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zjFusm*-Rt0ga>wTJ%Js!RRAek+O1yT9prG%GLl3L0GF#5q=!Tn8+8CB0H%=ABA)N! zE(arZbrWH&pL)RVy8{tZN_b0mcR65yb^&BvRp7Q~8&<#(JynkA#S#!G@M8dhNRfU_ zHW_2J5UCR2+YYwA|59mFq18TZ0{mJp0UrXc*4}*IBy&Q7(9VJPaFUN_w}~{$_|9p@enx33`U$(0D%bzGJgUN zh!mVilZ`)t6|^xLYXH^+iQs?$zA2pf093X%2CHxrK&+Lg72hp&`Jqzoi@;3xn*e z3nxxex0FLbk;K{FRvHRUJYj1q4LDkB8#W-o^n-XH*Vb|<)DHb2P?#MuBe2LFHbS7G zJ8TZ@Nnp0y0|7u|o6Nwf5O^7|RTd}`g(c2(Hp>T;!xGo1n`tm8a2VM(y8$%dc&KgV z;D9gesEb7r-^^`k4+gZi4S#@dA&C>7E#&~8*{)Nd9Ex~$*|u^RqH}MdArQbsb2|+J zLv5D@0a$4#4M{vLZEJhL=HYgk5fID{SrEWfYP((lOm4R&3;|$#8(sh!Vtb4LXh`DB zc&jW>;DE&)Gz?}3Ul?RN?qLWR6uW~j4A{`#-X09G-M4_cNX$0805ml5?76M-!GM6U zod$= 1$ +- *Composite search key* if $k > 1$ +- *unique key* if search key contains _candidate_ key of table +- index is stored as file +- *Clustered index*: Ordering of data is same as data entries + - key is known as *clustering key* + - Format 1 index is clustered index (Assume format 2 and 3 to be unclustered) + +== Tree based Index +- *root node* at level 0 +- Height of tree = no of levels of internal node +- *Leaf nodes* + - level h, where h is height of tree +- *internal nodes* store entries in form $(p_0, k_1, p_1, k_2, p_2, ..., p_n)$ + - $k_1 < k_2 < ... < k_n$ + - $p_i$ = disk page address +- *Order* of index tree + - Each non-root node has $m in [d, 2d]$ entries + - Root node has $m in [1, 2d]$ entries +- *Equality search*: At each _internal_ node $N$, find largest key $k_i$ in N, such that $k_i <= k$ + - if $k_i$ exists, go subtree $p_i$, else $p_0$ +- *Range search*: First matching record, and traverse doubly linked list +- *Min nodes at level* i is $2 times (d + 1)^(i-1), i >= 1$ +- *Max nodes at level* i is $(2d + 1)^(i)$ + +=== Operations (Right sibling first, then left) +=== Insertion ++ *Leaf node Overflow* + - Redistribute and then split + - *Split*: Create a new leaf $N$ with $d+1$ entries. Create a new index entry $(k, square.filled)$ where $k$ is smallest key in $N$ + - *Redistribute*: If sibling is not full, take from it. If given right, update right's parent pointer, else current node's parent pointer ++ *Internal node Overflow* + - Node has $2d+1$ keys. + - Push middle $(d+1)$-th key up to parent. +=== Deletion ++ *Leaf node* + - Redistribute then merge + - *Redistribution* + - Sibling must have $> d$ recordsto borrow + - Update parent pointers to right sibling's smallest key) + - *Merge* + - If sibling has $d$ entries, then merge + - Combine with sibling, and then remove parent node ++ *Internal Node Underflow* + - Let $N'$ be adjacent _sibling_ node of $N$ with $l, l > d$ entries + - Insert $(K, N' . p_i)$ into $N$, where $i$ is the leftmost(0) or rightmost entry(l) + - Replace $K$ in parent node with $N'.k_i$ + - Remove $(p_i, k_i)$ entry from $N'$ +=== Bulk Loading ++ Sort entries by search keys. ++ Load leaf pages with $2d$ entries ++ For each leaf page, insert index entry to rightmost parent page + + += Hash based Index +== Static Hashing +- Data stored in $N$ buckets, where hash function $h(dot)$ is used to id bucket + - record with key $k$ is inserted into $B_i, "where" i = h(k) mod N$ +- Bucket is primary data page with 0+ overflow data pages +== Linear Hashing +- Grows linearly by splitting buckets + - Systematic splitting: Bucket $B_i$ is split before $B_(i+1)$ +- Let $N_i = 2^i N_0$ be file size at beginning of round $i$ +- How to split bucket $B_i$ + - Add bucket $B_j$ (split image of $B_i$) + - Redistribute entries in $B_i$ between $B_i$ and $B_j$ + - `next++; if next == NLevel: (level++; next = 0)` +=== Performance +- Average: 1.2 IO for uniform data +- Worst Case: Linear in number of entries + +== Extensible