New Characterizations in Turnstile
Streams with Applications
Yuqing Ai
Wei Hu
Tsinghua University
Tsinghua University
Yi Li
David Woodruff
Facebook
IBM Almaden
Turnstile Streaming Model

Underlying 𝑛-dimensional vector 𝑥 initialized to 0

Stream of updates 𝑥 ← 𝑥 + 𝑒𝑖 or 𝑥 ← 𝑥 − 𝑒𝑖 for
standard unit vector 𝑒𝑖

At end of the stream, 𝑥 ∈ {−𝑚, … , −1, 0, 1, … , 𝑚}𝑛

Output an approximation to 𝑓(𝑥) w.h.p.

Goal: use as small space in bits as possible
Example: Estimating the ℓ2 -norm

Output 𝑍 with 1 − 𝜖 𝑥

Algorithm:
2
≤𝑍 ≤ 1+𝜖 𝑥
2
1. Let 𝑟 = 1/𝜖 2
2. Choose an 𝑟 × 𝑛 matrix 𝐴 of i.i.d. sign random
variables (+1 w.p. 1/2, −1 w.p. 1/2)
3. Maintain 𝐴𝑥 in the stream
4. Output
𝐴𝑥 2
𝑟
Generic Form

All known algorithms have the following generic
form (linear sketch):
1.
Sample a random matrix 𝐴
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?
The LNW Reduction

Yes! [Li, Nguyễn, Woodruff’14]

Theorem: for computing a function 𝑓 of 𝑥 in
−𝑚, … , 𝑚 𝑛 in the turnstile model, there is a
randomized algorithm which
1. samples a matrix 𝐴 and a vector 𝑞 uniformly
from 𝑂(𝑛 log 𝑚) instances
2. maintains (𝐴𝑥 mod 𝑞) in the stream
3. outputs a function of (𝐴𝑥 mod 𝑞)

Space complexity is optimal up to a constant
factor (not including the 𝑂(log 𝑛 + log log 𝑚) bits
for randomness)
Consequence
Input 𝑥
Create stream 𝑠(𝑥)
Input 𝑦
Create stream 𝑠(𝑦)
Lower Bound Technique
Streaming algorithm 𝒜
1. Run 𝒜 on 𝑠(𝑥), send state of 𝒜(𝑠(𝑥)) to Bob
2. Bob computes 𝒜(𝑠(𝑥), 𝑠(𝑦))
3. If Bob solves 𝑔(𝑥, 𝑦), space complexity of 𝒜 at
least the 1-way communication complexity of 𝑔
Consequence
Input 𝑥
Create stream 𝑠(𝑥)
Input 𝑦
Create stream 𝑠(𝑦)
The LNW reduction implies
If players can solve 𝑔(𝑥, 𝑦), then space of 𝒜 at least
the simultaneous communication complexity of 𝑔
Weaker model in which Alice and Bob simultaneously
send a message to a referee who outputs the answer
Our Result

Strengthen the LNW reduction from several
aspects:
◦ Remove the “box constraint”
◦ Generalize to the strict turnstile model
◦ Extend to multi-pass algorithms

Obtain new tight lower bounds
Strengthen the LNW Reduction

Remove the “box constraint”

Generalize to the strict turnstile model

Extend to multi-pass algorithms
The “Box Constraint”

The LNW reduction requires the algorithm to be
correct as long as 𝑥 ∈ −𝑚, … , 𝑚 𝑛 at the end of
the stream.

While processing the stream, may have 𝑥

The algorithm is not allowed to abort if this
happens. It must still be correct at the end of the
stream as long as 𝑥 ∈ −𝑚, … , 𝑚 𝑛 .

More natural requirement: the algorithm only needs
to be correct when 𝑥 belongs to −𝑚, … , 𝑚 𝑛 at all
time in the stream.
∞
≫𝑚
Stream Automaton
…
−𝑒𝑛
+𝑒𝑛
…
−𝑒1 , +𝑒2
…
Start
…
+𝑒1
+𝑒1
+𝑒5
−𝑒1
…
…
Path-Independent Automaton

Every 𝑥 ∈ ℤ𝑛 in a unique state
Path-Independent Automaton
−𝑒𝑛
+𝑒𝑛
…
−𝑒1 , +𝑒2
…
Start
…
+𝑒1
+𝑒1
+𝑒5
0 in two
different states
−𝑒1
…
…
Path-Independent Automaton

Every 𝑥 ∈ ℤ𝑛 in a unique state

Equivalent to 𝐴𝑥 mod 𝑞
Zero-Frequency Graph

For stream 𝜎, let freq 𝜎 ∈ ℤ𝑛 be the “net update”
to all coordinates.

Zero-freq graph: directed graph 𝐺 = (𝑉, 𝐸)
◦ 𝑉 = states of the automaton
◦ 𝑢, 𝑣 ∈ 𝐸 if there exists stream 𝜎 such that 𝑢 ⊕
𝜎 = 𝑣 and freq 𝜎 = 0

Terminal equivalence class: strongly connected
component in 𝐺 with no outgoing edge

Walk in G is a sequence of zero-frequency streams
The LNW Reduction
𝐺: zero-frequency graph of 𝒜old
 States of new automaton 𝒜new = terminal
equivalence classes in 𝐺



For a terminal equivalence class 𝐶 and an update 𝑒𝑖 ,
define transition as:
◦ Let 𝑣 ∈ 𝐶 be an arbitrary node
◦ Compute 𝑣 ⊕ 𝑒𝑖 using transition function of 𝒜old
◦ Walk from 𝑣 ⊕ 𝑒𝑖 in 𝐺 until reach a terminal
equivalence class 𝐶′
𝐶′ is unique
◦ Does not depend on 𝑣 or the walk
𝐶
Terminal
equivalence
class
𝑣
𝑒𝑖
freq(𝜎) = 0
Terminal
equivalence
class
𝐶′
The Box Constraint

