From b4cc3cb4dd87d970abe09e118f620c3ecd7621f0 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Mo=C3=B3r=20Krist=C3=B3f?= Date: Wed, 12 Jul 2023 12:58:31 +0200 Subject: [PATCH] =?UTF-8?q?14.=20=C3=A9s=2015.=20t=C3=A9tel=20kieg=C3=A9sz?= =?UTF-8?q?=C3=ADtve=20(#13)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../14. Alapvető algoritmusok.pdf | Bin 548488 -> 666146 bytes .../14. Alapvető algoritmusok.tex | 77 +++- 14. Alapvető algoritmusok/img/brute_force.png | Bin 0 -> 13208 bytes 14. Alapvető algoritmusok/img/bucket.png | Bin 0 -> 28155 bytes .../img/counting_sort.png | Bin 0 -> 41521 bytes 14. Alapvető algoritmusok/img/naiv_table.png | Bin 0 -> 4877 bytes .../img/radix_example.png | Bin 0 -> 54382 bytes .../15. Adatszerkezetek és adattípusok.pdf | Bin 371593 -> 700232 bytes .../15. Adatszerkezetek és adattípusok.tex | 394 ++++++++++++++---- .../img/abrazolas.png | Bin 0 -> 22777 bytes .../img/absz_node_parenttel.png | Bin 0 -> 32040 bytes .../img/absztrakt_node.png | Bin 0 -> 26793 bytes .../img/b_plus.png | Bin 0 -> 76097 bytes .../img/bin_sreach_tree_example.png | Bin 0 -> 28803 bytes .../img/del.png | Bin 0 -> 14551 bytes .../img/grafikus_szoveges.png | Bin 0 -> 26308 bytes .../img/heap_sort.png | Bin 0 -> 11933 bytes .../img/inorder.png | Bin 0 -> 9670 bytes .../img/inorder_bin_search_tree.png | Bin 0 -> 10899 bytes .../img/insert.png | Bin 0 -> 16194 bytes .../img/konkret_abrazolas.png | Bin 0 -> 8986 bytes .../img/levelOrder.png | Bin 0 -> 22635 bytes .../img/mmm_rotation.png | Bin 0 -> 21170 bytes .../img/mmn_roatation.png | Bin 0 -> 22772 bytes .../img/mmp_rotation.png | Bin 0 -> 27913 bytes .../img/postorder.png | Bin 0 -> 9549 bytes .../img/ppm_rotation.png | Bin 0 -> 27484 bytes .../img/ppn_rotation.png | Bin 0 -> 21623 bytes .../img/ppp_rotation.png | Bin 0 -> 20795 bytes .../img/pr_add.png | Bin 0 -> 28505 bytes .../img/pr_max.png | Bin 0 -> 29580 bytes .../img/pr_queue.png | Bin 0 -> 60282 bytes .../img/pr_sink.png | Bin 0 -> 40194 bytes .../img/pr_sort.png | Bin 0 -> 19869 bytes .../img/preorder.png | Bin 0 -> 9017 bytes .../img/search.png | Bin 0 -> 11731 bytes 36 files changed, 388 insertions(+), 83 deletions(-) create mode 100644 14. Alapvető algoritmusok/img/brute_force.png create mode 100644 14. Alapvető algoritmusok/img/bucket.png create mode 100644 14. Alapvető algoritmusok/img/counting_sort.png create mode 100644 14. Alapvető algoritmusok/img/naiv_table.png create mode 100644 14. Alapvető algoritmusok/img/radix_example.png create mode 100644 15. Adatszerkezetek és adattípusok/img/abrazolas.png create mode 100644 15. Adatszerkezetek és adattípusok/img/absz_node_parenttel.png create mode 100644 15. Adatszerkezetek és adattípusok/img/absztrakt_node.png create mode 100644 15. Adatszerkezetek és adattípusok/img/b_plus.png create mode 100644 15. Adatszerkezetek és adattípusok/img/bin_sreach_tree_example.png create mode 100644 15. Adatszerkezetek és adattípusok/img/del.png create mode 100644 15. Adatszerkezetek és adattípusok/img/grafikus_szoveges.png create mode 100644 15. Adatszerkezetek és adattípusok/img/heap_sort.png create mode 100644 15. Adatszerkezetek és adattípusok/img/inorder.png create mode 100644 15. Adatszerkezetek és adattípusok/img/inorder_bin_search_tree.png create mode 100644 15. Adatszerkezetek és adattípusok/img/insert.png create mode 100644 15. Adatszerkezetek és adattípusok/img/konkret_abrazolas.png create mode 100644 15. Adatszerkezetek és adattípusok/img/levelOrder.png create mode 100644 15. Adatszerkezetek és adattípusok/img/mmm_rotation.png create mode 100644 15. Adatszerkezetek és adattípusok/img/mmn_roatation.png create mode 100644 15. Adatszerkezetek és adattípusok/img/mmp_rotation.png create mode 100644 15. Adatszerkezetek és adattípusok/img/postorder.png create mode 100644 15. Adatszerkezetek és adattípusok/img/ppm_rotation.png create mode 100644 15. Adatszerkezetek és adattípusok/img/ppn_rotation.png create mode 100644 15. Adatszerkezetek és adattípusok/img/ppp_rotation.png create mode 100644 15. Adatszerkezetek és adattípusok/img/pr_add.png create mode 100644 15. Adatszerkezetek és adattípusok/img/pr_max.png create mode 100644 15. 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Alapvető algoritmusok/14. Alapvető algoritmusok.tex b/14. Alapvető algoritmusok/14. Alapvető algoritmusok.tex index 808f05f..c0bf6ea 100644 --- a/14. Alapvető algoritmusok/14. Alapvető algoritmusok.tex +++ b/14. Alapvető algoritmusok/14. Alapvető algoritmusok.tex @@ -144,11 +144,70 @@ Lemma: Az $n!$ levelet tartalmazó tökéletes fák közül azokra a legkisebb a Tétel: $AO_R(n) = \Omega(n\log{n})$. \section{Rendezés lineáris időben (bucket-, leszámláló- és radix rendezés)} -TODO +\subsection{Bucket rendezés} +A bucket rendezés egy olyan rendezési algoritmus, amelyet általában egyenletesen elosztott értékekkel dolgozó adatsorok rendezésére használnak. +\begin{itemize} + \item A bemeneti adatsorban meghatározunk egy tartományt vagy intervallumot, amelyben az értékek eloszlása közel azonos. Ez a tartomány általában a minimális és maximális érték közötti intervallum. + \item Létrehozunk egy vagy több "vödröt" vagy "bucketet", amelyek a tartományon belül helyezkednek el. A vödrök száma lehet állandó vagy változó, és attól függ, hogy milyen módon osztjuk el a tartományt. + \item Az adatokat a megfelelő vödrökbe helyezzük a kulcsuk alapján. Ez történhet például a kulcsérték egészrészének kiszámításával vagy a hash függvény