Hashing +- Overflowed bucket is resolved by splitting overflowed bucket +- No overflow pages, and order in which buckets are split is random +- Directory of pointers to buckets, directory has $2^d$ entries + - $d$ is global depth of hashed file + - Each bucket maintains a local depth $l in [0, d]$ + - Entries in a bucket of local depth $l$: same last $l$ bits +=== Bucket Overflow +- Number of directory entries could be more than number of buckets +- Number of dir entries pointing to bucket = $2^(d-l)$ +- When bucket $B$ with depth $l$ overflows, + - Increment local depth of $B$ to $l+1$ + - Allocate split image $B'$ + - Redistribute entries between $B$ and $B'$ using $(l+1)$th bit +- if $l+1 > "global depth " d$ + - Directory is doubled in size, , global depth to $d+1$ + - New entries point to same bucket as corresponding entry +- if $l+1 <= "global depth " d$ + - Update dir entry corresponding to split bucket's directory entry to point to split image + +=== Bucket Deletion +- $B_i$ & $B_j$(with same local depth $l$ and differ only in $l$th bit) can be merged if entries fit bin bucket + - $B_i$ is deallocated, $B_j$'s local depth decremented by 1. Directory entries that point to $B_i$ points to $B_j$ +=== Performance +- At most 2 disk IOs for equality selection +- Collisions: If they have same hashed value. + - Need overflow pages if collisions exceed page capacity + +#colbreak() += Sorting +== Notation +#table( + columns: (auto, auto), + $|r|$, [pages for R], + $||r||$, [tuples in r], + $pi_L (R)$, [project column by list $L$ from $R$], + $pi_L^* (R)$, [project with duplicates], + + $b_d$, [Data records that can fit on page], + $b_i$, [Data entries that can fit on page], + $b_r$, [RIDs that can fit on page], + + +) +== External Merge Sort +- *File size*: $N$ pages +- Memory pages available: $B$ +- *Pass 0*: Create sorted runs + - Read and sort $B$ pages at a time +- *Pass i*: Use $B-1$ pages for input, 1 for output, performing $B-1$-way merge sort +- *Analysis* + - Sorted runs: $N_0 = ceil(N/B)$ + - Total passes: $ceil(log_(B-1) (N_0))+1$ + - Total I/O: $2 N (ceil(log_(B-1) (N_0))+1)$ +=== Optimized Merge Sort +- Read and write in blocks of $b$ pages + - Allocate 1 Block for output + - Remaining memory for input: $floor(B/b)-1$ blocks +- *Analysis* + - sorted runs: $N_0 = ceil(N/B)$ + - Runs Merged at each pass $F = floor(B/b)-1$ + - No of merge passes: $ceil(log_F (N_0))$(+1 for total) + - Total IO: $2 N (ceil(log_F (N_0))+1)$ +- *Sorting with B+ Trees*: IO Cost: $h$ + Scan of leaf pages + Heap access (If not covering index) + +== Projection +=== Sort based approach +- Extract attributes, Sort attributes, remove duplicates +- *Analysis* + + Extract Attributes: $|R|"(scan)" + |pi_L^*(R)| "(output)"$ + + Sort Attributes: + - $N_0 = ceil((|pi_L^*(R)|)/B)$ + - Merging Passes: $log_(B-1) (N_0)$ + - Total IO: $2 |pi_L^*(R)| (log_(B-1) (N_0)+1)$ + + Remove Duplicates: $|pi_L^*(R)|$ +=== Optimized approach +- Merge Split step 2 into Creating and Merging sorted runs, and merge into step 1 and 3 respectively +- *Analysis* + - *Step 1* + - $B-1$ pages for initial sorted run + - Sorted Runs: $N_0 = ceil((|pi^*_L (R)|) / (B-1))$ + - Create sorted run = $|R| + |pi^*_L (R)|$ + - *Step 2* + - Merging passes: $ceil(log_(B-1) (N_0))$ + - Cost of merging: $2 |pi^*_L (R)| ceil(log_(B-1) (N_0))$ + - Cost of merging excluding IO output: $(2 ceil(log_(B-1) (N_0))-1) |pi^*_L (R)| $ + +=== Hash based approach +- *Partitioning* + - Allocate 1 page for input, $B-1$ page for output. + - Read 1 page at a time, for each tuple, create projection, hash($h$) to distribute to $B-1$ buffers + - Flush to disk when full. +- *Duplicate Elimination* + - For