Turnstile Streaming Algorithms Might as Well Be Linear Sketches

Yi Li Huy L. Nguyen David Woodruff Max-Planck Princeton IBM Almaden

Turnstile Streaming Model

• Underlying n-dimensional vector x initialized to 0n

• Long stream of updates x à x + ei or x à x - ei for standard unit vector ei

• At end of the stream, x 2 {-m, -m+1, …, m-1, m}n for some bound m · poly(n)

• Output an approximation to f(x) whp

• Goal: use as little space (in bits) as possible

Example: Euclidean Norm

• Want to output Z with (1-Ɛ) |x|2 · Z · (1+Ɛ) |x|2

• Let r = 1/Ɛ2

• Choose an r x n matrix A of i.i.d. sign random variables (+1 w.pr. ½, -1 w.pr. ½)

• Maintain Ax in the stream

• Output |Ax|2

• Proof: Johnson-Lindenstrauss Lemma

Generic Features

Algorithm for 2-norm has the following form:

1. Choose a random matrix A independent of x

2. Maintain Ax in the stream

3. Output a function of Ax

Question (?!): does the optimal algorithm for approximating any function in the turnstile model have this form?

All known algorithms have this form

Some functions f(x) may be weird:

What is xx1?

Some functions f(x) may be weird:

What is xx1?

Our Result

• Yes, up to a factor of log n

• Theorem: for computing a relation f for x in {-m, -m+1, …, m}n in the turnstile model, there is a correct (whp) algorithm which:

1. samples an integer matrix A uniformly from O(n log m) hardwired matrices with poly(n) bounded integer entries, independent of x,

2. outputs a function of Ax

Logarithm of the number of states of Ax, for x in {-m, -m+1, …, m}n, plus amount of randomness, is optimal up to a log n factor

Consequences

a 2 {0,1}n

Create stream s(a)

b 2 {0,1}n

Create stream s(b)

Lower Bound Technique1. Run Alg on s(a), transmit state of Alg(s(a)) to Bob

2. Bob computes Alg(s(a), s(b))

3. If Bob solves g(a,b), space complexity of Alg at least the 1-way communication complexity of g

Consequences

a 2 {0,1}n

Create stream s(a)b 2 {0,1}n

Create stream s(b)

Our main theorem implies:If players can solve g(a,b), then space of Alg at least the simultaneous communication complexity of g

Weaker public-coin model in which Alice and Bob simultaneously send a message to a referee

Non-Uniformity Restriction

• Careful wording: “samples an integer matrix A uniformly from O(n log m) hardwired matrices, with poly(n) bounded entries, independent of x”

• Algorithm is non-uniform– Output of each state for each A also hardwired

• Alternatively, allow algorithm to use more space to process a stream update, provided it only retains Ax and its randomness– Regenerate A during each stream update

Comment on the Model

• For each random seed, algorithm is a deterministic automaton with a finite number of states

• Main theorem only requires correctness for

x 2 {-m, -m+1, …, m}n

It counts the number of states as x varies in this range

• While processing the stream, may have |x|1 > m

• The algorithm can’t abort if this happens. It must still be correct at the end of the stream for x in {-m, -m+1, …, m}n

Related Work

• Ganguly

– Specific to heavy hitters problem

– Holds only for deterministic algorithms

Talk Outline

• Proof Overview

1. Reduction to path-independent automata

2. From path-independent automata to linear sketches

• Applications and Open Questions

Start

+e1

-e1, +e2

…-en

-e1

+e1

+en

…

+e5

…

…

… …

Stream Automaton for Fixed Randomness

Streaming algorithm only depends on x, not how it got there

Streaming algorithm only depends on x, not how it got there

0n in two different states

0n in two different states

Path-Independent Automaton

• Each x 2 Zn in a unique state

• Undirected connected graph

• Goal: for each randomness, can we modify the automaton to make it path-independent?

• Rule out algorithms that e.g., an algorithm that stores the last 5 stream updates

Strategy

For stream σ, freq(σ) 2 Zn is “net update” to each coordinate

Idea: 1. if in a state s, and update by a stream σ, with freq(σ) = 0, answers ought to be similar

2. collapse all states s, s’ for which s+σ = s’ and freq(σ) = 0 for some stream σ

Issue: how to formally define states, transition and output function of new automaton?

Intuitively makes things

path-independent

Intuitively makes things

path-independent

Zero-Frequency Graph• Directed multi-graph G = (V,E)

• V = states of old automaton Aold (for fixed randomness)

• (s,t) 2 E for each stream σ of finite length with s+σ=t and freq(σ) = 0

• Terminal equivalence class: strongly connected component with no outgoing edge– Walk in G eventually reaches a terminal equivalence class (Walk in G is a long sequence of zero-streams)

– States of new automaton Anew = terminal equivalence classes

New Transition Function

• Suppose in terminal equivalence class C

• Given an update ei

• Let v 2 C be an arbitrary node

• Compute v+ei using transition function of Aold

• Walk from v+ei until reach terminal equivalence class C’

• C’ is unique• Does not depend on choice of v• Only one terminal equivalence class reachable in any walk

Terminal equivalence

class

Terminal equivalence

class

u v

+ei +ei

Terminal equivalence

class

Terminal equivalence

class

freq(σ) = 0 freq(σ’) = 0

Terminal equivalence

class

Terminal equivalence

class

w

-ei

x

y

Contradiction: zero frequencypath w-x-v-y

Output Function of Anew

• In each terminal equivalence class C, sample node u from stationary distribution from random walk in C (add self-loops)– Output of Anew on C = Output of Aold on u

• If v is starting vertex of Aold,

– take a random walk in G from v

– let starting vertex of Anew be terminal equivalence class C reached

• Why is it correct?

Correctness

• Let ¦ be an arbitrary distribution on streams ¾

• Choose fixed randomness so Aold correct on ¦’: – Long sequence of zero frequency streams, – Followed by ¾ sampled from ¦,– Followed by long sequence of zero frequency streams

• Output of Anew on ¦ statistically close to output of Aold on ¦’

• => for every ¦ there is an Anew correct on ¦

• Can show Anew is path-independent (lying a little..)

Path Independence to Linear Sketches

• M = {x 2 Zn such that x in same state as 0n}

• States of automaton are cosets of Zn/M

• Use lattice tools…

Applications and Open Questions

• Simpler proof of existing lower bounds– No communication complexity

• Many dimension lower bounds known for sketching norms over the reals– Matrix norms, etc.– Do these give turnstile streaming lower

bounds with finite precision?