * 4 <= cache.len and iterator.length() * 4 <= items.len) {
iterator.begin();
while (!iterator.finished()) {
// merge A1 and B1 into the cache
var A1 = iterator.nextRange();
var B1 = iterator.nextRange();
var A2 = iterator.nextRange();
var B2 = iterator.nextRange();
if (lessThan(context, items[B1.end - 1], items[A1.start])) {
// the two ranges are in reverse order, so copy them in reverse order into the cache
const a1_items = items[A1.start..A1.end];
@memcpy(cache[B1.length()..][0..a1_items.len], a1_items);
const b1_items = items[B1.start..B1.end];
@memcpy(cache[0..b1_items.len], b1_items);
} else if (lessThan(context, items[B1.start], items[A1.end - 1])) {
// these two ranges weren't already in order, so merge them into the cache
mergeInto(T, items, A1, B1, cache[0..], context, lessThan);
} else {
// if A1, B1, A2, and B2 are all in order, skip doing anything else
if (!lessThan(context, items[B2.start], items[A2.end - 1]) and !lessThan(context, items[A2.start], items[B1.end - 1])) continue;
// copy A1 and B1 into the cache in the same order
const a1_items = items[A1.start..A1.end];
@memcpy(cache[0..a1_items.len], a1_items);
const b1_items = items[B1.start..B1.end];
@memcpy(cache[A1.length()..][0..b1_items.len], b1_items);
}
A1 = Range.init(A1.start, B1.end);
// merge A2 and B2 into the cache
if (lessThan(context, items[B2.end - 1], items[A2.start])) {
// the two ranges are in reverse order, so copy them in reverse order into the cache
const a2_items = items[A2.start..A2.end];
@memcpy(cache[A1.length() + B2.length() ..][0..a2_items.len], a2_items);
const b2_items = items[B2.start..B2.end];
@memcpy(cache[A1.length()..][0..b2_items.len], b2_items);
} else if (lessThan(context, items[B2.start], items[A2.end - 1])) {
// these two ranges weren't already in order, so merge them into the cache
mergeInto(T, items, A2, B2, cache[A1.length()..], context, lessThan);
} else {
// copy A2 and B2 into the cache in the same order
const a2_items = items[A2.start..A2.end];
@memcpy(cache[A1.length()..][0..a2_items.len], a2_items);
const b2_items = items[B2.start..B2.end];
@memcpy(cache[A1.length() + A2.length() ..][0..b2_items.len], b2_items);
}
A2 = Range.init(A2.start, B2.end);
// merge A1 and A2 from the cache into the items
const A3 = Range.init(0, A1.length());
const B3 = Range.init(A1.length(), A1.length() + A2.length());
if (lessThan(context, cache[B3.end - 1], cache[A3.start])) {
// the two ranges are in reverse order, so copy them in reverse order into the items
const a3_items = cache[A3.start..A3.end];
@memcpy(items[A1.start + A2.length() ..][0..a3_items.len], a3_items);
const b3_items = cache[B3.start..B3.end];
@memcpy(items[A1.start..][0..b3_items.len], b3_items);
} else if (lessThan(context, cache[B3.start], cache[A3.end - 1])) {
// these two ranges weren't already in order, so merge them back into the items
mergeInto(T, cache[0..], A3, B3, items[A1.start..], context, lessThan);
} else {
// copy A3 and B3 into the items in the same order
const a3_items = cache[A3.start..A3.end];
@memcpy(items[A1.start..][0..a3_items.len], a3_items);
const b3_items = cache[B3.start..B3.end];
@memcpy(items[A1.start + A1.length() ..][0..b3_items.len], b3_items);
}
}
// we merged two levels at the same time, so we're done with this level already
// (iterator.nextLevel() is called again at the bottom of this outer merge loop)
_ = iterator.nextLevel();
} else {
iterator.begin();
while (!iterator.finished()) {
const A = iterator.nextRange();
const B = iterator.nextRange();
if (lessThan(context, items[B.end - 1], items[A.start])) {
// the two ranges are in reverse order, so a simple rotation should fix it
mem.rotate(T, items[A.start..B.end], A.length());
} else if (lessThan(context, items[B.start], items[A.end - 1])) {
// these two ranges weren't already in order, so we'll need to merge them!
const a_items = items[A.start..A.end];
@memcpy(cache[0..a_items.len], a_items);
mergeExternal(T, items, A, B, cache[0..], context, lessThan);
}
}
}
} else {
// this is where the in-place merge logic starts!
