diff --git a/ProninVV/aufgabe-1-data-structures/report/document.aux b/ProninVV/aufgabe-1-data-structures/report/document.aux index ce8bce9..e3bbcaa 100644 --- a/ProninVV/aufgabe-1-data-structures/report/document.aux +++ b/ProninVV/aufgabe-1-data-structures/report/document.aux @@ -10,4 +10,6 @@ \@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Двоичное дерево поиска}{3}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {3}Методика эксперимента}{3}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {4}Результаты и анализ}{3}{}\protected@file@percent } -\gdef \@abspage@last{8} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Зависимость времени выполнения операций в BST от объема данных}}{4}{}\protected@file@percent } +\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Сравнение времени выполнения операций (в секундах) в зависимости от объема данных $N$}}{9}{}\protected@file@percent } +\gdef \@abspage@last{10} diff --git a/ProninVV/aufgabe-1-data-structures/report/document.pdf b/ProninVV/aufgabe-1-data-structures/report/document.pdf index cbaed04..2563511 100644 Binary files a/ProninVV/aufgabe-1-data-structures/report/document.pdf and b/ProninVV/aufgabe-1-data-structures/report/document.pdf differ diff --git a/ProninVV/aufgabe-1-data-structures/report/document.synctex.gz b/ProninVV/aufgabe-1-data-structures/report/document.synctex.gz index c40314e..e06b0e8 100644 Binary files a/ProninVV/aufgabe-1-data-structures/report/document.synctex.gz and b/ProninVV/aufgabe-1-data-structures/report/document.synctex.gz differ diff --git a/ProninVV/aufgabe-1-data-structures/report/document.tex b/ProninVV/aufgabe-1-data-structures/report/document.tex index 256db59..4a6acf7 100644 --- a/ProninVV/aufgabe-1-data-structures/report/document.tex +++ b/ProninVV/aufgabe-1-data-structures/report/document.tex @@ -52,55 +52,85 @@ Замеры производились для наборов данных объемом $N=500, 1000, 2000, 5000, 10000$ элементов. Использовались два сценария: перемешанные (\textit{shuffled}) и отсортированные по алфавиту (\textit{sorted}) записи. Каждая операция выполнялась 5 раз с последующим вычислением среднего арифметического значения с помощью функции \texttt{time.perf\_counter()}. \section{Результаты и анализ} - Было проведено 5 опытов. + Было проведено серию опытов для $N$ от 500 до 10000. \subsection*{1. Бинарное дерево поиска (BST) и влияние порядка} \begin{figure}[H] - \includegraphics[scale=0.7]{plots/Tree.eps} + \centering + \includegraphics[scale=0.7]{plots/TTree1.eps} + \caption{Зависимость времени выполнения операций в BST от объема данных} \end{figure} \begin{figure}[H] - \includegraphics[scale=0.7]{plots/Tre1.eps} + \centering + \includegraphics[scale=0.7]{plots/TTree2.eps} \end{figure} \begin{itemize} - \item \textbf{Деградация на отсортированных данных:} При вставке отсортированных данных время увеличилось с \textbf{0.12с} (1000 эл.) до \textbf{13.24с} (10000 эл.). Рост более чем в 100 раз при увеличении данных в 10 раз указывает на сложность $O(n^2)$ для заполнения. Дерево выродилось в список. - \item \textbf{Эффективность на перемешанных данных:} На \texttt{shuffled} данных вставка 10000 элементов заняла всего \textbf{0.03с}. Это подтверждает логарифмическую сложность $O(\log n)$ для сбалансированного дерева. + \item \textbf{Деградация на отсортированных данных:} При вставке отсортированных данных время увеличилось с \textbf{0.124с} ($N=1000$) до \textbf{13.27с} ($N=10000$). Рост времени в 100 раз при увеличении объема данных в 10 раз четко указывает на квадратичную сложность $O(n^2)$ для процесса заполнения всей структуры. Дерево