the data stream space complexity of cascaded norms t.s. jayram david woodruff ibm almaden
TRANSCRIPT
The Data Stream Space Complexity of Cascaded Norms
T.S. JayramDavid Woodruff
IBM Almaden
Data streams Algorithms access data in a sequential fashion
One pass / small space Need to be randomized and approximate [FM, MP, AMS]
Algorithm MainMemory
2 3 4 16 0 100 5 4 501 200 401 2 3 6 0
Frequency Moments and Norms Stream defines updates to a set of
items 1,2,…,d. fi = weight of item i positive-only vs. turnstile model
k-th Frequency Moment Fk = i |fi|k
p-th Norm: Lp = kfkp = (i |fi|p)1/p
Maximum frequency: p=1 Distinct Elements: p=0 Heavy hitters Assume length of stream and
magnitude of updates is · poly(d)
Classical Results
Approximating Lp and Fp is the same problem
For 0 · p · 2, Fp is approximable in O~(1) space (AMS, FM, Indyk, …)
For p > 2, Fp is approximable in
O~(d1-2/p) space (IW) this is best-possible (BJKS, CKS)
Cascaded Aggregates
Stream defines updates to pairs of items in {1,2,…n} x {1,2,…,d} fij = weight of item (i,j)
Two aggregates P and Q
0
BBB@
f 11 f 12 : : : f 1d
f 21 f 22 : : : f 2d...
......
...f n1 f n2 : : : f nd
1
CCCA
Q PP ± Q
P ± Q = cascaded aggregate
0
BBB@
Q(Row1)Q(Row2)
...Q(Rown)
1
CCCA
Motivation
Multigraph streams for analyzing IP traffic [Cormode-Muthukrishnan]
Corresponds to P ± F0 for different P’s F0 returns #destinations accessed by
each source Also introduced the more general
problem of estimating P ± Q Computing complex join estimates Product metrics [Andoni-Indyk-Krauthgamer]
Stock volatility, computational geometry, operator norms
k
n
n1-2/k d
1
k=p
0 1 2 1
0
1
2
p
n1-2/k d1-2/p
n1-1/k
£(1)
?
£(1)
d1-2/p d
n1-1/k
The Picture
Estimating Lk ± Lp
We give a 1-pass O~(n1-2/kd1-2/p) space algorithm when k ¸ p
We also provide a matching lower bound based on multiparty disjointness
We give a 1-pass O~(n1-2/kd1-2/p) space algorithm when k ¸ p
We also provide a matching lower bound based on multiparty disjointness
We give the (n1-1/k) bound for Lk ± L0 and Lk ± L1
Õ(n1/2) for L2 ± L0 without deletions [CM]Õ(n1-1/k) for Lk ± Lp for any p in {0,1} in turnstile [MW]
We give the (n1-1/k) bound for Lk ± L0 and Lk ± L1
Õ(n1/2) for L2 ± L0 without deletions [CM]Õ(n1-1/k) for Lk ± Lp for any p in {0,1} in turnstile [MW][Ganguly] (without
deletions)[Ganguly] (without deletions)
Follows from techniques of[ADIW]
Follows from techniques of[ADIW] Our upper
bound Our upper bound
Our Problem: Fk ± Fp
Fk ± Fp (M) = i (j |fij|p)k
= i Fp(Row i)k
0
BBB@
f 11 f 12 : : : f 1d
f 21 f 22 : : : f 2d...
......
...f n1 f n2 : : : f nd
1
CCCA
M =
High Level Ideas: Fk ± Fp
1. We want the Fk-value of the vector (Fp(Row 1), …, Fp(Row n))
2. We try to sample a row i with probability / Fp(Row i)
3. Spend an extra pass to compute Fp(Row i)
4. Could then output Fp(M) ¢ Fp(Row i)k-1
(can be seen as a generalization of [AMS])
How do we do the sampling efficiently??
How do we do the sampling efficiently??
