Something for almost nothing:
Advances in sublinear time
algorithms
Ronitt Rubinfeld
MIT and Tel Aviv U.
Algorithms for REALLY big data
No time
What can we hope to do without viewing
most of the data?
Small world phenomenon
• The social network is a
graph:
– “node’’ is a person
– “edge’’ between people that
know each other
• “6 degrees of separation’’
• Are all pairs of people
connected by path of
distance at most 6?
Vast data
• Impossible to access all of it
• Accessible data is too enormous to be
viewed by a single individual
• Once accessed, data can change
The Gold Standard
• Linear time algorithms:
– for inputs encoded by n
bits/words, allow cn time
steps (constant c)
• Inadequate…
What can we hope to do without viewing
most of the data?
• Can’t answer “for all” or “exactly” type statements:
• are all individuals connected by at most 6 degrees of
separation?
• exactly how many individuals on earth are left-handed?
• Compromise?
• is there a large group of individuals connected by at most 6
degrees of separation?
• approximately how many individuals on earth are left-handed?
What types of approximation?
Property Testing
• Does the input object have crucial properties?
• Example Properties:
•
•
•
•
•
•
Clusterability,
Small diameter graph,
Close to a codeword,
Linear or low degree polynomial function,
Increasing order
Lots and lots more…
“In the ballpark” vs. “out of the ballpark”
tests
• Property testing: Distinguish inputs that have specific property
from those that are far from having that property
• Benefits:
– Can often answer such questions much faster
– May be the natural question to ask
• When some “noise” always present
• When data constantly changing
• Gives fast sanity check to rule out very “bad” inputs (i.e., restaurant
bills)
• Model selection problem in machine learning
Examples
• Can test if a function is a homomorphism in CONSTANT
TIME [Blum Luby R.]
• Can test if the social network has 6 degrees of separation
in CONSTANT TIME [Parnas Ron]
Constructing a property tester:
• Find characterization of property that is
• Efficiently (locally) testable
• Robust • objects that have the property satisfy characterization,
• and objects far from having the property are unlikely to
PASS
Usually the
bigger
challenge
Example: Homomorphism property of
functions
• A “bad” testing characterization:
∀x,y f(x)+f(y) = f(x+y)
• Another bad characterization:
For most x f(x)+f(1) = f(x+1)
• Good characterization:
For most x,y f(x)+f(y) = f(x+y)
Example: 6 degrees of separation
• A “bad” testing characterization:
For every node, all other nodes within distance 6.
• Another bad one:
For most nodes, all other nodes within distance 6.
• Good characterization:
For most nodes, there are many other nodes within distance 6.
An example in depth
Monotonicity of a sequence
• Given: list y1 y2 ... yn
• Question: is the list sorted?
• Clearly requires n steps – must look at each yi
Monotonicity of a sequence
• Given: list y1 y2 ... yn
• Question: can we quickly test if the list close to sorted?
What do we mean by ``quick’’?
• query complexity measured in terms of list size n
• Our goal (if possible):
• Very small compared to n, will go for clog n
What do we mean by “close’’?
Definition: a list of size n is .99-close to sorted if can delete at
most .01n values to make it sorted. Otherwise, .99-far.