выродилось в линейный список, и поиск места вставки стал занимать $O(n)$ вместо ожидаемого $O(\log n)$. + \item \textbf{Эффективность на перемешанных данных:} На \texttt{shuffled} данных вставка 10000 элементов заняла всего \textbf{0.031с}. Это подтверждает логарифмическую сложность $O(\log n)$ для операций в дереве при случайном распределении ключей. \end{itemize} \subsection*{2. Хеш-таблица: Стабильность и скорость} - \begin{figure}[h] - \includegraphics[scale=0.7]{plots/hasht.eps} + \begin{figure}[H] + \centering + \includegraphics[scale=0.7]{plots/Thasht1.eps} + \end{figure} + + \begin{figure}[H] + \centering + \includegraphics[scale=0.7]{plots/Thasht2.eps} \end{figure} \begin{itemize} - \item \textbf{Чувствительность к порядку:} Хеш-таблица показала идентичные результаты как на \texttt{shuffled}, так и на \texttt{sorted} данных (около \textbf{0.16с} для 10000 вставок). Это объясняется тем, что хеш-функция распределяет ключи по бакетам независимо от их исходного порядка. - \item \textbf{Превосходство:} На больших объемах хеш-таблица оказалась самой быстрой структурой для поиска и удаления ($\approx 0.001$с), что подтверждает среднюю сложность $O(1)$. - \item \textbf{Замечание} так как таблица содержит списки со вставкой в конец, при вставке наблюдается отклонение от линейной зависимости в сторону квадратичной + \item \textbf{Чувствительность к порядку:} Хеш-таблица показала идентичные результаты как на \texttt{shuffled}, так и на \texttt{sorted} данных (около \textbf{0.165с} -- \textbf{0.167с} для 10000 вставок). Это объясняется тем, что хеш-функция распределяет ключи по бакетам независимо от их исходного порядка, предотвращая деградацию структуры. + \item \textbf{Превосходство:} На больших объемах хеш-таблица оказалась самой быстрой структурой для поиска и удаления ($\approx 0.001$с при $N=10000$), что подтверждает теоретическую среднюю сложность $O(1)$. + \item \textbf{Замечание:} Так как реализация использует списки для разрешения коллизий со вставкой в конец, при заполнении таблицы наблюдается рост времени вставки, стремящийся к квадратичному, однако абсолютные значения остаются на порядки ниже, чем у выродившегося BST. \end{itemize} + \subsection*{3. Связный список: Линейная зависимость} + \begin{figure}[H] + \centering + \includegraphics[scale=0.7]{plots/Tlinklist1.eps} + \end{figure} - \subsection*{3. Связный список: Стабильная медлительность} - - \begin{figure}[h] - \includegraphics[scale=0.7]{plots/llist.eps} + \begin{figure}[H] + \centering + \includegraphics[scale=0.7]{plots/Tlinklist2.eps} \end{figure} \begin{itemize} - \item \textbf{Поиск и удаление:} Связный список показал худшие результаты среди всех структур на случайных данных. Время поиска при 10000 элементах (\textbf{0.03с}) на два порядка медленнее, чем у хеш-таблицы. Это подтверждает линейную сложность $O(n)$. - \item \textbf{Вставка:} Вставка в конец без указателя на хвост дает $O(n^2)$ при заполнении (\textbf{3.04с} на 10000 эл.), что сопоставимо с выродившимся деревом. + \item \textbf{Поиск и удаление:} Связный список показал худшие результаты среди всех структур на случайных данных. Время поиска при 10000 элементах (\textbf{0.029с}) значительно медленнее, чем у BST на перемешанных данных (\textbf{0.0002с}). Это подтверждает линейную сложность $O(n)$. + \item \textbf{Вставка:} Вставка (вероятно, в конец или с сохранением порядка) дает $O(n^2)$ при заполнении (\textbf{2.83с} -- \textbf{3.00с} на 10000 эл.). Характер роста времени при переходе от $N=5000$ (\textbf{0.71с}) к $N=10000$ подтверждает квадратичную зависимость. \end{itemize} \subsection*{Вывод: