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1
Sublinear AlgorithmsLecture 1
Sofya RaskhodnikovaPenn State University
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OrganizationalCourse webpage:http://www.cse.psu.edu/~sxr48/sublinear-course/
Office hours:Tuesdays, 2:30PM-3:30PM, IST 343F
Evaluation• class participation• solutions to about 4 homework assignments• taking lecture notes about 2-3 times per person• the final project
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Tentative TopicsIntroduction, examples and general techniques.
Sublinear-time algorithms for• graphs• strings• geometric properties of images• basic properties of functions• algebraic properties and codes• metric spaces• distributions
Tools: probability, Fourier analysis, combinatorics, codes, …
Sublinear-space algorithms: streaming3
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Tentative PlanIntroduction, examples and general techniques.
Lecture 1. Background. Testing properties of images and lists.
Lecture 2. (Next week) Properties of functions and graphs. Sublinear approximation.
Lecture 3-5. Background in probability. Techniques for proving hardness. Other models for sublinear computation.
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Motivation for Sublinear-Time Algorithms
Massive datasets• world-wide web• online social networks• genome project• sales logs• census data• high-resolution images• scientific measurementsLong access time• communication bottleneck (slow connection)• implicit data (an experiment per data point)
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What Can We Hope For?
• What can an algorithm compute if it– reads only a sublinear portion of the data?– runs in sublinear time?
• Some problems have exact deterministic solutions
• For most interesting problems algorithms must be– approximate– randomized
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A Sublinear-Time Algorithm
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B L A - B L A - B L A - B L A - B L A - B L A - B L A - B L A
approximate answer
sublinear-time algorithm
? L? B ? L ? A
Quality of approximation
Resources• number of queries• running time
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Types of Approximation
Classical approximation• need to compute a value
output should be close to the desired value example: average
Property testing• need to answer YES or NO
Intuition: only require correct answers on two sets of instances that are very different from each other
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Classical Approximation
A Simple Example
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Approximate Diameter of a Point Set [Indyk]
Input: points, described by a distance matrix – is the distance between points and – satisfies triangle inequality and symmetry(Note: input size is )
Let be indices that maximize .
Maximum is the diameter.• Output: such that
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Algorithm and Analysis
1. Pick arbitrarily2. Pick to maximize 3. Output
• Approximation guarantee (triangle inequality) (choice of + symmetry of )
• Running time: 𝑖
𝑗
𝑘
ℓ
A rare example of a deterministic sublinear-time algorithm
Algorithm
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Property Testing
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Property Testing: YES/NO Questions
Does the input satisfy some property? (YES/NO)
“in the ballpark” vs. “out of the ballpark”
Does the input satisfy the property or is it far from satisfying it?
• sometimes it is the right question (probabilistically checkable proofs (PCPs))• as good when the data is constantly changing (WWW)• fast sanity check to rule out inappropriate inputs (airport security questioning)
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Property Tester
Close to YES
Far from YES
YES
Reject with probability 2/3
Don’t care
Accept with probability
Property Tester Definition
Probabilistic Algorithm
YES Accept with probability
Reject with probability 2/3
NO
far = differs in many places - ( fraction of places)
𝜀
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Randomized Sublinear Algorithms
Toy Examples
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Test
Property Testing: a Toy Example
Input: a string Question: Is ?
Requires reading entire input.Approximate version: Is or
does it have 1’s (“errors”)?
1. Sample positions uniformly and independently at random2. If 1 is found, reject; otherwise, accept
Analysis: If , it is always accepted. If is -far, Pr[error] = Pr[no 1’s in the sample]
If a test catches a witness with probability , then iterations of the test catch a witness with probability
Used:
Witness Lemma
0 0 0 1 … 0 1 0 0
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Randomized Approximation: a Toy Example
Input: a string Goal: Estimate the fraction of 1’s in (like in polls)It suffices to sample positions and output the average to get the
fraction of 1’s (i.e., additive error ) with probability ¸ 2/3
= value of sample . Then E[Y] (fraction of 1’s in )
Let be independently distributed random variables in [0,1] and let (sample sum). Then .