segítségével. + \item Minden vödörben levő adatsort rendezünk. Ez a rendezési lépés lehet bármely más rendezési algoritmus alkalmazása, például beszúrási rendezés vagy gyorsrendezés. + \item Az egyes vödrökből származó rendezett adatsorokat összevonjuk, hogy előálljon a rendezett kimeneti adatsor. +\end{itemize} +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{img/bucket.png} +\end{figure} + +\subsection{Leszámláló rendezés} +A leszámláló rendezés egy hatékony és stabil (az azonos értékű elemek sorrendjét megőrző) lineáris időkomplexitású rendezési algoritmus. +\begin{itemize} + \item Először meghatározzuk a bemeneti adatsor legnagyobb és legkisebb értékét, hogy meghatározzuk a tartományt, amelyben az értékek eloszlanak. Ez a lépés szükséges ahhoz, hogy létrehozhassunk egy leszámláló tömböt, amely tartalmazza a különböző értékek előfordulási számát. + \item Létrehozunk egy leszámláló tömböt, amelynek mérete a tartomány méretével egyezik meg. A leszámláló tömb minden eleme kezdetben 0. + \item Végigmegyünk a bemeneti adatsoron, és minden elem előfordulását megszámoljuk a leszámláló tömbben. Az elemeket a leszámláló tömb indexével azonosítjuk. + \item A leszámláló tömbben frissítjük az elemek előfordulási számát azzal, hogy hozzáadjuk az előző elem előfordulási számát. Ez a lépés segít abban, hogy meghatározzuk az adott elem végleges helyét a rendezett adatsorban. + \item Végigmegyünk újra a bemeneti adatsoron, és minden elemet a leszámláló tömb alapján beszúrunk a rendezett adatsor megfelelő pozíciójába. Eközben a leszámláló tömbből csökkentjük az adott elem előfordulási számát, hogy kezeljük az esetleges azonos értékeket. + \item Az eredményül kapott rendezett adatsor a bemeneti adatsor rendezett változata. +\end{itemize} +\begin{figure}[H] + \centering + \includegraphics[width=0.6\textwidth]{img/counting_sort.png} +\end{figure} + +\subsection{Radix rendezés} +A radix rendezés egy hatékony, stabil és nem összehasonlító rendezési algoritmus, amelyet általában egész számok rendezésére alkalmaznak. +\begin{itemize} + \item Először meghatározzuk a bemeneti adatsor legnagyobb értékét. Ez szükséges ahhoz, hogy meghatározzuk, hány számjegyet kell figyelembe venni a rendezés során. + \item Kezdjük a rendezést a legkisebb számjegytől (legkevésbé jelentős) a legnagyobbig (legjelentősebb). Ez lehet az egységjegy, tizedesjegy, százalékjegy stb. + \item A bemeneti adatsort rendezzük a jelenlegi számjegy szerint. Ez általában egy stabilitást megőrző rendezési algoritmus, például beszúrási rendezés vagy leszámláló rendezés alkalmazásával történik. + \item A rendezett adatsort átrendezzük és elhelyezzük az eredményül kapott sorrendben. Ez a lépés a stabilitást biztosítja, hogy az azonos értékű elemek sorrendje ne változzon. + \item Ismételjük meg a 2-4 lépéseket az összes számjegyre, a legkevésbé jelentőstől a legjelentősebbig. Ezáltal az adatsorban a rendezés minden számjegy szerint megtörténik. + \item Az eredményül kapott rendezett adatsor a bemeneti adatsor rendezett változata. +\end{itemize} +\begin{figure}[H] + \centering + \includegraphics[width=0.6\textwidth]{img/radix_example.png} + \caption{Példa, mivel jobban érthető mint az algo} +\end{figure} \section{Adattömörítések} \subsection{Naiv adattömörítés} -TODO +A tömörítendő szöveget karakterenként, fix hosszúságú bitsorozatokkal kódoljuk. +$\sum$ = $\sigma$ 1, $\sigma$2, . . . , $\sigma$i az ábécé. + +Egy-egy karakter $\lceil$lg d$\rceil$ bittel kódolható, ui. $\lceil$lg d$\rceil$ de biten 2 +\textsuperscript{$\lceil$lg d$\rceil$} de különböző bináris kód ábrázolható, és 2\textsuperscript{$\lceil$ lg d$\rceil$} $\geqslant$ d > 2\textsuperscript{$\lceil$ lg d$\rceil$-1} +, azaz $\lceil$lg d$\rceil$ biten ábrázolható +d-féle különböző kód, de eggyel kevesebb biten már nem. +$In : \sum \langle \rangle$ a tömörítendő szöveg. n = |In| jelöléssel n * $\lceil$lg d$\rceil$ bittel kódolható. +Pl. az ABRAKADABRA szövegre d = 5 és n = 11, ahonnét a tömörített +kód hossza $11 * \lceil lg 5 \rceil = 11 * 3 = 33$ bit. (A 3-bites kódok közül tetszőleges 5 +kiosztható az 5 betűnek.) A tömörített fájl a kódtáblázatot is tartalmazza. +A fenti ABRAKADABRA szöveg kódtáblázata lehet pl. a következő: + +\begin{figure}[H] + \centering + \includegraphics[width=0.3\textwidth]{img/naiv_table.png} +\end{figure} \subsection{Huffman-algoritmus} A Huffman-algoritmussal való tömörítés lényege, hogy a gyakrabban előforduló elemeket (karaktereket) rövidebb, míg a ritkábban előfordulókar hosszabb kódszavakkal kódoljuk. @@ -256,7 +315,19 @@ Az LZW (Lempel-Ziv-Welch) tömörítésnek a lényege, hogy egy szótárat bőv \end{description} \section{Mintaillesztés} \subsection{Brute-force mintaillesztés} -TODO +A brute force algoritmus egy egyszerű, de nem mindig hatékony módszer, amelyet problémák megoldására alkalmaznak. A brute force megközelítés során az algoritmus minden lehetséges lehetőséget kipróbál a probléma megoldására, majd megtalálja a helyes eredményt. + +\begin{itemize} + \item Meghatározzuk a probléma lehetséges megoldási területét. Ez lehet egy adott adatsor, egy meghatározott intervallum, vagy egyéb feltétel alapján meghatározott értékek. + \item Generáljuk az összes lehetséges kombinációt vagy lehetőséget a megoldási területen. Ez lehet például egy iteráció, amely végigmegy minden lehetséges értéken vagy kombináción. + \item Ellenőrizzük minden lehetséges kombináció vagy lehetőség esetén, hogy az adott megoldás kielégíti-e a probléma feltételeit vagy kritériumait. + \item Tároljuk el a helyes megoldásokat, vagy válasszuk ki a legjobb eredményt a probléma alapján. +\end{itemize} + +\begin{figure}[H] + \centering + \includegraphics[width=0.4\textwidth]{img/brute_force.png} +\end{figure} \subsection{Knuth-Morris-Pratt algoritmus} A Knuth-Morris-Pratt eljárásnak a Brute-Force (hasonlítsuk össze, toljunk egyet, stb..) módszerrel szemben az az előnye, hogy egyes esetekben, ha a mintában vannak ismétlődő elemek, akkor egy tolásnál akár több karakternyit is ugorhatunk. diff --git a/14. Alapvető algoritmusok/img/brute_force.png b/14. 