From 1954078478a588f0ce92766d298632d64b911dfd Mon Sep 17 00:00:00 2001 From: Robert Jeutter Date: Fri, 21 May 2021 08:46:20 +0200 Subject: [PATCH] Kapitel 5 --- Assets/DBimpl-Baumverfahren.png | Bin 0 -> 36967 bytes Assets/DBimpl-LSM-baum.png | Bin 0 -> 49010 bytes Assets/DBimpl-b+-baum-blattebenen.png | Bin 0 -> 70155 bytes Assets/DBimpl-b+-baum-traversieren.png | Bin 0 -> 72256 bytes Assets/DBimpl-b-baum-beispiel-2.png | Bin 0 -> 89359 bytes Assets/DBimpl-b-baum-beispiel-3.png | Bin 0 -> 61618 bytes Assets/DBimpl-b-baum-beispiel.png | Bin 0 -> 51833 bytes Assets/DBimpl-b-baum-lookup.png | Bin 0 -> 55906 bytes Assets/DBimpl-csb-baum.png | Bin 0 -> 8924 bytes Assets/DBimpl-präfix-b-baum.png | Bin 0 -> 30737 bytes Assets/Dbimpl-b+-baum-aufbau.png | Bin 0 -> 41363 bytes Assets/Dbimpl-mehr-attribut-b-baum.png | Bin 0 -> 34165 bytes Datenbank Implementierungstechniken.md | 310 +++++++++++++++++++++++-- 13 files changed, 292 insertions(+), 18 deletions(-) create mode 100644 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Sonst nächste freie Seite holen @@ -894,11 +894,11 @@ Operationen ### Sequenzielle Speicherung - sortiertes Speichern der Datensätze - | | | | | - | --- | --- | --- | --- | --- - 4711 | Gamma | Delta | ... | 2.5.1945 - 5588 | Beta | Alpha | ... | 7.3.1978 - 8832 | Max | Mustermann | ... | 9.1.2003 + | | | | | + | ---- | ----- | ---------- | --- | -------- | + | 4711 | Gamma | Delta | ... | 2.5.1945 | + | 5588 | Beta | Alpha | ... | 7.3.1978 | + | 8832 | Max | Mustermann | ... | 9.1.2003 | Sequenzielle Datei: Operationen - insert: Seite suchen, Datensatz einsortieren => beim Anlegen oder sequenziellen Füllen einer Datei jede Seite nur bis zu gewissem Grad (etwa 66%) füllen @@ -980,4 +980,278 @@ Probleme statischer Verfahren - Erweiterung von Hashverfahren um Anpassung des Bildbereichs = dynamische Hashverfahren - Behandlung von zusammengesetzten Schlüsseln = multidimensionale Zugriffsverfahren, z.B. multidimensionale Bäume oder raumfüllende Kurven -# Baumbasierte Indexstrukturen \ No newline at end of file +# Baumbasierte Indexstrukturen +## Baumverfahren +- Stufenanzahl dynamisch verändern +- wichtigste Baumverfahren: B-Bäume und ihre Varianten +- B-Baum-Varianten sind noch "allgegenwärtiger" in heutigen Datenbanksystemen als SQL +- SQL nur in der relationalen und objektrelationalen Datenbanktechnologie verbreitet; B-Bäume überall als Grundtechnik eingesetzt + +Baumverfahren: Überblick +![](Assets/DBimpl-Baumverfahren.png) + +## B-Bäume +- Ausgangspunkt: ausgeglichener, balancierter Suchbaum +- Ausgeglichen oder balanciert: alle Pfade von der Wurzel zu den Blättern des Baumes gleich lang +- Hauptspeicher-Implementierungsstruktur: binäre Suchbäume, beispielsweise AVL-Bäume von Adelson-Velskii und Landis +- Datenbankbereich: Knoten der Suchbäume zugeschnitten auf Seitenstruktur des Datenbanksystems +- mehrere Zugriffsattributwerte auf einer Seite +- Mehrwegebäume + +Prinzip des B-Baumes +- B-Baum von Bayer (B für balanciert, breit, buschig, Bayer, NICHT: binär) +- dynamischer, balancierter Indexbaum, bei dem jeder Indexeintrag auf eine Seite der Hauptdatei zeigt +- Mehrwegebaum völlig ausgeglichen, wenn + 1. alle Wege von Wurzel bis zu Blättern gleich lang + 2. jeder Knoten gleich viele Indexeinträge +- vollständiges Ausgleichen zu teuer, deshalb B-Baum-Kriterium: + Jede Seite außer der Wurzelseite enthält zwischen $m$ und $2m$ Einträge. + +Definition B-Baum +- Ordnung eines B-Baumes ist minimale Anzahl der Einträge auf den Indexseiten außer der Wurzelseite +- Bsp.: B-Baum der Ordnung 8 fasst auf jeder inneren Indexseite zwischen 8 und 16 Einträgen +- Def.: Ein Indexbaum ist ein B-Baum der Ordnung $m$ , wenn er die folgenden Eigenschaften erfüllt: + 1. Jede Seite enthält höchstens $2m$ Elemente. + 2. Jede Seite, außer der Wurzelseite, enthält mindestens $m$ Elemente. + 3. Jede Seite ist entweder eine Blattseite ohne Nachfolger oder hat $i+1$ Nachfolger, falls $i$ die Anzahl ihrer Elemente ist. + 4. Alle Blattseiten liegen auf der gleichen Stufe. + +Eigenschaften des B-Baumes +- $n$ Datensätze in der Hauptdatei => in $log_m(n)$ Seitenzugriffen von der Wurzel zum Blatt +- Durch Balancierungskriterium wird Eigenschaft nahe an der vollständigen Ausgeglichenheit erreicht (1. Kriterium vollständig erfüllt, 2. Kriterium näherungsweise) +- Kriterium garantiert 50% Speicherplatzausnutzung +- einfache, schnelle Algorithmen zum Suchen, Einfügen und Löschen von Datensätzen (Komplexität von $O(log_m(n))$) + +Suchen in B-Bäumen +- lookup wie in statischen Indexverfahren +- Startend auf Wurzelseite Eintrag im B-Baum ermitteln, der den gesuchten Zugriffsattributwert $w$ überdeckt => Zeiger verfolgen, Seite nächster Stufe laden +- Suchen: 38, 20, 6 +- ![](Assets/DBimpl-b-baum-lookup.png) + +Einfügen in B-Bäumen +- Einfügen eines Wertes _w_ + - mit lookup entsprechende Blattseite suchen + - passende Seite $n<2m$ Elemente, _w_ einsortieren + - passende Seite $n=2m$ Elemente, neue Seite erzeugen, + - ersten _m_ Werte auf Originalseite + - letzten _m_ Werte auf neue Seite + - mittleres Element auf entsprechende Indexseite nach oben + - eventuell dieser Prozess rekursiv bis zur Wurzel + +Einfügen in einen B-Baum: Beispiel +![](Assets/DBimpl-b-baum-beispiel.png) +![](Assets/DBimpl-b-baum-beispiel-2.png) + +Löschen in B-Bäumen +- bei weniger als $m$ Elementen auf Seite: Unterlauf +- Löschen eines Wertes $w$ : Bsp.: 24; 28, 38, 35 + - mit lookup entsprechende Seite suchen + - $w$ auf Blattseite gespeichert => Wert löschen, eventuell Unterlauf behandeln + - $w$ nicht auf Blattseite gespeichert => Wert löschen, durch lexikographisch nächstkleineres Element von einer Blattseite ersetzen, eventuell auf Blattseite => Unterlauf +- Unterlaufbehandlung + - Ausgleichen mit der benachbarten Seite (benachbarte Seite $n$ Elemente mit $n>m$) + - oder Zusammenlegen zweier Seiten zu einer (Nachbarseite $n=m$ Elemente), das "mittlere" Element von Indexseite darüber dazu, auf Indexseite eventuell => Unterlauf +- Einfügen des Elementes 22; Löschen von 22 + ![](Assets/DBimpl-b-baum-beispiel-3.png) + +Komplexität der Operationen +- Aufwand beim Einfügen, Suchen und Löschen im B-Baum immer $O(log_m(n))$ Operationen +- entspricht genau der "Höhe" des Baumes +- Konkret: Seiten der Größe 4 KB, Zugriffsattributwert 32 Bytes, 8-Byte-Zeiger: zwischen 50 und 100 Indexeinträge pro Seite; Ordnung dieses B-Baumes 50 +- 1.000.000 Datensätze: $log_{50}(1000000) = 4$ Seitenzugriffe im schlechtesten Fall +- Wurzelseite jedes B-Baumes normalerweise im Puffer: 3 Seitenzugriffe + +## B+-Baum +Varianten +- B+-Bäume: nur Blattebene enthält Daten -> Baum ist hohl +- B∗-Bäume: Aufteilen von Seiten