From 3746571f3095a73f6cf3120ff77b6b55347b8055 Mon Sep 17 00:00:00 2001 From: Robert Jeutter Date: Fri, 21 May 2021 11:15:49 +0200 Subject: [PATCH] Kapitel 6 --- Assets/DBimpl-erweiterbares-hashing-2.png | Bin 0 -> 47201 bytes Assets/DBimpl-erweiterbares-hashing-3.png | Bin 0 -> 44177 bytes Assets/DBimpl-erweiterbares-hashing.png | Bin 0 -> 34302 bytes Assets/DBimpl-grid-beispiel-1.png | Bin 0 -> 8196 bytes Assets/DBimpl-grid-beispiel-5.png | Bin 0 -> 19916 bytes Assets/DBimpl-grid-beispiel-6.png | Bin 0 -> 26178 bytes Assets/DBimpl-grid-files.png | Bin 0 -> 22951 bytes Assets/DBimpl-grid-start.png | Bin 0 -> 5159 bytes Assets/DBimpl-hashing-split.png | Bin 0 -> 43489 bytes Assets/DBimpl-lineares-hashen.png | Bin 0 -> 41112 bytes Assets/DBimpl-lineares-hashing-problem.png | Bin 0 -> 39537 bytes Assets/DBimpl-spiral-hashing-2.png | Bin 0 -> 20052 bytes Assets/DBimpl-spiral-hashing.png | Bin 0 -> 24366 bytes Assets/DBimpl-trie.png | Bin 0 -> 93594 bytes Assets/Dbimpl-grid-beispiel-2.png | 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Zugriff über Adressberechnung aus Schlüssel +- linearer Adressraum der Grösse $n$ + - Adressierung in einem Array +- Ziel: direkter Zugriff in $O(1)$ statt logarithmisch wie bei Bäumen + +Hashverfahren +- Schlüsseltransformation und Überlaufbehandlung +- DB-Technik: + - Disk-basiert: Bildbereich entspricht Seiten-Adressraum + - Hauptspeicher: Adresse in einem Array (Hauptspeicheradresse plus Offset) +- Dynamik: dynamische Hashfunktionen oder Re-Hashen + +Grundprinzipien +- Basis-Hashfunktion: $h(k)= k mod m$ + - $m$ oft Primzahl da besseres Verhalten bei Kollisionen + - oder $m=2^k$ aufgrund einfacher Berechnungen +- Überlauf-Behandlung + - Überlaufseiten als verkettete Liste + - lineares Sondieren + - quadratisches Sondieren + - doppeltes Hashen + - ... + +Hashverfahren für blockorientierte Datenhaltung +![](Assets/Dbimpl-hashverfahren.png) + +Operationen und Zeitkomplexität +- lookup, modify, insert, delete +- lookup benötigt maximal $1+ #B(h(w))$ Seitenzugriffe +- $#B(h(w))$ Anzahl der Seiten (inklusive der Überlaufseiten) des Buckets für Hash-Wert $h(w)$ +- Untere Schranke 2 (Zugriff auf Hashverzeichnis plus Zugriff auf erste Seite) + +Statisches Hashen: Probleme +- mangelnde Dynamik +- Vergrößerung des Bildbereichs erfordert komplettes Neu-Hashen +- Wahl der Hashfunktion entscheidend; + - Bsp.: Hash-Index aus 100 Buckets, Studenten über 6-stellige MATRNR (wird fortlaufend vergeben) hashen + - ersten beiden Stellen: Datensätze auf wenigen Seiten quasi sequenziell abgespeichert + - letzten beiden Stellen: verteilen die Datensätze gleichmäßig auf alle Seiten +- Sortiertes Ausgeben einer Relation nicht unterstützt + +## Hash-Funktionen +- klassisch, etwa Divisions-Rest-Methode + $h() = x mod m$ +- zusammengesetzt, etwa $h(k)= h_2 (h_1 (k))$ (siehe später Spriralhashen) +- ordnungserhaltend + $k_1 < k_2 => ( h(k_1) = h(k_2) \vee h(k_1) < h(k_2))$ +- dynamisch (siehe später) +- mehrdimensional (siehe später) +- materialisiert (etwa Dictionary Encoding, siehe später) + +Ordnungserhaltenes Hashen +- Schlüsselwerte werden als 8-Bit-Integer-Werte ohne Vorzeichen kodiert und sind