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Chap 12 Indexing and Hashing

Chap 12 Indexing and Hashing. 中文版:第 8 章 “索引和雜湊”. Chapter 12: Indexing and Hashing. Basic Concepts Ordered Indices B + -Tree Index Files B-Tree Index Files Static Hashing Dynamic Hashing Comparison of Ordered Indexing and Hashing Index Definition in SQL Multiple-Key Access.

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Chap 12 Indexing and Hashing

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  1. Chap 12Indexing and Hashing 中文版:第8章 “索引和雜湊”

  2. Chapter 12: Indexing and Hashing • Basic Concepts • Ordered Indices • B+-Tree Index Files • B-Tree Index Files • Static Hashing • Dynamic Hashing • Comparison of Ordered Indexing and Hashing • Index Definition in SQL • Multiple-Key Access

  3. 計算機的基本功能單元 Recall 算術與邏輯 輸入 記憶體 輸出 控制 中央處理器(CPU) I/O

  4. 指令的編碼格式 Recall 位移量 立即 定址模式 OP 碼

  5. Recall 指令的運作 (複雜指令型) m2 m3 int a, b, c; ….. a=b+c; ADDM ADDER a  m1 (3F55C2) b  m2 (3F55C4) c  m3 (3F55C6) m1 CPU 可以直接定址 16M 記憶體 的複雜指定集(CISC)處理器 ADDM m1, m2 , m3 e.g. 3B-22-A0-00-3F-55-C2-3F-55-C4-3F-55-C6

  6. Recall 指令的運作 (精簡指定型) int a, b, c; a=b+c; MOV ADD BX a  m1 (4EF55C2) b  m2 (4EF55C4) c  m3 (4EF55C6) ADDER CX m AR CPU MOV BX, m1 MOV CX, m2 ADD BX, CX MOV m1, AR 精簡指令型 或是 以有限狀態機或微程式工作的 複雜指定型之實際分解動作

  7. Recall Storage Hierarchy

  8. Recall Magnetic Hard Disk Mechanism 2007 Nobel AWD on Physics -- Albert Fert and Peter Grünberg Subject to “GMR” – Giant Magneto-resistance (巨磁電阻效應)

  9. Recall Storage Access • A database file is partitioned into fixed-length storage units called blocks. Blocks are units of both storage allocation and data transfer. • Database system seeks to minimize the number of block transfers between the disk and memory. We can reduce the number of disk accesses by keeping as many blocks as possible in main memory. • Buffer– portion of main memory available to store copies of disk blocks. • Buffer manager – subsystem responsible for allocating buffer space in main memory. IBM: DB2 server = Hardware + OS + Database (另例: Google) 如果非專用伺服器,可想像成每次磁碟重組後所保持的最佳狀態

  10. Recall Free Lists in File Organization • Store the address of the first deleted record in the file header. • Use this first record to store the address of the second deleted record, and so on • Can think of these stored addresses as pointerssince they “point” to the location of a record. • More space efficient representation: reuse space for normal attributes of free records to store pointers. (No pointers stored in in-use records.)

  11. Sequential Records in File • Deletion – use pointer chains • Insertion –locate the position where the record is to be inserted • if there is free space insert there • if no free space, insert the record in an overflow block • In either case, pointer chain must be updated • Need to reorganize the file from time to time to restore sequential order

  12. Chapter 12: Indexing and Hashing • Basic Concepts • Ordered Indices • B+-Tree Index Files • B-Tree Index Files • Static Hashing • Dynamic Hashing • Comparison of Ordered Indexing and Hashing • Index Definition in SQL • Multiple-Key Access

  13. Example: Dense Index Files • Dense index — Index record appears for every search-key value in the file. Linked List, Array, OR B+ Tree • Index on “Table’s Search Key”: 不必每列資料都指到; • 讓 index entries 結構性地整批讀入Buffer 之中協助工作; • 觀察:連 Join 動作都變快速了!(merge sort)