vermeiden durch "Shuffle" +- Präfix-B-Bäume: Zeichenketten als Zugriffsattributwerte, nur Präfix indexieren + +B+-Baum: Motivation +- B-Baum: Wie/wo werden Nicht-Schlüsseldaten (Tupel, TIDs) gespeichert? + - Zusammen mit Schlüsseln in allen Knoten? + - Problem beim Traversieren des Baumes, z.B. bei Bereichsanfragen + ![](Assets/DBimpl-b+-baum-traversieren.png) + +B+-Baum +- in der Praxis am häufigsten eingesetzte Variante des B-Baumes: + - effizientere Änderungsoperationen, insb. Löschen + - Verringerung der Baumhöhe +- Änderungen gegenüber B-Baum + - in inneren Knoten nur noch Zugriffsattributwert und Zeiger auf nachfolgenden Seite der nächsten Stufe; nur Blattknoten enthalten neben Zugriffsattributwert die Daten (Datensätze bzw. Verweise auf Datensätze in der Hauptdatei) + - Knoten der Blattebene sind untereinander verkettet für effiziente Unterstützung von Bereichsanfragen + +B+-Baum: Aufbau +![](Assets/Dbimpl-b+-baum-aufbau.png) + +Ordnung; Operationen +- Ordnung für B+-Baum: $(x,y), x$ Mindestbelegung der Indexseiten, $y$ Mindestbelegung der Blattseiten +- delete gegenüber B-Baum effizienter ("Ausleihen" eines Elementes von der Blattseite entfällt) +- Zugriffsattributwerte in inneren Knoten können sogar stehenbleiben +- häufig als Primärindex eingesetzt +- B+-Baum ist dynamische, mehrstufige, indexsequenziellen Datei + +B+-Baum: Blattebene mit Verweisen +![](Assets/DBimpl-b+-baum-blattebenen.png) + +B+-Baum: Datenstrukturen +```cpp +BPlusBranchNode =record of + nkeys: int; + ptrs: array[0 .. nkeys ] of PageNum; + keys: array[1 .. nkeys ]of KeyType; + level: int; +/* Level= 1 zeigt an, dass die ptrs auf Blätter zeigen */ +end; +``` +```cpp +BPlusLeafNode = record of + nkeys: int; + keys: array[1 .. nkeys ]of KeyType; + payload: array[1 .. nkeys ]of LoadType ; /* Daten bzw. TIDs */ + nextleaf: PageNum; +end; +``` + +Operationen im B+-Baum +- lookup: wie im B-Baum jedoch immer bis zur Blattebene +- $search(u,o)$: Lookup für unteren Wert $u$ , Traversieren auf der Blattebene bis zum oberen Wert $o$ +- $insert$: ähnlich zum B-Baum + - im Fall des Split bei Überlaufbehandlung wird nur ein Separatorschlüssel im Elternknoten eingefügt + - z.B. Kopie des kleinsten Schlüsselwertes des "rechten" Kindknotens oder geeigneter Wert zwischen beiden Knoten +- $delete$: ähnlich zum B-Baum, jedoch + - Löschen der Daten zunächst nur auf Blattebene + - bei Unterlauf: entweder Ausgleich mit Nachbarknoten oder Vereinigen mit Nachbarknoten; ggf. Anpassen (Ausgleich) oder Löschen (Vereinigen) des Separatorschlüssels + - alternativ: Unterlauf akzeptieren und durch spätere inserts oder Reorganisation auflösen + +Weitere Entwurfsentscheidungen für B+-Baum +- Schlüsselsuche im Knoten: sequenzielle Suche vs. binäre Suche vs. Interpolation Search +- Knotengröße: Maximieren der Anzahl der Vergleiche (Knotennutzen) pro I/O-Zeiteinheit + +| Seitengröße (KB) | Sätze/Seite | Knotennutzen | I/O-Zeit (ms) | Nutzen/Zeit | +| ---------------- | ----------- | ------------ | ------------- | ----------- | +| 4 | 143 | ≈7 | 5,020 | 1,427 | +| 16 | 573 | ≈ 9 | 5,080 | 1,804 | +| 64 | 2.294 | ≈ 11 | 5,320 | 2,098 | +| 128 | 4.588 | ≈ 12 | 5,640 | 2,157 | +| 256 | 9.175 | ≈ 13 | 6,280 | 2,096 | +| 1024 | 36.700 | ≈ 15 | 10,120 | 1,498 | +| 4096 | 146.801 | ≈ 17 | 25,480 | 0,674 | +[Literatur](G. Graefe: Modern B-Tree