A Simplicial Complex Model for Large Shared Scienti c Data ... · A Simplicial Complex Model for Large Shared Scienti c Data Repositories Introduction Motivation Motivation We want

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A Simplicial Complex Model for Large Shared Scientific Data Repositories

A Simplicial Complex Model for Large SharedScientific Data Repositories

Zihong Yuan, Stephane BressanBill Howe, David Maier

School of ComputingNational University of Singapore

IMS, 20 Aug 2015

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Introduction

Motivation

Motivation

We want to construct a simplicial complex based model and adatabase management system using this model to managescientific data.

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Introduction

Motivation

A Model for relational database management

▶ In 1970, E. F. Codd proposed the relational database modelfor large databanks in his seminal CACM article:

Codd, E. F. (1970). “A relational model of data forlarge shared data banks”. Communications of theACM 13 (6): 377.

▶ In 1981, E. F. Codd received the Turing award

“For his and continuing contributions to the theoryand practice of database management systems.”

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Introduction

Motivation






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Introduction

Motivation






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Introduction

Motivation






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Introduction

Motivation


The relational model is:

▶ rigorous: It is based on a formal concept: the relation.

▶ simple: The relation is a basic mathematical concept.

▶ practical: Relations are implemented and viewed as tables(similar to Excel spreadsheets).

▶ effective: The model has been used to manage business datafor the last 45 years.

▶ efficient: Relational database applications are implementedwith platforms called relational database management systems(RDBMS).

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Introduction

Motivation








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Introduction

Motivation








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Introduction

Motivation








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Introduction

Motivation








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Introduction

Motivation

Relational database management systems (RDBMS)

▶ RDBMS efficiently implement the primitives for the storageand manipulation of relations.

▶ Some popular RDBMS are Oracle, MySQL, IBM DB2,Microsoft SQL Server, Sybase.

▶ The language for programming RDBMS is an internationalstandard: SQL.

▶ RDBMS are used for managing payroll, logistics, bankaccounts, airline reservations, sales, online shopping and mostof the business applications managing data.

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Introduction

Motivation






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Introduction

Motivation






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Introduction

Motivation






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Introduction

Motivation

Why a Logical Model?

“Future users of large databanks must be protected fromhaving to know how the data is organized in themachine” E.F. Codd

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Introduction

Motivation

Real World

id employee

01 Dave

02 Jones

03 SmithThe relational model mediatesbetween the real world and the

machine.

▶ It must be effective:represent the real workaccurately.

▶ It must be efficient: thedata is well organized forfast manipulation.

Machine

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Introduction

Motivation

Real World

id employee

01 Dave

02 Jones


machine.



Machine

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Introduction

Motivation

Real World

id employee

01 Dave

02 Jones


machine.



Machine

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Introduction

Motivation

Three levels of a RDBMS

▶ Conceptual level: Data in the real world.

▶ Logical level: An intermediary representation that mediatesthe users’ views and interactions between real world andmachine.

▶ Physical level: How the data is stored in the machine.

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Introduction

Motivation





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Introduction

Motivation





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Introduction

Motivation

Example (conceptual model)

Employees Workfor Departments

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Introduction

Motivation

Example (logical model)

Employees

id employee

01 Dave

02 Jones

03 Smith

Workfor

eid did

01 d01

01 d02

02 d02

03 d02

Departments

id department

d01 marketing

d02 manufacture

Operations over tables (relational algebra): union, intersection,selection, projection, product, join.

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Introduction

Motivation

Example (a query in SQL)

▶ A query: How many employees work in the marketingdepartment?

▶ SQL command:SELECT COUNT (DISTINCT e.id)FROM employee e, workfor w, department dWHERE e.id = w.eid AND d.id = w.did AND d.department= “marketing”

▶ In relational algebra:employee ▷◁ workfor ▷◁ σdepartment=“marketing”(department)

▶ In a RDBMS, we have the model (table), operations(relational algebra) and a language (SQL).

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Introduction

Motivation






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Introduction

Motivation






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Introduction

Motivation






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Introduction

Motivation

Scientific data

a Singapore map, a human brain, a fighter

▶ In science and engineering, data are spatial, spatio-temporalor of other multi-dimensional structures.

