Turnstile Streaming Algorithms
Might as Well Be Linear Sketches
Yi Li
Max-Planck
Huy L. Nguyen
David Woodruff
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.
Some functions f(x) may be
Choose a random
weird: matrix A independent of x
2.
xx1?
Maintain AxWhat
in theis 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
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 Technique
1. 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 1way 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
Stream Automaton for Fixed
Randomness
…
-en
+en
…
-e1, +e2
Start
+e1
…
Streaming
algorithm only
depends on x, not …
how it got there
+e1
0n in two
+edifferent
5
states
-e1
…
…
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
Intuitively
makes things
For stream σ, freq(σ) 2 Zn is “net update”
to each
pathcoordinate
independent
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?
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
x
Terminal
equivalence
class
u
+ei
v
Contradiction:
zero frequency
path w-x-v-y
+ei
-ei
freq(σ) = 0
w
Terminal
equivalence
class
freq(σ’) = 0
y
Terminal
equivalence
class
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?