// 1. pull out two internal buffers each containing √A unique values
// 1a. adjust block_size and buffer_size if we couldn't find enough unique values
// 2. loop over the A and B subarrays within this level of the merge sort
// 3. break A and B into blocks of size 'block_size'
// 4. "tag" each of the A blocks with values from the first internal buffer
// 5. roll the A blocks through the B blocks and drop/rotate them where they belong
// 6. merge each A block with any B values that follow, using the cache or the second internal buffer
// 7. sort the second internal buffer if it exists
// 8. redistribute the two internal buffers back into the items
var block_size: usize = math.sqrt(iterator.length());
var buffer_size = iterator.length() / block_size + 1;
// as an optimization, we really only need to pull out the internal buffers once for each level of merges
// after that we can reuse the same buffers over and over, then redistribute it when we're finished with this level
var A: Range = undefined;
var B: Range = undefined;
var index: usize = 0;
var last: usize = 0;
var count: usize = 0;
var find: usize = 0;
var start: usize = 0;
var pull_index: usize = 0;
var pull = [_]Pull{
Pull{
.from = 0,
.to = 0,
.count = 0,
.range = Range.init(0, 0),
},
Pull{
.from = 0,
.to = 0,
.count = 0,
.range = Range.init(0, 0),
},
};
var buffer1 = Range.init(0, 0);
var buffer2 = Range.init(0, 0);
// find two internal buffers of size 'buffer_size' each
find = buffer_size + buffer_size;
var find_separately = false;
if (block_size <= cache.len) {
// if every A block fits into the cache then we won't need the second internal buffer,
// so we really only need to find 'buffer_size' unique values
find = buffer_size;
} else if (find > iterator.length()) {
// we can't fit both buffers into the same A or B subarray, so find two buffers separately
find = buffer_size;
find_separately = true;
}
// we need to find either a single contiguous space containing 2√A unique values (which will be split up into two buffers of size √A each),
// or we need to find one buffer of < 2√A unique values, and a second buffer of √A unique values,
// OR if we couldn't find that many unique values, we need the largest possible buffer we can get
// in the case where it couldn't find a single buffer of at least √A unique values,
// all of the Merge steps must be replaced by a different merge algorithm (MergeInPlace)
iterator.begin();
while (!iterator.finished()) {
A = iterator.nextRange();
B = iterator.nextRange();
// just store information about where the values will be pulled from and to,
// as well as how many values there are, to create the two internal buffers
// check A for the number of unique values we need to fill an internal buffer
// these values will be pulled out to the start of A
last = A.start;
count = 1;
while (count < find) : ({
last = index;
count += 1;
}) {
index = findLastForward(T, items, items[last], Range.init(last + 1, A.end), find - count, context, lessThan);
if (index == A.end) break;
}
index = last;
if (count >= buffer_size) {
// keep track of the range within the items where we'll need to "pull out" these values to create the internal buffer
pull[pull_index] = Pull{
.range = Range.init(A.start, B.end),
.count = count,
.from = index,
.to = A.start,
};
pull_index = 1;
if (count == buffer_size + buffer_size) {
// we were able to find a single contiguous section containing 2√A unique values,
// so this section can be used to contain both of the internal buffers we'll need
buffer1 = Range.init(A.start, A.start + buffer_size);
buffer2 = Range.init(A.start + buffer_size, A.start + count);
break;
} else if (find == buffer_size + buffer_size) {
// we found a buffer that contains at least √A unique values, but did not contain the full 2√A unique values,
// so we still need to find a second separate buffer of at least √A unique values
buffer1 = Range.init(A.start, A.start + count);
find = buffer_size;
} else if (block_size <= cache.len) {
// we found the first and only internal buffer that we need, so we're done!
buffer1 = Range.init(A.start, A.start + count);
break;
} else if (find_separately) {
// found one buffer, but now find the other one
buffer1 = Range.init(A.start, A.start + count);
find_separately = false;
} else {
// we found a second buffer in an 'A' subarray containing √A unique values, so we're done!
