P ARALLEL S KYLINE Q UERIES Foto Afrati Paraschos Koutris Dan Suciu Jeffrey Ullman University of Washington.

PARALLEL SKYLINE QUERIES

Foto AfratiParaschos KoutrisDan SuciuJeffrey Ullman

University of Washington

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WHAT IS THE SKYLINE?

• A d-dimensional set R• A point x dominates x’ if forall k: x(k) ≤ x’(k)• The skyline of R are all non-dominated points of R

domination

skyline

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CONTRIBUTIONS

• We design algorithms for Skyline Queries based on two parallel models:– MP: perfect load balancing [Koutris, Suciu ‘11]

– GMP: weaker load balancing [Afrati, Ullman ’10]

• We present 3 algorithms with theoretical guarantees for:– # synchronization steps– load balance

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PREVIOUS APPROACHES

• Several efficient algorithms for skyline queries exist in the literature

• Parallel algorithms use various partitionings:– Grid-based partitioning [WZFZAA ’06]

– Random partitioning [CRZ ’07]

– Angle-based space partitioning [VDK ’08]

– Hyperplane projections [KYZ ’11]

• Previous approaches typically require a logarithmic number of communication steps: our algorithms achieve 1 or 2 steps

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MASSIVELY PARALLEL MODELS

• P servers: R partitioned into R1,R2,…, RP

• n = |R|• The algorithm alternates between communication

and computation steps• MP model: each node holds O(n/P) data • GMP model: each node holds O(Pε * n/P) where 0

≤ ε < 1 – ε = 0 : GMP = MP– ε = 1 : GMP = sequential computation in one node

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AN EXAMPLE• How do we compute set intersection in one step

in the MP model?• Hash each value x (from R or S) to a server

Communication Phase send tuple R(x) to server @h(x) send tuple S(x) to server @h(x)Computation Phase output a tuple only if it occurs twice

Intersection Q(x):-R(x),S(x)

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THE BROADCAST STEP• In addition to regular communication steps, we

allow broadcast steps:• the data exchanged is independent of n

• Known results:• Q(x,y)=R(x),S(x,y) can be computed in 1 MP step iff

a broadcast step is allowed [Koutris, Suciu ‘11]

• Q(x,y)=R(x),S(x,y),T(y) can not be computed in 1 MP step [Koutris, Suciu ‘11] , but can be in 1 GMP step with ε=1/2 [Afrati, Ullman ‘10]

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• Broadcast• Grid-based partitioning into cells• Pre-processing the cells to compute the relaxed

skyline

• Communication• Careful distribution of the cells (with their data) to

the servers

• Computation: • Local computation of the skyline at each server

OUTLINE OF OUR APPROACH

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BUCKETIZING• Partition R into M buckets across some dimension,

such that each partition contains O(n/M) points • Equivalently, compute (M+1) partition points:

-∞ = b0 , b1 , … , bM = +∞

Algorithm:Local: each server evenly partitions its data to M bucketsBroadcast: servers exchange MxP partition pointsLocal: each server picks every P-th value as partition point

bucketize acrossdimension 1

bucketize acrossdimension 2

M=P or P1/(d-1)

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CELLS• A cell is an intersection of buckets from all dimensions• Every point belongs in exactly one cell• Every cell holds O(n/P) data (and not O(n/Pd) !!)

In each cell, we can keep only candidates for skyline points

candidate rejected

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• Broadcast• Grid-based partitioning into cells• Pre-processing the cells to compute the relaxed

skyline

• Communication• Careful distribution of the cells (with their data) to

the servers

• Computation: • Local computation of the skyline at each server

OUTLINE OF OUR APPROACH

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CELLS• We are interested in the non-empty cells• Any cell that is strictly dominated by another does not

contribute to the skyline

no points belong in thefinal skyline

strictdomination

domination

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RELAXED SKYLINE OF CELLS• The relaxed skyline consists of the non-empty cells

that are not strictly dominated by non-empty cells• We focus on the relaxed skyline of non-empty cells

skyline

relaxed skyline

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ON RELAXED SKYLINES• To compute the skyline points of a cell B, we need to

compare with cells that:• belong in the relaxed skyline• weakly dominate B (have one common coordinate)

cell B

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• Broadcast• Grid-based partitioning into cells• Pre-processing the cells to compute the relaxed

skyline

• Communication• Careful distribution of the cells (with their data) to

the servers

• Computation: • Local computation of the skyline at each server

OUTLINE OF OUR APPROACH

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A NAÏVE APPROACH• Try the following:• Partition into P buckets (M=P)• Allocate cells in the relaxed skyline to servers + cells

that weakly dominate them: O(n/P) data per cell• Locally compute the skyline points

• This works if the relaxed skyline is small• But the relaxed skyline can have as many as

Ω(Pd-1) cells for dimension d

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A 1-STEP ALGORITHM• Choose a coarser bucketization (<P buckets)• This gives a weak load-balanced algorithm with

maximum load of O( (n/P) P(d-2)/(d-1) )

• ε = (d-2)/(d-1) (ε=0 implies GMP=MP)

Corollary. For d=2 dimensions, we obtain a perfectly load balanced algorithm for MP

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A 2-STEP ALGORITHM• Step 1: group the cells in the relaxed skyline by

bucket for every dimension

Server 1

Server 2

Server 1 Server 2

…

…

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A 2-STEP ALGORITHM• For each bucket, compute the local skyline• A point is a skyline point iff it is a local skyline point in

every one of the d buckets• Step 2: intersect the local skylines This point is in the skyline of

the y-bucket, but not the x-bucket

x-bucket

y-bucket

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A 1-STEP ALGORITHM FOR 3D

Key idea: to reject this point, we only need the minimum x-coordinate from cell B

cell B

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A 1-STEP ALGORITHM FOR 3D• The observation reduces the number of points that

need to be communicated• With smart partitioning, we can achieve perfect load-

balance in 1 step• However, the property holds only for 2 and 3

dimensions

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CONLUSION

3 algorithms for Skyline Queries:• 2 step + perfect load balance• 1 step + some replication• 1 step + perfect load balance for d < 4

Open Questions• Can we compute the skyline in 1 step with perfect load

balance for >3 dimensions?• A more general question: what classes of queries can

we compute in the MP model with perfect or weaker load balance guarantees?

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Thank you!

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INTERIOR CELLS• Two cells are co-linear if they share exactly two

coordinates• A cell i is interior if every colinear cell in Sr(J) belongs in

the same hyperplane as i. Else, it is a corner cell. • Interior cells are easy to handle: we can send the whole

plane to a single processor

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CORNER CELLS• We group the corner cells into lines• Border cells are the minimal/maximal cells of each line• Fact: lines meet only on border cells• Grouping: each line is a group, a cell is assigned to the

lexicographically first line it belongs to

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Assigning the groups• We have two ways to assign groups to servers• The first is deterministic and greedily assigns a

group to any server that is not overloaded (M=P)• The second is randomized and sends each group

randomly to some server (M = P log P)

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About the MP model• [KS11] A dichotomy result on Conjunctive

Queries that can be computed in 1 step with perfect load balancing

• Easy Queries: – Q(x,y,z) :- R(x,y) , S(y,z)– Q(x,y,z,) :- R(x), S(x,y), T(x,y,z)

• Hard Queries:– Q(x,y) :- R(x), S(x,y), T(y)– Q(x,y) :- R(x), S(x), T(y)