CSIS7101 – Advanced Database Technologies

CSIS7101 – Advanced Database Technologies

Spatio-Temporal Data (Part 1)

On Indexing Mobile Objects

Kwong Chi Ho LeoWong Chi Kwong SimonLui, Tak Sing Arthur

CSIS7101 - Advanced Database Technologies

Spatio-Temporal Data (Part 1) 2

Content

1 Introduction 2 Indexing Mobile Objects 3 Indexing in 2 Dimensions 4 Summary 5 Reference



1. Introduction

• Traditional Database Model assume that data stored in the database remain constant.

• Not appropriate for applications with continuously changing data.

• Better approach: Object location as a function of time f(t), and update the database only when the parameters of f change.

• This approach introduces a novel problem since the database is not directly storing data values but function to compute these values.



2. Indexing Mobile Objects 2.1 Partition the mobile objects into two categories:

Let’s say objects are moving on an 1-dimensional plane.1. Object with low speed v 0.2. Objects with speed between v min and v max. (i.e. Moving

Objects) 2.2 Space-Range Representation 2.3 Dual Space-Time Representation 2.4 Simplex Range Searching 2.5 Achieve Logarithmic Query Time



2. Indexing Mobile Objects

2.2 Space-Range Representation Plot the trajectories y(t) = v t + a of the mobile objects as line

in time-location (t,y) plane. The query is expressed as 2-dimensional interval [(y1q, y2q),

(t1q, t2q)].



2.2 Space-Range Representation Shortcomings:

Minimum Bounding Rectangle (MBR) much larger than a Line.

Object retains trajectory until updated, all lines extend to “infinity”.

Mapping a line segment as a point in 4 dimensions, will not work.

SAM can only access queries until end of the current session.



2.3 Dual Space-Time Representation Map a line from primal plane (t, y) to a point in the dual

plane. Hough-X plane:

y(t) = v t + a is represented by a point (v, a) in dual space. Hough-Y plane:

y(t) = vt + a is transformed to t = 1/v y(t) – a/v. In dual space, the coordinates are n = 1/v and b = -a/v.



2.3 Dual Space-Time Representation Hough-X Dual Space

Query is transformed in a polygon query in the dual space. This polygon uses a linear constraint polygon.



2.3 Dual Space-Time Representation The 1-dimensional MOR query in dual Hough-X is

expressed as For v > 0 the query is : Q = C1 C2 C3 C4, where

C1 = v v min,

C2 = v v max,

C3= a + t2qv > y1q, and

C4 = a + t1qv <= y2q.

For v < 0. the query is : Q = D1 D2 D3 D4, whereD1 = v -v min,

D2 = v -v max,

D3 = a + t1qv y1q, and

D4 = a + t2qv y2q.



2.3 Dual Space-Time Representation

Problem: range of a = [-v max x t now , y max - v min x t now ] Time is increasing, values of the intercepts are not bounded. V max is significant, values of intercept become very large, and

potentially a problem (unbounded range of real numbers)

To solve: Assuming that when an object crosses a border it issues an

update (deletion or reflection). With minimal speed, all objects updated their motion information

at least once during a T period time instant,

where T period = y max / v min.



2.3 Dual Space-Time Representation Two distinct index structures. Each object stores only

once.[0, T period] and [T period, 2T period]

1st index: The intercept is computed using the line t = 0.2nd index: The intercept is computed using the line t = T period.

With this, the intercept will always have the values between 0 and v max × T period.

Both indices are used in database query.



2.3 Dual Space-Time Representation

Keeping on that, after time 2T period, the first index is deleted and a new index is created. We then have

[T period, 2Tperiod] and [2Tperiod, 3Tperiod]

With this method, intercept is bounded, and Performance of index structure remains asymptotically the same.



2.4 Simplex range searching The dual space-time representation transforms the

indexing mobile objects on a line to simplex range searching in two dimensions.

Theorem 1 (Lower Bound)Simplex reporting in d-dimensions with a query time of O (n δ +

k), where N is the number of points, n = N/B, K is the number of the reported points, k = K/B and 0 < δ ≤ 1, requires space Ω (n d(1- δ) – є ) disk blocks, for any fixed є.



2.4 Simplex range searching Two approaches:

1. Point Access Methods2. Query Approximation Algorithm



2.4.1 Point Access Method Point Access Method

Using R-trees to answer simplex range queries, the 1-dimensional MOR query can be answered in the dual Hough-X space.



2.4.2 Query Approximation Algorithm Query Approximation Algorithm

Using Hough-Y dual plane



2.4.2 Query Approximation Algorithm Since rectangular query is an efficient access method,

the query is approximated by a rectangular one.

The query rectangle

The query area is enlarged by area E = E1 + E2, and



2.4.2 Query Approximation Algorithm

E is based on yr (where the b coordinate is computed) and the query interval (y1q, y2q), which is unknown.