buffer2 = Range.init(A.start, A.start + count);
break;
}
} else if (pull_index == 0 and count > buffer1.length()) {
// keep track of the largest buffer we were able to find
buffer1 = Range.init(A.start, A.start + count);
pull[pull_index] = Pull{
.range = Range.init(A.start, B.end),
.count = count,
.from = index,
.to = A.start,
};
}
// check B for the number of unique values we need to fill an internal buffer
// these values will be pulled out to the end of B
last = B.end - 1;
count = 1;
while (count < find) : ({
last = index - 1;
count += 1;
}) {
index = findFirstBackward(T, items, items[last], Range.init(B.start, last), find - count, context, lessThan);
if (index == B.start) break;
}
index = last;
if (count >= buffer_size) {
// keep track of the range within the items where we'll need to "pull out" these values to create the internal buffe
pull[pull_index] = Pull{
.range = Range.init(A.start, B.end),
.count = count,
.from = index,
.to = B.end,
};
pull_index = 1;
if (count == buffer_size + buffer_size) {
// we were able to find a single contiguous section containing 2√A unique values,
// so this section can be used to contain both of the internal buffers we'll need
buffer1 = Range.init(B.end - count, B.end - buffer_size);
buffer2 = Range.init(B.end - buffer_size, B.end);
break;
} else if (find == buffer_size + buffer_size) {
// we found a buffer that contains at least √A unique values, but did not contain the full 2√A unique values,
// so we still need to find a second separate buffer of at least √A unique values
buffer1 = Range.init(B.end - count, B.end);
find = buffer_size;
} else if (block_size <= cache.len) {
// we found the first and only internal buffer that we need, so we're done!
buffer1 = Range.init(B.end - count, B.end);
break;
} else if (find_separately) {
// found one buffer, but now find the other one
buffer1 = Range.init(B.end - count, B.end);
find_separately = false;
} else {
// buffer2 will be pulled out from a 'B' subarray, so if the first buffer was pulled out from the corresponding 'A' subarray,
// we need to adjust the end point for that A subarray so it knows to stop redistributing its values before reaching buffer2
if (pull[0].range.start == A.start) pull[0].range.end -= pull[1].count;
// we found a second buffer in an 'B' subarray containing √A unique values, so we're done!
buffer2 = Range.init(B.end - count, B.end);
break;
}
} else if (pull_index == 0 and count > buffer1.length()) {
// keep track of the largest buffer we were able to find
buffer1 = Range.init(B.end - count, B.end);
pull[pull_index] = Pull{
.range = Range.init(A.start, B.end),
.count = count,
.from = index,
.to = B.end,
};
}
}
// pull out the two ranges so we can use them as internal buffers
pull_index = 0;
while (pull_index < 2) : (pull_index += 1) {
const length = pull[pull_index].count;
if (pull[pull_index].to < pull[pull_index].from) {
// we're pulling the values out to the left, which means the start of an A subarray
index = pull[pull_index].from;
count = 1;
while (count < length) : (count += 1) {
index = findFirstBackward(T, items, items[index - 1], Range.init(pull[pull_index].to, pull[pull_index].from - (count - 1)), length - count, context, lessThan);
const range = Range.init(index + 1, pull[pull_index].from + 1);
mem.rotate(T, items[range.start..range.end], range.length() - count);
pull[pull_index].from = index + count;
}
} else if (pull[pull_index].to > pull[pull_index].from) {
// we're pulling values out to the right, which means the end of a B subarray
index = pull[pull_index].from + 1;
count = 1;
while (count < length) : (count += 1) {
index = findLastForward(T, items, items[index], Range.init(index, pull[pull_index].to), length - count, context, lessThan);
const range = Range.init(pull[pull_index].from, index - 1);
mem.rotate(T, items[range.start..range.end], count);
pull[pull_index].from = index - 1 - count;
}
}
}
// adjust block_size and buffer_size based on the values we were able to pull out
buffer_size = buffer1.length();
block_size = iterator.length() / buffer_size + 1;
// the first buffer NEEDS to be large enough to tag each of the evenly sized A blocks,
// so this was originally here to test the math for adjusting block_size above
// assert((iterator.length() + 1)/block_size <= buffer_size);
// now that the two internal buffers have been created, it's time to merge each A+B combination at this level of the merge sort!