Sorted: 1 2 4 5 7 11 14 19 20 21 23 38 39 45
Close: 1 4 2 5 7 11 14 19 20 39 23 21 38 45
1 4
5 7 11 14 19 20
23
38 45
Far: 45 39 23 1 38 4 5 21 20 19 2 7 11 14
1
4 5
7 11 14
Requirements for algorithm:
• pass sorted lists
• if list passes test, can change at most .01 fraction of list to
make it sorted
An attempt:
• Proposed algorithm:
• Pick random i and test that yi≤yi+1
• Bad input type:
• 1,2,3,4,5,…j, 1,2,3,4,5,….j, 1,2,3,4,5,…j, 1,2,3,4,5,…,j
• Difficult for this algorithm to find “breakpoint”
• But other algorithms work well
yi
i
A second attempt:
• Proposed algorithm:
• Pick random i<j and test that yi≤yj
• Bad input type:
• n/m groups of m elements
m,m-1,m-2,…,1, 2m,2m-1,2m-2,…m+1, 3m,3m-1,3m-2,…,
• must pick i,j in same group
• need at least (n/m)1/2 choices to do this
yi
i
A test that works
• The test: (for distinct yi)
• Test several times:
•
•
•
•
•
Pick random i
Look at value of yi
Do binary search for yi
Does the binary search find any inconsistencies? If yes, FAIL
Do we end up at location i? If not FAIL
• Pass if never failed
• Running time: O(log n) time
• Why does this work?
• If list is in order, then test always passes
• If the test passes on choice i and j, then yi and yj are in correct order
• Since test usually passes, most yi’s in the right order
Many more properties studied!
• Graphs, functions, point sets, strings, …
• Amazing characterizations of problems testable in
graph and function testing models!
Properties of functions
• Linearity and low total degree polynomials
[Blum Luby R.] [Bellare Coppersmith Hastad Kiwi Sudan] [R. Sudan] [Arora Safra]
[Arora Lund Motwani Sudan Szegedy] [Arora Sudan] ...
• Functions definable by functional equations –
trigonometric, elliptic functions [R.]
• All locally characterized affine invariant function
classes! [Bhattacharyya Fischer Hatami Hatami Lovett]
• Groups, Fields [Ergun Kannan Kumar R. Viswanathan]
• Monotonicity [EKKRV] [Goldreich Goldwasser Lehman Ron
Samorodnitsky] [Dodis Goldreich Lehman Ron Raskhodnikova
Samorodnitsky][Fischer Lehman Newman Raskhodnikova R. Samorodnitsky]
[Chakrabarty Seshadri]…
• Convexity, submodularity
[Parnas Ron R.] [Fattal Ron] [Seshadri
Vondrak]…
• Low complexity functions, Juntas
[Parnas Ron Samorodnitsky] [Fischer Kindler Ron Safra Samorodnitsky]
[Diakonikolas Lee Matulef Onak R. Servedio Wan]…
Properties of sparse and general graphs
• Easily testable dense and hyperfinite graph properties are
completely characterized! [Alon Fischer Newman
Shapira][Newman Sohler]
• General Sparse graphs: bipartiteness, connectivity,
diameter, colorability, rapid mixing, triangle free,…
[Goldreich Ron] [Parnas Ron] [Batu Fortnow R. Smith White]
[Kaufman Krivelevich Ron] [Alon Kaufman Krivelevich Ron]…
• Tools: Szemeredi regularity lemma, random walks, local
search, simulate greedy, borrow from parallel algorithms
Some other combinatorial properties
• Set properties – equality, distinctness,...
• String properties – edit distance, compressibility,…
• Metric properties – metrics, clustering, convex hulls,
embeddability...
• Membership in low complexity languages –
regular languages, constant width branching programs,
context-free languages, regular tree languages…
• Codes – BCH and dual-BCH codes, Generalized ReedMuller,…
“Traditional” approximation
• Output number close to value of the optimal solution (not
enough time to construct a solution)
• Some examples:
•
•
•
•
•
Minimum spanning tree,
vertex cover,
max cut,
positive linear program,
edit distance, …
Example: Vertex Cover
• Given graph G(V,E), a vertex cover (VC) C is a
subset of V such that it “touches” every edge.
• What is minimum size of a vertex cover?
• NP-complete
• Poly time multiplicative 2-approximation based on
relationship of VC and maximal matching
Vertex Cover and Maximal Matching
• Maximal Matching:
• M ⊆ E is a matching if no node in in more than one edge.
• M is a maximal matching if adding any edge violates the
matching property
• Note: nodes matched by M form a pretty good
Vertex Cover!
