two decades of algorithms for the frechet distanceadriemel/shonan2016.pdf · two decades of...

Two Decades of Algorithmsfor the Frechet distance

Anne Driemel

based on work of Bringmann, Künnemann, Har-Peled, Wenk,Agarwal, Fox, Pan, Ying and others.

May 30, 2016

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/ department of mathematics and computer science

Overview of this talk

I Sequence Alignment MeasuresI Quadratic lower bounds based on SETHI Strongly subquadratic-time algorithms under input assumptionsI Partial distance measures based on Fréchet distanceI Data structures for Fréchet queriesI Median curves and clustering under the Fréchet distance

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I Sequence Alignment Measures• for Strings: LCS, Edit distance• for Time Series: Dynamic time warping, Discrete Fréchet distance• for Curves: Continuous Fréchet distance

I Quadratic lower bounds based on SETHI Strongly subquadratic-time algorithms under input assumptionsI Partial distance measures based on Fréchet distanceI Data structures for Fréchet queriesI Median curves and clustering under the Fréchet distance

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Sequences, Time series and Trajectories

I Examples of sequences:• DNA sequencesa

• Text documentsI Examples of time series:

• Stock prices• Sensor data (weather,

manufacturing, IT infrastructure ..)• Click streams and search trends

I Examples of trajectories:• Trajectories of taxis, migrating

animals, sports players, ...• Gesture-trajectories on

touchscreensa[D’haeseleer, 2006]

Anne Driemel, TU Dortmund

1

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Sequence Alignment Measures—Strings

Let A and B be sequences of elements from a fixed alphabet.

LCS:Longest common subsequence of Aand B.

A = jellybeans

B = bellybutton

Edit distance:The minimal number of operationsneeded to transform A into B usingthe basic edit operations (insert,delete, and replace).

A = jellybeans→ bellybeans→ . . . → bellybuttons →bellybutton = B

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Sequence Alignment Measures—Traversals

j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11 12

10

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1),Traversal:

12

10

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2),Traversal:

12

10

replace

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal:

12

10

replace

match

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal: (4, 4),

12

10

replace

match

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal: (4, 4), (5, 5),

12

10

replace

match

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal: (4, 4), (5, 5), (6, 6),

12

10

replace

match

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal: (4, 4), (5, 5), (6, 6), (7, 7),

12

10

replace

match

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal: (4, 4), (5, 5), (6, 6), (7, 7), (8, 7),

12

10

deletion

replace

match

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal: (4, 4), (5, 5), (6, 6), (7, 7), (8, 7),

(9, 7),

12

10

deletion

replace

match

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal: (4, 4), (5, 5), (6, 6), (7, 7), (8, 7),

(9, 7), (9, 8),

12

10

deletion

replace

match

insertion

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal: (4, 4), (5, 5), (6, 6), (7, 7), (8, 7),

(9, 7), (9, 8), (9, 8),

12

10

deletion

replace

match

insertion

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal: (4, 4), (5, 5), (6, 6), (7, 7), (8, 7),

(9, 7), (9, 8), (9, 8), (9, 10),

12

10

deletion

replace

match

insertion

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal: (4, 4), (5, 5), (6, 6), (7, 7), (8, 7),

(9, 7), (9, 8), (9, 8), (9, 10), (9, 11),

12

10

deletion

replace

match

insertion

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal: (4, 4), (5, 5), (6, 6), (7, 7), (8, 7),

(9, 7), (9, 8), (9, 8), (9, 10), (9, 11), (10, 12)

12

10

deletion

replace

match

insertion

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j1 e2 l3 l4 y5 b6 e7 a8 n9 s10

b1 e2 l3 l4 y5 b6 u7 t8 t9 o10 n11

((1, 1), (2, 2), (3, 3),Traversal: (4, 4), (5, 5), (6, 6), (7, 7), (8, 7),

(9, 7), (9, 8), (9, 8), (9, 10), (9, 11), (10, 12)

12

10

(11, 12))

deletion

replace

match

insertion

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Sequence Alignment Measures—Time Series

Let X and Y be sequences of elements from a metric space.

