Efficient Top-k Queries in Large-Scale Networks

1
Efficient Top-k Queries in Large-Scale Networks

• Pei Cao
• Cisco Systems, Inc.
• Consulting Faculty, Stanford University

2
Motivation

• Enterprise content delivery networks (CDNs)
• CE web cache and streaming media cache combined
• Number of branches 50 - 2000

Data Center
Central Manager
56Kbps,128kbps, DSL
Branch Offices
. . .
. . .
CE
CE
CE
3
Top-k Queries in CDNs

• Example queries
• Across all CEs, which URLs are accessed most
  often?
• Across all CEs, which domains consume the most
  storage?
• Across all CEs, which cached objects produced the
  biggest bandwidth savings?
• etc.

4
Definitions

• a network of m nodes, connected to a central
  manager (CM)
• each node i has a reverse-sorted list of (
  x, Vi(x) )
• an objects sum
• V(x) V1(x)V2(x)Vm(x)
• Problem find the k objects with highest sums
• Goal answer this question with minimum network
  traffic
• ? A generic problem in distributed systems

5
Existing Methods

• Naïve Algorithm
• Each node sends the full list of objects and
  their values to the Central Manager
• Threshold Algorithm (TA)
• Proposed by multiple groups in the database
  research community

6
The Threshold Algorithm (TA)

• Example find top 2 objects with max sums in
  three columns

Node 1
Node 2
Node 3
Central Manager (CM)
?
(A, 10) (C, 8) (E, 8) (F, 8) (B, 7) (D, 5) (J,
1) (K, 1) . . .
(B, 10) (D, 9) (F, 8) (H, 6) (G, 5) (C, 1) (A,
1) . . .
(C, 10) (A, 9) (G, 8) (J, 7) (F, 6) (D, 4) (B,
1) . . .
T 30 V(A)20, V(C)19, V(B)18
?
T 26 V(A)20, V(C)19,
?
T 24 V(F)22, V(A)20,
?
T 21 V(F)22, V(A)20,
?
T 18 V(F)22, V(A)20,
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Adapting TA for Distributed Environments

• Consists of multiple rounds, each round having
  two round trips
• Round-trip 1 sorted access CM asks for the
  next B objects on the lists and nodes respond
• Round-trip 2 random lookup CM sends a list of
  object names to nodes and nodes supply values
• B k
• Issues
• of rounds unpredictable
• O(m2) network traffic

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New Algorithm Three-Phase Uniform Threshold
(TPUT)

• Motivation terminate in a fixed number of round
  trips regardless of input
• Operates in three phases
• Lower-bound estimation
• Pruning
• Final lookup

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Partial Sums and Upper Bounds

• Partial sum PS(x) ?Vi(x)
• Upper bound U(x) ?Ui(x)

Vi(x), if x has been reported by node i to CM
Vi(x)
0, otherwise
Vi(x), if x has been reported by node i to CM
Ui(x)
Ti, otherwise
Ti Node i sends all objects with values gt Ti
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Examples
Node 2
Node 3
Node 1
CM
(B, 10) (D, 9) (F, 8) (H, 6) (G, 5) (C, 1) (A,
1) . . .
(C, 10) (A, 9) (G, 8) (J, 7) (F, 6) (D, 4) (B,
1) . . .
PS(A) 10 0 9 19 U(A) 10 9 9
28 PS(B) 0 10 0 10 U(B) 8 10 9
27
(A, 10) (C, 8) (E, 8) (F, 8) (B, 7) (D, 5) (J,
1) . . .
?
For any object O, PS(O) V(O) U(O)
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Steps in TPUT

• Phase 1
• Manager ? Nodes start top-k query
• Nodes ? Manager here are my top-k objects
• Manager
• Calculate partial sums of all objects
• Take the kth partial sum E1 (E1 E) set t
  E1/m
• Phase 2
• Manager ? Nodes send me all objects with value
  t
• Nodes ? Manager here they are
• Manager
• Calculate partial sums again take the kth
  partial sum E2 (E1 E2 E)
• Calculate upper bounds of all objects
• S objects whose upper bounds are E2
• Phase 3
• Manager ? Nodes here is S send me all objects
  in S
• Nodes ? Manager here they are

