AK: Introduce cutoff to insertion sort for Quicksort

Implement insertion sort in AK. The cutoff value 7 is a magic number
here, values [5, 15] should work well. Main idea of the cutoff is to
reduce recursion performed by quicksort to speed up sorting
of small partitions.

AK/InsertionSort.h

@@ -0,0 +1,45 @@
+/*
+ * Copyright (c) 2022, Marc Luqué <marc.luque@outlook.com>
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ */
+
+#pragma once
+
+#include <AK/Concepts.h>
+#include <AK/StdLibExtras.h>
+
+namespace AK {
+
+// Standard Insertion Sort, with `end` inclusive!
+template<typename Collection, typename Comparator, typename T = decltype(declval<Collection>()[declval<int>()])>
+void insertion_sort(Collection& col, ssize_t start, ssize_t end, Comparator comparator)
+requires(Indexable<Collection, T>)
+{
+    for (ssize_t i = start + 1; i <= end; ++i) {
+        for (ssize_t j = i; j > 0 && comparator(col[j], col[j - 1]); --j)
+            swap(col[j], col[j - 1]);
+    }
+}
+
+template<typename Collection, typename Comparator, typename T = decltype(declval<Collection>()[declval<int>()])>
+void insertion_sort(Collection& collection, Comparator comparator)
+requires(Indexable<Collection, T>)
+{
+    if (collection.size() == 0)
+        return;
+    insertion_sort(collection, 0, collection.size() - 1, move(comparator));
+}
+
+template<typename Collection, typename T = decltype(declval<Collection>()[declval<int>()])>
+void insertion_sort(Collection& collection)
+requires(Indexable<Collection, T>)
+{
+    if (collection.size() == 0)
+        return;
+    insertion_sort(collection, 0, collection.size() - 1, [](auto& a, auto& b) { return a < b; });
+}
+
+}
+
+using AK::insertion_sort;

AK/QuickSort.h

@@ -1,11 +1,13 @@
 /*
  * Copyright (c) 2018-2020, Andreas Kling <kling@serenityos.org>
+ * Copyright (c) 2022, Marc Luqué <marc.luque@outlook.com>
  *
  * SPDX-License-Identifier: BSD-2-Clause
  */
 
 #pragma once
 
+#include <AK/InsertionSort.h>
 #include <AK/StdLibExtras.h>
 
 namespace AK {
@@ -14,10 +16,24 @@ namespace AK {
  * pivot quick_sort below. The other quick_sort below should only be used when
  * you are stuck with simple iterators to a container and you don't have access
  * to the container itself.
+ *
+ * We use a cutoff to insertion sort for partitions of size 7 or smaller.
+ * The idea is to avoid recursion for small partitions.
+ * The value 7 here is a magic number. According to princeton's CS algorithm class
+ * a value between 5 and 15 should work well in most situations:
+ * https://algs4.cs.princeton.edu/23quicksort/
  */
+
+static constexpr int INSERTION_SORT_CUTOFF = 7;
+
 template<typename Collection, typename LessThan>
 void dual_pivot_quick_sort(Collection& col, int start, int end, LessThan less_than)
 {
+    if ((end + 1) - start <= INSERTION_SORT_CUTOFF) {
+        AK::insertion_sort(col, start, end, less_than);
+        return;
+    }
+
     while (start < end) {
         int size = end - start + 1;
         if (size > 3) {

Tests/AK/CMakeLists.txt

@@ -38,6 +38,7 @@ set(AK_TEST_SOURCES
     TestIPv4Address.cpp
     TestIPv6Address.cpp
     TestIndexSequence.cpp
+    TestInsertionSort.cpp
     TestIntegerMath.cpp
     TestIntrusiveList.cpp
     TestIntrusiveRedBlackTree.cpp

Tests/AK/TestInsertionSort.cpp

@@ -0,0 +1,57 @@
+/*
+ * Copyright (c) 2022, Marc Luqué <marc.luque@outlook.com>
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ */
+
+#include <LibTest/TestCase.h>
+
+#include <AK/InsertionSort.h>
+#include <AK/Random.h>
+#include <AK/Vector.h>
+
+static constexpr int const n = 10;
+static constexpr int const iterations = 10;
+static constexpr int const seed = 1337;
+
+TEST_CASE(sorts_ascending)
+{
+    srand(seed);
+
+    for (int i = 0; i < iterations; ++i) {
+        Vector<int, n> ints;
+        for (int j = 0; j < n; ++j)
+            ints.append(get_random<int>());
+        Vector<int, n> ints_copy = ints;
+
+        insertion_sort(ints);
+        insertion_sort(ints_copy);
+
+        for (int j = 0; j < n - 1; ++j) {
+            EXPECT(ints[j] <= ints[j + 1]);
+            EXPECT_EQ(ints[j], ints_copy[j]);
+            EXPECT_EQ(ints[j + 1], ints_copy[j + 1]);
+        }
+    }
+}
+
+TEST_CASE(sorts_decending)
+{
+    srand(seed);
+
+    for (int i = 0; i < iterations; ++i) {
+        Vector<int, n> ints;
+        for (int j = 0; j < n; ++j)
+            ints.append(get_random<int>());
+        Vector<int, n> ints_copy = ints;
+
+        insertion_sort(ints, [](auto& a, auto& b) { return a > b; });
+        insertion_sort(ints_copy, [](auto& a, auto& b) { return a > b; });
+
+        for (int j = 0; j < n - 1; ++j) {
+            EXPECT(ints[j] >= ints[j + 1]);
+            EXPECT_EQ(ints[j], ints_copy[j]);
+            EXPECT_EQ(ints[j + 1], ints_copy[j + 1]);
+        }
+    }
+}