Review – Estimating Fp [IW]
Level sets:
Level t is good if |St|(1+ε)2t ¸ F2/B
Items from such level sets are also good
St = f i j (1+ ²)t · jf i j · (1+ ²)t+1g
²-Histogram [IW]
Finds approximate sizes s’t of level sets For all St, s’t · (1+ε)|St|
For good St, s’t ¸ (1- ε)|St|
Also provides O~(1) random samples from each good St
Space: O~(B)
Sampling Rows According to Fp value Treat n x d matrix M as a vector:
Run ε-Histogram on M for certain B Obtain (1§ε)-approximation st’ to |St| for good t
Fk ± Fp(M’) ¸ (1-ε) Fk ± Fp(M), where M’ is M restricted to good items (Holder’s inequality)
To sample, Choose a good t with probability
st’(1+ε)pt/Fp’(M),
where Fp’(M) = sumgood t st’ (1+ε)pt
Choose random sample (i, j) from St
Let row i be the current sample
Pr[row i] = t [st’(1+ε)pt/Fp’(M)]¢[|St Å row i|/|St|]
¼ Fp(row i)/Fp(M)
Pr[row i] = t [st’(1+ε)pt/Fp’(M)]¢[|St Å row i|/|St|]
¼ Fp(row i)/Fp(M) Problems1. High level algorithm requires
many samples (up to n1-1/k) from the St, but [IW] just gives O~(1).
Can’t afford to repeat in low space
2. Algorithm may misclassify a pair (i,j) into St when it is in St-1
Problems1. High level algorithm requires
many samples (up to n1-1/k) from the St, but [IW] just gives O~(1).
Can’t afford to repeat in low space
2. Algorithm may misclassify a pair (i,j) into St when it is in St-1
High Level Ideas: Fk ± Fp
1. We want the Fk-value of the vector (Fp(Row 1), …, Fp(Row n))
2. We try to sample a row i with probability / Fp(Row i)
3. Spend an extra pass to compute Fp(Row i)
4. Could then output Fp(M) ¢ Fp(Row i)k-1
(can be seen as a generalization of [AMS])
How do we avoid an extra pass??
How do we avoid an extra pass??
Avoiding an Extra Pass Now we can sample a Row i / Fp(Row i)
We design a new Fk-algorithm to run on(Fp(Row 1), …, Fp(Row n))
which only receives IDs i with probability / Fp(Row i)
For each j 2 [log n], algorithm does:1. Choose a random subset of n/2j rows2. Sample a row i from this set with Pr[Row i] / Fp(Row i)
We show that O~(n1-1/k) oracle samples is enough to estimate Fk up to 1§ε
New Lower Bounds
Alice Bob
n x d matrix A n x d matrix B
NO instance: for all rows i, ¢(Ai, Bi) · 1
YES instance: there is a unique row j for which¢(Aj, Bj) = d, and for all i j, ¢(Ai, Bi) · 1
We show distinguishing these cases requires (n/d) randomized communication CC
Implies estimating Lk(L0) or Lk(L1) needs (n1-1/k) space
Information Complexity Paradigm [CSWY, BJKS]: the information cost IC is the
amount of information the transcript reveals about the inputs
For any function f, CC(f) ¸ IC(f)
Using their direct sum theorem, it suffices to show an (1/d) information cost of a protocol for deciding if ¢(x,y) = d or ¢(x,y) · 1
Caveat: distribution is only on instances where ¢(x,y) · 1
Working with Hellinger Distance Given the prob. distribution vector ¼(x,y) over transcripts of an
input (x,y) Let Ã(x,y)¿ = ¼(x,y)¿
1/2 for all ¿
Information cost can be lower bounded by ¢(u,v) = 1 kÃ(u,u) - Ã(u,v)k2
Unlike previous work, we exploit the geometry of the squared Euclidean norm (useful in later work [AJP])
Short diagonals property:¢(u,v) = 1 kÃ(u,u) - Ã(u,v)k2 ¸ (1/d) ¢(u,v) = d kÃ(u,u) - Ã(u,v)k2
a
b
c
d
ef
a2 + b2 + c2 + d2 ¸ e2 + f2
Open Problems
Lk ± Lp estimation for k < p
Other cascaded aggregates, e.g. entropy
Cascaded aggregates with 3 or more stages