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Example
Node 2
Node 3
Node 1
CM
(B, 10) (D, 9) (F, 8) (H, 6) (G, 5) (C, 1) (A,
1) . . .
(C, 10) (A, 9) (G, 8) (J, 7) (F, 6) (D, 4) (B,
1) . . .
(A, 10) (C, 8) (E, 8) (F, 8) (B, 7) (D, 5) (J,
1) . . .
S(F) 22 S(A) 20 S(C) 19 Top 2 objects
are F and A.
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Improving the Pruning Power

• Set t (E1/m) a, where 0ltalt1

. . .
E2/m
t
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Compression via Hashing

• Problem object IDs can be too long
• Solution send hashed keys of object IDs
• Node report to CM (hash(o), V(o))
• If hash(o1)hash(o2), then V max(V(o1), V(o2))
• Candidate set S is a set of hashed keys
• Size of key log(total of objects in all
  nodes)
• Effect
• Algorithm is still correct
• However, might need an additional round trip

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Evaluating TPUT Algorithm

• Trace-driven simulation
• Optimality analysis

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Trace Data for Simulations
NLANR-10 daily web access from 10 NLANR proxies
Worldcup-30 2-hr logs from 30 WorldCup web servers
DEC-64 split 1-day DEC proxy traces into 64 sub-traces by client IP
DEC-128 split 2-day DEC proxy traces into 128 sub-traces by client IP
NLANR-203 split NLANR traces into 203 sub proxy traces by client IP
Berkeley-512 Split one week UCB traces into 512 sub traces by client IP
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Performance Metrics

• Communication costs
• Unicast-bytes
• Multicast-bytes
• Messages are all compressed by gzip

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Results on Unicast-Bytes
m10
m30
m64
m128
m203
m512
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Number of Objects Looked-Up
Trace K10 TA K10 TPUT/0.5 K100 TA K100 TPUT/0.5
NLANR-10 166 18 1486 176
WorldCup-30 46 12 238 101
DEC-64 3164 31 9817 244
DEC-128 6928 28 26680 250
NLANR-203 5576 28 43954 238
Berkeley-512 47899 41 180550 132
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Results on Multicast-Bytes
m10
m30
m64
m128
m203
m512
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Optimality Analysis

• Main results
• TPUT is instance optimal for data sets with a
  log-log slope function C(n)
• Zipf distribution C(n) n
• Zipf distribution opt-ratio (m-1)2m km
• Setting alt1 reduces cost qualitatively.
• Zipf distribution opt-ratio (m-1)?O(vm )
  km/a

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General Instance Optimality

• Definition
• An algorithm R is instance-optimal with
  optimality ratio C1, if exists C2, such that for
  any data series D, and any algorithm A,
• cost(R, D) C1 cost(A, D) C2
• cost is amount of network traffic
• TA is instance optimal with opt-ratio O(m2)

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Worst Cases for Fixed Number Round-Trip Algorithms

• TPUT is not general instance optimal
• Nor can any algorithm that terminates in a fixed
  number of round trips regardless of input

Finding obj with highest sum
Node 1 (A, 1) (C, 1) (X1, 0.6) (X2,
0.6) . . . (Xn, 0.6) (B, 0.5) . .
Node 2 (B, 1) (D, 0.2) . . . . . . . . .
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Log-Log Slope Function

• L(j) is the value at position j in a
  reverse-sorted list
• The list satisfies log-log slope function C(n),
  if, for all jk, L(jC(n)) lt L(j)/n
• For Zipf-like distribution L(j) 1/j?, C(n)
  n1/?.

List Position 1
. . . .
. Position j . .
. . .
. . Position jC(n) .
. . .
. . .
L(j)
lt L(j)/n
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Properties of the Two Lower Bounds

• Let E be the true bottom
• E1 E/m
• E2 gt E/2
• E2 E1
• E2 gt E E1(m-1)/m
• For any x, V(x) PS(x) lt (m-1)t? V(x) PS(x)
  lt (m-1) E1/m? E E2 lt E1 (m-1)/m
• E2 gt (m/(2m-1))E

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Restricted Instance Optimality of TPUT (a1)

• Assume D is a collection of m lists all following
  log-log slope function C(n), then for any
  algorithm A,
• cost(TPUT,D) cost(A,D) ((m-1)C(2m) C(m)k)
• Proof assume the optimal algorithm for D stops
  at position bi on list i, then L(bi) lt E
• The number of objects in S from node i is bi
  C(2m)
• Each node sends C(m) k objects in round-trip
  2