▶ For the management and manipulation of such data inscientific applications, table is not a natural choice.

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Introduction

Motivation

Scientific data

a Singapore map, a human brain, a fighter

▶ In science and engineering, data are spatial, spatio-temporalor of other multi-dimensional structures.

▶ For the management and manipulation of such data inscientific applications, table is not a natural choice.

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Introduction

Motivation

The problem

▶ Can we devise a logical model that effectively representsscientific data?

▶ Can we design and implement a data management systemthat efficiently manipulates scientific data in this model?

▶ The above model and system should work effectively andefficiently for managing both topological information andgeometric information.

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Introduction

Motivation

The problem




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Introduction

Motivation

The problem




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Introduction

Motivation

Topological information vs. geometric information

▶ Topological: connectedness, homology, homotopy, · · · .

▶ Geometric: length, angle, direction, area, volume, · · · .▶ Why we need separate topological information from the

geometric information?

▶ For some scientific data (e.g. social network), it is difficult todefine a suitable metric.

▶ In the research, getting topological information can be seen asa first step. If we only consider the topological information,we can delete unnecessary geometric information to acceleratethe computation.

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Introduction

Motivation


▶ Topological: connectedness, homology, homotopy, · · · .▶ Geometric: length, angle, direction, area, volume, · · · .

▶ Why we need separate topological information from thegeometric information?



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Introduction

Motivation


▶ Topological: connectedness, homology, homotopy, · · · .▶ Geometric: length, angle, direction, area, volume, · · · .▶ Why we need separate topological information from the




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Introduction

Motivation






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Introduction

Motivation






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Introduction

Related Work

Related work

▶ We are not the 1st people to see this problem. Up to now,there are several DBMSs for scientific data.

(1) Relational DBMS + spatial features▶ For the logical level, still store data in tables. SQL+ spatial

construct (e.g. Oracle Spatial Data Cartridge).▶ For the physical level, spatial index (e.g. R-tree).▶ Tables can not reflect the spatial structures of data, it is not a

natural view of scientific data.

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Introduction

Related Work

Related work


(1) Relational DBMS + spatial features

▶ For the logical level, still store data in tables. SQL+ spatialconstruct (e.g. Oracle Spatial Data Cartridge).

▶ For the physical level, spatial index (e.g. R-tree).▶ Tables can not reflect the spatial structures of data, it is not a


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Introduction

Related Work

Related work



construct (e.g. Oracle Spatial Data Cartridge).

▶ For the physical level, spatial index (e.g. R-tree).▶ Tables can not reflect the spatial structures of data, it is not a


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Introduction

Related Work

Related work



construct (e.g. Oracle Spatial Data Cartridge).▶ For the physical level, spatial index (e.g. R-tree).

▶ Tables can not reflect the spatial structures of data, it is not anatural view of scientific data.

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Introduction

Related Work

Related work



construct (e.g. Oracle Spatial Data Cartridge).▶ For the physical level, spatial index (e.g. R-tree).▶ Tables can not reflect the spatial structures of data, it is not a


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Introduction

Related Work

(2) Array DBMS

▶ An array database represent everything as a homogeneouscollections of data items (often called pixels, voxels, etc). Inthe database jargon, this is called a regular grid.

a regular grid, an irregular lake

▶ Array database only consider points, no faces.

▶ It can not model irregular grids efficiently.

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Introduction

Related Work

(2) Array DBMS





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Introduction

Related Work

(2) Array DBMS





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Introduction

Related Work

(2) Array DBMS





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Introduction

Related Work

(3) Grid DBMS (B. Howe, D. Maier, 2004)

▶ Represent data as cell complexes.

▶ Data bounded to higher dimensional cells.

A Singapore weather map

▶ Cells in cell complexes are too general for most applications.

17 / 41

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Introduction

Related Work






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Introduction

Related Work






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Introduction

Related Work






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Introduction

Related Work

Other related work

▶ In computational geometry, E. Brisson introduces the regularCW-complex (cell-tuple data structure) to represent ageometric object (1993).

▶ In geographic information system (GIS), F. Penninga, P. J. M.Van Oosterom investigate 3-dimensional simplicial complex torepresent the geographic information (2008).