0 0 0 1 … 0 1 0 0
Hoeffding Bound
Apply Hoeffding Bound with substitute
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Property Testing
Simple Examples
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Testing Properties of Images
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Pixel Model
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Query: point
Answer: color of
Input: matrix of pixels(0/1 values for black-and-white pictures)
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Testing if an Image is a Half-plane [R03]
A half-plane or -far from a half-plane?
O(1/ time
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Half-plane Instances
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A half-plane -far from a half-plane
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Half-plane Instances
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A half-plane -far from a half-plane
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Half-plane Instances
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A half-plane -far from a half-plane
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Half-plane Instances
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A half-plane -far from a half-plane
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Half-plane Instances
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A half-plane -far from a half-plane
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Half-plane Instances
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A half-plane -far from a half-plane
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Half-plane Instances
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A half-plane -far from a half-plane
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Strategy
“Testing by implicit learning” paradigm
• Learn the outline of the image by querying a few pixels.• Test if the image conforms to the outline by random sampling,
and reject if something is wrong.
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Half-plane Test
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Claim. The number of sides with different corners is 0, 2, or 4.
Algorithm1. Query the corners.
? ?
? ?
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Half-plane Test: 4 Bi-colored Sides
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Claim. The number of sides with different corners is 0, 2, or 4.
Analysis• If it is 4, the image cannot be a half-plane.
Algorithm1. Query the corners.2. If the number of sides with different corners is 4, reject.
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Half-plane Test: 0 Bi-colored Sides
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Claim. The number of sides with different corners is 0, 2, or 4.
Analysis• If all corners have the same color, the image is a
half-plane if and only if it is unicolored.
Algorithm1. Query the corners.2. If all corners have the same color , test if all pixels have color (as in Toy Example 1).
?
?
??
?
?
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Half-plane Test: 2 Bi-colored Sides
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Claim. The number of sides with different corners is 0, 2, or 4.
Algorithm1. Query the corners.2. If # of sides with different corners is 2, on both sides find 2 different
pixels within distance by binary search.3. Query pixels from 4. Accept iff all pixels are white and all pixels are black.
Analysis• The area outside of has pixels. • If the image is a half-plane, W contains only
white pixels and B contains only black pixels.• If the image is -far from half-planes, it has
wrong pixels in • By Witness Lemma, samples suffice to catch a
wrong pixel.
? ?𝜀𝑛 /2
??𝜀𝑛 /2
𝑊
𝐵
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Testing if an Image is a Half-plane [R03]
A half-plane or -far from a half-plane?
O(1/ time
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Other Results on Properties of Images
• Pixel ModelConvexity [Berman Murzabulatov R]Convex or -far from convex?
O(1/ time
Connectedness [Berman Murzabulatov R]Connected or -far from connected?
O(1/ time
Partitioning [Kleiner Keren Newman 10]Can be partitioned according to a template or is -far?
time independent of image size• Properties of sparse images [Ron Tsur 10]
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Testing if a List is Sorted
Input: a list of n numbers x1 , x2 ,..., xn
• Question: Is the list sorted?Requires reading entire list: (n) time
• Approximate version: Is the list sorted or ²-far from sorted? (An ² fraction of xi ’s have to be changed to make it sorted.)
[Ergün Kannan Kumar Rubinfeld Viswanathan 98, Fischer 01]: O((log n)/²) time (log n) queries• Attempts: 1. Test: Pick a random i and reject if xi > xi+1 .
Fails on: 1 1 1 1 1 1 1 0 0 0 0 0 0 0 Ã 1/2-far from sorted
2. Test: Pick random i < j and reject if xi > xj.
Fails on: 1 0 2 1 3 2 4 3 5 4 6 5 7 6 Ã 1/2-far from sorted
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Is a list sorted or ²-far from sorted?