• Vertex cover must include at least one node in each edge
of M
• Union of nodes of M form a vertex cover
• So |M| ≤ VC ≤ 2 |M|
“Classical” approximation examples
• Can get CONSTANT TIME approximation for vertex cover
on sparse graphs!
• Output y which is at most 2∙ OPT + ϵn
How?
• Oracle reduction framework [Parnas Ron]
•
•
Construct “oracle” that tells you if node u in 2-approx vertex
cover
Use oracle + standard sampling to estimate size of cover
But how do you implement the oracle?
Implementing the oracle – two
approaches:
• Sequentially simulate computations of a local
distributed algorithm [Parnas Ron]
• Figure out what greedy maximal matching
algorithm would do on u [Nguyen Onak]
Greedy algorithm for maximal matching
• Algorithm:
• ←∅
• For every edge (u,v)
• If neither of u or v matched
• Add (u,v) to M
• Output M
• Why is M maximal?
• If e not in M then either u or v already matched by earlier
edge
Implementing the Oracle via Greedy
• To decide if edge e in matching:
• Must know if adjacent edges that come before e
in the ordering are in the matching
• Do not need to know anything about edges
coming after
• Arbitrary edge order can have long
dependency chains!
1
2
4
8
25
36
47
88
89
Odd or even
steps from
beginning?
110 111 112 113
Breaking long dependency chains
[Nguyen Onak]
• Assign random ordering to edges
• Greedy works under any ordering
• Important fact: random order has short
dependency chains
Better Complexity for VC
• Always recurse on least ranked edge first
• Heuristic suggested by [Nguyen Onak]
• Yields time poly in degree [Yoshida Yamamoto Ito]
• Additional ideas yield query complexity nearly
linear in average degree for general graphs [Onak Ron
Rosen R.]
Further work
• More complicated arguments for maximum
matching, set cover, positive LP… [Parnas Ron + Kuhn
Moscibroda Wattenhofer] [Nguyen Onak] [Yoshida Yamamoto Ito]
Can dependence be
made poly in average
degree?
• Even better results for hyperfinite graphs [Hassidim
Kelner Nguyen Onak][Newman Sohler]
• e.g., planar
No samples
What if data only accessible via random
samples?
Distributions
Play the lottery?
Is the lottery unfair?
• From Hitlotto.com: Lottery experts agree,
past number histories can be the key to
predicting future winners.
True Story!
• Polish lottery Multilotek
• Choose “uniformly” at random distinct 20 numbers
out of 1 to 80.
• Initial machine biased
• e.g., probability of 50-59 too small
• Past results:
http://serwis.lotto.pl:8080/archiwum/wyniki_wszystkie.php?id_gra=2
Thanks to Krzysztof Onak (pointer) and Eric Price (graph)
New Jersey Pick 3,4 Lottery
• New Jersey Pick k ( =3,4) Lottery.
• Pick k digits in order.
• 10k possible values.
• Assume lottery draws iid
• Data:
• Pick 3 - 8522 results from 5/22/75 to 10/15/00
• 2-test gives 42% confidence
• Pick 4 - 6544 results from 9/1/77 to 10/15/00.
• fewer results than possible values
• 2-test gives no confidence
Distributions on BIG domains
• Given samples of a distribution, need to know, e.g.,
•
•
•
•
entropy
number of distinct elements
“shape” (monotone, bimodal,…)
closeness to uniform, Gaussian, Zipfian…
• No assumptions on shape of distribution
• i.e., smoothness, monotonicity, Normal distribution,…
• Considered in statistics, information theory, machine
learning, databases, algorithms, physics, biology,…
Key Question
• How many samples do you need in terms of
domain size?
• Do you need to estimate the probabilities of each
domain item?
• Can sample complexity be sublinear in size of the
domain?
Rules out standard statistical techniques,
learning distribution
Our Aim:
Algorithms with sublinear sample complexity
Similarities of distributions
Are p and q close or far?