each partition $R_i$, create hash table, hash each tuple with hash function $h' != h$ to bucket $B_j$ if $t in.not B_j$ +- *Partition Overflow*: hash table for $pi^*_L (R_i)$ is larger than memory pages allocated for $pi_L (R)$ + +- *Analysis* + - IO Cost (no partition overflow) : $|R| + 2|pi^*_L (R)|$ + - Partitioning Phase: $|R| + |pi^*_L (R)|$ + - Duplicate Elimination: $|pi^*_L (R)|$ + - To Avoid partition overflows: + - $|R_i| = (|pi^*_L (R)|) / (B-1)$ + - $B > "size of hash table", |R_i| times f$ + - $B > sqrt(f times |pi^*_L (R)|)$ + += Selection +- *Conjunct*: $1>=$ terms connected by $or$ +- *CNF predicate*: $1>=$ conjuncts connected by $and$ +- *Covered Conjunct* - predicate $p_i$ is covered conjunct if each attribute in $p_i$ is in key $K$ or include column of Index $I$ + - $p = ("age" > 5) and ("height" = 180) and ("level" = 3)$ + - $I_1 "key" = ("level", "weight", "height")$ + - $p_c "wrt" I_1 = ("height" = 180) and ("level" = 3)$ +- *Primary Conjunct* + - $I$ matches $p$ if attributes in $p$ form prefix of $K$ and all comparison operators are equality except last + - $p_p$ is largest subset of conjuncts in $p$ such that $I$ matches $p_p$ +- $sigma_p (R)$: Select rows from $R$ that satisfy predicate $p$ +- Access Path: way of accessing data records / entries + - *Table Scan*: Scan all data pages (Cost: $|R|$) + - *Index Scan*: Scan index pages + - *Index Combination*: Combine from multiple index scans + - Scan/Combination can be followed by RID lookup to retrieve data +- *Index only plan*: Query where it does not need to access any data tuples in $R$ +- *Covering Index*: $I$ is covering index if all of $R$s attribute in query is part of the key / include columns of $I$ + +== B+ Trees +- For Index Scan + RID Lookup, many matching RIDs could refer to same page + - Sort matching RIDs before performing lookup: Avoid retrieving same page + +=== Analysis +#{ +let nin = [$N_"internal"$] +let nle = [$N_"leaf"$] +let nlo = [$N_"lookup"$] +let nso = [$N_"sort"$] +let nco = [$N_"combine"$] +[ +Cost of index scan = $nin + #nle + nlo$ +- *#nin*: No of internal nodes accessed + - Height of B+ tree index + - $ "height(est)" = cases( + ceil(log_F (ceil( (||R||) / b_d))) &"if index is clustered", + ceil(log_F (ceil( (||R||) / b_i))) &"otherwise", +) $ +- *#nlo*: Data pages accessed for RID lookups + - If $I$ is covering index for $sigma_p (R), nlo = 0$ + - else $nlo = ||sigma_p_c (R)||$ + - If matching RIDs are sorted before RID lookup + - $nlo = nso + min{||sigma_p_c (R)||, |R|}$ +- *#nso*: sorting matching RIDs + - $nso = 0 "if" ceil( (||sigma_p_c (R)||) / b_r ) <= B$ (if RIDs can fit into $B$) + - $ nso = 2 ceil((||sigma_p_c (R)||) / b_r) ceil(log_(B-1) (N_0)), N_0 = ceil(ceil((||sigma_p_c (R)||) / b_r) / B) $ + - Sorting with External Merge Sort + - #nso does'nt include read IO for pass 0 as its included in #nin and #nle + - #nso does'nt incldue write IO for final merging pass as RID is used for lookup +- *#nle*: Leaf pages scanned for evaluating $sigma_p (R)$ + - $nle = ceil((||sigma_p_p (R)||)/b_d)$ if clustered + - $nle = ceil((||sigma_p_p (R)||)/b_i)$ if unclustered +- *Index Combination* +- Cost = $nin^p + nle^p + nin^q + nle^q + nco + nlo$ + - #nco: IO cost to compute join of $pi_p$ $pi_q$ + - If $min{|pi_X_p (S_p)|, |pi_X_q (S_q)|} <= B$ + - One of the join operands can fit in mem, then $nco = 0$ +== Hash based Index Scan +- Cost: $N_"dir" + N_"bucket" + nlo$ + - $N_"dir"$: no of directory pages accessed (1 if extensible hash, 0 otherwise) + - $N_"bucket"$: max no of index's primary/overflow pages accessed + - $nlo= nso + min{||sigma_p_c (R)||, |R|}$ if I is not covering index for $sigma_p (R)$ +]}