spring 2016 lecture 4: segmentationkbuchin/teaching/2ima20/... · 2. prove that optimal solutions...

2IMA20

Algorithms for Geographic Data

Spring 2016

Lecture 4: Segmentation

Motivation: Geese Migration

Two behavioural types

stopover

migration flight

Input:

GPS tracks

expert description of behaviour

Goal: Delineate stopover sites of migratory geese

Abstract / general purpose questions

Single trajectory

simplification, cleaning

segmentation into semantically meaningful parts

finding recurring patterns (repeated subtrajectories)

Two trajectories

similarity computation

subtrajectory similarity

Multiple trajectories

clustering, outliers

flocking/grouping pattern detection

finding a typical trajectory or computing a mean/median

trajectory

visualization

Problem

For analysis, it is often necessary to break a trajectory into pieces

according to the behaviour of the entity (e.g., walking, flying, …).

Input: A trajectory T, where each point has a set of attribute values,

and a set of criteria.

Attributes: speed, heading, curvature…

Criteria: bounded variance in speed, curvature, direction,

distance…

Aim: Partition T into a minimum number of subtrajectories (so-called

segments) such that each segment fulfils the criteria.

“Within each segment the points have similar attribute values”

Criteria-Based Segmentation

Goal: Partition trajectory into a small number of segments

such that a given criterion is fulfilled on each segment

Example

Trajectory T sampled with equal time intervals

Criterion: speed cannot differ more than a factor 2?

Example



Criterion: direction of motion differs by at most 90⁰?

speed

Example




speedspeed

heading

Example




speedspeed

headingspeed &

heading

Decreasing monotone criteria

Definition: A criterion is decreasing monotone, if it holds on a

segment, it holds on any subsegment.

Examples: disk criterion (location), angular range (heading),

speed…

Theorem: A combination of conjunctions and disjunctions of

decreasing monotone criteria is a decreasing monotone criterion.

Greedy Algorithm

Observation:

If criteria are decreasing monotone, a greedy strategy works.

TU/e Prerequisites

Designing greedy algorithms

1. try to discover structure of optimal solutions: what properties do optimal solutions have ?

what are the choices that need to be made ?

do we have optimal substructure ?

optimal solution = first choice + optimal solution for subproblem

do we have greedy-choice property for the first choice ?

2. prove that optimal solutions indeed have these properties

prove optimal substructure and greedy-choice property

3. use these properties to design an algorithm and prove correctness

proof by induction (possible because optimal substructure)

TU/e Prerequisites

Lemma: Let k be the smallest index such that 𝜏 𝑡0, 𝑡𝑘 fails for a monotone

decreasing criterion 𝜏. If there is a solution, then there is an optimal solution

including 𝜏 𝑡0, 𝑡𝑘−1 as segment.

Proof. Let OPT be an optimal solution for A. If OPT includes ai then the

lemma obviously holds, so assume OPT does not include ai.

We will show how to modify OPT into a solution OPT* such that

(i) OPT* is a valid solution

(ii) OPT* includes ai

(iii) size(OPT*) ≤ size(OPT)

Thus OPT* is an optimal solution including 𝜏 𝑡0, 𝑡𝑘−1 , and so the lemma

holds. To modify OPT we proceed as follows.

standard text you can

basically use in proof for any

greedy-choice property quality OPT* ≥ quality OPT

here comes the modification, which is problem-specific

Greedy Algorithm

Observation:

If criteria are decreasing monotone, a greedy strategy works.

For many decreasing monotone criteria Greedy requires O(n) time,

e.g. for speed, heading…

Greedy Algorithm

Observation:

For some criteria, iterative double & search is faster.

Double & search: An exponential search followed by a binary search.

Criteria-Based Segmentation

Boolean or linear combination of decreasing monotone criteria

Greedy Algorithm

incremental in O(n) time or constant-update criteria e.g. bounds

on speed or heading

double & search in O(n log n) time for non-constant update

criteria

[M.Buchin et al.’11]

Motivation: Geese Migration

Two behavioural types

stopover

migration flight

Input:

GPS tracks

expert description of behaviour

Goal: Delineate stopover sites of migratory geese

Case Study: Geese Migration

Data

Spring migration tracks

White-fronted geese

4-5 positions per day

March – June

Up to 10 stopovers during spring migration

Stopover: 48 h within radius 30 km

Flight: change in heading <120°

Kees

Adri

Comparison

manual

computed

Evaluation

Few local differences:

Shorter stops, extra cuts in computed segmentation

A combination of decreasing and increasing monotone criteria

stopover migration flight

Criteria

Within

radius 30km

At least

48hAND

Change in

heading <120°OR

Decreasing and Increasing Monotone Criteria

Observation: For a combination of decreasing and increasing

monotone criteria the greedy strategy does not always work.

Example: Min duration 2 AND Max speed range 4

speed1

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Non-Monotone Segmentation

Many Criteria are not (decreasing) monotone:

Minimum time

Standard deviation

Fixed percentage of outliers

For these Aronov et al. introduced the start-stop diagram

Example: Geese Migration

Start-Stop Diagram

input trajectory compute start-stop diagram compute segmentation

Algorithmic approach

Start-Stop Diagram

Given a trajectory T over time interval

I = {t0,…,t} and criterion C

The start-stop diagram D is (the upper

diagonal half of) the n x n grid,

where each point (i,j) is associated to

segment [ti,tj] with

• (i,j) is in free space if C holds on [ti,tj]

• (i,j) is in forbidden space if C does not

hold on [ti,tj]

A (minimal) segmentation of T

corresponds to a (min-link) staircase in D

free space

forbidden space

staircase

Start-Stop Diagram

A (minimal) segmentation of T corresponds

to a (min-link) staircase in D.

