Communication Steps For Parallel Query Processing ParaschosKoutris Paul Beame Dan Suciu University of Washington PODS 2013
Motivation • Understand the complexity of parallel query processing on big data • Focus on shared-nothingarchitectures • MapReduceis such an example • Dominating parameters of computation: • Communication cost • Number of communication rounds
Computation Models • The MapReducemodel • [Afratiet al., 2012] tradeoff between reducer size (input size of a reducer) and replication rate (in how many reducers a tuple is sent) • The MUD model [Feldman et al., 2010] • (Massive, Unordered, Distributed) model • The MRC model [Karloff et al., 2010] • MapReduce computation + load balancing
The MPC Model • N: total input size (in bits) • p:number of servers • Servers have unlimited computational power • Computation proceeds in synchronous rounds: • Local computation • Global communication Server 1 Input N . . . . . . . . . . . . Server p Round 1 Round 2
MPC Parameters • Each server receives in total a bounded number of bits: O(N/p× pε) 0 ≤ ε < 1 • Complexity parameters: • Number of computation roundsr • Space exponent ε(governs data replication) What are the space exponent/round tradeoffs for query processing in the MPC model ?
Our Results • ONE ROUND: • Lower bounds on the space exponent for any (randomized) algorithm that computes a Conjunctive Query • The lower bound holds for a class of inputs (matching databases), for which we show tight upper bounds • MULTIPLE ROUNDS: • Almost tight space exponent/round tradeoffs for tree-like Conjunctive Queries under a weaker communication model
Outline • Warm-up: The Triangle Query • One Communication Round • Multiple Communication Rounds
Conjunctive Queries • We mainly study full Conjuctive Queries w/o self-joins: Q(x, y, z, w, v) = R(x,y,z), S(x,w,v), T(v,z) • The hypergraphof the query Q: • Variables as vertices • Atoms as hyperedges S x w R y v z T
The Triangle Query (1) • Find all trianglesQ(x,y,z) = R(x,y), S(y,z),T(z,x) • 2-round Algorithm: • ROUND 1: [R hash-join S] • R(a, b) h(b) • S(b, c) h(b) • JoinlocallyU(a, b, c) = {R(a, b), S(b, c)} • ROUND 2: [T’ hash-join T] • U(a, b, c) h(c) • T(c, a) h(c) • JoinlocallyQ(a,b,c) = {U(a, b ,c), T(c, a)} • Replicationε = 0
The Triangle Query (2) • 1-round Algorithm:[Ganguly ’92, Afrati ’10, Suri’11] • The p servers form a cube: [p1/3] ×[p1/3] × [p1/3] • Sendeachtupleto servers: • R(a, b) (h1(a), h2(b), - ) • S(b, c) (-, h2(b), h3(c) ) eachtuplereplicatedp1/3times • T(c, a) (h1(a), -, h3(c) ) • Replicationε = 1/3 • (h1(a), h2(b), h3(c))
Lower Bound For Triangles (1) • Theorem: No (randomized) algorithm can compute triangles in one round with space exponent ε < 1/3 • Say that R, S,T are random permutations over [n]2 • Expected #triangles = 1 Lemma: For any deterministic algorithm and ε=0, the p servers report in expectation O(1/p1/2) tuples • Each relation contains N = (n logn) bits of information • Any server knows a 1/pfraction of input: N/pbits
Lower Bound For Triangles (2) • axy = Pr[server knows tuple R(x,y)] • axy ≤ 1/n • ∑x,yaxy = O(n/p) • Similarly for S(y,z), T(z,x): byz, czx • Friedgut’s inequality: ∑x,y,zaxybyzczx ≤ (∑x,yaxy2)1/2 (∑y,zbyz2)1/2 (∑z,xczx2)½ • #know-triangles = O(1/p3/2) • Summing over all servers, O(1/p1/2) known output tuples