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Effect of alt1

• Property
• If object x appears in n nodes in Phase 2 and
  U(x) E2, then its average value in those nodes
  R(x) E2 (1-a)/n
• Let li the num of objects in S that appear in
  exactly i nodes in Phase 2, then
• 1l1 2l2 3l3 mlm C(m (1a)/a)
  ?bi
• l1 l2 li C( i (1 a)/(1-a)) ?bi
• Size of S is l1 l2 lm

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Analysis of alt1

• Whats the maximum l1l2 lm under the
  following constraints?
• 1l12l2 3l3 mlm C(m (1a)/a) B
• l1 C(1ß) B
• l1l2 C(2ß) B
• ...
• l1l2 lm C( m ß) B
• where ß (1a)/(1-a), B ?bi
• Solution maximize l1, l2, , ld, and
  set ld1, ld2, , lm to 0
• Li C(i ß) B C((i-1) ß) B
• d C(d ß) B - ?C(i ß) B C(m (1a)/a) B
• Candidate set size S C(d ß) B

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a For Zipf Distributions

• For Zipf distribution, where C(n) n, size of
  candidate set S is cvm B
• ? Optimality ratio for TPUT with alt1 is (m-1)
  c vm m/a k

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TPUT for Hierarchical Networks
Phase 1 Lower-Bound Estimation
Phase 2 Selection by value Pruning
Phase 3 Final lookup
S
tE/m a
. . .
S
t (E/mn) a
. . .
. . .
. . .
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Summary and Future Work

• TPUT should be used for top-k queries in
  distributed networks
• TPUT is instance-optimal under the log-log slope
  function assumption
• Introducing alt1 improves performance
  significantly
• Future work
• Evaluating TPUT for hierarchical and P2P networks
• Distributed algorithms for other aggregate
  statistics

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Backup Slides
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Bandwidth Consumption of Threshold Algorithm
Trace Raw Data K10 TA UniCast K10 TA MultiCast K100 TA UniCast K100 TA UniCast
NL-10 26MB 56.3KB 25.9KB 318KB 132KB
WC-30 426KB 31KB 22KB 96KB 80KB
DEC-64 7.4MB 1.7MB 160KB 4.6MB 359KB
DEC-128 15MB 7.2MB 419KB 24.6MB 1.2MB
NL-203 44MB 22MB 1.2MB 143MB 4.2MB
UCB-512 78MB 423MB 16.1MB 1.47GB 31MB
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Bandwidth Consumption of TPUTHash
Trace Raw Data K10 TPUT-H UniCast K10 TPUT-H MultiCast K100 TPUT-H UniCast K100 TPUT-H UniCast
NL-10 26MB 8KB 7KB 52KB 49KB
WC-30 426KB 44KB 38KB 99KB 89KB
DEC-64 7.4MB 64KB 59KB 322KB 300KB
DEC-128 15MB 161KB 150KB 870KB 828KB
NL-203 44MB 154KB 139KB 764KB 687KB
UCB-512 78MB 1.03MB 978KB 15.8MB 15.3MB
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Unicast-Bytes for Top-100 Objects
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Multicast-Bytes for Top-100 Objects
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Varying a
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Fixed-Number Round Trip Algorithms

• Criteria by which a node decides to send objects
• By position
• By name
• By value
• Any fixed-number round trip algorithm must
  include a by value operation
• Any algorithm, if include by value operation,
  wont be instance optimal

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TA Running over Networks
Node 2
Node 3
Node 1
CM
(B, 10) (D, 9) (F, 8) (H, 6) (G, 5) (C, 1) (A,
1) . . .
(C, 10) (A, 9) (G, 8) (J, 7) (F, 6) (D, 4) (B,
1) . . .
T 26 looks up A, B, C, D ? V(A)20, V(C)19
cant stop
(A, 10) (C, 8) (E, 8) (F, 8) (B, 7) (D, 5) (J,
1) . . .
?
T 21 looks up E, F, G, H, J ? V(F)22,
V(A)20 cant stop
?
?
T 10 stop
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TPUT

• Phase 3
• Manager ? Nodes here is S send me all objects
  in S
• Nodes ? Manager here they are
• Manager calculate sums for objects in S select
  the top k objects