Multiversion Concurrency Control

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Presentation transcript:

Multiversion Concurrency Control Chapter 5 송 인선

Contents 1. Goal and Overview 2. Multiversion Schedules 3. Multiversion Serializability 3.

1. Goal and Overview Main Idea of Multiversion Concurrency Control One copy of each data item Overwriting and read-from relationship Add more copy of each data item Keep the old version of a data item reasonable extension of serializability theory the presentation focus on.. Accurate Notation of Multiversion Concurrency Control Define the Multiversion Schedule, MCSR and MVSR Dealing with possible issues of Multiversion Concurrency Control

2. Multiversion Schedules Example 5.1 𝒔 𝟏 = r1(x) w1(x) r2(x) w2(y) r1(y) w1(z) c1 c2 𝒔 𝟏  CSR (cyclic conflict between t1 and t2) but, schedule would be tolerable if r1(y) could read the old version y0 of y to be consistent with r1(x), s would then be equivalent to a serial s‘ = t1 t2 𝒔 𝟏 𝒔 𝟐 T1 T2 𝑟𝑒𝑎𝑑 𝑥 𝑤𝑟𝑖𝑡𝑒 𝑥 𝑟𝑒𝑎𝑑(𝑦) 𝑤𝑟𝑖𝑡𝑒 𝑧 𝐶 1 𝑤𝑟𝑖𝑡𝑒 𝑦 𝐶 2 T1 T2 𝑟𝑒𝑎𝑑 𝑥 𝑤𝑟𝑖𝑡𝑒 𝑥 𝑟𝑒𝑎𝑑( 𝑦 0 ) 𝑤𝑟𝑖𝑡𝑒 𝑧 𝐶 1 𝑤𝑟𝑖𝑡𝑒 𝑦 𝐶 2 𝒔 𝟐 ∈ CSR

2. Multiversion Schedules Definition 5.1 < Version Function > Let s be a history with initial transaction t0 and final transaction t. A version function for s is a function h, which associates with each read step of s a previous write step on the same data item, and the identity for writes. h(ri(x)) = wj(x) for some wj(x) <s ri(x), and ri(x) reads xj (Read Step) h(wi(x)) = wi(x), and wi(x) writes xi. (Write Step)

2. Multiversion Schedules Definition 5.2 < Multiversion schedule > Let T = {t1, ... , tn} be a (finite) set of transactions. 1. A multiversion history for transactions T ={t1, ..., tn} is a pair m=(op(m), <m) where <m is an order on op(m) and op(m) = i=1..n h(op(ti)) for some version function h, for all t  T and all p, q  op(ti): p <t q  h(p) <m h(q), if h(rj(x)) = rj(xi), ij, and cj is in m then ci is in m and ci <m cj. 2. A multiversion schedule is a prefix of a multiversion history. Example 5.2 totally ordered multiversion schedule for the history from Example 5.1 with a version function h s = r1(x) w1(x) r2(x) w2(y) r1(y) w1(z) c1 c2 m = r1(x0) w1(x1) r2(x1) w2(y2) r1(y0) w1(z1) c1 c2 : h(r1(y0) ) = w0(y0)

2. Multiversion Schedules Definition 5.1 < Monoversion Schedule > A multiversion schedule is a monoversion schedule if its version function maps each read to the last preceding write on the same data item. Example m = r1(x0) w1(x1) r2(x1) w2(y2) r1(y2) w1(z1) c1 c2 s = r1(x) w1(x) r2(x) w2(y) r1(y) w1(z) c1 c2 effect of a monoversion schedule a single version of each data item is kept order of steps version function of a monoversion schedule is uniquely determined

3.1 Multiversion View Serializability Example 5.3 s = w0(x) c0 w1(x) c1 r2(x) w2(y) c2 : 3 different conflicts m = w0(x0) c0 w1(x1) c1 r2(x0) w2(y2) c2 : only 1 conflict Definition 5.4 < Reads-From Relation > Let m be a multiversion schedule, ti, tj ∈ trans(m). The reads-from relation of m is defined by RF(m) = {(ti, x, tj) | rj(xi)  op(m)}.

3.1 Multiversion View Serializability Definition 5.5 < View Equivalence > Let m and m` be two multiversion schedule such that trans(m`) = trans(m). m and m` are view equivalent, abbreviated m v m`, if RF(m) = RF(m`). difference between multiversion and monoversion of view equivalence : multiversion schedules m and m‘ with trans(m)=trans(m‘) always have the same final writes (‮since writes produce versions that are not erased anymore) consideration of final write operations is now irrelevant ! but, this no longer holds if the number of versions can be stored is limited!

