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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)

- 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

- 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

- 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)

w

Array: CM[i,j]

d

Mergeable Summaries

7

5

1

- 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

6

4

2

1

Mergeable Summaries

- 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 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

- 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

- 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

- 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

- 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

Buffer

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

- 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