From f9bbb8d83a389ce5533a1d0b0525cc5653cd8591 Mon Sep 17 00:00:00 2001 From: ip-ram Date: Tue, 20 Jan 2026 09:34:05 +0500 Subject: [PATCH] =?UTF-8?q?=D0=90=D0=BB=D0=B3=D0=BE=D1=81=D1=8B?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .gitignore | 1 + Архив/sort_Insertion.gif | Bin 0 -> 91950 bytes Архив/sort_bucket.png | Bin 0 -> 98631 bytes Архив/sort_qsort.gif | Bin 0 -> 93016 bytes Архив/sort_shell.gif | Bin 0 -> 276994 bytes Архив/биекция.png | Bin 0 -> 12325 bytes Архив/иньекция.png | Bin 0 -> 11879 bytes Архив/сюрьекция.png | Bin 0 -> 11838 bytes .../Сортировка вставками.md | 20 ++++++++++++++ .../Сортировка пузырьком.md | 2 +- математика/Биективнотсь.md | 3 +++ математика/Блочная сортировка.md | 25 ++++++++++++++++++ математика/Быстрая сортировка.md | 22 +++++++++++++++ математика/Иньективность.md | 3 +++ математика/Сортировка Шелла.md | 21 +++++++++++++++ математика/Сюрьективность.md | 3 +++ 16 files changed, 99 insertions(+), 1 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ztp{Es54i|KyC-sbs2cyV-m&bjiB5U2R=hYSufUhbB^XWgCReDhu!s!`tjrt(j0-2 z8~NEks)D`4;#{AU4k78GYK8sNCy}fd{aYMN<5%XF&}aML&TS@^6ZumltLmSSidTUjN|$gUxqAS)wwirWEMhw-@2ZIz z{Bz<7@l9nKc99)8Rm4kFbL_5+fOYUR`I{gj0+#pH*ej5+DkVTTd*B_0Z!fN`q&cR8 zksvyaKK6eNJUg|$*~CdaD6%}Ln!LwO&pLaPgWpm%9W_O?7dG&vKX`xFfXd2eCV!IVfMalsCD3|u7Z=V@D|(7%Bwa*Fdd_h(o|qtd4zQXXmixBF+82ff1%~>!8JkreSO_4VM<8(65ae4=G`C~ih7#RXj{%g QctfPFqW!c=$tvRi0N1BGu>b%7 literal 0 HcmV?d00001 diff --git a/Языки программирования/Сортировка вставками.md b/Языки программирования/Сортировка вставками.md index f54e9dd..6efcfb2 100644 --- a/Языки программирования/Сортировка вставками.md +++ b/Языки программирования/Сортировка вставками.md @@ -1,2 +1,22 @@ Это алгоритм, в котором элементы входной последовательности просматриваются по одному, и каждый новый поступивший элемент размещается в подходящее место среди ранее упорядоченных элементов. [Сложность алгоритма](Оценка%20сложности%20алгоритма): $O(n^2)$. #### Алгоритм +На вход алгоритма подаётся последовательность $n$ чисел: $a_1,a_2,...,a_n$. Сортируемые числа также называют ключами. Входная последовательность на практике представляется в виде массива с $n$ элементами. На выходе алгоритм должен вернуть перестановку исходной последовательности $a^`_1,a^`_2,...,a^`_n$, чтобы выполнялось следующее соотношение $a_1^`\leqslant a_2^`\leqslant ... \leqslant a_n^`$ . + +В начальный момент отсортированная последовательность пуста. На каждом шаге алгоритма выбирается один из элементов входных данных и помещается на нужную позицию в уже отсортированной последовательности до тех пор, пока набор входных данных не будет исчерпан. В любой момент времени в отсортированной последовательности элементы удовлетворяют требованиям к выходным данным алгоритма. + +Данный алгоритм можно ускорить при помощи использования бинарного поиска для нахождения места текущему элементу в отсортированной части. Проблема с долгим сдвигом массива вправо решается при помощи смены указателей. + +#### Пример +![[sort_Insertion.gif]] +#### В виде кода +```python +def insertion_sort(arr): + for i in range(1, len(arr)): + key = arr[i] + j = i - 1 + while j >= 0 and arr[j] > key: + arr[j + 1] = arr[j] + j -= 1 + arr[j + 1] = key + return arr +``` \ No newline at end of file diff --git a/Языки программирования/Сортировка пузырьком.md b/Языки программирования/Сортировка пузырьком.md index 82d875f..37103c8 100644 --- a/Языки программирования/Сортировка пузырьком.md +++ b/Языки программирования/Сортировка