3.1 Multiversion View Serializability Example 5.4 m = w0(x0) w0(y0) c0 r3(x0) w3(x3) c3 w1(x1) c1 r2(x1) w2(y2) c2 m‘ = w0(x0) w0(y0) c0 w1(x1) c1 r2(x1) r3(x0) w2(y2) w3(x3) c3 c2 Then m v m‘ It is not sufficient to require that given multiversion schedule m is view equivalent to serial multiversion schedule m`. The reason is that even a serial m` does not necessarily have a reads-from relation that is compatible with a serial monoversion schedule. Example 5.5 m = w0(x0) w0(y0) c0 r1(x0) r1(y0) w1(x1) w1(y1) c1 r2(x0) r2(y1) c2 s = w0(x) w0(y) c0 r1(x) r1(y) w1(x) w1(y) c1 r2(x) r2(y) c2 RF(m) ≠ RF(s)

3.1 Multiversion View Serializability Definition 5.6 < Multiversion view serializability > Let m be a multiversion history. Then m is called multiversion view serializable if there exists a serial monoversion history m` for the same set of transactions such that m v m'. Let MVSR denote the class of all multiversion view-serializable histories. for m  MVSR, there is always a serial monoversion history s for trans(m) = trans(s) such that m v s Example 5.6 Let m = w0(x0) w0(y0) c0 w1(x1) c1 r2(x1) r3(x0) w3(x3) c3 w2(y2) c2 m‘ = w0(x0) w0(y0) c0 r3(x0) w3(x3) c3 w1(x1) c1 r2(x1) w2(y2) c2 Then we have m v m‘. s = w0(x) w0(y) c0 r3(x) w3(x) c3 w1(x) c1 r2(x) w2(y) c2 Moreover, m’ is vew equivalent to history m‘ v s

3.2 Testing Membership in MVSR Theorem 5.1 VSR  MVSR Every view-serializable history can be perceived as a monoversion history and hence as a specific multiversion history (in MVSR) Example m = w0(x0) w0(y0) c0 r1(x0) w2(x2) w2(y2) c2 r1(y0) c1 s = w0(x) w0(y) c0 r1(x) w2(x) w2(y) c2 r1(y) c1 s  VSR, but m  MVSR ( m v <t0t1t2> ) MVSR is a relaxation of conditions over VSR, in other words, with additional restrictions (in VSR) over MVSR

3.2 Testing Membership in MVSR Theorem 5.2 Deciding if a given mv history m is in MVSR is NP-complete. The conflict graph of a multiversion schedule m is a directed graph G(m) with transactions as nodes and an edge from ti to tj if rj(xi)  op(m). Theorem 5.3 For any two multiversion schedules m, m‘ : m v m‘  G(m) = G(m‘). Example (T5.3) is not true m = w1(x1) r2(x0) w1(y1) r2(y1) c1 c2 m‘ = w1(x1) w1(y1) c1 r2(x1) r2(y0) c2 G(m) = G(m‘), but not m v m‘

3.2 Testing Membership in MVSR Definition 5.7 < Version Order > A version order for a data item x is any nonreflexive & total ordering of all versions of x that are written by operations in m. A version order << for m is the union of all version orders of data items written by ops in m. wi(xi) preceded wj(xj) in the schedule : xi << xj Example 5.7 m = w0(x0) w0(y0) w0(z0) c0 r1(x0) r2(x0) r2(z0) r3(z0) w1(y1) w2(x2) w3(y3) w3(z3) c1 c2 c3 r4(x2) r4(y3) r4(z3) c4 A version order for m could be x0 << x2, y0 << y1 << y3, z0 << z3

3.2 Testing Membership in MVSR Definition 5.7 < Multiversion serialization graph(MVSG) > For a given schedule m and a version order <<, the multiversion serialization graph MVSG (m, <<) of m is a graph with transactions as nodes and the following edges: (i) all edges of G(m) are in MVSG(m, <<) (i.e., for rk(xj) in op(m) there is an edge from tj to tk) (ii) for rk(xj), wi(xi) in op(m): if xi << xj then there is an edge from ti to tj (iii) for rk(xj), wi(xi) in op(m): if xj << xi then there is an edge from tk to ti