gleichmässig im Bereich $0...2^8-1$ verteilt. +- Die Extraktion der ersten drei Bits ergibt eine ordnungserhaltende Hashfunktion für den Bereich $0...2^3-1$. +- Sind die Schlüsselwerte nicht gleichverteilt, etwa weil es sich um fortlaufend vergebene Nummern handelt, ist das Ergebnis zwar weiterhin ordnungserhaltend, aber die Hash-Tabelle ist sehr ungleichmäßig gefüllt. + +## Hardware-sensitives Hashen +Neue Hardware und Hash-Funktionen +- Beobachtung: Hashen mit klassischem Sondieren ungünstig für neue Hardware + - schwer parallelisierbar + - Clustern von Werten verletzt Nähe der Werte (bei Cache Lines) +- Varianten versuchen beide Punkte anzugehen + - Cuckoo-Hashing + - optimiertes lineares Sondieren + - Hopscotch-Hashing + - Robin-Hood-Hashing + +Cuckoo-Hashen +- Kuckucks-Hashen +- soll Parallelität erhöhen im Vergleich zu linearem Sondieren +- Idee: Zwei Tabellen mit zwei Hash-Funktionen + - im Fall einer Kollision in einer Tabelle wird in der zweiten Tabelle gesucht + - ist dort der Platz belegt, wird der dortige Eintrag verdrängt in die jeweils andere Tabelle + - _Kuckuck wirft Ei aus dem Nest_ + - dies wird solange gemacht bis ein freier Platz gefunden wird +- Beispiel + - zwei einfache Hash-Funktionen, die jeweils die letzte beziehungsweise vorletzte Dezimalstelle einer Zahl extrahieren + $h_1(k) = k mod 10$ + $h_2(k) = (k/10) mod 10$ + - Bei einer Suche muss immer in beiden Tabellen nachgeschaut werden, also $T_1[h_1(k)] = k\vee T_2[h_2(k)] = k$. + - Wir fügen die Zahlen $433, 129$ und $555$ in die Tabelle $T_1$ ein. Beim Einfügen von $783$ ist der Platz in Tabelle $T_1$ belegt, so dass diese Zahl in $T_2$ gespeichert werden muss. Wird nun mit $103$ eine weitere Zahl eingefügt, die mit $433$ unter $h_1$ kollidiert, ist dies mit $h_2$ weiterhin möglich. + +Cuckoo Beispiel +- Ergebnis des Einfügens von 433, 129 , 555 , 783 , 103 +| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | +| ----- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +| $T_1$ | | | | 433 | | 555 | | | | 129 | +| $T_2$ | 103 | | | | | | | | 783 | +- Wird nun die Zahl $889$ eingefügt, so sind beide möglichen Positionen belegt. $889$ kann in $T_1$ die dort stehende Zahl $129$ verdrängen, die in $T_2$ an der Position $T_2[2]$ gespeichert werden kann. +| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | +| ----- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +| $T_1$ | | | | 433 | | 555 | | | | 129 | +| $T_2$ | 103 | | 129 | | | | | | 783 | +- Wird nun $789$ eingefügt, sind wiederum beide Positionen belegt. Das Verdrängen von $889$ aus $T_1$ würde zu einem kaskadierenden Verdrängen führen: $889$ würde in $T_2$ dann $783$ verdrängen, das wiederum $433$ in $T_1$ verdrängen würde. Dies würde gehen da $433$ an der Stelle $T_2[3]$ Platz hätte +| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | +| ----- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +| $T_1$ | | | | 783 | | 555 | | | | 129 | +| $T_2$ | 103 | | 129 | 433 | | | | | 783 | + +Weitere Prinzipien der Optimierung +- Lokalität von Datenzugriffen verringert die Wahrscheinlichkeit von Cache Misses +- Blockung von Daten kann an die Grösse von Cache-Lines (64 