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Adatszerkezetek és adattípusok/15. Adatszerkezetek és adattípusok.tex b/15. Adatszerkezetek és adattípusok/15. Adatszerkezetek és adattípusok.tex index 6853b38..6a881c7 100644 --- a/15. Adatszerkezetek és adattípusok/15. Adatszerkezetek és adattípusok.tex +++ b/15. Adatszerkezetek és adattípusok/15. Adatszerkezetek és adattípusok.tex @@ -26,13 +26,14 @@ \begin{document} \maketitle -TODO: Erősen hiányos! - \begin{tetel}{Adatszerkezetek és adattípusok} - Tömb, verem, sor, láncolt listák; bináris fa, általános fa, bejárások, ábrázolások; bináris kupac, prioritásos sor; bináris kereső fa és műveletei, AVL fa, B+ fa; hasító táblák, hasító függvények, kulcsütközés és feloldásai: láncolással, nyílt címzéssel, próbasorozat; gráfok ábrázolásai. + Tömb, verem, sor, láncolt listák; bináris fa, általános fa, bejárások, ábrázolások; bináris +kupac, prioritásos sor; bináris kereső fa és műveletei, AVL fa, B+ fa; hasító táblák, hasító +függvények, kulcsütközés és feloldásai: láncolással, nyílt címzéssel, próbasorozat; gráfok +ábrázolásai. \end{tetel} -\section{Egyszerű adattípusok ábrázolásai, műveletei és fontosabb alkalmazásai} +\section{Egyszerű adattípusok} \subsection{Adattípus} @@ -63,104 +64,337 @@ TODO: Erősen hiányos! \includegraphics[width=0.3\textwidth]{img/Torol.jpg} \caption{Lista műveletei} \end{figure} - \item \textit{Fa}: Egyszerű, bináris és speciális (kupac, bináris keresőfa, AVL-fa). A bináris fát rekurzívan definiáljuk: $t \in T(E)$ [bin. fák típusérték halmaza(alaptípus)] $\iff$ $t$ üres fa (jele: $\Omega$), vagy $t$-nek van gyökéreleme, $bal(t)$, $jobb(t)$ részfája. Láncoltan ábrázoljuk, tömbösen csak teljes fák, illetve kupac esetén. - \item \textit{Kupac}: Olyan bináris fa, melynek alakja majdnem teljes és balra rendezett. Tömbösen ábrázoljuk, mert pointeresen a bonyolult lépkedést nem teszi lehetővé, tömbösen indexösszefüggésekkel könnyen megoldható. - \begin{figure}[H] - \centering - \includegraphics[width=0.3\textwidth]{img/KupacbaBeszur.jpg} - \caption{Kupac műveletei} - \end{figure} - \item \textit{Hasítótábla} - \item \textit{Gráf} [Nem egyszerű adattípus.] \end{itemize} -\subsection{Ábrázolásai} +\section{Fák és bejárásaik, ábrázolásaik} -Absztrakciós szintek: -\begin{enumerate} - \item \textit{absztrakt adattípus (ADT)}: specifikáció szintje, itt nincs szerkezeti összefüggés, csak matematikai fogalmak, műveletek logikai axiómákkal vagy előfeltételekkel. - \begin{itemize} - \item algebrai (axiomatikus) specifikáció, példa: $Verembe : V \times E \to V$. Axióma, példa: $Felso(Verembe(v,e)) = e$ - \item funkcionális (elő- és utófeltételes) specifikáció, példa: (elsőbbségi) sor $S(E), s \in S$ egy konkrét sor, $s = \{(e_1, t_1), ..., (e_n, t_n)\}$, $n \geq 0$. Ha $n=0$, akkor a sor üres. \\ - $\forall i,j \in [1..n]: i \neq j \to t_i \neq t_j$. \\ - $Sorbol: S \to S \times E$, $\mathcal{D}_{Sorbol} = S \backslash \{ures\}$. Előfeltétel: $Q = (s = s' \wedge s' \neq \emptyset)$, utófeltétel: $R = (s = s' \backslash \{(e,t)\} \wedge (e,t) \in s' \wedge \forall i (e_i, t_i) \in s' : t \leq t_i)$. - \end{itemize} - \item \textit{absztrakt adatszerkezet (ADS)}: kognitív pszichológia szintje, ábrák. Az alapvető szerkezeti tulajdonságokat tartalmazza (nem mindet). Ennek a szintnek is része a műveletek halmaza. Példák: az ábra egy irányított gráf, művelet magyarázata, adatszerkezet definiálása. - \begin{figure}[H] - \centering - \includegraphics[width=1\textwidth]{img/ads.png} - \caption{ADS} - \end{figure} - \item \textit{ábrázolás/reprezentáció}: döntések (tömbös vagy pointeres ábrázolás), a nyitva hagyott szerkezeti kérdések. Egy adatszerkezetet többféle reprezentációval is meg lehet valósítani (pl. prioritásos sor lehet rendezetlen tömb, rendezett tömb, kupac). - \begin{itemize} - \item \textit{tömbös ábrázolás}: takarékos ábrázolás, elhelyezése, tetszőleges rákövetkezések, bejárások, de ezeket meg kell adni. - \item \textit{pointeres ábrázolás}: minden pointer egy összetett rekord elejére mutat. - \end{itemize} - \item \textit{implementálás} - \item \textit{fizikai szint} -\end{enumerate} +\subsection{Bináris fa} -\subsection{Műveletei} +A fákat nagy méretű adathalmazok és multihalmazok ábrázolására, +de egyéb adatreprezentációs célokra is gyakran használjuk. +A (bináris) fák esetében minden adatelemnek vagy szokásos nevén csúcsnak +legfeljebb kettő rákövetkezője van: egy bal és/vagy egy +jobb rákövetkezője. +\textit{Fogalmak:} \begin{itemize} - \item Üres adatszerkezet létrehozása - \item Annak lekérdezése, hogy üres-e az adatszerkezet - \item Elem berakása, itt ellenőrizni kell, hogy nem telt-e még meg - \item Elem kivétele vagy törlése, itt ellenőrizni kell, hogy nem üres-e - \item Adott tulajdonságú elem (például maximum, veremben a felső) lekérdezése, itt is ellenőrizni kell, hogy üres-e az adatszerkezet - \item Bejárások (preorder, inorder, postorder, szintfolytonos), listáknál az első, előző vagy következő elemre lépés - \item Elem módosítása bizonyos adatszerkezeteknél (pl. listák) + \item \textit{gyerek}: a csúcs bal vagy jobb rákövetkezője + \item \textit{szülő}: a gyerekek csúcsa + \item \textit{testvérek}: azonos csúcs gyerekei + \item \textit{levél}: gyerek nélküli szülő + \item \textit{gyökércsúcs}: nincs szülője + \item \textit{belső csúcs}: nem-levél csúcs + \item \textit{leszármazottak}: egy csúcs gyerekei és annak gyerekei + \item \textit{ősök}: egy csúcs szülője és annak szülei \end{itemize} -\subsection{Fontosabb alkalmazásai} +\subsection{Általános fa} +A bináris fa fogalma általánosítható. Ha a fában egy tetszőleges csúcsnak legfeljebb r rákövetkezője van, r-áris fáról beszélünk. Egy csúcs gyerekeit és a hozzájuk tartozó részfákat ilyenkor [0..r)-beli szelektorokkal szokás sorszámozni. Ha egy csúcsnak nincs i-edik gyereke (i $\epsilon$ [0..r)), akkor az i-edik +részfa üres. Így tehát a bináris fa és a 2-áris fa lényegében ugyanazt jelenti, azzal, +hogy itt a left $\sim$ 0 és a right $\sim$ 1 szelektor-megfeleltetést alkalmazzuk. -\textit{Prioritásos sor}: nagygépes programfuttatásnál az erőforrásokat a prioritás arányában osszuk el, adott pillanatban a maximális prioritásút válasszuk. Sürgősségi ügyeleten, gráfalgoritmusoknál is alkalmazható. -\textit{B-fa}: ipari méretekben adatbázisokban használják. +A fa szintjeit a következők képpen határozzuk meg. A gyökér van a nulladik szinten. Az +i-edik szintű csúcsok gyerekeit az (i + 1)-edik szinten találjuk. A fa magassága egyenlő a legmélyebben fekvő levelei szintszámával. Az üres fa magassága h($\oslash$) = -1. -\section{A hatékony adattárolás és visszakeresés néhány megvalósítása (bináris keresőfa, AVL-fa, 2-3-fa és B-fa, hasítás (,,hash-elés”))} +Az itt tárgyalt fákat gyökeres fáknak is nevezik, mert tekinthetők olyan irányított gráfoknak, amiknek az élei a gyökércsúcstól a levelek felé vannak +irányítva, a gyökérből minden csúcs pontosan egy úton érhető el. -\subsection{Bináris keresőfa} -Nincsenek benne azonos kulcsok, a követendő elv: ,,kisebb balra, nagyobb jobbra". Inorder bejárással növekvő kulcssorozatot kapunk. \\ -Műveletigénye fa magasságú nagyságrendű. \\ -Az a cél, hogy a bináris keresőfa ne nyúljon meg láncszerűen, erre jó az AVL-fa és a 2-3-fa. +\subsection{Bejárásaik} +A fákkal dolgozó programok gyakran kapcsolódnak a négy klasszikus +bejárás némelyikéhez, amelyek adott sorrend szerint bejárják a fa csúcsait, +és minden csúcsra ugyanazt a műveletet hívják meg, amivel kapcsolatban +megköveteljük, hogy futási ideje $\Theta$ (1) legyen (ami ettől még persze összetett +művelet is lehet). A *f csúcs feldolgozása lehet például f → key kiíratása. +Üres fára mindegyik bejárás az üres program. +\textit{Nemüres r-áris fákra} +\begin{itemize} + \item \textit{Preorder}: először a fa gyökerét dolgozza fel, majd sorban bejárja a 0..r - 1-edik részfákat; + \begin{figure}[H] + \centering + \includegraphics[width=0.3\textwidth]{img/preorder.png} + \caption{Preorder bejárás} + \end{figure} + \item \textit{Inorder}: először bejárja a nulladik részfát, ezután a fa gyökerét dolgozza fel, majd sorban bejárja az 1..r - 1-edik részfákat; + \begin{figure}[H] + \centering + \includegraphics[width=0.3\textwidth]{img/inorder.png} + \caption{Inorder bejárás} + \end{figure} + \item \textit{Postorder}: előbb sorban bejárja a 0..r - 1-edik részfákat, és a fa gyökerét csak a részfák után dolgozza fel + \begin{figure}[H] + \centering + \includegraphics[width=0.3\textwidth]{img/postorder.png} + \caption{Postorder bejárás} + \end{figure} + \item \textit{LevelOrder}: a csúcsokat a gyökértől kezdve szintenként, minden szintet balról jobbra bejárva dolgozza fel. + \begin{figure}[H] + \centering + \includegraphics[width=0.3\textwidth]{img/levelOrder.png} + \caption{LevelOrder bejárás} + \end{figure} +\end{itemize} + +Az első három bejárás tehát nagyon hasonlít egymásra. Nevük megsúgja, +hogy a gyökércsúcsot a részfákhoz képest mikor dolgozzák fel. + +\subsection{Ábrázolásaik} +\subsubsection{Grafikus} +A legtermészetesebb és az egyik leggyakrabban használt a bináris fák láncolt +ábrázolása: \begin{figure}[H] \centering - \includegraphics[width=0.3\textwidth]{img/Beszur.jpg} - \includegraphics[width=0.3\textwidth]{img/Kovetkezo.jpg} - \caption{Bináris keresőfa műveletei} + \includegraphics[width=0.6\textwidth]{img/abrazolas.png} + \caption{Ugyanaz a bináris fa grafikus és láncolt ábrázolással.} \end{figure} -\subsection{AVL-fa} -Cél: a $t$ bináris keresőfa magasságának $\log_2(n)$ közelében tartása, azaz $h(t) \leq c \cdot \log_2(n)$, ahol $c$ elfogadhatóan kicsi. Az ilyen fát kiegyensúlyozottnak nevezzük. \\ -\textit{AVL}: Adelszon-Velszkij, Landisz 1962-ben alkották meg. \\ -A $t$ bináris keresőfát egyúttal AVL-fának nevezzük $\iff$ $t$ minden $x$ csúcsára $|h(bal(x))-h(jobb(x))| \leq 1$. \\ -Minden csúcsnak van egy címkéje $+,-,=$ (gyerekek magasságának különbsége). A beszúrás helyétől felfelé ellenőrizzük ezeket, és ha kell, akkor módosítjuk. Ha valahol $++$ vagy $--$ alakul ki, akkor ott elromlik az AVL-tulajdonság, egy vagy több forgatással vagy átkötéssel konstans műveletigénnyel helyre lehet hozni. \\ -Többféle séma is van: $(++,+), (++,-), (++,=)$ és a tükörképeik. +\subsubsection{Absztrakt} +Az üres fa reprezentációja a $\oslash$ pointer, jelölése tehát ugyanaz, mint az +absztrakt fáknál. A bináris fa csúcsait pl. az alábbi