Dynamic time warping:The minimum summed costover all possible traversalsT = ((a1, b1), . . . , (at, bt)).

minT

∑16i6t

d(X[ai], Y[bi])

Discrete Fréchet distance:The minimum bottleneck-costover all possible traversalsT = ((a1, b1), . . . , (at, bt)).

minT

max16i6t

d(X[ai], Y[bi])

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Sequence Alignment Measures—Curves

Let X and Y be parametrized curves in IRd.

Continuous Fréchet distance:The minimum bottleneck-cost over allpossible monotone bijections f, g.

inff,g

sup16t60

‖X(f(t)) − Y(g(t))‖

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Computation of Sequence Alignments

I Basic quadratic-time algorithms:• Sequence alignment: simple dynamic programming• DTW, Discrete Fréchet: shortest path in a graph• Continuous Fréchet distance [Alt and Godau, 1995]

I Slightly below-quadratic time algorithms in word RAM model:• Discrete Fréchet distance [Agarwal et al., 2014]

• O(n2 log logn/ logn) time• Continuous Fréchet distance [Buchin et al., 2014a]

• O(n2(log logn)2/ logn) time (decision problem)

I ..many heuristics/ filtering techniques:• Dynamic time warping [Keogh and Ratanamahatana, 2005]• Edit distance [Gravano et al., 2001] (see also [Chen et al., 2005])

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SETH-Hardness of alignment problems

Strong Exponential Time Hypothesis (SETH):For no ε > 0, there exists an algorithm for k-SAT that runs in timeO(2(1−ε)N), for any k > 3, whereN is the number of variables.

I If SETH is true, then the following alignment measures cannot be computed intimeO(n2−ε) for some fixed ε > 0 ("strongly subquadratic" time).

• Fréchet distance [Bringmann, 2014, Bringmann and Mulzer, 2016]• Edit distance [Backurs and Indyk, 2015]• Dynamic time warping [Abboud et al., 2015, Bringmann and Künnemann, 2015b]• LCS [Abboud et al., 2015, Bringmann and Künnemann, 2015b]

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k-SATN variables,M clauses

→

OrthogonalVectorssets of vectors ofsize |A| = |B| = n =2N/2 and dimensiond =M

→SequencealignmentSequences oflengthO(nd)

↓

SETH is false ←Algorithm withrunning timeO(2(1−ε

′)N), ε ′ = ε2

←Assume Algorithmwith running timeO((nd)2−ε) forsome fixed ε > 0.

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I k-SAT Problem:• GivenN variables x1, . . . , xN andM CNF-clauses• Does there exist a truth assignment that satisfies all clauses?

I Orthogonal Vectors Problem:• Given sets of vectorsA and B from 0, 1d

• Does there exist a pair ai ∈ A, bj ∈ B with 〈ai, bj〉 = 0?I Reduction [Williams, 2004]:

• Split the set of variables and consider 2N/2 truth assignments• Record for each truth assignment z:

• az[i] = 0 iff z applied to x1, . . . , xN/2 satisfies ith clause• bz[i] = 0 iff z applied to xN/2+1, . . . , xN satisfies ith clause

• Yields two sets of vectorsA and B with size 2N/2 and dimensionM

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• Does there exist a pair ai ∈ A, bj ∈ B with 〈ai, bj〉 = 0?

I Reduction [Williams, 2004]:• Split the set of variables and consider 2N/2 truth assignments• Record for each truth assignment z:



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• Does there exist a pair ai ∈ A, bj ∈ B with 〈ai, bj〉 = 0?I Reduction [Williams, 2004]:

• Split the set of variables and consider 2N/2 truth assignments• Record for each truth assignment z:



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I Reduction to Sequence Alignment [Bringmann and Künnemann, 2015b]• Let δ be a sequence alignment measure• Creates sequence X from setA and sequence Y from set B• Coordinate gadgets: encode 0X, 1X, 0Y , 1Y as substring s.t. ∃ρ0, ρ1 :

• ρ0 = δ(0X, 1Y) = δ(1X, 0Y) = δ(0X, 0Y)• ρ1 = δ(1X, 1Y)• ρ0 < ρ1

• Alignment gadgets: force optimal matching to use substrings atomically• Example encoding ai = 0101 and bj = 1010