  14. Basic Concepts • Indexing mechanisms used to speed up access to desired data. • E.g., author catalog in library • Search Key - attribute to set of attributes used to look up records in a file. • An index fileconsists of records (called index entries) of the form • Index files are typically much smaller than the original file • Two basic kinds of indices: • Ordered indices: search keys are stored in sorted order • Hash indices: search keys are distributed uniformly across “buckets” using a “hash function”. 在 Table/ Index file 之中,都如此稱之 pointer search-key

  15. Index Evaluation Metrics • Access types supported efficiently. E.g., • records with a specified value in the attribute (特定值) • or records with an attribute value falling in a specified range of values. (特定範圍) • Access time • Insertion time • Deletion time • Space overhead

  16. Ordered Indices • In an ordered index, index entries are stored sorted on the search key value. E.g., author catalog in library. • Index-sequential file: ordered sequential file with a primary index. (the earliest idea; c.f. B+ Tree file) • Primary index: in a sequentially ordered file, the index whose search key specifies the sequential order of the file. • Also called clustering index • The search key of a primary index is usually but not necessarily the primary key. • Secondary index:an index whose search key specifies an order different from the sequential order of the file. Also called non-clustering index.

  17. Dense Index Files • Dense index — Index record appears for every search-key value in the file.

  18. Sparse Index Files • Sparse Index: contains index records for only some search-key values. • Applicable when records are sequentially ordered on search-key • To locate a record with search-key value K we: • Find index record with largest search-key value < K • Search file sequentially starting at the record to which the index record points

  19. Sparse Index Files (Cont.) • Compared to dense indices: • Less space and less maintenance overhead for insertions and deletions. • Generally slower than dense index for locating records. • Good tradeoff: sparse index with an index entry for every block in file, corresponding to least search-key value in the block.

  20. Multilevel Index • If primary index does not fit in memory, access becomes expensive. • Solution: treat primary index kept on disk as a sequential file and construct a sparse index on it. • outer index – a sparse index of primary index • inner index – the primary index file • If even outer index is too large to fit in main memory, yet another level of index can be created, and so on. • Indices at all levels must be updated on insertion or deletion from the file.

  21. Multilevel Index (Cont.)

  22. Index Update: Deletion • If deleted record was the only record in the file with its particular search-key value, the search-key is deleted from the index also. (直接作用在資料列,Index只是輔助) • Single-level index deletion: • Dense indices – deletion of search-key:similar to file record deletion. • Sparse indices – • if an entry for the search key exists in the index, it is deleted by replacing the entry in the index with the next search-key value in the file (in search-key order). • If the next search-key value already has an index entry, the entry is deleted instead of being replaced.

  23. Index Update: Insertion • Single-level index insertion: • Perform a lookup using the search-key value appearing in the record to be inserted. • Dense indices – if the search-key value does not appear in the index, insert it. • Sparse indices – if index stores an entry for each block of the file, no change needs to be made to the index unless a new block is created. • If a new block is created, the first search-key value appearing in the new block is inserted into the index. • Multilevel insertion (as well as deletion) algorithms are simple extensions of the single-level algorithms

  24. Secondary Indices (Example) • Secondary indices have to be dense !!! • Index record points to a bucket that contains pointers to all the actual records with that particular search-key value. Secondary index on balance field of account (Primary Index 不需要bucket 因為它有”原始建檔鏈”為依靠。) (如果 Primary Index 又建在 Primary Key 則根本不重複,更無需bucket)

  25. Status-index S CITY-index Athens S1 London Smith 20 10 S2 20 Paris Jones 10 London 20 S3 Paris London Blake 30 30 S4 London Paris 20 Clark 30 Athens S5 Adams 30 Paris Other Possible Definition • Primary index : index on primary key. <e.g> s# • Secondary index: index on other field. <e.g> city • A given table may have any number of indexes. 參考書 (C. J. Date) 書上的另一種定義 注意:(1) Primary Index 強制定義在 Primary Key ; (2) 每筆都指(e.g. London),允許重複值,未設 bucket. (3) 容許重複值,此時看似方便,B+ tree 則麻煩且浪費。