osztály objektumaiként +ábrázolhatjuk, ahol a BinTree absztrakt típus reprezentációja egyszerűen a +Node*. -\subsection{2-3-fa és B-fa} +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{img/absztrakt_node.png} +\end{figure} -2-3-fa kis méretben az elmélet számára jó, a B-fa a gyakorlati változat adatbázisban. \\ -$t$ 2-3-fa $\iff$ minden belső csúcsnak 2 vagy 3 gyereke van, a levelek azonos szinten helyezkednek el, adatrekordok csak a levelekben vannak, belső pontokban kulcsok és mutatók, levelekben a kulcsok balról jobbra nőnek. \\ -Ha 4 gyerek lenne a beszúrás után, akkor csúcsot kell vágni. Ha törlésnél 1 gyerek lenne valahol, akkor csúcsösszevonásokat és gyerekátadást alkalmazunk. \\ -B-fa nagyobb méretű, itt két határ között mozog a gyerekszám: $\lceil\frac{r}{2}\rceil$ és $r$, ahol $50 \leq r \leq 1000$. +Néha hasznos, ha a csúcsokban van egy parent szülő pointer is, mert a fában +így felfelé is tudunk haladni. +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{img/absz_node_parenttel.png} +\end{figure} -\subsection{Hasítás} +\subsubsection{Zárójelezett} -Kulcsos rekordokat tárol. +Tetszőleges nemüres bináris fa zárójeles, azaz szöveges alakja: + ( balRészFa Gyökér jobbRészFa ) +Az üres fát az üres string reprezentálja. A könnyebb olvashatóság kedvéért +többféle zárójelpárt is használhatunk. A zárójeles ábrázolás lexikai elemei: +nyitózárójel, csukó zárójel és a csúcsok címkéi. +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{img/grafikus_szoveges.png} + \caption{Ugyanaz a bináris fa grafikus és szöveges ábrázolással.} +\end{figure} + +\section{Kupac és sor} +A bináris kupac egy konkrét adatszerkezet, +amelyet a prioritásos sor implementálására használnak. +A prioritásos sor általánosabb fogalom, amely többféle adatszerkezetet +vagy algoritmust foglalhat magában a prioritás alapján történő rendezés +és az elemek elérése szempontjából. + +\subsection{Bináris kupac} +Egy teljes bináris fa, +amelyben az elemek prioritása a csúcsokban található kulcsok alapján van meghatározva. +A bináris kupac hatékonyan támogatja a prioritásos sor műveleteket, +például az elem beszúrását és eltávolítását a legmagasabb prioritással. +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{img/konkret_abrazolas.png} + \caption{Példa} +\end{figure} + +\subsection{Metódusai} +\subsubsection{Hozzáadás} +Új elemet hozzáadunk a kupachoz a levélszintre (a legközelebbi szabad pozícióba). +Ezután a kupacban felfelé haladva összehasonlítjuk az új elemet az ő szülőjével. +Ha az új elem prioritása nagyobb, akkor felcseréljük az elemeket, és folytatjuk a felfelé mozgást a egyel magasabb szintre. +Ezt addig ismételjük, amíg az új elem prioritása nem megfelelő a bináris kupac tulajdonságainak. +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{img/pr_add.png} + \caption{Az add(x) algoritmusa} +\end{figure} + +\subsubsection{Törlés} +A legmagasabb prioritással rendelkező elem mindig a kupac gyökerén található. +Eltávolítjuk(remMax) ezt az elemet, amely a legfelső elem a kupacban. +Ilyenkor a kupac üres lesz, vagy helyettesíteni kell a gyökérlemet az aljában található legutolsó elemmel. +Ha a kupac nem üres, a helyettesítő elemet lefelé mozgatjuk a kupacban, hogy helyreállítsuk a kupac tulajdonságait. +Az elemet összehasonlítjuk a gyermekével, majd a kisebb prioritású gyermekkel cseréljük(sink), ha az nagyobb. +Ezt a folyamatot addig ismételjük, amíg az elem megfelelő helyre nem kerül a kupacban. + +\begin{figure}[H] + \centering + \includegraphics[width=0.4\textwidth]{img/pr_max.png} + \caption{A remMax() algoritmusa} +\end{figure} + +\begin{figure}[H] + \centering + \includegraphics[width=0.4\textwidth]{img/pr_sink.png} + \caption{A sink(x) algoritmusa} +\end{figure} + +\section{Különleges fák} +\subsection{Bináris kereső fa} +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{img/bin_sreach_tree_example.png} + \caption{Példa bináris kereső fára} +\end{figure} +A bináris kereső fa egy olyan adatszerkezet, amelyet a rendezett adatok hatékony tárolására és lekérdezésére használnak. +\textit{Tulajdonságai} \begin{itemize} - \item \textit{Hasítás láncolással}: a kulcsütközést láncolással oldja fel. Van egy hasítófüggvény: $h: U \to [0..m-1]$, elvárás vele kapcsolatban, hogy gyorsan számolható és egyenletes legyen. $m$-et úgy választjuk meg $n$ nagyságrendjének ismeretében, hogy $\alpha = \frac{n}{m}$ lesz a várható listahossz, ha egyenletes hasítást feltételezünk.\\ - Például kétirányú listát használhatunk a hasításhoz. Műveletek: beszúrás, keresés, törlés. \\ - Gyakorlatban érdemes $m$-et úgy megválasztani, hogy olyan prímszám legyen, ami nem esik 2-hatvány közelébe. - \item \textit{Hasítás nyitott/nyílt címzéssel}: A kulcsokat lehessen egészként értelmezni, ekkor vannak jó hasítófüggvények. \\ - Próbálkozás általános képlete: $h(k) + h_i(k)$ $(mod$ $M)$, $0 \leq i \leq M-1$. Egész addig alkalmazza, amíg üres helyet nem talál. - \begin{enumerate} - \item \textit{Lineáris próba}: $h_i(k) = -i$ $(mod$ $M)$, egyesével balra lépegetve keressük az üres helyet. Hátránya az elsődleges csomósodás, ez jelentős lassulást okoz beszúrásnál és keresésnél. - \item \textit{Négyzetes próba}: $h_i(k) = (-1)^i(\lceil\frac{i}{2}\rceil)^2$ $(mod$ $M)$, a négyzetszámokkal lépegetünk balra-jobbra, ezek az eltolások kiadják $\{0,1,...,M-1\}$-et. Hátrány: másodlagos csomósodás. - \item \textit{Kettős hash-elés}: $h_i(k) =-ih'(k)$ $(mod$ $M)$, $h'(k)$ a $k$-hoz tartozó egyedi lépésköz, $(h'(k),M)=1$ relatív prímek. Ha az $M$ elég nagy, akkor nincs csomósodás. - \end{enumerate} - \item \textit{Hasítófüggvények}: Leggyakoribb: $k$ egész, kongruencia reláció. Általánosan: $h(k) = (ak +b$ $(mod$ $p))$ $(mod$ $M)$, az univerzális hasítás családja. Tapasztalat: $k$ egyenletesen hasít. + \item \textit{Rendezettség}: Minden csúcsnak van egy kulcsa, amelyek alapján eldönthető, hogy a bal részfában (kisebb mint a csúcs) vagy a jobb részfában (nagyobb mint a csúcs) helyezkedik el. + \item \textit{Gyors keresés}: Lehetővé teszi a gyors keresést a rendezettségnek köszönhetően. Tehát, ha egy elemet keresünk és a csúcstól indulunk egyből tudjuk, hogy jobbra vagy balra van a keresett elem. + \begin{figure}[H] + \centering + \includegraphics[width=0.4\textwidth]{img/search.png} + \caption{Keresés bináris kereső fában} + \end{figure} + \item \textit{Beszúrás és törlés}: Beszúrásnál az új elemet a megfelelő helyre illesztjük, a rendezettséget figyelembe véve. Törlésnél pedig a struktúrát kell megváltoztatni, hogy a rendezettség megmaradjon. + \begin{figure}[H] + \centering + \includegraphics[width=0.4\textwidth]{img/insert.png} + \caption{Beszúrás bináris kereső fába} + \end{figure} + \begin{figure}[H] + \centering + \includegraphics[width=0.4\textwidth]{img/del.png} + \caption{Törlés bináris kereső fából} + \end{figure} + \item \textit{In-order bejárás}: Ezt a bejárást használva, megkapjuk a bináris kereső fa elemeit rendezett sorrendben. + \begin{figure}[H] + \centering + \includegraphics[width=0.4\textwidth]{img/inorder_bin_search_tree.png} + \caption{In-order bejárás bináris kereső fában} + \end{figure} \end{itemize} +\subsection{AVL fa} +Az AVL fa egy speciális bináris kereső fa. Célja az, hogy fenntartsa az egyensúlyt a fa minden csomópontjában, +hogy a keresési, beszúrási és törlési műveletek hatékonyan végezhetők legyenek. +Tehát a bináris kereső fa tulajdonságain felül van még kettő: a magasság és az önkiegyensúlyozás. + +\subsubsection{Magasságtulajdonság} +Az AVL fa az egyensúly fenntartása érdekében használja a magasságtulajdonságot. Minden csomópont rendelkezik egy magasságértékkel, amely a csomóponttól a legtávolabbi levél magasságát jelenti. +Az AVL fában az összes csomópont magasságkülönbsége legfeljebb 1 lehet, vagyis az AVL fa egyensúlyban van. + + +\subsubsection{Önkiegyensúlyozás} +Ha egy beszúrási vagy törlési művelet után az AVL fa egyensúlyát megsértik, akkor a fa kiegyensúlyozására szolgáló rotációs műveleteket végeznek. +A rotációk célja, hogy a magasságkülönbséget korrigálják és visszaállítsák az AVL fa egyensúlyát. + +\subsubsection{Rotációk} +Annak függvényében, hogy melyik oldal változott és mennyire az alábbi rotációkat kell használni. +Ha a bal oldali részfa süllyed egy szintet akkor - jelölést kap, ha a jobb akkor + jelölést. +A ++/-- azt jelenti, hogy az egyik oldalon a részfa magassága kettővel nagyobb mint a másikon. +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{img/ppp_rotation.png} + \includegraphics[width=0.5\textwidth]{img/ppm_rotation.png} + \includegraphics[width=0.5\textwidth]{img/mmm_rotation.png} + \includegraphics[width=0.5\textwidth]{img/mmp_rotation.png} + \includegraphics[width=0.5\textwidth]{img/ppn_rotation.png} + \includegraphics[width=0.5\textwidth]{img/mmn_roatation.png} +\end{figure} + +\subsection{B+ fa} +A B+ fa, amiben minden csúcs legfeljebb d mutatót (4 $\leq$ d), és legfeljebb d-1 kulcsot tartalmaz,ahol d a fára jellemző állandó, a B+ fa fokszáma. +Úgy tekintjük, hogy a belső csúcsokban mindegyikreferencia +két kulcs "között" van, azaz egy olyan részfa gyökerére mutat, amiben minden érték a kétkulcs között található. + +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{img/b_plus.png} + \caption{4-es fokszámú B+ fa} +\end{figure} + +Tetszőleges d-ed fokú B+ fa a következő invariánsokat teljesíti, ahol 4 $\leq$ d állandó: +\begin{itemize} + \item Minden levélben legfeljebb d-1 kulcs, és ugyanennyi, a megfelelő adatrekordrahivatkozó mutató található. + \item A gyökértől mindegyik levél ugyanolyan távol található. + \item Minden belső csúcsban eggyel több mutató van, mint kulcs, ahol d a felső határ a mutatók számára. + \item Minden Cs belső csúcsra, ahol k a Cs csúcsban a kulcsok száma: az első gyerekhez tartozórészfában minden kulcs kisebb, mint a Cs első kulcsa; az utolsó gyerekhez tartozó részfábanminden kulcs nagyobb-egyenlő, mint a Cs utolsó kulcsa; és az i-edik gyerekhez tartozó részfában(2 $\leq$ i $\leq$ k) lévő tetszőleges r kulcsra Cs.kulcs[i-1] $\leq$ r < Cs.kulcs[i]. + \item A gyökércsúcsnak legalább két gyereke van. Kivéve, ha ez a fa egyetlen csúcsa. + \item Minden, a gyökértől különböző belső csúcsnak legalább d/2 alsó egész-rész gyereke van. + \item Minden levél legalább d/2 alsó egész-rész kulcsot tartalmaz. A B+ fa által reprezentált adathalmaz minden kulcsa megjelenik valamelyik levélben, balról jobbraszigorúan monoton növekvő sorrendben. +\end{itemize} + +\subsubsection{Beszúrás} +Ha a fa üres, hozzunk létre egy új levélcsúcsot, és a beszúrandókulcs/mutató pár a tartalma! +Különben keressük meg a kulcsnak megfelelő levelet! Ha a levélben márszerepel a kulcs, a beszúrás sikertelen. Egyébként: +\begin{itemize} + \item Ha a csúcsban van üres hely, szúrjuk be a megfelelő kulcs/mutató párt kulcs szerint rendezetten ebbe a csúcsba! + \item Ha a csúcs már tele van, vágjuk szét két csúccsá, középen és osszuk el a d darab kulcsot egyenlően a kétcsúcs között! Ha a létre jött csúcs egy levél, vegyük