Efficient Top-k Queries in Large-Scale Networks

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Efficient Top-k Queries in Large-Scale Networks

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

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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
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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.

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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

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Existing Methods

• Each node sends the full list of objects and
  their values to the Central Manager
• Pro simple to implement works fine when the
  number of objects is small
• Con when the number of objects is large,
  consumes too much network bandwidth
• Use the threshold algorithm (TA)
• Proposed by multiple groups in the database
  research community

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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 has 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

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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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Problems with TA in Large Networks

• Num of round-trips required vary by data input
• High bandwidth consumption when number of nodes
  is large
• In round trip 2, the list of random-lookup
  objects are the union of all objects sent by m
  nodes in round trip 1
• In round trip 2, the list goes to all m nodes

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New Algorithm Two-Phase Uniform Threshold (TPUT)

• Motivation algorithm should terminate in a fixed
  (and small) number of round trips
• Operates in two phases
• Phase 1 get a lower-bound estimate on the bottom
  value in the top-k set (i.e. the true bottom,
  denoted as E)
• Phase 2 all nodes send objects who sums are
  potentially higher than the lower bound CM
  aggregates the info, refines the estimate,
  determines the candidates, and looks up
  candidates in all nodes

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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)
Li, otherwise
Li is the lowest value that node I has reported
to CM
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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
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(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

• Round-trip 1
• Manager ? Nodes start top-k query
• Nodes ? Manager here are my top-k objects
• Manager
• Calculate partial sums of all objects and sort
  them
• Take the kth value, call it E1 E1 E
• set t E1/m
• Round-trip 2
• Manager ? Nodes send me all objects with value
  t
• Nodes ? Manager here they are
• Manager
• Calculate partial sums of all objects and sort
  them take the kth value, call it E2 E1 E2
  E
• For each object, calculate its upper bound
  select those objects whose upper bounds are E2
  call the set S

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TPUT

• Round-trip 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

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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) . . .
PS(A) 19 PS(C) 18 ? E1 18 t 6
(A, 10) (C, 8) (E, 8) (F, 8) (B, 7) (D, 5) (J,
1) . . .
?
PS(F) 22 PS(A) 19 ? E2 19 U(H) 18, U(J)
19 ? H and J are out! S (A, B, C, D, E, F, G)
?
S(F) 22 S(A) 20 S(C) 19 Top 2 objects
are F and A.
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Improving the Pruning Power

• Observation if E2 E1, then no object can be
  pruned away
• Solution set t (E1/m) a, where 0ltalt1
• Effect
• Increases traffic in round-trip 2
• But shrinks the set of candidates, hence reduce
  traffic in round-trip 3
• Optimal alpha depends on data set
• Default alpha 0.5

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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( of objects in all nodes)
• Effect
• maintains correctness of pruning by upper bounds
• 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 web accesses from 30 servers
  hosting 1997 WorldCup
• DEC-64 split one-day DEC traces into 64
  sub-traces by client IP
• Simulating an enterprise with 64 branch offices
• DEC-128 split two-day DEC traces into 128
  sub-traces by client IP
• Simulating an enterprise with 128 branch offices
• NLANR-208 split NLANR traces into 208 sub proxy
  traces by client IP
• Simulating an enterprise CDN of 208 nodes
• Berkeley-512 split one week UCB traces into 512
  sub traces
• Simulating a 512 branch office with 16 people per
  branch

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Performance Metrics

• Communication costs
• Messages are always compressed by gzip
• Unicast-bytes assuming CM communicates with
  nodes via uni-cast
• Multicast-bytes assuming CM broadcasts to nodes

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Results on Unicast-Bytes
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Results on Multicast-Bytes
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Optimality Analysis

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

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General Instance Optimality

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

• E1 E/m, where E is the true bottom
• E2 gt E/2
• E2 E1
• 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 E
  E1(m-1)/m
• E2 gt (m/(2m-1))E
• Consequently
• Since L(k) E1 in every node, each node sends at
  most kC(m) to manager in round trip 2
• A candidate in round trip 3 has average value
  RgtE/2m

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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 candidates in round-trip 3 is
  bi C(2m)

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Effect of alt1

• Intuition
• if an object appears in few nodes and still
  makes the cut, then its average value must be
  high
• if an object has a small value and makes the
  cut, then it must appear in many nodes
• Let li be the num of objects that appear in
  exactly i nodes from round-trip 2, then
• 1l1 2l2 3l3 mlm C(m (1a)/a)
  ?bi
• For each i, If an object appears in less than i
  nodes and still makes the cut, then its average
  value R E2 (1-a)/I? l1 l2 li C( i
  (1 a)/(1-a)) ?bi
• Size of candidate set 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
• Lj 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 is O(vm) ?bi
• ? Optimality ratio for TPUT with alt1 is (m-1)
  c vm mk
• Optimal a depends on m, but should gt 1/3
  default 0.5

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Summary and Open Questions

• TPUT algorithm works well for top-k queries in
  distributed networks
• Introducing a0.5 improve performance
  significantly
• TPUT is instance-optimal under log-log slope
  function assumption
• Easy to extend the algorithm to hierarchical
  networks
• Open question
• Is TPUT instance optimal compared with all fixed
  round trip algorithms over all data sets?

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Performance of Threshold Algorithm
Trace Raw Data K10 TA UniCast K10 TA MultiCast K100 TA UniCast K100 TA UniCast
NL-10 26MB
WC-30
DEC-64
DEC-128
NL-208
UCB-512
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Unicast-Bytes for Top-100 Objects
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Multicast-Bytes for Top-100 Objects
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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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Why Uniform Threshold?
Node 2
Node 3
Node 1
CM
(B, 10) (D, 9)
(C, 10) (A, 9)
T 8 9 9 26 E1 18 ? Could set a per-node
ti Li E1/T
(A, 10) (C, 8)
?

• Benefit of uniform threshold E2 gt E/2, where E
  is the true bottom
• E2 E1
• E2 E (m-1)/m E1
• because V(x)-PS(x) lt (m-1)t for all x
• ? E2 (m/(2m-1)) E

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