0X 1X 0X 1X 0X 0X 0X 0X 0X 1X

0Y 1Y 0Y1Y 1Y

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0X 1X 0X 1X 0X 0X 0X 0X 0X 1X

0Y 1Y 0Y1Y 1Y

(d+ 1) · ρ0

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0X 1X 0X 1X 0X 0X 0X 0X 0X 1X

0Y 1Y 0Y1Y 1Y

d · ρ0 + ρ1

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0X 1X 0X 1X 0X 0X 0X 0X 0X 1X

0Y 1Y 0Y1Y 1Y

non-atomic matching

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“This edit distance is something that I’ve been trying to get betteralgorithms for since I was a graduate student, in the mid-’90s. I certainlyspent lots of late nights on that — without any progress whatsoever. So atleast now there’s a feeling of closure. The problem can be put to sleep.”Piotr Indyk (MIT News, June 10, 2015)

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I Sequence Alignment MeasuresI Quadratic lower bounds based on SETHI Strongly subquadratic-time algorithms under input assumptions

• Realistic Input Models• Useful Tools and Techniques

• Distance matrix• Free space diagram• Curve simplification• Well-separated pair decomposition

• (1+ ε)-Approximation Algorithms

I Partial distance measures based on Fréchet distanceI Data structures for Fréchet queriesI Median curves and clustering under the Fréchet distance

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Zoo of Realistic Input Models

Type of curve Definition Illustration

κ-straight[Alt et al., 2004]

For any two points p, q on X:|X[p, q]| 6 κ‖p− q‖

p

p

X

κ-bounded[Alt et al., 2004]

For any two points p, q on X:X[p, q] ⊆ b(p, r) ∪ b(q, r)where r = κ‖p− q‖/2

p

p

X

r

backbone curves[Aronov et al., 2006]

For any two vertices v, u of X:‖v − u‖ > 1 and any edge of X haslength in [c1, c2] for constants c1, c2

κ-packed curves[Driemel et al., 2012]

For any ball b(p, r):|X[p, q] ∩ b(p, q)| 6 κr X

r

p

κ-low-density curves[de Berg et al., 2002]

The number of edges of X intersectingany smaller ball is at most κ

r

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Zoo of Realistic Input Models

Idea of κ-packed curves: Capture natural families of curves

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Tools: Distance Matrix

I Two sequences X and YI Matrix entry δi,j stores distance

between ith element of X and jthelement of Y

I Traversal of X and Y corresponds tomonotone path through entries

I DTW = shortest path invertex-weighted graph

X

Y

xi

yj

δij

i

j δij

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Tools: Distance Matrix

I Two sequences X and YI Matrix entry δi,j stores distance

between ith element of X and jthelement of Y

I Traversal of X and Y corresponds tomonotone path through entries

I DTW = shortest path invertex-weighted graph

X

Y

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Tools: Free space diagram

Due to [Alt and Godau, 1995]:I Parametric space of X and YI δ-free space = Pairs of points on X

and Y which are within distance δI Free space complexityN6δ:

Number of non-empty cellsI Fréchet distance d(X, Y) 6 δ

iff there exists a montone path πI Testing for π takesO(N6δ) time

using a BFS on the reachable cells

X

Y

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X

Y

px

py

X(px)

Y(py)

p

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X

Y

≤ δ

X(px)

Y(py)

p

px

py

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X

Y

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δ

X

Y

π

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X

Y

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Tools: Curve Simplification

I Many more sophisticated algorithms:[Godau, 1991, Agarwal et al., 2005a, Abam et al., 2010, Agarwal et al., 2005b]

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δ


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Tools: Approximate Pairwise Distances

I s-well-separated pair• A and B are well-separated, iff there exist

two balls b(pA, r) and b(pB, r):(i)A ⊆ b(pA, r)(ii) B ⊆ b(pB, r)(iii) ‖pA − pB‖ > (s+ 2)r

I s-well-separated pair decomposition (WSPD)• For any set of points P one can compute a set

of pairs (A1, B1), . . . , (Am, Bm):(i) (Ai, Bi) are s-well-separated(ii) any p, q ∈ P covered by unique (Ai, Bi).