a létrejött második csúcs legkisebb értékének másolatát, és ismételjük meg ezt a beszúró algoritmust, hogy beszúrjuk azt a szülő csúcsba! Ha a csúcs nemlevél, vegyük ki a középső értéket a kulcsok elosztása során, és ismételjük meg ezt a beszúróalgoritmust, hogy beszúrjuk ezt a középső értéket a szülő csúcsba! (Ha kell, a szülő csúcsot előbblétrehozzuk. Ekkor a B+ fa magassága nő.) +\end{itemize} + +\subsubsection{Törlés} +Keressük meg a törlendő kulcsot tartalmazó levelet! Ha ilyen nincs, a törlés meghiúsul. +\begin{itemize} + \item Ha keresés során megtalált levélcsúcs egyben a gyökércsúcs is: + \begin{itemize} + \item Töröljük a megfelelő kulcsot és a hozzá tartozó mutatót a csúcsból! + \item Ha a csúcs tartalmaz még kulcsot, kész vagyunk. + \item Különben töröljük a fa egyetlen csúcsát, és üres fát kapunk. + \end{itemize} + \item A keresés során megtalált levélcsúcs nem a gyökércsúcs: + \begin{itemize} + \item Töröljük a megfelelő kulcsot és a hozzá tartozó mutatót a levélcsúcsból! + \item Ha a levélcsúcs még tartalmaz elég kulcsot és mutatót, hogy teljesítse az invariánsokat, készvagyunk. + \item Ha a levélcsúcsban már túl kevés kulcs van ahhoz, hogy teljesítse az invariánsokat, de a következő,vagy a megelőző testvérének több van, mint amennyi szükséges, osszuk el a kulcsokat egyenlően közte és a megfelelő testvére között! Írjuk át a két testvér közös szülőjében a két testvérhez tartozóhasító kulcsot a két testvér közül a második minimumára! + \item Ha a levélcsúcsban már túl kevés kulcs van ahhoz, hogy teljesítse az invariánst, és a következő,valamint a megelőző testvére is a minimumon van, hogy teljesítse az invariánst, akkor egyesítsükegy vele szomszédos testvérével! Ennek során a két testvér közül a (balról jobbra sorrend szerinti)másodikból a kulcsokat és a hozzájuk tartozó mutatókat sorban átmásoljuk az elsőbe, annak eredetikulcsai és mutatói után, majd a második testvért töröljük. Ezután meg kell ismételnünk a törlőalgoritmust a szülőre, hogy eltávolítsuk a szülőből a hasító kulcsot (ami eddig elválasztotta a mostegyesített levélcsúcsokat), a most törölt második testvérre hivatkozó mutatóval együtt. + \end{itemize} + \item Belső — a gyökértől különböző — csúcsból való törlés: + \begin{itemize} + \item Töröljük a belső csúcs éppen most egyesített két gyereke közti hasító kulcsot és az egyesítés sorántörölt gyerekére hivatkozó mutatót a belső csúcsból! + \item Ha a belső csúcsnak van még floor(d/2) gyereke, (hogy teljesítse az invariánsokat) kész vagyunk. + \item Ha a belső csúcsnak már túl kevés gyereke van ahhoz, hogy teljesítse az invariánsokat, de a következő, vagy a megelőző testvérének több van, mint amennyi szükséges, osszuk el a gyerekeketés a köztük levő hasító kulcsokat egyenlően közte és a megfelelő testvére között, a hasító kulcsokközé a testvérek közti (a közös szülőjükben lévő) hasító kulcsot is beleértve! A gyerekek és ahasító kulcsok újraelosztása során, a középső hasító kulcs a testvérek közös szülőjében a kéttestvérhez tartozó régi hasító kulcs helyére kerül úgy, hogy megfelelően reprezentálja a köztükmegváltozott vágási pontot! (Ha a két testvérben a gyerekek összlétszáma páratlan, akkor azújraelosztás után is annak a testvérnek legyen több gyereke, akinek előtte is több volt!) + \item Ha a belső csúcsnak már túl kevés gyereke van ahhoz, hogy teljesítse az invariánst, és a következő,valamint a megelőző testvére is a minimumon van, hogy teljesítse az invariánst, akkor egyesítsükegy vele szomszédos testvérével! Az egyesített csúcsot a két testvér közül a (balról jobbra sorrendszerinti) elsőből hozzuk létre. Gyerekei és hasító kulcsai először a saját gyerekei és hasító kulcsaiaz eredeti sorrendben, amiket a két testvér közti (a közös szülőjükben lévő) hasító kulcs követ, ésvégül a második testvér gyerekei és hasító kulcsai jönnek, szintén az eredeti sorrendben. Ezutántöröljük a második testvért. A két testvér egyesítése után meg kell ismételnünk a törlő algoritmusta közös szülőjükre, hogy eltávolítsuk a szülőből a hasító kulcsot + (ami eddig elválasztotta a mostegyesített testvéreket), a most törölt második testvérre hivatkozó mutatóval együtt. + \end{itemize} + \item A gyökércsúcsból való törlés, ha az nem levélcsúcs: + \begin{itemize} + \item Töröljük a gyökércsúcs éppen most egyesített két gyereke közti hasító kulcsot és az egyesítés sorántörölt gyerekére hivatkozó mutatót a gyökércsúcsból! + \item Ha a gyökércsúcsnak van még 2 gyereke, kész vagyunk. + \item Ha a gyökércsúcsnak csak 1 gyereke maradt, akkor töröljük a gyökércsúcsot, és a megmaradtegyetlen gyereke legyen az új gyökércsúcs! (Ekkor a B+ fa magassága csökken.) + \end{itemize} +\end{itemize} + +\section{Hasító táblák} +A hasító tábla (hash table), más néven hasítótábla vagy hash map, egy hatékony adatszerkezet, amely kulcs-érték párokat tárol. +A hasító tábla kulcsokat használ a tárolt értékekhez való gyors hozzáférésre. +A hasító tábla alapvetően egy tömb, amely az adatokat ún. hash funkció segítségével tárolja és keresi. +\subsection{Hasító függvények} +Amikor egy kulcshoz tartozó értéket hozzá szeretnénk adni a hasító táblához, +először egy hash funkciót alkalmazunk a kulcsra. A hash funkció egy olyan algoritmus, amely egyedi hash kódot generál a kulcs alapján. +A hash kód egy indexet határoz meg a tömbben, ahol az értéket tárolni fogjuk. +\begin{itemize} + \item \textit{Tárolás:} Az előző lépésben generált hash kód alapján elhelyezzük az értéket a hasító táblában