Mergeable summaries
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Mergeable Summaries. Pankaj K. Agarwal (Duke) Graham Cormode (AT&T) Zengfeng Huang (HKUST) Jeff Phillips (Utah) Zhewei Wei (HKUST) Ke Yi (HKUST). Mergeability. Ideally, summaries could behave like simple aggregates: associative, commutative

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Mergeable Summaries

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Mergeable summaries

Mergeable Summaries

Pankaj K. Agarwal (Duke)

Graham Cormode (AT&T)

Zengfeng Huang (HKUST)

Jeff Phillips (Utah)

Zhewei Wei (HKUST)




  • Ideally, summaries could behave like simple aggregates: associative, commutative

    • Oblivious to the computation tree (size and shape)

  • Error is preserved

  • Size is bounded

    • Ideally, independent of base data size

    • At least sublinear in base data

    • Rule out “trivial” solution of keeping union of input

Mergeable Summaries

Models of summary construction

Models of summary construction

  • Offline computation: e.g. sort data, take percentiles

  • Streaming: summary merged with one new item each step

  • One-time merge:

  • Equal-weight merges: only merge summaries of same weight

  • (Full) mergeability: allow arbitrary merging schemes

    • Data aggregation in sensor networks

    • Large-scale distributed computation (MUD, MapReduce)

Mergeable Summaries

Merging sketches

Merging: sketches

  • Example: most sketches (random projections) are mergeable

  • Count-Min sketch of vector x[1..U]:

    • Creates a small summary as an array of w  d in size

    • Use d hash functions h to map vector entries to [1..w]

    • Estimate x[i] = minj CM[ hj(i), j]

  • Trivially mergeable: CM(x + y) = CM(x) + CM(y)


Array: CM[i,j]


Mergeable Summaries

Summaries for heavy hitters

Summaries for heavy hitters




  • Misra-Gries (MG) algorithm finds up to k items that occur more than 1/k fraction of the time in a stream

  • Keep k different candidates in hand. For each item in stream:

    • If item is monitored, increase its counter

    • Else, if < k items monitored, add new item with count 1

    • Else, decrease all counts by 1





Mergeable Summaries

Streaming mg analysis

Streaming MG analysis

  • N = total size of input

  • M = sum of counters in data structure

  • Error in any estimated count at most (N-M)/(k+1)

    • Estimated count a lower bound on true count

    • Each decrement spread over (k+1) items: 1 new one and k in MG

    • Equivalent to deleting (k+1) distinct items from stream

    • At most (N-M)/(k+1) decrement operations

    • Hence, can have “deleted” (N-M)/(k+1) copies of any item

  • Set k=1/, the summary estimates any count with error N

Mergeable Summaries

Merging two mg summaries

Merging two MG Summaries

  • Merging alg:

    • Merge the counter sets in the obvious way

    • Take the (k+1)th largest counter = Ck+1, and subtract from all

    • Delete non-positive counters

    • Sum of remaining counters is M12

  • This alg gives full mergeability:

    • Merge subtracts at least (k+1)Ck+1from counter sums

    • So (k+1)Ck+1 (M1 + M2 – M12)

    • By induction, error is ((N1-M1) + (N2-M2) + (M1+M2–M12))/(k+1)=((N1+N2) –M12)/(k+1)

Mergeable Summaries

Q uantiles classical results

-quantiles: classical results

  • The classical merge-reduce algorithm

    • Input: two -quantiles of size 1/

    • Merge: combine the two -quantiles, resulting in an -quantiles summary of size 2/

    • Reduce: take every other element, size getting down to 1/, error increasing to 2 (the -approximation of an -approximation is a (2)-approximation)

  • Can also be used to construct -approximations in higher dimensions

    • Error proportional to the number of levels of merges

    • Final error is  log n, rescale  to get it right

  • This is not mergeable!

Mergeable Summaries

Q uantiles new results

-quantiles: new results

  • The classical merge-reduce algorithm

    • Input: two -quantiles of size 1/

    • Merge: combine the two -quantiles, resulting in an -quantiles summary of size 2/

    • Reduce: pair up consecutive elements and randomly take one, size getting down to 1/, error stays at!

  • Summary size = O(1/ log1/2 1/) gives an -quantiles summary w/ constant probability

    • Works only for same-weight merges

  • The same randomized merging algorithm has been suggested (e.g., Suri et al. 2004), but previous analysis showed that error could still increase

Mergeable Summaries

Unequal weight merges

Unequal-weight merges

  • Use equal-size merging in a standard logarithmic trick:

  • Merge two summaries as binary addition

  • Fully mergeable quantiles, in O(1/ log n log1/2 1/)

    • n = number of items summarized, not known a priori

  • But can we do better?

Wt 32

Wt 16

Wt 8

Wt 4

Wt 2

Wt 1

Mergeable Summaries

Hybrid summary

Hybrid summary

  • Observation: when summary has high weight, low order blocks don’t contribute much

    • Can’t ignore them entirely, might merge with many small sets

  • Hybrid structure:

    • Keep top O(log 1/) levels as before

    • Also keep a “buffer” sample of (few) items

    • Merge/keep equal-size summaries, and sample rest into buffer

  • Analysis rather delicate:

    • Points go into/out of buffer, but always moving “up”

    • Size O(1/ log1.5(1/)) gives a correct summary w/ constant prob.

Wt 32

Wt 16

Wt 8


Mergeable Summaries

Other mergeable summaries

Other mergeable summaries

  • -approximations in constant VC-dimension d of sizeO(-2d/(d+1) log 2d/(d+1)+1(1/))

  • -kernels in d-dimensional space in O((1-d)/2)

    • For “fat” point sets: bounded ratio between extents in any direction

Mergeable Summaries

Open problems

Open Problems

  • Better bounds than O(1/ log1.5(1/)) for 1D?

  • Deterministic mergeable -approximations?

    • Heavy hitters are deterministically mergeable (0 dim)

    • dim ≥ 1 ?

  • -kernels without the fatness assumption?

  • -nets?

    • Random sample of size O(1/ log(1/)) suffices

    • Smaller sizes (e.g., for rectangles)

    • Deterministic?

Mergeable Summaries

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