  26. Primary and Secondary Indices • Indices offer substantial benefits when searching for records. • BUT: Updating indices imposes overhead on database modification --when a file is modified, every index on the file must be updated, • Sequential scan using primary index is efficient, but a sequential scan using a secondary index is expensive • Each record access may fetch a new block from disk • Block fetch requires about 5 to 10 milliseconds • versus about 100 nanoseconds for memory access

  27. Chapter 12: Indexing and Hashing • Basic Concepts • Ordered Indices • B+-Tree Index Files • B-Tree Index Files • Static Hashing • Dynamic Hashing • Comparison of Ordered Indexing and Hashing • Index Definition in SQL • Multiple-Key Access

  28. B+-Tree Index Files B+-tree indices are an alternative to indexed-sequential files. • Disadvantage of indexed-sequential files • performance degrades as file grows, since many overflow blocks get created. • Periodic reorganization of entire file is required. • Advantage of B+-treeindex files: • automatically reorganizes itself with small, local, changes, in the face of insertions and deletions. • Reorganization of entire file is not required to maintain performance. • (Minor) disadvantage of B+-trees: • extra insertion and deletion overhead, space overhead. • Advantages of B+-trees outweigh disadvantages • B+-trees are used extensively

  29. Example of a B+-tree B+-tree for account file (n = 3) • 此處設 n=3 以方便講解,想像 n=100 • 通常設 n= (block size)/ (entry size), 如此則 (block size) = n*(entry size) = (node size)

  30. B+-Tree Index Files (Cont.) A B+-tree is a rooted tree satisfying the following properties: • All paths from root to leaf are of the same length • Each node that is not a root or a leaf has between n/2 and n children.( n/2-1 ~ n-1 key values) • A leaf node has between(n–1)/2 and n–1 values • Special cases: • If the root is not a leaf, it has at least 2 children. • If the root is a leaf (that is, there are no other nodes in the tree), it can have between 0 and (n–1) values. 低於下限使合併進而減少樹高, 高於上限使分裂進而增大樹高。

  31. B+-Tree Node Structure • Typical node • Ki are the search-key values • Pi are pointers to children (for non-leaf nodes) or pointers to records or buckets of records (for leaf nodes). • The search-keys in a node are ordered K1 < K2 < K3 < . . .< Kn–1 • 本書範例,起始值令 i = 1 • 如用 C. J. Date 之定義,則會有 Ki-1≤ Ki的情形。

  32. Leaf Nodes in B+-Trees Properties of a leaf node: • For i = 1, 2, . . ., n–1, pointer Pi either points to a file record with search-key value Ki, or to a bucket of pointers to file records, each record having search-key value Ki. Only need bucket structure if search-key does not form a primary key. • If Li, Lj are leaf nodes and i < j, Li’s search-key values are less than Lj’s search-key values • Pn points to next leaf node in search-key order P1 K1 P2 K2 P3 *(Pi) = Ki

  33. Non-Leaf Nodes in B+-Trees • Non leaf nodes form a multi-level sparse index on the leaf nodes. For a non-leaf node with m pointers: • All the search-keys in the subtree to which P1 points are less than K1 • For 2  i  n – 1, all the search-keys in the subtree to which Pi points have values greater than or equal to Ki–1 and less than Ki • All the search-keys in the subtree to which Pn points have values greater than or equal to Kn–1 *(Pj) < Kj≤ *(Pj+1)

  34. Example of a B+-tree *(Pj) < Kj≤ *(Pj+1) *(Pi) = Ki Brighton Downtown Mianus Perryridge Redwood Round Hill B+-tree for account file (n = 3) P 所指為 H.D. 上的儲存位置 or say Binary file 的位元序位。

  35. Example of B+-tree • Leaf nodes must have between 2 and 4 values ((n–1)/2 and n –1, with n = 5). • Non-leaf nodes other than root must have between 3 and 5 children ((n/2 and n with n =5). • Root must have at least 2 children. B+-tree for account file (n = 5)