• Can be computed withm = O(sdn) in timeO(n logn) [Callahan and Kosaraju, 1995]

≥ sr

r

r

A

B

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(1+ ε)-Approximation—Fréchet distance

I Packing lemma from [Driemel et al., 2012]• Consider the pairs of edges that are within

distance δ to each other• We charge every pair to the shorter edge• Let u be such a shorter edge• Assume that ‖u‖ > εδ• We bound the charges to u:

‖b ∩ Y‖‖u‖ 6

κr

‖u‖ 6 O(κε

)

where r = 32‖u‖+ δ

• In totalO(κn/ε) pairs of edges that containpotential matches

X

Y

u

δ

b

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I Packing lemma from [Driemel et al., 2012]• Consider the pairs of edges that are within

distance δ to each other• We charge every pair to the shorter edge• Let u be such a shorter edge• Assume that ‖u‖ > εδ• We bound the charges to u:

‖b ∩ Y‖‖u‖ 6

κr

‖u‖ 6 O(κε

)

where r = 32‖u‖+ δ

• In totalO(κn/ε) pairs of edges that containpotential matches

X

Y

u

δ

b

‖u‖

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I Decision Algorithm:• Simplify the curves with parameter εδ• Compute reachability in Free space

diagram to answer either:(i) d(X, Y) 6 (1+ ε)δ(ii) d(X, Y) > δ

• By the packing lemma, this takesO(κn/ε) time

I Main Algorithm:• Compute candidate values using

WSPD on vertices• Binary search using decision

algorithm

π

σ

≤ µ

dF(π′, σ′)

≤ µ

π′

σ′

dF(π, σ)

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I Result for various realistic input models [Driemel et al., 2012]

Type of curve Running time

κ-packed O(κn/ε+ κn logn)

κ-straight Same as 2κ-packed

κ-bounded O((κ/ε)dn+ κdn logn

)

O(1)-low density O(n2(d−1)/d

ε2+ n2(d−1)/d logn

)

c-packed & closed O(c2n/ε2 + c2n logn

)

I Slightly faster by factor 1√ε

: [Bringmann and Künnemann, 2015a]

I For κ-packed curves this is optimal unless SETH is false [Bringmann, 2014]I Previous similar approach [Aronov et al., 2006]

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(1+ ε)-Approximation—DTW

Due to [Agarwal et al., 2016]:I ’Plateau’ rectangles

• Decomposition of curves based on WSPD• Uniform δij inside a rectangle

I Algorithm• Truncate WSPD-quadtree usingn-approximation

• YieldsO(logn) levels of the tree• Cover optimal path with ’plateau’ rectangles• Compute paths ending on rectangle

boundaries only

i1 i2 i3 i4

j1

j2

X

Y

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(1+ ε)-Approximation—DTW

I Result for various realistic input models [Agarwal et al., 2016]

Type of curve Running time

κ-packed O((κn/ε) logn)

κ-bounded O((κ/ε)dn logn

)

backbone sequences O(ε−1n(2−(1/d)) logn

)

I Algorithm can be extended to version of Edit distance:

minC

∑16i6t

d(X[ai], Y[bi]) + g(m+ n− 2|C|)

where C is a subset of a traversal and g(·) is a gap penalty

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Partial distance measures for curves

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X

Y

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Partial MatchingGiven two polygonal curves X and Y and a parameter δ > 0, does there exista contiguous part X of X, such that the Fréchet distance between X and Y isat most δ.

Y

X

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Partial AlignmentGiven two polygonal curves X and Y and a parameter δ > 0, maximize thelength of subcurves of X and Y that are within distance δ to each otherunder the Fréchet distance.

X

Y

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k-Shortcut-Fréchet distanceGiven two polygonal curves X and Y and an integer parameter k, computethe minimal Fréchet distance between Y and any k-shortcut curve of X.

k-Shortcut-CurveX becomes a k shortcut curve of X by replacing at most k disjoint subcurvesby straight-line edges (shortcuts).