пузырьком.md @@ -1,4 +1,4 @@ -сортировка простыми обменами, метод сортировки обменами. По сравнению с другими алгоритмами считается простейшим для понимания и реализации. Эффективен для массивов небольшого размера. n - размер массива, количество элементов массива. [Сложность алгоритма](Оценка%20сложности%20алгоритма): $O(n^2)$. +По сравнению с другими алгоритмами считается простейшим для понимания и реализации. Эффективен для массивов небольшого размера. n - размер массива, количество элементов массива. [Сложность алгоритма](Оценка%20сложности%20алгоритма): $O(n^2)$. #### Алгоритм Выполняется некоторое количество проходов по массиву - начиная от начала массива, перебираются пары соседних элементов массива. Если 1-й элемент пары больше 2-го, элементы переставляются (выполняется обмен). diff --git a/математика/Биективнотсь.md b/математика/Биективнотсь.md new file mode 100644 index 0000000..b25e431 --- /dev/null +++ b/математика/Биективнотсь.md @@ -0,0 +1,3 @@ +Свойство [функции](функция), которое объединяет в себе [[Сюрьективность]] и [[Иньективность]] одновременно, для любого x существует только один y. +$\forall x\in X\exists !y\in Y:(x,y)\in F$ +![[биекция.png]] \ No newline at end of file diff --git a/математика/Блочная сортировка.md b/математика/Блочная сортировка.md new file mode 100644 index 0000000..1ea092e --- /dev/null +++ b/математика/Блочная сортировка.md @@ -0,0 +1,25 @@ +Алгоритм  в котором сортируемые элементы распределяются между конечным числом отдельных блоков так, чтобы все элементы в каждом следующем по порядку блоке были всегда больше (или меньше), чем в предыдущем. Каждый блок затем сортируется отдельно, либо рекурсивно тем же методом, либо другим. Затем элементы помещаются обратно в массив. Этот тип сортировки может обладать линейным временем исполнения. [Сложность алгоритма](Оценка%20сложности%20алгоритма): $O(n)$. +#### Алгоритм +Если входные элементы подчиняются равномерному закону распределения, то математическое ожидание времени работы алгоритма карманной сортировки является линейным. Это возможно благодаря определенным предположениям о входных данных. При карманной сортировке предполагается, что входные данные равномерно распределены на отрезке \[0, 1). + +Идея алгоритма заключается в том, чтобы разбить отрезок \[0, 1) на $n$ одинаковых отрезков (карманов), и разделить по этим карманам $n$ входных величин. Поскольку входные числа равномерно распределены, предполагается, что в каждый карман попадет небольшое количество чисел. Затем последовательно сортируются числа в карманах. Отсортированный массив получается путём последовательного перечисления элементов каждого кармана. +#### Пример +![[sort_bucket.png]] +#### В виде кода +```python +def block_sort(arr, block_size=5): + if not arr: + return arr + min_val = min(arr) + max_val = max(arr) + block_count = (max_val - min_val) // block_size + 1 + blocks = [[] for _ in range(block_count)] + for x in arr: + index = (x - min_val) // block_size + blocks[index].append(x) + result = [] + for block in blocks: + block.sort() + result.extend(block) + return result +``` \ No newline at end of file diff --git a/математика/Быстрая сортировка.md b/математика/Быстрая сортировка.md new file mode 100644 index 0000000..8257ce5 --- /dev/null +++ b/математика/Быстрая сортировка.md @@ -0,0 +1,22 @@ +Один из самых быстрых известных универсальных алгоритмов сортировки массивов: в среднем [Сложность алгоритма](Оценка%20сложности%20алгоритма) $O(n\log(n)⁡)$ обменов при упорядочении $n$ элементов, из-за наличия ряда недостатков на практике обычно используется с некоторыми доработками. +#### Алгоритм +Алгоритм состоит из трёх шагов: +1. Выбрать элемент из массива. Назовём его опорным. +2. Разбиение: перераспределение элементов в массиве таким образом, что элементы, меньшие опорного, помещаются перед ним, а большие или равные — после. +3. Рекурсивно применить первые два шага к двум подмассивам слева и справа от опорного элемента. Рекурсия не применяется к массиву, в котором только один элемент или отсутствуют элементы. + +В ранних реализациях, как правило, опорным выбирался первый элемент, что снижало производительность на отсортированных массивах. Для улучшения эффективности может выбираться средний, случайный элемент или (для больших массивов) медиана первого, среднего и последнего элементов. Медиана всей последовательности является оптимальным опорным элементом, но её вычисление слишком трудоёмко для использования в сортировке. +#### Пример +![[sort_qsort.gif]] +#### В виде кода +```python +def quicksort(arr): + if len(arr) <= 1: + return arr + pivot = arr[len(arr) // 2] + left = [x for x in arr if x < pivot] + middle = [x for x in arr if x == pivot] + right = [x for x in arr if x > pivot] + return quicksort(left) + middle + quicksort(right) + +``` \ No newline at end of file diff --git a/математика/Иньективность.md b/математика/Иньективность.md new file mode 100644 index 0000000..988af2e --- /dev/null +++ b/математика/Иньективность.md @@ -0,0 +1,3 @@ +Свойство [функции](функция), которое означает что одинаковый результат возможен только при одинаковых входных данных, для любого x существует только один y. +$\forall x_1,x_2\in X:(x_1\not =x_2=>f(x_1)\not =f(x_2))$ +![[иньекция.png]] \ No newline at end of file diff --git a/математика/Сортировка Шелла.md b/математика/Сортировка Шелла.md new file mode 100644 index 0000000..8274038 --- /dev/null +++ b/математика/Сортировка Шелла.md @@ -0,0 +1,21 @@ +Усовершенствованный вариант сортировки вставками. Идея метода Шелла состоит в сравнении элементов, стоящих не только рядом, но и на определённом расстоянии друг от друга. Иными словами — это сортировка вставками с предварительными «грубыми» проходами. [Сложность алгоритма](Оценка%20сложности%20алгоритма) $O(n^2)$ +#### Алгоритм +При сортировке Шелла сначала сравниваются и сортируются между собой значения, стоящие один от другого на некотором расстоянии $d$. После этого процедура повторяется для некоторых меньших значений $d$, а завершается сортировка Шелла упорядочиванием элементов при $d=1$. Эффективность сортировки Шелла в определённых случаях обеспечивается тем, что элементы «быстрее» встают на свои места. +#### Пример +![[sort_shell.gif]] +#### В виде кода +```python +def shell_sort(data: list[int]) -> list[int]: + last_index = len(data) + step = len(data)//2 + while step > 0: + for i in range(step, last_index, 1): + j = i + delta = j - step + while delta >= 0 and data[delta] > data[j]: + data[delta], data[j] = data[j], data[delta] + j = delta + delta = j - step + step //= 2 + return data +``` \ No newline at end of file diff --git a/математика/Сюрьективность.md b/математика/Сюрьективность.md new file mode 100644 index 0000000..9327f64 --- /dev/null +++ b/математика/Сюрьективность.md @@ -0,0 +1,3 @@ +Свойство [функции](функция), которое означает что при любом входе получишь значение, для любого x существует хотя бы один y. +$\forall x\in X\exists y\in Y:f(x)=y$ +![[сюрьекция.png]] \ No newline at end of file