18 / 41

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Introduction

Related Work

Other related work

▶ In computational geometry, E. Brisson introduces the regularCW-complex (cell-tuple data structure) to represent ageometric object (1993).

▶ In geographic information system (GIS), F. Penninga, P. J. M.Van Oosterom investigate 3-dimensional simplicial complex torepresent the geographic information (2008).

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Introduction

Our Model

Simplicial complex based DBMS

▶ Can we use simplicial complexes for grid database? Anyadvantage?

▶ Comparing to cells in cell complexes, simplexes are easier torepresent in computers.

▶ It is easy to compute the homology of a simplicial complex(simplicial homology).

▶ We believe that cell complexes are too general to be effectiveand efficient, and we believe that simplicial complexes hasbetter properties.

19 / 41

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Introduction

Our Model






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Introduction

Our Model






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Introduction

Our Model






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Mathematical Concepts

Abstract simplicial complexes

Abstract simplicial complex

▶ A abstract simplicial complex is a pair (V,Σ), where V is afinite set, and Σ is a family of subsets of V such thatσ ∈ Σ, τ ⊆ σ implies τ ∈ Σ.

▶ Elements in V are called vertices. Elements in Σ are calledsimplexes. ∀σ, τ ∈ Σ, if σ ⊆ τ , then σ is a face of τ , and τis a coface of σ.

▶ The inclusion relation gives a partial order over Σ. Themaximal elements (maximal simplexes) determine the wholeset Σ.

▶ dimension, star, closed star, link,· · · .

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Homology of abstract simplicial complexes

▶ Given a total order (V,⩽) over V . For an elementσ = {v0, v1, · · · , vk} ∈ Σ with v0 < · · · < vk, define the faceoperator di(0 ⩽ i ⩽ k) as

di{v0, · · · , vk} = {v0, · · · , vi, · · · , vk}.

▶ With the face operator, it is easy to formulate the chaincomplex of (V,Σ). Thus we can compute the homology of(V,Σ).

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Homology of abstract simplicial complexes

▶ Given a total order (V,⩽) over V . For an elementσ = {v0, v1, · · · , vk} ∈ Σ with v0 < · · · < vk, define the faceoperator di(0 ⩽ i ⩽ k) as

di{v0, · · · , vk} = {v0, · · · , vi, · · · , vk}.

▶ With the face operator, it is easy to formulate the chaincomplex of (V,Σ). Thus we can compute the homology of(V,Σ).

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Geometric simplicial complexes


▶ A k-simplex σ = [v0, · · · , vk] in Rn is the convex hull ofk + 1 points v0, · · · , vk satisfying v1 − v0, · · · , vk − v0 arelinearly independent.

▶ A simplicial complex K is a finite set (enough forcomputers) of simplexes satisfying:

(1) Any face of a simplex from K is also in K.(2) The intersection of any two simplexes σ1, σ2 ∈ K is either

empty or a face of σ1 and σ2.

▶ Given a geometric simplicial complex K, there is a canonicalabstract simplicial complex defined by K. Given an abstractsimplicial complex L = (V,Σ), its realization |L| exists.

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(1) Any face of a simplex from K is also in K.

(2) The intersection of any two simplexes σ1, σ2 ∈ K is eitherempty or a face of σ1 and σ2.


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Simplicial maps

Simplicial maps

▶ Let K = (V,Σ), L = (V ′,Σ′) be two simplicial complexes. Amap f : V → V ′ is called a simplicial map if∀σ ∈ Σ, f(σ) ∈ Σ′. Here f(σ) = {f(v)|v ∈ σ}.

▶ Every continuous map between triangulable spaces could beapproximated by a simplicial map.

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Simplicial maps

Simplicial maps

▶ Let K = (V,Σ), L = (V ′,Σ′) be two simplicial complexes. Amap f : V → V ′ is called a simplicial map if∀σ ∈ Σ, f(σ) ∈ Σ′. Here f(σ) = {f(v)|v ∈ σ}.

▶ Every continuous map between triangulable spaces could beapproximated by a simplicial map.

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Construct simplicial complexes


Let X be a topological space or a point cloud (a finite set ofpoints). There are various ways to construct a simplicial complex.