3.2 Testing Membership in MVSR Example 5.8 m = w0(x0) w0(y0) w0(z0) c0 r1(x0) r2(x0) r2(z0) r3(z0) w1(y1) w2(x2) w3(y3) w3(z3) c1 c2 c3 r4(x2) r4(y3) r4(z3) c4 version order for m : x0 << x2, y0 << y1 << y3, z0 << z3 Edges that do not occur in G(m) are the following : by (iii) (t1, t2) : r1(x0), w2(x2)  op(m), and x0 << x2 by (ii) (t1, t3) : w1(y1), r4(y3)  op(m), and y1 << y3 by (iii) (t2, t3) : r2(z0), w3(z3)  op(m), and z0 << z3 t0 t1 t2 t3 t4 t0 t1 t2 t3 t4

3.2 Testing Membership in MVSR Theorem 5.4 m is in MVSR iff there exists a version order << such that MVSG(m, <<) is acyclic. Proof (If) Since MVSG(m,<<) is acyclic, it can be sorted topologically into a serial history m' such that RF(m)=RF(m'), i.e., m v m'. Suppose that m' contains rk(xj), wi(xi), k ≠ j, i ≠ j, i ≠ k. if xi << xj : ti → tj, ti <m‘ tj, if xj<< xi : tk → ti, tk <m‘ ti so, no transaction writes x between tj and tk in m‘, version number can be dropped from m‘ without changing the reads-from relation. (only if) Let MV(m,<<) denote the graph that contains version order edge only. If trans(m)=trans(m‘), then MV(m,<<)=MV(m‘,<<) for every version order << . Let m  MVSR, and m‘ be a serial monoversion history for the same transactions such that m v m‘ G(m) = G(m‘) by Thm. 5.3, and G(m‘) is acyclic(by m‘ is serial), hence so is G(m). Now define a specific version order xi <<0 xj : ti <m‘ tj (i.e. if ti → tj is in G(m‘)) Thus, if MV(m‘,<<0) is added to G(m‘), the result graph MVSG(m‘,<<0) remains acyclic. And the same holds MV(m‘,<<0)=MV(m,<<0) is added to G(m)[=G(m‘)],

3.3 Multiversion Conflict Serializability Definition 5.9 < Multiversion conflict > A multiversion conflict in multiversion schedule m is a pair of steps ri(xj) and wk(xk) such that ri(xj) <m wk(xk). convention : wr operation pairs on the same version the above : rw operation pairs on the same data item Definition 5.10 < Muliversion reducibility > A multiversion history is multiversion reducible if it can be transformed into a serial monoversion history by exchanging the order of adjacent steps other than multiversion conflict pairs. Definition 5.11 < Muliversion conflict serializability > A multiversion history m is multiversion conflict serializable if there is a serial monoversion history for the same set of transactions in which all pairs of operations in multiversion conflict occur in the same order as in m. MCSR denotes the class of all multiversion conflict serializable histories.

3.3 Multiversion Conflict Serializability Theorem 5.5 A multiversion history is multiversion reducible iff it is multiversion conflict serializable. Theorem 5.6 MCSR  MVSR Example m = w0(x0) w0(y0) w0(z0) c0 r2(y0) r3(z0) w3(x3) c3 r1(x3) w1(y1) c1 w2(x2) c2 r(x2) r(y1) r(z0) c m  MCSR, but m  MVSR (m v t0t3t2t1t)

3.3 Multiversion Conflict Serializability Definition 5.12 < Multiversion conflict graph> For a multiversion schedule m the multiversion conflict graph is a graph with transactions as nodes and an edge from ti to tk if there are steps ri(xj) and wk(xk) such that ri(xj) <m wk(xk). Theorem 5.7 A multiversion history is MCSR iff its multiversion conflict graph is acyclic. MCSR has a polynomial membership test, and all practical mv concurrency control protocols fall into this class.

3.3 Multiversion Conflict Serializability All histories MVSR MCSR VSR CSR 𝑠 1 𝑠 5 𝑠 4 𝑠 3 𝑠 2 S1 = r1(x) r2(x) w1(x) w2(x) c1 c2 S2 = w1(x) c1 r2(x) r3(y) w3(x) w2(y) c2 c3 S3 = w1(x) c1 r2(x) r3(y) w3(x) w2(y) c2 c3 w4(x) c4 S4 = r1(x) w1(x) r2(x) r2(y) w2(y) r1(y) w1(y) c1 c2 S5 = r1(x) w1(x) r2(x) w2(y) c2 w1(y) w3(y) c1 c3