Techniques, Foundations and Trends in Databases, 2010) + +- Konsistenzprüfung während der Traversierung + - mögliche Inkonsistenzen durch Speicherfehler, konkurrierende Änderungen + Implementierungsfehler, ... +- Cache-Optimierung: Knotenstruktur, Kompression (Präfix-/Suffix Truncation) +- Pointer Swizzling: zur Ersetzung der logischen Seitennummern durch physische Adressen +- Knotengröße: bei großen Puffern kann großer Teil der inneren Knoten im Puffer gehalten werden -> größere Knoten (im MB-Bereich) sinnvoll +- Verzögern des Splits bei Überlaufbehandlung: Ausgleichen mit Nachbarknoten soweit möglich; verbessert Auslastung der Knoten +- zusätzliche Verweise zwischen Knoten + - z.B. zwischen inneren Knoten der gleichen Ebene, zu Elternknoten, etc. + - ermöglichen Konsistenzchecks, aber erschweren Sperren für Nebenläufigkeitskontrolle + +## Weitere Varianten +Präfix-B-Baum +- B-Baum über Zeichenkettenattribut + - lange Schlüssel in inneren Knoten -> hoher Speicherplatzbedarf + - vollständige Schlüssel eigentlich nicht notwendig, da nur "Wegweiser" +- Idee: Verwaltung von Trennwerten -> Präfix-B-Baum + - in inneren Knoten nur Trennwerte, die lexikographisch zwischen den Werten liegen + - möglichst kurze Trennwerte, z.B. kürzester eindeutiger Präfix + ![](Assets/DBimpl-präfix-b-baum.png) + - aber: Beispiel "Vandenberg" und "Vandenbergh" + +Mehr-Attribut-B-Baum +- B-Baum ist eindimensionale Struktur, jedoch können mehrere Attribute als kompositer Schlüssel indexiert werden + `create index NameIdx on KUNDE(Name, Vorname)` +- allerdings: Attribute werden bei partial-match-Anfragen nicht gleich behandelt! +- Alternative: raumfüllende Kurven -> multidimensionale Indexstrukturen + ![](Assets/Dbimpl-mehr-attribut-b-baum.png) + +## Optimierungen für moderne Hardware +Optimierungspotential +- Verbesserung der Cache-Trefferrate + - Organisation der Datenstruktur entsprechend Cacheline (Größe, Anordnung der Daten) + - In-Place-Update im Hauptspeicher -> Cache-Invalidierung: verändertes Update Handling + - Pointer Swizzling +- Berücksichtigung der Storage-Eigenschaften + - SSD vs. HDD + - Bevorzugung sequentieller Schreiboperationen + - spezielle Berücksichtigung für Main-Memory-Indexe +- Synchronisation in Multicore-Umgebungen + - Lock/Latch-freie Operationen + + +CSB+-Baum +- =Cache Sensitive B+ Tree (Rao, Ross: Making B+-Trees Cache Conscious in Main Memory, SIGMOD 2000) +- "Cache-Freundlichkeit" durch + 1. Platzsparen im Knoten -> mehr relevante Daten im Cache + 2. Eliminieren von Zeigern für 1. und Reduzierung von Zeigerarithmetik +- Ansatz: veränderte Struktur der inneren Knoten + - Zeiger auf ersten Kindknoten + - alle Kindknoten eines Knotens sind in einem zusammenhängenden Speicherbereich (Knotengruppe) allokiert und werden über einenOffsetadressiert + ![](Assets/DBimpl-csb-baum.png) + +Bw-Baum +- = Buzzword Tree (Levandoski, Lomet, Sengupta: The Bw-Tree: A B-tree for New Hardware Platforms, ICDE 2013) +- Ziele: Cache-Freundlichkeit, Multicore-CPUs, Flash-Speichereigenschaften +- Techniken + - überwiegend Latch-freie Operationen, stattdessen atomare CAS-Instruktionen + - spezifische Struktur-Modifikationsoperationen: Folge von atomaren Modifikationen, Blink-ähnliche Struktur + - Delta-Updates und Log Structured Storage (LSS) + - Änderungen an Seiten/Knoten werden nicht direkt