iterator.begin();
while (!iterator.finished()) {
A = iterator.nextRange();
B = iterator.nextRange();
// remove any parts of A or B that are being used by the internal buffers
start = A.start;
if (start == pull[0].range.start) {
if (pull[0].from > pull[0].to) {
A.start += pull[0].count;
// if the internal buffer takes up the entire A or B subarray, then there's nothing to merge
// this only happens for very small subarrays, like √4 = 2, 2 * (2 internal buffers) = 4,
// which also only happens when cache.len is small or 0 since it'd otherwise use MergeExternal
if (A.length() == 0) continue;
} else if (pull[0].from < pull[0].to) {
B.end -= pull[0].count;
if (B.length() == 0) continue;
}
}
if (start == pull[1].range.start) {
if (pull[1].from > pull[1].to) {
A.start += pull[1].count;
if (A.length() == 0) continue;
} else if (pull[1].from < pull[1].to) {
B.end -= pull[1].count;
if (B.length() == 0) continue;
}
}
if (lessThan(context, items[B.end - 1], items[A.start])) {
// the two ranges are in reverse order, so a simple rotation should fix it
mem.rotate(T, items[A.start..B.end], A.length());
} else if (lessThan(context, items[A.end], items[A.end - 1])) {
// these two ranges weren't already in order, so we'll need to merge them!
var findA: usize = undefined;
// break the remainder of A into blocks. firstA is the uneven-sized first A block
var blockA = Range.init(A.start, A.end);
var firstA = Range.init(A.start, A.start + blockA.length() % block_size);
// swap the first value of each A block with the value in buffer1
var indexA = buffer1.start;
index = firstA.end;
while (index < blockA.end) : ({
indexA += 1;
index += block_size;
}) {
mem.swap(T, &items[indexA], &items[index]);
}
// start rolling the A blocks through the B blocks!
// whenever we leave an A block behind, we'll need to merge the previous A block with any B blocks that follow it, so track that information as well
var lastA = firstA;
var lastB = Range.init(0, 0);
var blockB = Range.init(B.start, B.start + @min(block_size, B.length()));
blockA.start += firstA.length();
indexA = buffer1.start;
// if the first unevenly sized A block fits into the cache, copy it there for when we go to Merge it
// otherwise, if the second buffer is available, block swap the contents into that
if (lastA.length() <= cache.len) {
const last_a_items = items[lastA.start..lastA.end];
@memcpy(cache[0..last_a_items.len], last_a_items);
} else if (buffer2.length() > 0) {
blockSwap(T, items, lastA.start, buffer2.start, lastA.length());
}
if (blockA.length() > 0) {
while (true) {
// if there's a previous B block and the first value of the minimum A block is <= the last value of the previous B block,
// then drop that minimum A block behind. or if there are no B blocks left then keep dropping the remaining A blocks.
if ((lastB.length() > 0 and !lessThan(context, items[lastB.end - 1], items[indexA])) or blockB.length() == 0) {
// figure out where to split the previous B block, and rotate it at the split
const B_split = binaryFirst(T, items, items[indexA], lastB, context, lessThan);
const B_remaining = lastB.end - B_split;
// swap the minimum A block to the beginning of the rolling A blocks
var minA = blockA.start;
findA = minA + block_size;
while (findA < blockA.end) : (findA += block_size) {
if (lessThan(context, items[findA], items[minA])) {
minA = findA;
}
}
blockSwap(T, items, blockA.start, minA, block_size);
// swap the first item of the previous A block back with its original value, which is stored in buffer1
mem.swap(T, &items[blockA.start], &items[indexA]);
indexA += 1;
// locally merge the previous A block with the B values that follow it
// if lastA fits into the external cache we'll use that (with MergeExternal),
// or if the second internal buffer exists we'll use that (with MergeInternal),
// or failing that we'll use a strictly in-place merge algorithm (MergeInPlace)
if (lastA.length() <= cache.len) {
mergeExternal(T, items, lastA, Range.init(lastA.end, B_split), cache[0..], context, lessThan);
} else if (buffer2.length() > 0) {
mergeInternal(T, items, lastA, Range.init(lastA.end, B_split), buffer2, context, lessThan);
} else {
mergeInPlace(T, items, lastA, Range.init(lastA.end, B_split), context, lessThan);
}
if (buffer2.length() > 0 or block_size <= cache.len) {
// copy the previous A block into the cache or buffer2, since that's where we need it to be when we go to merge it anyway
if (block_size <= cache.len) {
@memcpy(cache[0..block_size], items[blockA.start..][0..block_size]);
} else {
blockSwap(T, items, blockA.start, buffer2.start, block_size);
}
// this is equivalent to rotating, but faster
// the area normally taken up by the A block is either the contents of buffer2, or data we don't need anymore since we memcopied it
// either way, we don't need to retain the order of those items, so instead of rotating we can just block swap B to where it belongs
blockSwap(T, items, B_split, blockA.start + block_size - B_remaining, B_remaining);
} else {
// we are unable to use the 'buffer2' trick to speed up the rotation operation since buffer2 doesn't exist, so perform a normal rotation
mem.rotate(T, items[B_split .. blockA.start + block_size], blockA.start - B_split);
}
// update the range for the remaining A blocks, and the range remaining from the B block after it was split
lastA = Range.init(blockA.start - B_remaining, blockA.start - B_remaining + block_size);
lastB = Range.init(lastA.end, lastA.end + B_remaining);
// if there are no more A blocks remaining, this step is finished!