Bytes) angepasst werden, und erhöht den Durchsatz +- Parallelisierung für SIMD basierend auf einer Vektorisierung der Daten kann insbesondere SIMD-basierte Co-Prozessoren gut ausnutzen, aber greift auch bei MICs + +Optimiertes lineares Sondieren +- Lineares Sondieren ist gut geeignet, um den Sondierungsvorgang auf Vektoren zu parallelisieren + - Suchschlüssel kann in einen Vektor der Länge $n$ an alle Positionen kopiert werden + - beginnend ab derem initialen Sondierungspunkt $h(k)$ können dann Vektoren jeweils mit Vektoren aus der Hash-Tabelle verglichen werden, also zuerst mit + $H[h)k),...,h(k)+n - 1]$, dann mit $H[h(k) + n,...,h(k) + 2n-1 ]$, etc. + - Vergleich kann parallel erfolgen; muss sowohl auf Vorhandensein von $k$ als auch auf Existenz einer leeren Position prüfen + +Hopscotch-Hashen +- Hopscotch: _Himmel und Hölle_ beziehungsweise _wild herumhopsen_ +- Begrenzung des Sondierungsraum auf eine (konstante) Länge +- Idee: + - beim Einfügen erfolgt die Suche (parallel) in der festen Nachbarschaft + - wird Schlüssel $k$ nicht gefunden und existiert kein freier Slot in der festen Nachbarschaft, dann wird versucht, $k$ mit einem anderen Schlüssel aus der festen Nachbarschaft zu tauschen + - dafür wird die nächste freie Stelle gesucht; von dieser wird rückwärts in Richtung $h(k)$ gesucht und jeder Eintrag $k′$ untersucht + - wenn die aktuelle freie Stelle noch in der festen Nachbarschaft von $k′$ liegt, wird getauscht: $k′$ springt auf die freie Stelle + +Robin-Hood-Hashen +- Robin-Hood: _Nimm von den Reichen gib es den Armen_ +- Basisidee: in der Situation, dass beim Sondieren für $k$ ein Platz bereits mit einem Element $k′$ besetzt ist, wird der nächste Sondierungsschritt mit demjenigen Element weitergeführt, das die kleinere Distanz zum eigentlichen Hash-Wert $h(k)$ bzw. $h(k′)$ hat + +## Dynamische Hash-Verfahren +Lineares Hashen +- Folge von Hash-Funktionen, die wie folgt charakterisiert sind: + - $h_i$: dom(Primärschlüssel) ->$\{0,..., 2^i \times N\}$ ist eine Folge von Hash-Funktionen mit $i\in\{0,1,2,...\}$ und $N$ als Anfangsgröße des Hash-Verzeichnisses + - Wert von $i$ wird auch als Level der Hash-Funktion bezeichnet + - $dom(Primärschlüssel)$ wird im folgenden als $dom(Prim)$ abgekürzt +- Für diese Hash-Funktionen gelten die folgenden Bedingungen: + - $h_{i+1}(w) = h_i(w)$ für etwa die Hälfte aller $w\in dom(Prim)$ + - $h_{i+1}(w) = h_i(w) + 2^i\times N$ für die andere Hälfte + - Bedingungen sind zum Beispiel erfüllt, wenn $h_i(w)$ als $w mod(2^i\times N)$ gewählt wird + - Darstellung durch Bit-Strings, Hinzunahme eines Bits verdoppelt Bildbereich + +Prinzip lineares Hashen +- für ein $w$ höchstens zwei Hash-Funktionen zuständig, deren Level nur um 1 differiert, Entscheidung zwischen diesen beiden durch Split-Zeiger + - $sp$ Split-Zeiger (gibt an, welche Seite als nächstes geteilt wird) + - $lv$ Level (gibt an, welche Hash-Funktionen benutzt werden) +- Aus Split-Zeiger und Level läßt sich die Gesamtanzahl $Anz$ der belegten Seiten wie folgt berechnen: + - $Anz = 2^{lv} + sp$ +- Beide Werte werden am Anfang mit 0 initialisiert. +![](Assets/DBimpl-lineares-hashen.png) + +Lookup +- $$s := h_{lv}(w)$; +- if $s < sp$ +- then $s := h_{lv + 1}(w)$; +- zuerst Hash-Wert mit der "kleineren" Hash-Funktion bestimmen +- liegt dieser unter dem Wert des Split-Zeigers => größere Hash-Funktion verwenden + +Splitten einer Seite +1. Die Sätze der Seite (Bucket), auf die $sp$ zeigt, werden mittels $h_{lv+1}$ neu verteilt (ca. die Hälfte der Sätze wird auf Seite (Bucket) unter Hash-Nummer $2^{lv}*N +sp$ verschoben) +2. Der Split-Zeiger wird weitergesetzt: $sp:=sp +1;$ +3. Nach Abarbeiten eines Levels wird wieder bei Seite 0 begonnen; der Level wird um 1 erhöht: + +``` +if sp = 2^{lv} * N then + begin + lv := lv + 1 ; + sp := 0 + end; +``` +![](DBimpl-hashing-split.png) + + +Problem lineares Hashen +![](Assets/DBimpl-lineares-hashing-problem.png) + +Erweiterbares Hashen +- Problem: Split erfolgt an fester Position, nicht dort wo Seiten überlaufen +- Idee: binärer Trie zum Zugriff auf Indexseiten +- Blätter unterschiedlicher Tiefe + - Indexseiten haben Tiefenwert + - Split erfolgt bei Überlauf +- aber: Speicherung nicht als Trie, sondern als Array + - entspricht vollständigem Trie mit maximaler Tiefe + - "shared" Seiten als Blätter + - Array der Grösse 2 _d_ für maximale Tiefe _d_ + - erfordert nun nur einen Speicherzugriff! + - bei Überlauf: Indexgrösse muss möglicherweise verdoppelt werden! +- Ausgangslage: + - Einfügen von 00111111 würde Überlauf bei erreichter maximaler Tiefe erzeugen + ![](Assets/DBimpl-erweiterbares-hashing.png) + - Verdopplung der Indexgrösse + ![](Assets/DBimpl-erweiterbares-hashing-2.png) + - nun möglich: Split der Seite + ![](Assets/DBimpl-erweiterbares-hashing-3.png) + +Variante: Array als Trie gespeichert +![](Assets/DBimpl-trie.png) + +Spiral-Hashen +- Problem: zyklisch erhöhte Wahrscheinlichkeit des Splittens +- Lösung: unterschiedliche Dichte der Hashwerte + - Interpretation der Bit-Strings als binäre Nachkommadarstellung einer Zahl zwischen $0.0$ und $1.0$ + - Funktion von $[0.0,1.0] -> [0.0,1.0]$ so dass Dichte gleichmässig verteilter Werte nahe $1.0$ doppelt so gross ist wie nahe $0.0$ +- Umverteilung mittels Exponentialfunktion +- Funktion $exp(n)$ $exp(n) = 2^n - 1$ erfüllt die Bedingungen +- insbesondere gilt $2^0 - 1 = 0$ und $2^1 - 1 = 1$ +- Hashfunktion exhash + $exhash(k) = exp(h(k)) = 2^{h(k)} - 1$ +- Wirkung der verwendeten Hashfunktion im Intervall $0.0$ bis $1.0$ + | $n$ | $2^n-1$ | + | ----- | ----------- | + | $0.0$ | $0.0$ | + | $0.1$ | $0.0717735$ | + | $0.2$ | $0.1486984$ | + | $0.3$ | $0.2311444$ | + | $0.4$ | $0.3195079$ | + | $0.5$ | $0.4142136$ | + | $0.6$ | $0.5157166$ | + | $0.7$ | $0.6245048$ | + | $0.8$ | $0.7411011$ | + | $0.9$ | $0.866066$ | + | $1.0$ | $1.0$ | +- Spiralförmiges Ausbreiten + - Ausgangslage: 4 Seiten der Tiefe 2 + ![](Assets/DBimpl-spiral-hashing.png) +- Spiralförmiges Ausbreiten + - Split der Seite mit der höchsten Dichte + - Ergebnis: 5 Seiten, davon 3 der Tiefe 2 und 2 der Tiefe 3 + ![](Assets/DBimpl-spiral-hashing-2.png) + +## Grid-File +Grid-Files +- bekannteste und von der Technik her attraktive mehrdimensionale Dateiorganisationsform +- eigene Kategorie: Elemente der Schlüsseltransformation wie bei Hashverfahren und Indexdateien wie