lévő tömb megfelelő indexénél. Ha két vagy több kulcsnak véletlenül ugyanaz a hash kódja, akkor azt hash ütközésnek nevezzük. + \item \textit{Hash ütközések kezelése:} A hash ütközések kezelése kritikus szerepet játszik a hasító tábla hatékonyságában. + \begin{itemize} + \item \textit{Láncolás:} Minden hash kódhoz egy "lánc" tartozik, amely az ütköző kulcsokat tárolja egy adatszerkezetben (általában egy láncolt lista vagy tömb). Ha egy új kulcsnak azonos hash kódja van, a megfelelő láncra kerül, és az érték hozzáadódik a láncolt listához vagy tömbhöz. + \item \textit{Nyílt címzés:} Az ütköző kulcsokat közvetlenül a tömbben tároljuk, anélkül, hogy külön adatszerkezetet használnánk. Ha egy új kulcsnak ütközik a hash kódja, különböző üres helyeket próbálunk meg keresni a tömbben, amíg üres helyet találunk a tároláshoz. + \end{itemize} + \item \textit{Keresés:} A keresés műveletekor ismét alkalmazzuk a hash funkciót a keresett kulcsra, és meghatározzuk az érték tárolásának helyét a hasító táblában. Ha láncolást használunk, akkor végigmegyünk a láncolt listán, hogy a kulcshoz tartozó értéket keresük a hasító táblában, alkalmazzuk a hash funkciót a keresett kulcsra, és meghatározzuk az érték tárolásának helyét a hasító táblában. Ha láncolást használunk, akkor végigmegyünk a láncolt listán vagy a tömbben, és összehasonlítjuk a kulcsokat, hogy megtaláljuk a megfelelő értéket. + \item \textit{Törlés:} A törlés művelete hasonló a kereséshez. Először alkalmazzuk a hash funkciót a kulcsra, és meghatározzuk az érték tárolásának helyét a hasító táblában. Ha láncolást használunk, akkor végigmegyünk a láncolt listán vagy a tömbben, megtaláljuk a megfelelő értéket, és töröljük azt. +\end{itemize} +A hasító tábla előnyei közé tartozik a gyors adatelérés. Ha a hash funkció jól tervezett és a hash ütközéseket hatékonyan kezeljük, a keresési, beszúrási és törlési műveletek átlagos időkomplexitása O(1) közelíthető, ami nagyon hatékony. +Azonban a hasító tábla néhány korlátot is felvet. Először is, a hash funkció nem mindig garantálja a teljesen egyedi hash kódokat, így lehetőség van a hash ütközések előfordulására. Ezt a megfelelő ütközéskezeléssel kell kezelni. Másodsorban, a hasító tábla fogyasztja a memóriát, különösen, ha nagy méretű tömböt kell tartalmaznia. Az adatmennyiség és a hash funkció hatékonysága közötti egyensúly megtalálása fontos szempont a hatékonyság szempontjából. + +\subsection{Próbasorozat} +A próbasorozat egy adatszerkezetekben, például hasító táblákban vagy hasábokban alkalmazott módszer, amelyet a kulcsok egyedi helyét meghatározására használnak. +Amikor egy kulcsot beszúrunk vagy keresünk egy hasító táblában vagy hasábokban, a próbasorozatot használjuk annak meghatározására, hogy hol található a kulcs tárolási helye a táblában. +A próbasorozat olyan sorozat vagy sorrend, amelyet a hash kódhoz vagy a kulcs alapján generálunk. +A leggyakoribb próbasorozatok közé tartoznak: +\begin{itemize} + \item \textit{Lineáris próbasorozat:} A kiválasztott hash kód vagy index már foglalt a táblában, az algoritmus egymás után következő indexeket próbál meg, amíg üres helyet nem talál. Például, ha az eredeti hash kód 5, és az 5-ös index már foglalt, az algoritmus a 6-os, majd a 7-es indexet próbálja meg, és így tovább. + \item \textit{Négyzetes próbasorozat:} Az algoritmus négyzet alakban növeli az indexet az ütközés esetén. Például, ha az eredeti hash kód 5, és az 5-ös index már foglalt, az algoritmus a $(5+1^2 = 6)$, majd a $(5+2^2 = 9)$ indexet próbálja meg, és így tovább. + \item \textit{Dupla hasításos próbasorozat:} Az algoritmus egy második hash funkciót használ az ütközés esetén a következő index meghatározására. A második hash funkció különböző hash kódot generál, amely segít elkerülni az összeomlást és a ciklusokat az indexek között. +\end{itemize} + +\section{Gráfok ábrázolásai} +\begin{itemize} + \item \textit{Szomszédsági mátrix:} A szomszédsági mátrix egy n x n méretű mátrix, ahol n a gráf csúcsainak száma. Az i. sor és j. oszlop eleme az i. és j. csúcs közötti élek jelenlétét vagy súlyát jelzi. Ha az élek irányítottak, akkor a mátrix nem szimmetrikus. Ez a reprezentáció hatékonyan tárolja a gráf szerkezetét, de erőforrásigénye négyzetes arányban nő a csúcsok számával. + \item \textit{Éllista:} Az éllista egy olyan lista, amely az összes él adatait tartalmazza. Minden élhez tartozik a kezdőcsúcs, a végcsúcs és esetleges további tulajdonságok, például súly. Ez a reprezentáció kevés memóriát igényel, de a gráf szerkezetének lekérdezésekor időigényesebb lehet. + \item \textit{Szomszédsági lista:} A szomszédsági lista minden csúcshoz tartozó listát tartalmaz, amelyben felsorolják a csúcs közvetlenül szomszédos csúcsait vagy éleit. Ez a reprezentáció általában hatékonyabb a ritka gráfoknál, mivel csak a ténylegesen szomszédos csúcsokat tárolja. Azonban a gráf szerkezetének lekérdezésekor lineáris időigényű lehet. + \item \textit{Incidencia mátrix:} Az incidencia mátrix egy n x m méretű mátrix, ahol n a csúcsok száma, m pedig az élek száma. Az i. sor és j. oszlop eleme 1, ha az i. csúcs érintett az j. élben, különben 0 vagy más jelölés lehet. Ez a reprezentáció hatékonyan tárolja a gráf szerkezetét és az élek attribútumait, de az erőforrásigénye arányos a csúcsok és élek számával. +\end{itemize} + + \end{document} \ No newline at end of file diff --git a/15. Adatszerkezetek és adattípusok/img/abrazolas.png b/15. 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