blockA.start += block_size;
if (blockA.length() == 0) break;
} else if (blockB.length() < block_size) {
// move the last B block, which is unevenly sized, to before the remaining A blocks, by using a rotation
// the cache is disabled here since it might contain the contents of the previous A block
mem.rotate(T, items[blockA.start..blockB.end], blockB.start - blockA.start);
lastB = Range.init(blockA.start, blockA.start + blockB.length());
blockA.start += blockB.length();
blockA.end += blockB.length();
blockB.end = blockB.start;
} else {
// roll the leftmost A block to the end by swapping it with the next B block
blockSwap(T, items, blockA.start, blockB.start, block_size);
lastB = Range.init(blockA.start, blockA.start + block_size);
blockA.start += block_size;
blockA.end += block_size;
blockB.start += block_size;
if (blockB.end > B.end - block_size) {
blockB.end = B.end;
} else {
blockB.end += block_size;
}
}
}
}
// merge the last A block with the remaining B values
if (lastA.length() <= cache.len) {
mergeExternal(T, items, lastA, Range.init(lastA.end, B.end), cache[0..], context, lessThan);
} else if (buffer2.length() > 0) {
mergeInternal(T, items, lastA, Range.init(lastA.end, B.end), buffer2, context, lessThan);
} else {
mergeInPlace(T, items, lastA, Range.init(lastA.end, B.end), context, lessThan);
}
}
}
// when we're finished with this merge step we should have the one
// or two internal buffers left over, where the second buffer is all jumbled up
// insertion sort the second buffer, then redistribute the buffers
// back into the items using the opposite process used for creating the buffer
// while an unstable sort like quicksort could be applied here, in benchmarks
// it was consistently slightly slower than a simple insertion sort,
// even for tens of millions of items. this may be because insertion
// sort is quite fast when the data is already somewhat sorted, like it is here
sort.insertion(T, items[buffer2.start..buffer2.end], context, lessThan);
pull_index = 0;
while (pull_index < 2) : (pull_index += 1) {
var unique = pull[pull_index].count * 2;
if (pull[pull_index].from > pull[pull_index].to) {
// the values were pulled out to the left, so redistribute them back to the right
var buffer = Range.init(pull[pull_index].range.start, pull[pull_index].range.start + pull[pull_index].count);
while (buffer.length() > 0) {
index = findFirstForward(T, items, items[buffer.start], Range.init(buffer.end, pull[pull_index].range.end), unique, context, lessThan);
const amount = index - buffer.end;
mem.rotate(T, items[buffer.start..index], buffer.length());
buffer.start += (amount + 1);
buffer.end += amount;
unique -= 2;
}
} else if (pull[pull_index].from < pull[pull_index].to) {
// the values were pulled out to the right, so redistribute them back to the left
var buffer = Range.init(pull[pull_index].range.end - pull[pull_index].count, pull[pull_index].range.end);
while (buffer.length() > 0) {
index = findLastBackward(T, items, items[buffer.end - 1], Range.init(pull[pull_index].range.start, buffer.start), unique, context, lessThan);
const amount = buffer.start - index;
mem.rotate(T, items[index..buffer.end], amount);
buffer.start -= amount;
buffer.end -= (amount + 1);
unique -= 2;
}
}
}
}
// double the size of each A and B subarray that will be merged in the next level
if (!iterator.nextLevel()) break;
}
}
// merge operation without a buffer
fn mergeInPlace(
comptime T: type,
items: []T,
A_arg: Range,
B_arg: Range,
context: anytype,
comptime lessThan: fn (@TypeOf(context), lhs: T, rhs: T) bool,
) void {
if (A_arg.length() == 0 or B_arg.length() == 0) return;
// this just repeatedly binary searches into B and rotates A into position.
