1. New Characterizations in Turnstile Streams with Applications
Yuqing Ai
Tsinghua University
Wei Hu
Tsinghua University
Yi Li
Facebook
David Woodruff
IBM Almaden

2. Turnstile Streaming Model
Underlying 𝑛-dimensional vector 𝑥 initialized to 0
Stream of updates 𝑥←𝑥+ 𝑒 𝑖 or 𝑥←𝑥− 𝑒 𝑖 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

3. Example: Estimating the ℓ 2 -norm
Output 𝑍 with 1−𝜖 𝑥 2 ≤𝑍≤ 1+𝜖 𝑥 2
Algorithm:
Let 𝑟=1/ 𝜖 2
Choose an 𝑟×𝑛 matrix 𝐴 of i.i.d. sign random variables (+1 w.p. 1/2, −1 w.p. 1/2)
Maintain 𝐴𝑥 in the stream
Output 𝐴𝑥 𝑟

4. Generic Form
All known algorithms have the following generic form (linear sketch):
Sample a random matrix 𝐴
Maintain 𝐴𝑥 in the stream
Output a function of 𝐴𝑥
Question (?!): does the optimal algorithm for approximating any function in the turnstile model have this form?

5. 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
samples a matrix 𝐴 and a vector 𝑞 uniformly from 𝑂(𝑛 log 𝑚 ) instances
maintains (𝐴𝑥 mod 𝑞) in the stream
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)

6. Consequence Input 𝑥 Input 𝑦 Create stream 𝑠(𝑥) Create stream 𝑠(𝑦)
Lower Bound Technique
Streaming algorithm 𝒜
Run 𝒜 on 𝑠(𝑥), send state of 𝒜(𝑠(𝑥)) to Bob
Bob computes 𝒜(𝑠(𝑥), 𝑠(𝑦))
If Bob solves 𝑔(𝑥,𝑦), space complexity of 𝒜 at least the 1-way communication complexity of 𝑔

7. Consequence Input 𝑥 Input 𝑦 Create stream 𝑠(𝑥) 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

8. 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

9. Strengthen the LNW Reduction
Remove the “box constraint”
Generalize to the strict turnstile model
Extend to multi-pass algorithms

10. 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 ∞ ≫𝑚
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.

11. Stream Automaton … + 𝑒 𝑛 … − 𝑒 𝑛 … − 𝑒 1 , + 𝑒 2 … + 𝑒 1 + 𝑒 1 + 𝑒 5
Start …
+ 𝑒 1
+ 𝑒 1
+ 𝑒 5
− 𝑒 1 … …

12. Path-Independent Automaton
Every 𝑥∈ ℤ 𝑛 in a unique state

13. Path-Independent Automaton
+ 𝑒 𝑛 …
− 𝑒 𝑛 …
− 𝑒 1 , + 𝑒 2
Start …
+ 𝑒 1
+ 𝑒 1
0 in two different states
+ 𝑒 5
− 𝑒 1 … …

14. Path-Independent Automaton
Every 𝑥∈ ℤ 𝑛 in a unique state
Equivalent to 𝐴𝑥 mod 𝑞

15. Zero-Frequency Graph
For stream 𝜎, let freq 𝜎 ∈ ℤ 𝑛 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

16. The LNW Reduction 𝐺: zero-frequency graph of 𝒜 old
States of new automaton 𝒜 new = terminal equivalence classes in 𝐺
For a terminal equivalence class 𝐶 and an update 𝑒 𝑖 , define transition as:
Let 𝑣∈𝐶 be an arbitrary node
Compute 𝑣⊕ 𝑒 𝑖 using transition function of 𝒜 old
Walk from 𝑣⊕ 𝑒 𝑖 in 𝐺 until reach a terminal equivalence class 𝐶′
𝐶′ is unique
Does not depend on 𝑣 or the walk