выбор структуры данных} \begin{enumerate} - \item \textbf{Хеш-таблица} — лучший выбор для реальных задач «ключ-значение». Она обеспечивает стабильное $O(1)$ и не зависит от порядка входящих данных. - \item \textbf{BST} — эффективен только при условии случайного распределения данных или использовании самобалансирующихся деревьев. В противном случае велик риск деградации до скорости списка. - \item \textbf{Связный список} — в данной реализации неэффективен для поиска и массовой вставки. Его стоит использовать только для специфических задач (стеки, очереди), где работа идет преимущественно с головой списка за $O(1)$. + \item \textbf{Хеш-таблица} — наиболее универсальный выбор. Она обеспечивает стабильное $O(1)$ для поиска и не зависит от порядка входящих данных. + \item \textbf{BST} — крайне эффективен ($O(\log n)$) при случайном распределении данных, но без механизмов самобалансировки критически уязвим к отсортированным входным последовательностям, замедляясь до уровня списка. + \item \textbf{Связный список} — продемонстрировал самую низкую производительность на операциях поиска и массовой вставки. Его использование оправдано только в специфических сценариях (например, реализация стека), где работа ведется исключительно с головой списка за $O(1)$. \end{enumerate} + + \subsection*{Сводная таблица результатов} + \begin{table}[H] + \centering + \small + \begin{tabular}{|l|l|c|c|c|c|c|} + \hline + \textbf{Структура} & \textbf{Режим} & \textbf{Опер.} & \textbf{N=500} & \textbf{N=1000} & \textbf{N=5000} & \textbf{N=10000} \\ \hline + \multirow{3}{*}{LinkList} & Shuffled & Insert & 0.0066 & 0.0292 & 0.7089 & 2.8358 \\ + & Shuffled & Find & 0.0012 & 0.0026 & 0.0147 & 0.0289 \\ + & Sorted & Insert & 0.0065 & 0.0290 & 0.7637 & 3.0042 \\ \hline + \multirow{3}{*}{HashTable} & Shuffled & Insert & 0.0007 & 0.0022 & 0.0468 & 0.1670 \\ + & Shuffled & Find & 0.0001 & 0.0002 & 0.0008 & 0.0014 \\ + & Sorted & Insert & 0.0007 & 0.0022 & 0.0448 & 0.1646 \\ \hline + \multirow{3}{*}{BinTree} & Shuffled & Insert & 0.0009 & 0.0021 & 0.0145 & 0.0309 \\ + & Shuffled & Find & 0.0001 & 0.0001 & 0.0002 & 0.0002 \\ + & Sorted & Insert & \textbf{0.0298} & \textbf{0.1239} & \textbf{3.3052} & \textbf{13.2706} \\ \hline + \end{tabular} + \caption{Сравнение минимального времени выполнения операций (в секундах) в зависимости от объема данных $N$} + \end{table} - - %\section{Заключение} - % Ответ на вопрос о выборе структуры в реальной жизни - + \end{document} diff --git a/ProninVV/aufgabe-1-data-structures/report/plots/llist.eps b/ProninVV/aufgabe-1-data-structures/report/plots/TLinklist1.eps similarity index 95% rename from ProninVV/aufgabe-1-data-structures/report/plots/llist.eps rename to ProninVV/aufgabe-1-data-structures/report/plots/TLinklist1.eps index 065d114..eec390a 100644 --- a/ProninVV/aufgabe-1-data-structures/report/plots/llist.eps +++ b/ProninVV/aufgabe-1-data-structures/report/plots/TLinklist1.eps @@ -1,8 +1,8 @@ %!PS-Adobe-3.0 EPSF-3.0 %%LanguageLevel: 3 -%%Title: llist.eps +%%Title: TLinklist1.eps %%Creator: Matplotlib v3.10.0, https://matplotlib.org/ -%%CreationDate: Sat 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a/ProninVV/aufgabe-1-data-structures/report/preambule.tex b/ProninVV/aufgabe-1-data-structures/report/preambule.tex index d8dc43d..a017e55 100644 --- a/ProninVV/aufgabe-1-data-structures/report/preambule.tex +++ b/ProninVV/aufgabe-1-data-structures/report/preambule.tex @@ -6,7 +6,7 @@ \usepackage[utf8]{inputenc} \usepackage{comment} - +\usepackage{multirow} \usepackage{fontspec} \setmainfont{Times New Roman}