X

X

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I Partial Matching• O(n2) time [Maheshwari et al., 2014]• Approach: ’free space map’

I Partial Alignment• O(n3 logn)-time exact algorithms for L1 and L∞ [Buchin et al., 2009]• O(n3/ε log(n/ε))-time numerical approximation for L2 [Maheshwari et al., 2014]• Exact computation not possible for L2 in the Algebraic Computation Model overthe Rational Numbers (ACMQ) [Maheshwari et al., 2014]

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I k-Shortcut-Fréchet distance• Vertex-shortcutting can be (3+ ε)-approximated [Driemel and Har-Peled, 2013]

• κ-packed curves with at most k shortcuts: O(κ2kn log3 n)• κ-packed curves with unbounded k: O(κ2n log3 n)

• Discrete Fréchet distance [Avraham et al., 2015]• Shortcuts on one curve only: O((n)6/5+ε)-time (randomized) algorithm• Shortcuts on both curves: O(n4/3 log3 n)-time algorithm

• Continuous shortcutting is NP-hard [Buchin et al., 2014b]

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Shortcut Fréchet distance

Find a path π in the parametric space:

I π has to stay inside the δ-free spaceI π has to be monotoneI π can use (affordable) shortcuts between

vertices of X

Definition (Tunnels and Gates)I Tunnel(p,q) matches a shortcut X[px, qx] to a

subcurve Y[py, qy]I Gates are the endpoints of a tunnelI Price is the Fréchet distance

Y

X

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vertices of X



Y

X

π

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vertices of X



p

q

Shortcut

Y

X

π

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vertices of X



p

q

Tunnel

Gates

Shortcut

Y

X

Subcurve

π

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Preprocessing:

I Balanced binary tree on edges of YI Every node: a subcurve of YI Store the Fréchet distance to the

shortcut in each node

Price of Tunnel(p,q):

I Find decomposition of subcurveY[py, qy] of sizem = O(logn)

I Compute Fréchet distance ∆ toX[px, qx] inO(m logm) time

I Return ∆+ max06i6m dF(nodei)

Y

⇒ 3-approximation, more work to achieve (1+ ε)-approximation

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Preprocessing:







Y

dF(nodei)

nodei


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Preprocessing:







Y

Y[py, qy]

X[px, qx]


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Preprocessing:







Y

∆

Y[py, qy]

X[px, qx]


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Preprocessing:







Y

∆

dF(nodei)

nodei

Y[py, qy]

X[px, qx]


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Data structures for Fréchet queries

Task: Preprocess a set of curves to answer ...I Nearest-Neighbor queries

• returns the curve closest to a query curveI Range queries

• returns all curves within certain distance toa query curve

I Range counting queries• returns the number of curves within certain

distance to a query curveI Distance queries

• returns the distance to a query curve

VariantsI ApproximationI Probabilistic (e.g, LSH)I Preprocess ’all subcurves

of a given curve’I Preprocess ’all paths in a

given graph’

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I Approximate-Nearest-Neighbor data structure [Indyk, 2002]:• for the discrete Fréchet distance• Performance

• Input: n sequences of maximum lengthm• ApproximationO((logn log logm)t−1) for any t > 1• Query timeO((m+ logn)O(t))• Space exponential in tm1/t

• Approach• Use ∞-product metric on subcurves• Store each input curve using different partitions• Optimal partition is used by the nearest neighbor• Defers alignment problem into the data structure

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I Approximate range counting data structure [de Berg et al., 2013]:• Preprocesses all subcurves of a given curve• Range counting• Queries with line segments• Performance

• Approximation factor 2+ 3√2

• Query timeO( n√s

polylogn)• SpaceO(s polylogn)• For any n 6 s 6 n2

• Approach• Simplify the curve with range parameter ε• Organize all subcurves of 1,2, and 3 edges

in partition tree [Chan, 2012]• Use filtering to allow variable range size ε

Anne Driemel, TU Dortmund

1

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I Approximate range searching data structure [Gudmundsson and Smid, 2015]:• Preprocess all paths of a geometric tree• Unit range testing• Assumptions: tree is κ-packed, edge lengthΩ(1)• Queries with line segments