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Start-Stop Diagram

Discrete case:

A non-monotone segmentation can

be computed in O(n2) time.

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Stable Criteria

Definition:

A criterion is stable if and only if 𝑖=𝑜𝑛 𝑣 𝑖 = 𝑂(𝑛) where

𝑣 𝑖 = number of changes of validity on segments [0,i], [1,i],…, [i-1,i]

Stable Criteria

Definition:

A criterion is stable if and only if 𝑖=𝑜𝑛 𝑣 𝑖 = 𝑂(𝑛) where

𝑣 𝑖 = number of changes of validity on segments [0,i], [1,i],…, [i-1,i]

Observations:

Decreasing and increasing monotone criteria are stable.

A conjunction or disjunction of stable criterion are stable.

Compressed Start-Stop Diagram

For stable criteria the start-stop diagram can be compressed by applying

run-length encoding.

Examples:

increasing monotonedecreasing monotone

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Computing the Compressed Start-Stop Diagram

For a decreasing criterion consider the algorithm:

ComputeLongestValid(crit C, traj T)

Algorithm:

Move two pointers i,j from n to 0 over the trajectory. For every

trajectory index j the smallest index i for which 𝜏 𝑡𝑖 , 𝑡𝑗 satisfies

the criterion C is stored.

i j

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For a decreasing criterion ComputeLongestValid(crit C, traj T):

Move two pointers i,j from n to 0

over the trajectory

i j

Requires a data structure for segment [ti,tj] allowing the operations

isValid, extend, and shorten, e.g., a balanced binary search tree on

attribute values for range or bound criteria.

Runs in O(nc(n)) time where c(n) is the time to update & query.

Analogously for increasing criteria.

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The start-stop diagram of a conjunction (or disjunction) of two stable

criteria is their intersection (or union).

The start-stop diagram of a

negated criteria is its inverse.

The corresponding compressed

start-stop diagrams can be

computed in O(n) time.

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Attributes and criteria

Examples of stable criteria

Lower bound/Upper bound on attribute

Angular range criterion

Disk criterion

Allow an approximate fraction of outliers

…

Computing the Optimal Segmentation

Observation: The optimal segmentation for [0,i] is either one segment,

or an optimal sequence of segments for [0,j<i] appended with a

segment [j,i], where j is an index such [j,i] is valid.

Dynamic programming algorithm

for each row from 0 to n

find white cell with min-link

That is, iteratively compute a table S[0,n]

where entry S[i] for row i stores

last: index of last link

count: number of links so far

runs in O(n2) time

S

nil, 0nil, ∞

nil, ∞nil, ∞

nil, ∞nil, ∞

0, 0+1=10, 1

7, 28, 3

8, 3

10, 4


T

[Alewijnse et al.’14]

More efficient dynamic programming algorithm for compressed

diagrams

Process blocks of white cells using a range query in a binary search

tree T (instead of table S) storing

index: row index



augmented by minimal count

in subtree

TU/e Prerequisites

Methodology for augmenting a data structure

1. Choose an underlying data structure.

2. Determine additional information to maintain.

3. Verify that we can maintain additional information

for existing data structure operations.

4. Develop new operations.

1. R-B tree

2. min-count[x]

3. theorem next

slide

4. count(block)

using range

query

TU/e Prerequisites

Theorem

Augment a R-B tree with field f, where f[x] depends only on

information in x, left[x], and right[x] (including f[left[x]] and

f[right[x]]). Then can maintain values of f in all nodes during

insert and delete without affecting O(log n) performance.

When we alter information in x, changes propagate only upward

on the search path for x …


T

[Alewijnse et al.’14]

More efficient dynamic programming algorithm for compressed

diagrams

Process blocks of white cells using a range query in a binary search

tree T (instead of table S) storing

index: row index



augmented by minimal count

in subtree

runs in O(n log n) time

Beyond criteria-based segmentation

When is criteria-based segmentation applicable?

What can we do in other cases?

Excursion: Movement Models

Movement Models

Movement data: sequence

of observations, e.g. (xi,yi,ti)

Linear movement realistic?

Movement Models




Space-time Prisms

Movement Models




Space-time Prisms

Random Motion Models

(e.g. Brownian bridges)

Brownian Bridges - Examples

Brownian bridges no bridges

Segment by diffusion coefficient

Summary

Greedy algorithm for decreasing monotone criteria O(n) or O(n log n)

time

[M.Buchin, Driemel, van Kreveld, Sacristan, 2010]

Case Study: Geese Migration

[M.Buchin, Kruckenberg, Kölzsch, 2012]

Start-stop diagram for arbitrary criteria O(n2) time

[Aronov, Driemel, van Kreveld, Löffler, Staals, 2012]

Compressed start-stop diagram for stable criteria O(n log n) time

[Alewijnse, Buchin, Buchin, Sijben, Westenberg, 2014]

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References

S. Alewijnse, T. Bagautdinov, M. de Berg, Q. Bouts, A. ten Brink, K.

Buchin and M. Westenberg. Progressive Geometric Algorithms.

SoCG, 2014.

M. Buchin, A. Driemel, M. J. van Kreveld and V. Sacristan.

Segmenting trajectories: A framework and algorithms using

spatiotemporal criteria. Journal of Spatial Information Science,

2011.

B. Aronov, A. Driemel, M. J. Kreveld, M. Loffler and F. Staals.

Segmentation of Trajectories for Non-Monotone Criteria. SODA,

2013.

M. Buchin, H. Kruckenberg and A. Kölzsch. Segmenting

Trajectories based on Movement States. SDH, 2012.

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References

spring 2016 lecture 4: segmentationkbuchin/teaching/2ima20/... · 2. prove that optimal solutions...

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