Outline • Warm-up: The Triangle Query • One Communication Round • Multiple Communication Rounds
Matching Databases • Every relation R(A1, …, Ak) contains exactly n tuples • Every attribute Aicontains each value in {1, …, n} only once • A matching database has no skew Relation R(X, Y, Z) 1 1 2 2 3 … … … n-1 n n
Fractional Vertex Cover • Vertex cover number τ:minimum number of variables that cover every hyperedge • Fractional vertex cover number τ*: minimum weight of variables such that each hyperedge has weight at least 1 Vertex Cover τ = 2 1/2 Fractional Vertex Cover τ* = 3/2 x 0 w 1/2 y 0 v 1/2 z Q(x, y, z, w, v) = R(x,y,z), S(x,w,v), T(v,z)
Lower Bounds • Theorem: Any randomized algorithm in the MPC model will fail to compute a Conjunctive Query Q with: • 1 round • ε < 1 – 1/ τ*(Q) • Input a matching database
Upper Bounds • Theorem: The HYPERCUBE (randomized) algorithm can compute any Conjunctive QueryQwith: • 1 round • ε≥ 1 – 1/ τ*(Q) • Input a matching database (no skew) • Exponentially small probability of failure (on input N)
HyperCube Algorithm • Q(x1,…, xk) = S1(…), …, Sl(…) • Compute τ* and minimum cover: v1,v2, …, vk • Assign to each variable xi a share exponent e(i) = vi / τ* • Assigneachof the p servers topoints on a k-dimensional hypercube: [p] = [pe(1)] × … × [pe(k)] • Hasheachtupleto the appropriatesubcube Q(x,y,z,w,v)=R(x,y,z),S(x,w,v),T(v,z) τ* = 3/2 : vx = vv = vz = ½ vy = vw = 0 e(x) = e(v) = e(z) = 1/3 e(y) = e(w) = 0 [p] =[p1/3]×[p0]×[p1/3]×[p0]×[p1/3] e.g. S(a,b,c) (hx(a), 1, -, 1, hv(c))
Examples • Cycle query: Ck(x1,…, xk) = S1(x1, x2), …, Sk(xk, x1) • τ*= k/2 • ε= 1 - 2/k • Star query: Tk(z, x1,…, xk) = S1(z, x1), …, Sk(z, xk) • τ*= 1 • ε= 0 • Line query: Lk(x0, x1,…, xk) = S1(x0, x1), …, Sk(xk-1, xk) • τ*=k/2 • ε= 1 - 1/ k/2
Outline • Warmup: The Triangle Query • One Communication Round • Multiple Communication Rounds
Multiple Rounds • Ourresultsapplyto a weakermodel (tuple-based MPC): • onlyjointuplescan be sent in rounds > 1 e.g.{R(a,b), S(b,c)} • routingofeachtuple tdependsonly on t • Theorem: For every tree-like query Q, any tuple-based MPC algorithm requires at least • log(diam(Q)) / log2/(1-ε) rounds • This bound agrees with the upper bound within 1 round • diam(Q): largestdistancebetweentwovertices in the hypergraph • tree-like queries: #variables + #atoms - Σ(arities) = 1
Example • Line query: Lk(x0,x1,…, xk) = S1(x0,x1), …, Sk(xk-1,xk) • tree-like: • #variables = k+1 • #atoms = k • Σ(arities) = 2k • diam(Lk) = k • For space exponent ε = 0, we need at least log(k)/log(2/(1-0)) = log(k) rounds diameter = k xk x1 x2 x3 x4 …
Connected Components • As a corollaryofourresults for multiplerounds, weobtainlowerboundsbeyondconjunctivequeries: Theorem: Any tuple-based MPC algorithm that computes the Connected Components in any undirected graph with space exponent any ε<1 requires requires Ω(logp)communicationrounds
Conclusions • Tightlower and upper boundsfor one communication round in the MPC model • The first lower bounds for multiple communication rounds • Connected components cannot be computed in a constant number of rounds • Open Problems: • Lower and upper bounds for skewed data • Lower bounds for > 1 rounds in the general model
Thank you !
