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Streaming space complexity of nearly all functions of one variable

Vladimir Braverman, Stephen Chestnut, David P. Woodruff, Lin F. Yang

January 7, 2016

A stream of m = 7 items from [n] = [4] 4, 2, 3, 2, 4, 2, 2 f =         f 2

i =0

A stream of m = 7 items from [n] = [4] 4, 2, 3, 2, 4, 2, 2 f =     1     f 2

i =

1

A stream of m = 7 items from [n] = [4] 2, 3, 2, 4, 2, 2 f =     1 1     f 2

i =

2

A stream of m = 7 items from [n] = [4] 3, 2, 4, 2, 2 f =     1 1 1     f 2

i =

3

A stream of m = 7 items from [n] = [4] 2, 4, 2, 2 f =     2 1 1     f 2

i =

6

A stream of m = 7 items from [n] = [4] 4, 2, 2 f =     2 1 2     f 2

i =

9

A stream of m = 7 items from [n] = [4] 2, 2 f =     3 1 2     f 2

i =

14

A stream of m = 7 items from [n] = [4] 2 f =     4 1 2     f 2

i =

21

A stream of m = 7 items from [n] = [4] f =     4 1 2     f 2

i =

21 How much storage for a streaming (1 ± ǫ)-approximation to

i f 2 i ?

Classify g : Z≥0 → R Is there a streaming (1 ± ǫ)-approximation for

i g(fi)

using only poly( 1

ǫ log nm) bits?

Previous works g(x) = 1(x = 0): [FM85],[KNW10] g(x) = xp: [F85],[AMS96],[IW05],[I06] g(x) = x log x: [CDM06],[CCM07],[HNO08] monotonic g: [BO10],[BC15] ǫ = Ω(

1 polylog(n))

m = poly(n) g(0) = 0 g(x) > 0, ∀x > 0

Recursive Subsampling [Indyk & Woodruff 2005]

An α-heavy hitter is any item i∗ such that g(fi∗) ≥ α

i g(fi).

Theorem (Braverman & Ostrovsky 2010)

ǫ2 log3 n-heavy hitters ⇒ (1 ± ǫ)-approximation to

• i

g(fi).

Recursive Subsampling [Indyk & Woodruff 2005]

An α-heavy hitter is any item i∗ such that g(fi∗) ≥ α

i g(fi).

Theorem (Braverman & Ostrovsky 2010)

ǫ2 log3 n-heavy hitters ⇒ (1 ± ǫ)-approximation to

• i

g(fi). Heavy hitters by CountSketch[Charikar, Chen & Farach-Colton 2002] Find i∗ such that f 2

i∗ ≥ α i f 2 i

Estimate fi∗ O(α−1 log2 n) bits.

Three properties are sufficient and almost necessary for ˜ O(1) bits g(x) x

Slow-jumping g(x) x x y g(x) g(y) g(y) g(x) y x 2 YES: g(x) = x2 log x NO: g(x) = x3

Slow-dropping g(x) x x g(x) y g(y) g(y) g(x) 1 YES: g(x) = Θ( 1 log x ) NO: g(x) = Θ(1 x )

Predictable g(x) x g(x) g(y)

• y−x≪x

g(y) = (1 ± ǫ)g(x)

• r

g(y − x) g(x) YES: g(x) = (2 + sin x)1(x > 0) NO: g(x) = (2 + sin x)x2

Predictable g(x) x g(x) g(y) g(y − x)

• y−x≪x

g(y) = (1 ± ǫ)g(x)

• r

g(y − x) g(x) YES: g(x) = (2 + sin x)1(x > 0) NO: g(x) = (2 + sin x)x2

Three properties are sufficient and almost necessary for O(1) bits slow-jumping

g(y) g(x)

y

x

2, slow-dropping g(y) g(x), and predictable whenever 0 < y − x ≪ x g(y) = (1 ± ǫ)g(x) or g(y − x) g(x). g(x) lower bound fails x3 Ω(n1/3) slow-jumping 1/x Ω(n) slow-dropping g(x) = (2 + sin x)x2 Ω(n) predictability

Almost necessary? 2−i(x) x 1

1 2 1 4

i(x) = max{j ∈ N : 2j divides x}