geometric matching on sequential data veli mäkinen ag genominformatik technical fakultät bielefeld...

Geometric Matching on Sequential Data

Veli Mäkinen

AG Genominformatik

Technical Fakultät

Bielefeld Universität

Stringology Haifa 2005 Geometric matching on sequential data 2

Introduction

Motivation: To study problems in the intersection of geometry and stringology.

Applications to time-series data.

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Three problems

1D point set matching under translations (Akutsu, COCOON’04).

1D point set matching under translations, scaling and noise (Böcker & Mäkinen, EuroCG’05)

2D point set matching under translations (Ukkonen & Lemström & Mäkinen, 2003 + Cieliebak & Mäkinen, 2005).

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1D point set matching under translations

Two point sets A and B of sizes m and n. Problem 1a: Find largest common point set

of f(A) and B over translations f. Problem 1b: Find largest common point set

of f(A) and a continuous subset of B. Let k be the number of unmatched points.

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Example

B

A

f(A)

Problem 1a: k=3Problem 1b: k=1

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Solutions

Trivial in O(m2n log n) time. Easy in O(mn log m) time. Akutsu gives an O(k3+n log n) time solution.

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Akutsu’s solution

Use differential encoding for A and B. A’=a2-a1,a3-a2,..., am-am-1,

B’=b2-b1,b3-b2,..., bn-bn-1.

Construct suffix tree T of A’#B’$. Preprocess T for LCA queries.

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Akutsu’s solution...

Let Jump(ai,bj)=h where h is largest integer such that,

Jump(ai,bj) can be computed O(1) time.

bj bj+h-1

ai ai+h-1

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Akutsu’s solution...

Observation: One of the first k+1 points in both A and B must match.

Each match defines a translation. For each translation, one needs at most k+1

queries to Jump() to find out whether there is large enough overlap.

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Akutsu’s solution...

Theorem 1: Problem 1a can be solved in O(k3+n log n) time and Problem 1b in O(k2n+n log n) time.

Akutsu also gives reductions from 2D/3D problems to 1D achieving good bounds.

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Three problems

1D point set matching under translations (Akutsu, COCOON’04).

1D point set matching under translations, scaling and noise (Böcker & Mäkinen, EuroCG’05)

2D point set matching under translations (Ukkonen & Lemström & Mäkinen, 2003 + Cieliebak & Mäkinen, 2005).

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Linear 1D point set matching

Let us consider generalization where we allow also scaling and noise.

We search for best linear mapping from point set A to point set B.- maximum number of points of A should move close to points of B.

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Example

A

B

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Example...

A

B

f(A)

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Linear 1D point set matching...

There is an optimum mapping such that two points of A are mapped exactly at -distance from some points of B.

One mapping fixes the translation, second the scale around the new origin defined by the translation.

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Example

2

A

B

f(A)

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Degenerate solution!

2B

A

f(A)

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One-to-one mapping

To avoid the degenerate solution, one needs a better definition for the mapping searched for.

Hence, we search for a mapping producing maximum size one-to-one matching between the points (Problem 2).

2 22 2 2 2

f(A)B

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Solving one-to-one case

Consider a fixed translation and scale. Construct a bipartite graph having edges

between points of f(A) and B that are at -distance.

Solve the maximum matching problem on this graph.

2 22 2 2 2

f(A)B

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Solving one-to-one case...

Repeating the algorithm on each relevant translation and scale gives the optimum solution.

The overall time complexity is O((mn)2 g(mn)) where g(x) is the complexity of the maximum matching algorithm on a graph with x edges.

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Solving one-to-one case faster

Consider a fixed translation, and sort the relevant scales from smallest to largest.

Observation [Alt et al. 88]: The graph Gi corresponding to ith scale differs from the graph Gi-1 of the (i-1)th scale by one edge.

The maximum matching on Gi can be found by searching for an augmenting path in Gi-1 added/deleted one edge.

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Solving one-to-one case faster..

Incremental computation gives O((mn)3) time solution.

Theorem 2: Problem 2 can be solved in O((mn)2(m+n)) time.

To obtain the result, we exploit the monotonicity of the match graph.

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Staircase property

fi(A)

B

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Greedy algorithm is enough

B

fi(A)

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scale i => scale i+1

B

fi+1(A)

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scale i+1

B

fi+1(A)

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scale i+1 => scale i+2

B

fi+2(A)

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scale i+2

B

fi+2(A)

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Observation - open question

Observation: With only translations and noise, we obtain O(mn(m+n)) time.

The staircase matrix changes only by one cell when moving from one scale to another.

Question: Can one update the greedy path incrementally?

O(1) solution for the above would imply that adding noise does not make the problem any harder.

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Three problems

1D point set matching under translations (Akutsu, COCOON’04).

1D point set matching under translations, scaling and noise (Böcker & Mäkinen, EuroCG’05)

2D point set matching under translations (Ukkonen & Lemström & Mäkinen, 2003 + Cieliebak & Mäkinen, 2005).

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2D point set matching

B

A f(A)

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Solutions

Easy in O(mn log m) time by constructing the set of mn translation vectors, sorting it, and finding maximum repeating element.

Possible also in O(mn) time by using naive string matching type algorithm.

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Naive point set matching

A

B

Remark: This is the fastest known algorithm for this problem!!

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Restricted case?

Would the problem become easier if there were no other points inside the area of matches?

f(A)

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Restricted case?

Restricted 1D case is extremely easy:- Exact string matching on the differentially encoded sequences.

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Easier on grid points

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Easier on grid points...

The problem becomes a special case of two-dimensional exact string matching.

Can be solved in O(N2) time on a text grid of size N£N and pattern grid of size M£M.

Notice that the run-length encoded representation of the rows of the matrix is of size O(n).

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Easier on grid points...

The algorithm of Amir & Landau & Sokol, 2002, for run-length compressed 2D search can be applied:- Time complexity O(M2+n). (can be reduced to O(m2+n)?)

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What about Bird-Baker?

Our idea to solve the problem is to modify Bird-Baker algorithm to work directly on point sets.

As a preliminary tool, we need an Aho-Corasick automaton that recognizes run-length encoded binary strings.

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Run-length encoding

5.7 12.2

3.1 9.3 ...

05.71012.2

...

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Modified Aho-Corasick automaton

Proposition: There is an automaton accepting a set of run-length encoded binary strings with the following properties:- O(m log m) construction time, where m is the number of 1-bits in the set.- Reading a fail-link in O(log m) time. - Scanning a string with n 1-bits in O(n log m) time.

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Bird-Baker on point sets

Now we can build our automaton on the rows of set A, scan it with the rows of set B.

Let R be the set of positions where a row of A was accepted inside the rows of B.

After sorting R by columns, we can test in O(|R|) time if any column of R contains the correct sequence of accepting states.

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Bird-Baker on point sets

The overall running time is O(n log m +|R| log |R|).

Unfortunately, there are examples where |R|=(mn) :-(

Hence, it is still open if (even) the restricted case has o(mn) solution or not.