Idea: Associate positions in the list with vertices of the directed line.
Construct a graph (2-spanner)• by adding a few “shortcut” edges (i, j) for i < j• where each pair of vertices is connected by a path of length at most 2
……
≤ n log n edges
1 2 3 … n-1 n
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Is a list sorted or ²-far from sorted?
Pick a random edge (xi ,xj) from the 2-spanner and reject if xi > xj.
1 2 5 4 3 6 7Analysis:• Call an edge (xi ,xj) violated if xi > xj , and good otherwise.• If xi is an endpoint of a violated edge, call it bad. Otherwise, call it good.
Proof: Consider any two good numbers, xi and xj.
They are connected by a path of (at most) two good edges (xi ,xk), (xk ,xj).
) xi ≤ xk and xk ≤ xj
) xi ≤ xj
12
12
5 4 3xi xj
xk
Claim 1. All good numbers xi are sorted.
Test [Dodis Goldreich Lehman Raskhodnikova Ron Samorodnitsky 99]
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Test [Dodis Goldreich Lehman Raskhodnikova Ron Samorodnitsky 99]
Is a list sorted or ²-far from sorted?
Pick a random edge (xi ,xj) from the 2-spanner and reject if xi > xj.
1 2 5 4 3 6 7Analysis:• Call an edge (xi ,xj) violated if xi > xj , and good otherwise.• If xi is an endpoint of a bad edge, call it bad. Otherwise, call it good.
Proof: If a list is ²-far from sorted, it has ¸ ² n bad numbers. (Claim 1)• Each violated edge contributes 2 bad numbers. • 2-spanner has ¸ ² n/2 violated edges out of · n log n.
12
12
5 4 3xi xj
xk
Claim 1. All good numbers xi are sorted. Claim 2. An ²-far list violates ¸ ² /(2 log n) fraction of edges in 2-spanner.
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Is a list sorted or ²-far from sorted?
Pick a random edge (xi ,xj) from the 2-spanner and reject if xi > xj.
1 2 5 4 3 6 7Analysis:• Call an edge (xi ,xj) violated if xi > xj , and good otherwise.
By Witness Lemma, it suffices to sample (4 log n )/² edges from 2-spanner.
Sample (4 log n)/ ² edges (xi ,xj) from the 2-spanner and reject if xi > xj.
Guarantee: All sorted lists are accepted.All lists that are ²-far from sorted are rejected with probability ¸2/3.Time: O((log n)/²)
12
12
5 4 3xi xj
xk
Test [Dodis Goldreich Lehman Raskhodnikova Ron Samorodnitsky 99]
Algorithm
Claim 2. An ²-far list violates ¸ ² /(2 log n) fraction of edges in 2-spanner.
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Generalization
Observation: The same test/analysis apply to any edge-transitive property of a list of
numbers that allows extension.
• A property is edge-transitive if1) it can be expressed in terms conditions on ordered pairs of numbers
2) it is transitive: whenever and satisfy (1), so does
• A property allows extension if3) any function that satisfies (1) on a subset of the numbers can be
extended to a function with the property
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12
x y z
12
x y
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Testing if a Function is Lipschitz [Jha R]
A function f : D R is Lipschitz if it has Lipschitz constant c:that is, if for all x,y in D,
distanceR(f(x),f(y)) ≤ distanceD(x,y).• can rescale by to get a Lipschitz function from a function with Lipschitz
constant
Consider f : {1,…,n} R:
The Lipschitz property is edge-transitive:
1. a pair (x,y) is good if |f(y)-f(x)| ≤ |y-x|2. (x,y) and (y,z) are good ) (x,z) is good It also allows extension for the range R.Testing if a function f : {1,…,n} R is Lipschitz takes O((log n )/²) time.
Does the spanner-based test apply if the range is with Euclidean distances? with Euclidean distances?
nodes = points in the domain; edges = points at distance 1node labels = values of the function
2 3 3 5421
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Properties of a List of n Numbers
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• Sorted or -far from sorted?