  36. Observations about B+-trees • Since the inter-node connections are done by pointers, “logically” close blocks need not be “physically” close (相接近). • The non-leaf levels of the B+-tree form a hierarchy of sparse indices. • The B+-tree contains a relatively small number of levels • Level below root has at least 2* n/2 values • Next level has at least 2* n/2 * n/2 values • .. etc. • If there are K search-key values in the file, the tree height is no more than  logn/2(K) • thus searches can be conducted efficiently. • Insertions and deletions to the main file can be handled efficiently, as the index can be restructured in logarithmic time (as we shall see). 建成樹狀的意義在此

  37. Queries on B+-Trees • Find all records with a search-key value of k. • N=root • Repeat • Examine N for the smallest search-key value > k. • If such a value exists, assume it is Ki. Then set N = Pi • Otherwise k Kn–1. Set N = Pn Until N is a leaf node • If for some i, key Ki = k follow pointer Pito the desired record or bucket. • Else no record with search-key value k exists.

  38. Queries on B+-Trees (Cont.) • If there are K search-key values in the file, the height of the tree is no more than logn/2(K). • A node is generally the same size as a disk block, typically 4 kilobytes • and n is typically around 100 (40 bytes per index entry). • With 1 million search key values and n = 100 • at most log50(1,000,000) = 4 nodes are accessed in a lookup. • Contrast this with a balanced binary tree with 1 million search key values — around 20 nodes are accessed in a lookup • above difference is significant since every node access may need a disk I/O, costing around 20 milliseconds | (Pi ,Ki)| ≈ 40 |node| = 40 n ≈ 4000 ≈ |block|

  39. Updates on B+-Trees: Insertion • Find the leaf node in which the search-key value would appear • If the search-key value is already present in the leaf node • Add record to the file • If necessary add a pointer to the bucket. • If the search-key value is not present, then • add the record to the main file (and create a bucket if necessary) • If there is room in the leaf node, insert (Pi, Ki) pair in the leaf node • Otherwise, split the node (along with the new (key-value, pointer) entry) as discussed in the next slide. Example on Page 41 Recall: *(Pi) = Ki

  40. Updates on B+-Trees: Insertion (Cont.) • Splitting a leaf node: • take the n (search-key value, pointer) pairs (including the one being inserted) in sorted order. Place the first n/2 in the original node, and the rest in a new node. • let the new node be p, and let k be the least key value in p. Insert (k,p) in the parent of the node being split. • If the parent is full, split it andpropagatethe split further up. • Splitting of nodes proceeds upwards till a node that is not full is found. • In the worst case the root node may be split increasing the height of the tree by 1. Result of splitting node containing Brighton and Downtown on inserting Clearview Next step: insert entry with (Downtown,pointer-to-new-node) into parent

  41. Updates on B+-Trees: Insertion (Cont.) (2) Insert “Clearview” (只有底層分裂) (1) Insert "Naples" (無結構變化)

  42. Updates on B+-Trees: Insertion (Cont.) n = 4 Downtown Mianus Redwood ….. (insert) Downtown Mianus Perryridge Redwood • 底層/中層 “分裂”動作: • 做一個 n+1大的Node, • 再把 P值兩等分切開, • 兩新Nodes 後半留空; • 注意— • 切兩等分,分段的 K值 • 將複製/搬移 到更上一層成為 新K值。 • Insert a new K • Create a new Root (if necessary) Perryridge Perryridge Redwood Downtown Mianus ….. (delete) Downtown Perryridge Redwood (3) Insert “Perryridge” (上傳造成中層分裂) then, delete “Mianus” or “any leaf K values” (向右合併刪除上層 K值)