If a constant c is use to keep equidistant yr, we can have observation index, y i = y max /c × i



2.4.2 Query Approximation Algorithm A given 1-dimensional query MOR query will be forwarded to the

index(es) that minimize E. Rectangle range search = simple range search on the b

coordinate axis. Each of the observation index can be a B+ tree. To process query interval [y1q, y2q], two cases will be considered,

Case 1: y2a – y1q ≤ y max / c The area is bounded by



2.4.2 Query Approximation Algorithm Case 2: y2q – y1q > y max / c

The query is decomposed into collection of smaller sub-queries: one sub-query per sub-terrain contained by the original query’s endpoints. The sub-queries can be answered with bounded E using an appropriate “ observation” index.

Lemma 1The one dimensional MOR query can be answered in time

O(log B n + (K + K’) / B), where K’ is the approximation error. The space used is O(c n) where c is a small constant, and the update is O(c log B n).



2.5 Achieve Logarithmic Query Time Restrict our queries to occur before the first time that a

point overtakes another. To index mobile objects in a bounded time interval T in the

future. Consider objects are moving on a line, Find all the objects

that lie in the segment [ y l , y r ] at time t q (i.e. t q = t1q = t2q).



2.5 Achieve Logarithmic Query Time Lemma 2

If we have the relative ordering of all the objects at time t q, the position of the objects at time t c that corresponds to the closest crossing event before t q, and the speed of the objects, we can find the objects that are in [y l, yr] in O(log2N + K) time.

Consider objects {p1, p2, ……. , p n}, pi has relative position y i and a velocity vi. At time t q, the position pi is pi = y i + vi × t q. The objects stored in the binary tree are sorted.

In O(log2N) time we can find the positions of yl and yr relative the objects at time t q, and we report the objects that lie between.



2.5 Achieve Logarithmic Query Time

e.g. An object list {p1, p2, p3, p4, p5, p6, p7}, and at time t q and say, [y l, yr] = [p3, p6]




We can find all the crossings of objects in time O (Nlog2N + Mlog2M), where M is the number of crosses in the time period [0,T].

Consider N Objects {p1, p2, ……, pN}, pi has relative position yi and a velocity vi at time 0. Another sorted order list {pt(1) , pt(2) , ……pt(N)} at time T. Then objects i and j cross if and only if t(j) < t(i).

Keep the objects in a linked list, in the same order they were at time t = 0. Scan the sorted lit of objects at time t = T. All the crossings can be reported in O(N + M). After all crossings are reported we can find when each occurs and sort them on their time attribute.



2.5 Achieve Logarithmic Query Time e.g.

Ordered list {p1, p2, p3, p4, p5, p6, p7} at time t = 0

Ordered list {p1, p3, p2, p6, p4, p5, p7} at time t = T

{ , p3, p2, p6, p4, p5, p7}

{ , p3, , p6, p4, p5, p7} crossing

{ , , , p6, p4, p5, p7}

{ , , , p6, , p5, p7} crossing

{ , , , p6, , , p7} crossing

{ , , , , , , p7}

{ , , , , , , }

3 Crossings ! M crossings define M ordered list of the N objects.




We store the O(M) order lists of N objects in O(n+m) blocks and perform a search on any list in O(log B(n+m)) I/O’s, where n = N/B and m = M/B.

Embed the binary tree structure of the list of N objects inside a B-tree. Each B tree use O(n) nodes where each node hold B entries.

A s node stores the evolution of trees B(0), B(t1), B(t2), …, B(t M).

Post the entries of s as changes in the history of the parent node p. When a new copy of node s is created, a new record is added on the “log” of p that has the same position l, but a pointer to the new copy of s and the current time.



2.5 Achieve Logarithmic Query Time Overall space O(n+m), and Query time O(log B(n+m))

Theorem 2:Given N objects and a time limit T, an one dimensional MOR1 query

can be answered in time log B (n +m) using space O(n +m), where m = M/B and M is the number of crosses of objects in the time limit T .



3. Indexing in Two Dimensions

The 1.5-dimensional problem Objects moving in the plane but their movement is

restricted on using a given collection of routes (roads) on the finite terrain.

Each predefined route can be represented as a sequence of connected (straight line) segments.

Indexing the line segment by standard SAM. Indexing the object moving on the given route is

an1-dimensional model.



3. Indexing in Two Dimensions

The 2-dimensional problem Decompose the motion of the object into two

independent motions, one in the x-axis and the other in the y-axis.

For each axis, use the methods for the 1-dimensional case and answer two 1-dimensional MOR queries. The algorithms for the 1-dimensional case can be used.

Take the intersection of the two answers to find the answer to the initial query.



4. Summary

One-dimensional case,We have:

A dynamic, external memory algorithm with guaranteed worst case performance and linear space.

A practical approximation algorithm also in the dynamic, external memory setting, which has linear space and expected logarithmic query time.

An algorithmic query time for a restricted version of the problem.

Two-dimensional case, Extending the techniques to 2-dimensional networks of 1-

dimensional routes as well objects moving on a plane.



5. Reference George Kollios, Dimitrios Gunopulos and

Vassilis Tsotras. On Indexing Mobile Objects. ACM PODS, 1999.



On Indexing Mobile Objects

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CSIS7101 – Advanced Database Technologies