ausgeführt, sondern in Delta-Records pro Knoten erfasst + - keine Synchronisation für Zugriff auf Seiten notwendig + - Seite kann auch nach Änderung im Cache verbleiben + - Seiten + Delta Records werden periodisch konsolidiert + - Log Structured Storage -> später! + - Mapping-Tabelle: logische Seitennummern in + - (a) Offset im Flash-Speicher + - (b) Zeiger im Hauptspeicher + +## LSM-Baum +- = Log Structured Merge Tree (O’Neil, Cheng, Gawlick, O’Neil: The log-structured merge-tree (LSM-tree). Acta Informatica. 33 (4): 351-385, 1996) +- Ziel: höherer Schreibdurchsatz durch Eliminierung verstreuter In-Place-Updates +- Einsatz in diversen NoSQL-Systemen: HBase, Cassandra, BigTable, LevelDB, RocksDB, ... +- Grundidee + - Batches von Schreiboperationen werden sequentiell in Indexdateien gespeichert, d.h. Sortieren vor Schreiben auf Externspeicher (Log Structured) + - Neue Updates werden in neuen Indexdateien gespeichert + - Indexdateien werden periodisch zusammengefügt (Merge) + - Leseoperationen müssen alle Indexdateien konsultieren +![](Assets/DBimpl-LSM-baum.png) + +LSM-Baum: Realisierung +- Main-Memory-Baum $C_0$ als Puffer, sortiert nach Schlüsseln, z.B. als AVL- oder RB-Baum + - bei Erreichen eines gegebenen Füllgrads -> Herausschreiben auf Disk (siehe unten) + - ergänzt um Write Ahead Logging auf Disk für Wiederherstellung nach Systemfehler +- mehrere Append-only, disk-basierte Indexe $C_1, C_2,...,$ ebenfalls sortiert nach Schlüssel (z.B. als B-Baum) + - effiziente Unterstützung von Scans (Schlüsselsuche) + - Merge in einem Schritt +- Aktualität der Indexe: $C_0,C_1,...,C_n$ + +LSM-Baum: Verdichtung +- wenn bestimmte Anzahl von Dateien erzeugt wurden (z.B. 5 Dateien je 100 Datensätze), werden diese in eine Datei gemischt (1 Datei mit 500 Sätzen) +- sobald 5 Dateien mit 500 Sätzen vorliegen, dann Mischen in eine Datei +- usw. +- Nachteil: große Anzahl von Dateien, die alle durchsucht werden müssen + +LSM-Baum: Ebenenweise Verdichtung +1. pro Ebene wird eine bestimmte Zahl von Dateien verwaltet, partitioniert nach Schlüsseln (keine Überlappung der Schlüssel) -> Suche nur in einer Datei notwendig +2. Ausnahme: erste Ebene (Überlappung erlaubt) +3. Mischen der Dateien jeweils in die nächsthöhere Ebene: Auswahl einer Datei und Aufteilen der Sätze -> Platz für neue Daten schaffen + +LSM-Baum: Lesezugriffe +- grundsätzlich: Verbesserung der Schreibperformance zulasten der Leseperformance + - Suche in allen Indexen $C_0,C_1,...,C_n$ notwendig +- Vermeiden unnötiger Lesevorgänge durch Bloom-Filter pro Index oder pro Run + - (probabilistische) Datenstruktur zum Feststellen, ob Objekt in einer Menge enthalten ist + - Bit-Feld: Objekt wird über $k$ Hashfunktionen auf $k$ Bits abgebildet, die auf 1 gesetzt werden + - Prüfen auf Enthaltensein: Hashfunktionen anwenden -> Wenn alle $k$ Bits $= 1$, dann ist Objekt enthalten + - aber falsch-positive Werte möglich! + - können ebenfalls durch Mischen kombiniert werden + - [Quelle](Bloom: Space/Time Trade-offs in Hash Coding with Allowable Errors. CACM, 13(7):422-426, 1970) + +## Zusammenfassung +- B+-Baum als "Arbeitspferd" für Indexing +- Standardoperationen: Suche, Einfügen, Löschen +- noch nicht betrachtet: Nebenläufigkeitskontrolle und Wiederherstellung im Fehlerfall +- diverse Varianten und Optimierungen +- LSM-Baum für schreibintensive Workloads