// the paper suggests using the 'rotation-based Hwang and Lin algorithm' here,
// but I decided to stick with this because it had better situational performance
//
// (Hwang and Lin is designed for merging subarrays of very different sizes,
// but WikiSort almost always uses subarrays that are roughly the same size)
//
// normally this is incredibly suboptimal, but this function is only called
// when none of the A or B blocks in any subarray contained 2√A unique values,
// which places a hard limit on the number of times this will ACTUALLY need
// to binary search and rotate.
//
// according to my analysis the worst case is √A rotations performed on √A items
// once the constant factors are removed, which ends up being O(n)
//
// again, this is NOT a general-purpose solution – it only works well in this case!
// kind of like how the O(n^2) insertion sort is used in some places
var A = A_arg;
var B = B_arg;
while (true) {
// find the first place in B where the first item in A needs to be inserted
const mid = binaryFirst(T, items, items[A.start], B, context, lessThan);
// rotate A into place
const amount = mid - A.end;
mem.rotate(T, items[A.start..mid], A.length());
if (B.end == mid) break;
// calculate the new A and B ranges
B.start = mid;
A = Range.init(A.start + amount, B.start);
A.start = binaryLast(T, items, items[A.start], A, context, lessThan);
if (A.length() == 0) break;
}
}
// merge operation using an internal buffer
fn mergeInternal(
comptime T: type,
items: []T,
A: Range,
B: Range,
buffer: Range,
context: anytype,
comptime lessThan: fn (@TypeOf(context), lhs: T, rhs: T) bool,
) void {
// whenever we find a value to add to the final array, swap it with the value that's already in that spot
// when this algorithm is finished, 'buffer' will contain its original contents, but in a different order
var A_count: usize = 0;
var B_count: usize = 0;
var insert: usize = 0;
if (B.length() > 0 and A.length() > 0) {
while (true) {
if (!lessThan(context, items[B.start + B_count], items[buffer.start + A_count])) {
mem.swap(T, &items[A.start + insert], &items[buffer.start + A_count]);
A_count += 1;
insert += 1;
if (A_count >= A.length()) break;
} else {
mem.swap(T, &items[A.start + insert], &items[B.start + B_count]);
B_count += 1;
insert += 1;
if (B_count >= B.length()) break;
}
}
}
// swap the remainder of A into the final array
blockSwap(T, items, buffer.start + A_count, A.start + insert, A.length() - A_count);
}
fn blockSwap(comptime T: type, items: []T, start1: usize, start2: usize, block_size: usize) void {
var index: usize = 0;
while (index < block_size) : (index += 1) {
mem.swap(T, &items[start1 + index], &items[start2 + index]);
}
}
// combine a linear search with a binary search to reduce the number of comparisons in situations
// where have some idea as to how many unique values there are and where the next value might be
fn findFirstForward(
comptime T: type,
items: []T,
value: T,
range: Range,
unique: usize,
context: anytype,
comptime lessThan: fn (@TypeOf(context), lhs: T, rhs: T) bool,
) usize {
if (range.length() == 0) return range.start;
const skip = @max(range.length() / unique, @as(usize, 1));
var index = range.start + skip;
while (lessThan(context, items[index - 1], value)) : (index += skip) {
if (index >= range.end - skip) {
return binaryFirst(T, items, value, Range.init(index, range.end), context, lessThan);
}
}
return binaryFirst(T, items, value, Range.init(index - skip, index), context, lessThan);
}
fn findFirstBackward(
comptime T: type,
items: []T,
value: T,
range: Range,
unique: usize,
context: anytype,
comptime lessThan: fn (@TypeOf(context), lhs: T, rhs: T) bool,
) usize {
if (range.length() == 0) return range.start;
const skip = @max(range.length() / unique, @as(usize, 1));
var index = range.end - skip;
while (index > range.start and !lessThan(context, items[index - 1], value)) : (index -= skip) {
if (index < range.start + skip) {
return binaryFirst(T, items, value, Range.init(range.start, index), context, lessThan);
}
}
return binaryFirst(T, items, value, Range.init(index, index + skip), context, lessThan);
}
fn findLastForward(
comptime T: type,