• Approximation factor√2+ ε

• Query timeO(polylogn)• SpaceO(n polylogn)

• Queries with polygonal curves of size k• Approximation factor 3• Query timeO(k polylogn)• SpaceO(n polylogn)

• Approach• Path decomposition of tree [Cole and Vishkin, 1988]• Build data structure of [Driemel and Har-Peled, 2013] for each path

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I Distance-queries data structure [Driemel and Har-Peled, 2013]:• Queries with line segments

• Approximation factor (1+ ε)• Query timeO(logn log logn)• SpaceO(n)

• Queries with polygonal curves of size k• Approximation factor 23• Query timeO(k2 logn log(k logn))• SpaceO(n logn)

• Used to determine prices of shortcuts

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Rangequeries

Distancequeries

NNqueries

polygonal curve geometric tree

approx-factor: (1 + ε)q-time: O(log n loglog n)

unit-range testing:(*)approx-factor: (

√2 + ε)

q-time: O(polylog n)

range counting:approx-factor: (2 + 3

√2)

q-time: O(

n√spolylog n

)

Query

polygonalcurve

polygonalcurve

linesegment

linesegment

Input

approx-factor: 23O(|Q|2 polylog(|Q|n)

)

unit-range testing:(*)approx-factor: 3

q-time: O (|Q| polylog n)

Type ofQuery

(*) using input assumptions

set of curves

approx-factor:O((log n log logm)(t−1))

q-time: O((m+ log n)O(t)

)

for any t > 1

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Median curves and clustering

Task: Summarize a set of curvesI Common alignment of k curves

• Find k reparametrizations that minimize the k-wise Fréchet distanceI Median curve of k curves

• Walking a pack of k dogs: Find optimal path of the dog walkerI Clustering of n curves

• Walking a pack of n dogs with a group of k dog walkers: Find k optimal paths andassignment of dogs to walkers

I Subcurve clustering• Find disjoint subcurves that together form a re-occurring pattern

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I Median curve of k curves• Walking a pack of k dogs: Find optimal path

of the dog walker• Minimum enclosing ball in metric space• Dynamic programming inO(nk) time and

space for curves of length n• Studied for different distance measures:

• Continuous Fréchet distance[Dumitrescu and Rote, 2004]

• Discrete Fréchet distance [Ahn et al., 2016]• Weak Fréchet distance

[Har-Peled and Raichel, 2013]• Dynamic time warping

[Petitjean and Gançarski, 2012]

VariantsI min sum of distances

(Fermat-Weber problem)

I constraints on center curve

• vertices of the input• complexity

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I Clustering of n curves• Walking a pack of n dogs with a group of k dog walkers: Find k optimal paths and

assignment of dogs to walkers• First studied by [Driemel et al., 2016]

• Constrain each center curve to use at most ` vertices• (k, `)-center: minimize maximum Fréchet distance to closest center• (k, `)-median: minimize sum of Fréchet distances to closest center

• Univariate curves:• (1+ ε)-approximation inO(nm logm) time (ignoring dependencies on k, `, ε)• m denotes the maximal length of the curves

• Curves in any dimension:• (k, `)-center: 8-approximation inO(knm` log(m`)) time• (k, `)-median: 65-approximation inO((nk+ k7 log5 n)m` log(m`)) time

• Sampling property for (1, `)-median (see also [Ackermann et al., 2010])

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I Subcurve clustering• Find disjoint subcurves that together form a re-occurring pattern• First studied by [Buchin et al., 2011]• Does there exists a set of subcurves S with:

• size of set |S| > k• length of a representative is at least `• common Fréchet distance at most ε

• 2-approximation algorithmO(`n2) time andO(n`2) space• GPU Implementation [Gudmundsson and Valladares, 2015]

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Summary

I Sequence Alignment Measures• for Strings: LCS, Edit distance• for Time Series: Dynamic time warping, Discrete Fréchet distance• for Curves: Continuous Fréchet distance

I Quadratic lower bounds based on SETHI Strongly subquadratic-time algorithms under input assumptionsI Partial distance measures based on Fréchet distanceI Data structures for Fréchet queriesI Median curves and clustering under the Fréchet distance

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References VI

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