For a stream 𝜎, define
|𝜎|max =
max
prefix 𝜔 of 𝜎
freq 𝜔
∞
𝜎 = (𝜎1 , 𝜎2 , … , 𝜎𝑘 ) on 𝒜new
𝜎′ = (… , 𝜎1 , … , 𝜎2 , … , 𝜎𝑘 , … ) on 𝒜old
𝜏1



𝜏2
𝜏3
𝜏4 𝜏5
𝜏6
…
𝜏1 , 𝜏2 , … are zero-frequency streams (walks in 𝐺)
Length of 𝜏𝑖 could be very large
When |𝜎|max ≤ 𝑚, |𝜎′|max could be very large
Zero-Freq Stream Length

𝐿: upper bound on the lengths of 𝜏𝑖 ’s

|𝜎|max ≤ 𝑚 ⟹ |𝜎′|max ≤ 𝑚 + 𝐿/2

Want 𝐿 ≤ 𝑚

Let s = # states in 𝒜old
Lemma: if there is a zero-freq stream from 𝑢 to 𝑣,
then there exists such a stream with length at most
𝑛
𝑠
poly 𝑛𝑠 ⋅ + 1

𝑛

𝐿 ≤ poly 𝑛𝑠 ⋅
𝑠
𝑛
+1
𝑛
Tightness of Our Bound


𝐿 ≤ poly 𝑛𝑠 ⋅
𝑠
𝑛
+1
Lower bound: 𝐿 ≥
𝑛
𝑠 Ω(𝑛)
𝑛
Removing the Box Constraint

Want 𝐿 ≤ 𝑚

𝐿 ≤ poly 𝑛𝑠 ⋅

𝑠 𝑐𝑛
𝐿≤𝑚 ⟸
𝑠
𝑛
+1
𝑛
≤ 𝑠 𝑐𝑛
≤ 𝑚 ⟸ log 𝑠 ≤
log 𝑚
𝑐𝑛
Space of 𝒜old
Application: Counting
𝑛=1
 Problem: output |𝑥| up to additive error 𝑚/4, while
𝑥 varies in {−𝑚, … , 𝑚}


𝑂(log 𝑚) space algorithm

Is there an Ω(log 𝑚) lower bound?
◦ For insertion streams, no: approximate counting
◦ For relative error, yes: but proof doesn’t apply
◦ For additive error… yes!
Application: Counting

Condition for removing box constraint: space ≤
log 𝑚
log 𝑚
=
𝑐𝑛
𝑐
log 𝑚
,
𝑐

Assume space ≤

𝐴𝑥 mod 𝑞 = (𝑎1 𝑥 mod 𝑞1 , 𝑎2 𝑥 mod 𝑞2 , … , 𝑎𝑟 𝑥 mod 𝑞𝑟 )
◦ Show lcm 𝑞1 , … , 𝑞𝑟 = Ω(𝑚)
 Cannot distinguish 𝑥, 𝑥 + lcm, 𝑥 + 2 ⋅ lcm, …
◦ Ω(𝑚) different states, Ω(log 𝑚) space
otherwise done
Application: Norm Estimation

Problem: for 𝑥 ∈ −𝑚, … , 𝑚 𝑛 , output 𝑥
1
additive error 𝑛1/𝑝 𝑚
𝑝
up to
4

Ω(log 𝑚) space lower bound

𝑂(log 𝑚 + log log 𝑛) space algorithm (1 ≤ 𝑝 ≤ 2)
[KNW’10]

Lower bound tight when log log 𝑛 = 𝑂 log 𝑚 ⟺
𝑛 ≤ exp poly(𝑚)
Strengthen the LNW Reduction

Remove the “box constraint”

Generalize to the strict turnstile model

Extend to multi-pass algorithms
The Strict Turnstile Model

The strict turnstile model: no negative coordinates,
i.e., 𝑥𝑖 ≥ 0 at all times in the stream

Dynamic graph streams: insertions and deletions of
edges
◦ Allow multi-graphs, but no negative edges

Generalize the LNW reduction to the strict turnstile
model
◦
◦
◦
◦
𝐿: upper bound on the length of zero-freq streams
Initialize all coordinates of 𝑥 to be 𝐿
Now the reduction guarantees 𝑥 is always nonnegative
Subtract 𝐿 from all coordinates at the end of the stream
Application: Maximum Matching

[AKLY’16]: For outputting an 𝑛𝜖 -approximate
maximum matching, space is Θ(𝑛2−3𝜖 )
◦ Lower bound only in simultaneous communication
model

Can apply our reduction
Strengthen the LNW Reduction

Remove the “box constraint”

Generalize to the strict turnstile model

Extend to multi-pass algorithms
Multi-Pass Algorithms

𝑝-pass automaton
◦ After 𝑖-th pass (𝑖 < 𝑝), output an automaton 𝒜𝑖+1
◦ Run 𝒜𝑖+1 on input stream in (𝑖 + 1)-st pass
◦ After 𝑝-th pass, output answer

Theorem: There is a 𝑝-pass automaton for which
each automaton in each pass is path-independent
◦ Space is optimal up to a constant factor
Conclusions

New progress on characterizing turnstile streaming
algorithms as linear sketches

Applications
◦ Optimal lower bounds for counting with additive
error, maximum matching in dynamic graph

Open questions
◦ Box constraint
◦ After removing box constraint, still have very long
streams
◦ Better reduction?
Thank you!