items: []T,
value: T,
range: Range,
unique: usize,
context: anytype,
comptime lessThan: fn (@TypeOf(context), lhs: T, rhs: T) bool,
) usize {
if (range.length() == 0) return range.start;
const skip = @max(range.length() / unique, @as(usize, 1));
var index = range.start + skip;
while (!lessThan(context, value, items[index - 1])) : (index += skip) {
if (index >= range.end - skip) {
return binaryLast(T, items, value, Range.init(index, range.end), context, lessThan);
}
}
return binaryLast(T, items, value, Range.init(index - skip, index), context, lessThan);
}
fn findLastBackward(
comptime T: type,
items: []T,
value: T,
range: Range,
unique: usize,
context: anytype,
comptime lessThan: fn (@TypeOf(context), lhs: T, rhs: T) bool,
) usize {
if (range.length() == 0) return range.start;
const skip = @max(range.length() / unique, @as(usize, 1));
var index = range.end - skip;
while (index > range.start and lessThan(context, value, items[index - 1])) : (index -= skip) {
if (index < range.start + skip) {
return binaryLast(T, items, value, Range.init(range.start, index), context, lessThan);
}
}
return binaryLast(T, items, value, Range.init(index, index + skip), context, lessThan);
}
fn binaryFirst(
comptime T: type,
items: []T,
value: T,
range: Range,
context: anytype,
comptime lessThan: fn (@TypeOf(context), lhs: T, rhs: T) bool,
) usize {
var curr = range.start;
var size = range.length();
if (range.start >= range.end) return range.end;
while (size > 0) {
const offset = size % 2;
size /= 2;
const mid_item = items[curr + size];
if (lessThan(context, mid_item, value)) {
curr += size + offset;
}
}
return curr;
}
fn binaryLast(
comptime T: type,
items: []T,
value: T,
range: Range,
context: anytype,
comptime lessThan: fn (@TypeOf(context), lhs: T, rhs: T) bool,
) usize {
var curr = range.start;
var size = range.length();
if (range.start >= range.end) return range.end;
while (size > 0) {
const offset = size % 2;
size /= 2;
const mid_item = items[curr + size];
if (!lessThan(context, value, mid_item)) {
curr += size + offset;
}
}
return curr;
}
fn mergeInto(
comptime T: type,
from: []T,
A: Range,
B: Range,
into: []T,
context: anytype,
comptime lessThan: fn (@TypeOf(context), lhs: T, rhs: T) bool,
) void {
var A_index: usize = A.start;
var B_index: usize = B.start;
const A_last = A.end;
const B_last = B.end;
var insert_index: usize = 0;
while (true) {
if (!lessThan(context, from[B_index], from[A_index])) {
into[insert_index] = from[A_index];
A_index += 1;
insert_index += 1;
if (A_index == A_last) {
// copy the remainder of B into the final array
const from_b = from[B_index..B_last];
@memcpy(into[insert_index..][0..from_b.len], from_b);
break;
}
} else {
into[insert_index] = from[B_index];
B_index += 1;
insert_index += 1;
if (B_index == B_last) {
// copy the remainder of A into the final array
const from_a = from[A_index..A_last];
@memcpy(into[insert_index..][0..from_a.len], from_a);
break;
}
}
}
}
fn mergeExternal(
comptime T: type,
items: []T,
A: Range,
B: Range,
cache: []T,
context: anytype,
comptime lessThan: fn (@TypeOf(context), lhs: T, rhs: T) bool,
) void {
// A fits into the cache, so use that instead of the internal buffer
var A_index: usize = 0;
var B_index: usize = B.start;
var insert_index: usize = A.start;
const A_last = A.length();
const B_last = B.end;
if (B.length() > 0 and A.length() > 0) {
while (true) {
if (!lessThan(context, items[B_index], cache[A_index])) {
items[insert_index] = cache[A_index];
A_index += 1;
insert_index += 1;
if (A_index == A_last) break;
} else {
items[insert_index] = items[B_index];
B_index += 1;
insert_index += 1;
if (B_index == B_last) break;
}
}
}
// copy the remainder of A into the final array
const cache_a = cache[A_index..A_last];
@memcpy(items[insert_index..][0..cache_a.len], cache_a);
}
fn swap(
comptime T: type,
items: []T,
order: *[8]u8,
x: usize,
y: usize,
context: anytype,
comptime lessThan: fn (@TypeOf(context), lhs: T, rhs: T) bool,
) void {
if (lessThan(context, items[y], items[x]) or ((order.*)[x] > (order.*)[y] and !lessThan(context, items[x], items[y]))) {
mem.swap(T, &items[x], &items[y]);
mem.swap(u8, &(order.*)[x], &(order.*)[y]);
}
}