17. Terminal equivalence class 𝐶𝑣 𝑒𝑖
freq(𝜎) = 0
Terminal equivalence class 𝐶′

18. The Box Constraint For a stream 𝜎, define
| 𝜎| max = max prefix 𝜔 of 𝜎 freq 𝜔 ∞
𝜏 1 , 𝜏 2 , … are zero-frequency streams (walks in 𝐺)
Length of 𝜏 𝑖 could be very large
When | 𝜎| max ≤𝑚, | 𝜎′| max could be very large
𝜎=( 𝜎 1 , 𝜎 2 , …, 𝜎 𝑘 ) on 𝒜 new
𝜎′=(… ,𝜎 1 ,…, 𝜎 2 , …, 𝜎 𝑘 , …) on 𝒜 old
𝜏 1
𝜏 2
𝜏 3
𝜏 4
𝜏 5
𝜏 6 …

19. Zero-Freq Stream Length
𝐿: upper bound on the lengths of 𝜏 𝑖 ’s
| 𝜎| max ≤𝑚 ⟹| 𝜎′| max ≤𝑚+𝐿/2
Want 𝐿≤𝑚
Let s = # states in 𝒜 old
Lemma: if there is a zero-freq stream from 𝑢 to 𝑣, then there exists such a stream with length at most poly 𝑛𝑠 ⋅ 𝑠 𝑛 +1 𝑛
𝐿≤poly 𝑛𝑠 ⋅ 𝑠 𝑛 +1 𝑛

20. Tightness of Our Bound
𝐿≤poly 𝑛𝑠 ⋅ 𝑠 𝑛 +1 𝑛
Lower bound: 𝐿≥ 𝑠 𝑛 Ω(𝑛)

21. Removing the Box Constraint
Want 𝐿≤𝑚
𝐿≤poly 𝑛𝑠 ⋅ 𝑠 𝑛 +1 𝑛 ≤ 𝑠 𝑐𝑛
𝐿≤𝑚 ⟸ 𝑠 𝑐𝑛 ≤𝑚 ⟸ log 𝑠 ≤ log 𝑚 𝑐𝑛
Space of 𝒜 old

22. 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!

23. Application: Counting
Condition for removing box constraint: space ≤ log 𝑚 𝑐𝑛 = log 𝑚 𝑐
Assume space ≤ log 𝑚 𝑐 , otherwise done
𝐴𝑥 mod 𝑞=( 𝑎 1 𝑥 mod 𝑞 1 , 𝑎 2 𝑥 mod 𝑞 2 , …, 𝑎 𝑟 𝑥 mod 𝑞 𝑟 )
Show lcm 𝑞 1 , …, 𝑞 𝑟 =Ω(𝑚)
Cannot distinguish 𝑥, 𝑥+lcm, 𝑥+2⋅lcm, …
Ω(𝑚) different states, Ω( log 𝑚 ) space

24. Application: Norm Estimation
Problem: for 𝑥∈ −𝑚, …, 𝑚 𝑛 , output 𝑥 𝑝 up to additive error 𝑛 1/𝑝 𝑚
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)

25. Strengthen the LNW Reduction
Remove the “box constraint”
Generalize to the strict turnstile model
Extend to multi-pass algorithms

26. The Strict Turnstile Model
The strict turnstile model: no negative coordinates, i.e., 𝑥 𝑖 ≥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

27. Application: Maximum Matching
[AKLY’16]: For outputting an 𝑛 𝜖 -approximate maximum matching, space is Θ ( 𝑛 2−3𝜖 )
Lower bound only in simultaneous communication model
Can apply our reduction

28. Strengthen the LNW Reduction
Remove the “box constraint”
Generalize to the strict turnstile model
Extend to multi-pass algorithms

29. Multi-Pass Algorithms
𝑝-pass automaton
After 𝑖-th pass (𝑖<𝑝), output an automaton 𝒜 𝑖+1
Run 𝒜 𝑖+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

30. 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!