• p is given via samples
• q is either
• known to the tester (e.g. uniform)
• given via samples
Is p uniform?
• Theorem: ([Goldreich Ron] [Batu
p
Fortnow R. Smith White] [Paninski])
samples
Test
Pass/Fail?
Sample complexity of
distinguishing
=
/2
1
from |  −  |1 >  is ( )
• Nearly same complexity to test
if p is any known distribution
[Batu Fischer Fortnow Kumar R.
1
White]:“Testing
identity”
−  1 = Σ  −

Upper bound for L2 distance [Goldreich Ron]
• L2 distance:
−
2
2
= ∑  − 
• ||p-U||22 = S(pi -1/n)2
= Spi2 - 2Spi /n + S1/n2
= Spi2 - 1/n
2
========
• Estimate collision probability to estimate L2
distance from uniform
Testing uniformity [GR, Batu et. al.]
• Upper bound: Estimate collision probability and use
known relation between between L1 and L2 norms
• Issues:
• Collision probability of uniform is 1/n
• Use O(sqrt(n)) samples via recycling
• Comment: [P] uses different estimator
• Easy lower bound: (n½)
• Can get  (n½/2) [P]
Back to the lottery…
plenty of samples!
Is p uniform?
• Theorem: ([Goldreich Ron][Batu
p
Fortnow R. Smith White] [Paninski])
samples
Test
Pass/Fail?
Sample complexity of
distinguishing
p=U
from |p-U|1> is (n1/2)
• Nearly same complexity to test
if p is any known distribution
[Batu Fischer Fortnow Kumar R.
White]: “Testing identity”
Testing closeness of two distributions:
Transactions of 20-30 yr olds
trend change?
Transactions of 30-40 yr olds
Testing closeness
p
q
Test
Pass/Fail?
Theorem: ([BFRSW] [P. Valiant])
Sample complexity of
distinguishing
p=q
from ||p-q||1 >
~ 2/3
is (n )
Why so different?
• Collision statistics are all that matter
• Collisions on “heavy” elements can hide collision
statistics of rest of the domain
• Construct pairs of distributions where heavy
elements are identical, but “light” elements are
either identical or very different
Additively estimate distance?
Output ||p-q||1 ±
need (n/log n) samples [G. Valiant P. Valiant]
Information theoretic quantities
• Entropy
• Support size
Information in neural spike trails
[Strong, Koberle, de Ruyter van Steveninck, Bialek ’98]
• Each application of stimuli
gives sample of signal (spike
trail)
Neural signals
time
• Entropy of (discretized) signal
indicates which neurons
respond to stimuli
Compressibility of data
Can we get multiplicative
approximations?
• In general, no….
• 0 entropy distributions are hard to distinguish
• What if entropy is bigger?
• Can g-multiplicatively approximate the entropy with Õ(n1/g2)
samples (when entropy >2g/) [Batu Dasgupta R. Kumar]
• requires (n1/g2) [Valiant]
• better bounds when support size is small [Brautbar
Samorodnitsky]
• Similar bounds for estimating support size [Raskhodikova Ron
R. Smith] [Raskhodnikova Ron Shpilka Smith]
More properties:
• Independence: [Batu Fischer Fortnow Kumar R. White]
• Limited Independence: [Alon Andoni Kaufman Matulef R.
Xie] [Haviv Langberg]
• K-flat distributions [Levi Indyk R.]
• K-modal distributions [Daskalakis Diakonikolas Servedio]
• Poisson Binomial Distributions [Daskalakis Diakonikolas
Servedio]
• Monotonicity over general posets [Batu Kumar R.]
[Bhattacharyya Fischer R. P. Valiant]
• Properties of multiple distributions [Levi Ron R.]
Conclusion:
• For many problems, we need a lot less time
and samples than one might think!
• Many cool ideas and techniques have been
developed
• Lots more to do!
Thank you!
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Sublinear time algorithms