• Lipschitz (does not change too drastically) or -far from satisfying the Lipschitz property?
O(log n/ time
This bound is tight [Chakrabarty Dixit Jha Seshadhri 15]
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Basic Properties of Functions
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f(000)
f(111) f(011)
f(100)
f(101)
f(110)f(010)
f(001)
Boolean Functions
Graph representation:-dimensional hypercube
• vertices: bit strings of length • edges: is an edge if can be obtained from by increasing one
bit from 0 to 1
• each vertex is labeled with
001001 011001
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Monotonicity of Functions
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[Goldreich Goldwasser Lehman Ron Samorodnitsky, Dodis Goldreich Lehman Raskhodnikova Ron Samorodnitsky Fischer Lehman Newman Raskhodnikova Rubinfeld Samorodnitsky]
• A function is monotone if increasing a bit of does not decrease .
• Is monotone or -far from monotone ( has to change on many points to become monontone)?
– Edge is violated by if .
Time: – , logarithmic in the size of the input, – for restricted class of tests– Recent: for nonadaptive tests
[Khot Minzer Safra 15, Chen De Servidio Tang 15]
0
0 0
01
1
1
1
1
1 0
00
0
1
1
monotone
-far from monotone
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Monotonicity Test [GGLRS, DGLRRS]
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Idea: Show that functions that are far from monotone violate many edges.
Analysis• If is monotone, EdgeTest always accepts. • If is -far from monotone, by Witness Lemma, it suffices to show that
fraction of edges (i.e., edges) are violated by .– Let denote the number of edges violated by .
Contrapositive: If , can be made monotone by changing values.
EdgeTest (, ε)1. Pick 2 edges uniformly at random from the hypercube.2. Reject if some is violated (i.e. . Otherwise, accept.
Repair Lemma can be made monotone by changing values.
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Repair Lemma: Proof Idea
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Proof idea: Transform f into a monotone function by repairing edges in one dimension at a time.
Repair Lemma can be made monotone by changing values.
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Repairing Violated Edges in One Dimension
0 0 0 0
1
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0
0
0
0
0
1
1
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1
Swapping horizontal dimension
Swap violated edges 10 in one dimension to 01.
Let = # of violated edges in dimension
Enough to prove the claim for squares
i
j
Claim. Swapping in dimension does not increase for all dimensions
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Proof of The Claim for Squares
• If no horizontal edges are violated, no action is taken.
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Swapping horizontal dimension
i
j
Claim. Swapping in dimension does not increase for all dimensions
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Proof of The Claim for Squares
• If both horizontal edges are violated, both are swapped, so the number of vertical violated edges does not change.
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Swapping horizontal dimension
i
j
01 10
1 0 0 1
Claim. Swapping in dimension does not increase for all dimensions
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Proof of The Claim for Squares
• Suppose one (say, top) horizontal edge is violated.• If both bottom vertices have the same label, the vertical edges
get swapped.
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i
j
Swapping horizontal dimension
1 0 0 1
𝒗𝒗 𝒗𝒗
Claim. Swapping in dimension does not increase for all dimensions
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Proof of The Claim for Squares
• Suppose one (say, top) horizontal edge is violated.• If both bottom vertices have the same label, the vertical edges
get swapped. • Otherwise, the bottom vertices are labeled 01, and the
vertical violation is repaired.55
i
j
Swapping horizontal dimension
1 0 0 1
10 10
Claim. Swapping in dimension does not increase for all dimensions
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Proof of The Claim for Squares
After we perform swaps in all dimensions:• becomes monotone• # of values changed:
• Improve the bound by a factor of 2.56
Claim. Swapping in dimension does not increase for all dimensions
Repair Lemma can be made monotone by changing values.
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Testing if a Functions is monotone
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Monotone or -far from monotone?
O(n/ time (logarithmic in the size
of the input)
0
0 0
01
1
1
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1 0
00
0
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monotone
-far from monotone