  43. Updates on B+-Trees: Insertion (Cont.) Yoyodome (4) Insert “Flamingo” (上傳造成中層分裂) n = 4 Zypher Downtown Naples Perryridge Perryridge Redwood Downtown Eaglenet Mianus Naples Oldriver Brighton Clearview Downtown Flamingo Naples Perryridge Naples Yoyodome Perryridge Zypher Downtown Flamingo Perryridge Redwood Downtown Eaglenet Flamingo Mianus Naples Oldriver Brighton Clearview

  44. Updates on B+-Trees: Insertion (Cont.) (5) (5) Review: “中層”分裂搬移到上層的 K值在 Delete 過程的重新分配時,又 會被找出來填補合併。e.g. Delete “Redwood” and “Round Hill” (6) Insert “Flamingo” and “Eaglenet” 的連續動作 (Flamingo 會分裂而且上傳,但因 Mianus又要分列出去, 這個 Flamingo 會再傳到最上層 Root) 見下頁

  45. Updates on B+-Trees: Insertion (Cont.) Flamingo Perryridge Downtown Flamingo Mianus Mianus Redwood Downtown Redwood Round Hill Downtown Eaglenet Flamingo Mianus Perryridge Brighton Clearview (6) Insert “Flamingo” and “Eaglenet”

  46. Insertion in B+-Trees (Cont.) • Splitting a non-leaf node: when inserting (k,p) into an already full internal node N • Copy N to an in-memory area M with space for n+1 pointers and n keys • Insert (k,p) into M • Copy P1,K1, …, K n/2-1,P n/2 from M back into node N • Copy Pn/2+1,Kn/2+1,…,Kn,Pn+1 from M into newly allocated node N’ • Insert (Kn/2,N’) into parent N • Read pseudocode in book! Mianus Downtown Mianus Perryridge Downtown Redwood

  47. Break

  48. Updates on B+-Trees: Deletion (**) 再重新評估是否少於下限 Remove (Pi, Ki) from a leaf node, Or (Ki, Pi+1) from an inner node. Na in [n/2 , n] D M,R Adequate entries Too few entries Nb < n/2 Case (1) (2) Merge (3) Redistribute 不影響 Tree 結構, 什麼都不必更動, 各 Node, P, K 如舊。 借用(or 重分配)可視為 "先合併再分裂"動作, 做一個 n+1大的Node, 再把 P值兩等分切開, 兩新Node 後半留空; (a) 底層直接重分配, 最後,複製兩等分分段的 K值到上層替換; (b) 中層要先把原先搬到上層的K值搬回來重分配,最後,搬移新的分段 K值到上層替換。 向左右合併皆可,“中層”合併則必須把它們原先搬到上層的 K值搬回來(*)。合併時 Node 只保留左邊,而右邊的 Node 取消後,其上層 P要被取消: (a) P在上層Node之左側: 直接向左合併或借用-- (3) 注意 – 不可能是全列最左; (b) P在Node之中間或右側: 刪除相鄰左邊的 K 值(*)(**) 理由 – 右邊node內會殘留 比左 K 值還大的entry Example for (1)(2)(3): 課堂解說 Downtown Mianus Redwood

  49. Updates on B+-Trees: Deletion • Find the record to be deleted, and remove it from the main file and from the bucket (if present) • Remove (search-key value, pointer) from the leaf node if there is no bucket or if the bucket has become empty • If the node has too few entries due to the removal, and the entries in the node and a sibling fit into a single node, then merge siblings: • Insert all the search-key values in the two nodes into a single node (the one on the left), and delete the other node. • Delete the pair (Ki–1, Pi), where Pi is the pointer to the deleted node, from its parent, recursively using the above procedure. Example on Page 43

  50. Updates on B+-Trees: Deletion • Otherwise, if the node has too few entries due to the removal, but the entries in the node and a sibling do not fit into a single node, then redistribute pointers: • Redistribute the pointers between the node and a sibling such that both have more than the minimum number of entries. • Update the corresponding search-key value in the parent of the node. • The node deletions may cascade upwards till a node which has n/2 or more pointers is found. • If the root node has only one pointer after deletion, it is deleted and the sole child becomes the root.

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