bei Baumverfahren kombiniert + - deshalb hier bei Hash-Verfahren betrachtet + +Grid-File: Zielsetzungen +- Prinzip der 2 Plattenzugriffe: Jeder Datensatz soll bei einer _exact-match_ -Anfrage in 2 Zugriffen erreichbar sein +- Zerlegung des Datenraums in Quader: _n_ -dimensionale Quader bilden die Suchregionen im Grid-File +- Prinzip der Nachbarschaftserhaltung: Ähnliche Objekte sollten auf der gleichen Seite gespeichert werden +- Symmetrische Behandlung aller Raum-Dimensionen: _partial-match_ -Anfragen ermöglicht +- Dynamische Anpassung der Grid-Struktur beim Löschen und Einfügen + +Prinzip der zwei Plattenzugriffe +- bei exact-match + 1. gesuchtes _k_ -Tupel auf Intervalle der Skalen abbilden; als Kombination der ermittelten Intervalle werden Indexwerte errechnet; Skalen im Hauptspeicher => noch kein Plattenzugriff + 2. über errechnete Indexwerte Zugriff auf das _Grid-Directory_ ; dort Adressen der Datensatz-Seiten gespeichert; erster _Plattenzugriff_. + 3. Der Datensatz-Zugriff: zweiter _Plattenzugriff_. + +Aufbau eines Grid-Files +![](Assets/DBimpl-grid-files.png) +- Grid: $k$ eindimensionale Felder (Skalen), jede Skala repräsentiert Attribut +- _Skalen_ bestehen aus Partition der zugeordneten Wertebereiche in Intervalle +- Grid-Directory besteht aus Grid-Zellen, die den Datenraum in Quader zerlegen +- Grid-Zellen bilden eine Grid-Region, der genau eine Datensatz-Seite zugeordnet wird +- Grid-Region: $k$-dimensionales, konvexes (Regionen sind paarweise disjunkt) + +Operationen +- Zu Anfang: Zelle = Region = eine Datensatz-Seite +- Seitenüberlauf: + - Seite wird geteilt + - falls zugehörige Gridregion aus nur einer Gridzelle besteht, muss ein Intervall auf einer Skala in zwei Intervalle unterteilt werden + - besteht Region aus mehreren Zellen, so werden diese Zellen in einzelne Regionen zerlegt +- Seitenunterlauf: + - Zwei Regionen zu einer zusammenfassen, falls das Ergebnis eine neue, konvexe Region ergibt + +Beispiel +- Start-Grid-File + ![](Assets/DBimpl-grid-start.png) + - Datensätze einfügen: $(45,D),(2,B),(87,S),(75,M),(55,K),(3,Y),(15,D),(25,K),(48,F)$ + - jede Seite des Grid-Files fasst bis zu drei Datensätze +- Eingefügt: $(45, D), (2, B), (87, S)$ + ![](Assets/DBimpl-grid-beispiel-1.png) +- Einfügen von $(75, M)$ erzwingt Split + ![](Assets/Dbimpl-grid-beispiel-2.png) +- Eingefügt: $(55, K)$ + ![](Assets/Dbimpl-grid-beispiel-3.png) +- Einfügen von $(3, Y)$ erzwingt wiederum einen Split + ![](Assets/Dbimpl-grid-beispiel-4.png) +- Eingefügt: (15, D), (25, K), + ![](Assets/DBimpl-grid-beispiel-5.png) +- Einfügen von (48, F) erzwingt wiederum einen Split + ![](Assets/DBimpl-grid-beispiel-6.png) + + +Buddy-System +- Beschriebenes Verfahren: Buddy-System (Zwillings-System) +- Die im gleichen Schritt entstandenen Zellen können zu Regionen zusammengefasst werden; Keine andere Zusammenfassung von Zellen ist im Buddy-System erlaubt +- Unflexibel beim Löschen: nur Zusammenfassungen von Regionen erlaubt, die vorher als Zwillinge entstanden waren +- Beispiel: $(15,D)$ löschen: Seiten 1 und 4 zusammenfassen; $(87,S)$ löschen, Seite 2 zwar unterbelegt, kann aber mit keiner anderen Seite zusammengefasst werden +