(1) Find a triangulation of X. In other words, find a simplicalcomplex K and a homeomorphism f : |K| → X.

▶ Every smooth manifold is triangulable.

(2) Using an open cover and consider its nerve.

▶ Let U = {Uα}α∈A be an open covering of X. The nerve ofU , denoted by NU , is the abstract simplicial complex withvertex set A, and where a family {α0, · · · , αk} spans ak-simplex if and only if Uα0 ∩ · · · ∩ Uαk

= ∅.

▶ Under “good” conditions, |NU| ≃ X.

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= ∅.


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= ∅.


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= ∅.


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= ∅.


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= ∅.


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(3) Cech complexLet X be a metric space. Let Bϵ(X) = {Bϵ(x)}x∈X . Thenerve of Bϵ(X), denoted by C(X, ϵ), is called the Cechcomplex of X.

(4) Vietoris-Rips complexLet X be a metric space with metric d. The Vietoris-Ripscomplex for X, attached to the parameter ϵ, denoted byV R(X, ϵ), is the simplicial complex whose vertex set is X,and where {x0, · · · , xk} spans a k-simplex if and only ifd(xi, xj) ⩽ ϵ for all 0 ⩽ i, j ⩽ k.

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(3) Cech complexLet X be a metric space. Let Bϵ(X) = {Bϵ(x)}x∈X . Thenerve of Bϵ(X), denoted by C(X, ϵ), is called the Cechcomplex of X.

(4) Vietoris-Rips complexLet X be a metric space with metric d. The Vietoris-Ripscomplex for X, attached to the parameter ϵ, denoted byV R(X, ϵ), is the simplicial complex whose vertex set is X,and where {x0, · · · , xk} spans a k-simplex if and only ifd(xi, xj) ⩽ ϵ for all 0 ⩽ i, j ⩽ k.

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(5) Witness complexLet X be a finite metric space, and let L ⊂ X be a finitelandmark set. Let ϵ > 0. For every point of x ∈ X, let mx

denote the distance from x to the set L. Then we define thestrong witness complex W s(X,L, ϵ) to be the complexwhose vertex set is L, and where a collection {l0, · · · , lk}spans a k-simplex if and only if there is a point x ∈ X (thewitness) so that

d(x, li) ⩽ mx + ϵ

for all i.

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Operations

Gridfields

What is a gridfield?

▶ A gridfield appears when data is bound to a grid (simplicialcomplex).

▶ A gridfield G is a 4-tuple (V,Σ, k, f), where G = (V,Σ) is agrid, k is a non-negative integer, and f : Σk → Rn is afunction from k-dimensional simplexes in Σ to Rn.

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Operations

Gridfields




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Operations

Gridfields




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Operations

Bind

Bind

▶ The bind operator constructs a gridfield from a gridG = (V,Σ), an integer k and a function f : Σk → Rn. Inother words, the bind operator associates data to eachk-simplex of G.

▶ Bind operation allow us to apply topological operations ongrids prior to binding data.

▶ Furthermore, we could apply geometric operations afterbinding the geometric information.

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Operations

Bind

Bind




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Operations

Bind

Bind




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Operations

Select

Select

▶ For each subgrid of a grid, there is a select operator(restriction).

▶ select(salt > 29)

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Operations

Select

Select

▶ For each subgrid of a grid, there is a select operator(restriction).

▶ select(salt > 29)

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Operations

Closed star, link

Closed star, link

a vertex, its closed star, its link.

▶ A useful concept to discuss the local neighborhood is theclosed star of a simplex.

▶ Given a simplicial complex (V,Σ), for a simplex τ ∈ Σ, theclosed star and link are

St τ = {subsets of σ|τ ⊆ σ, σ ∈ Σ},Lk τ = {complement set of τ in σ|τ ⊆ σ, σ ∈ Σ}.

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Operations

Closed star, link

Closed star, link





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Operations

Closed star, link

Closed star, link





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Operations

Aggregate (gluing)

Aggregate (gluing)

▶ Let K = (V,Σ), L = (V ′,Σ′) be two simplicial complexes. Asimplicial map f : V → V ′ is a quotient map if f issurjective. Then L is called a quotient of K.

▶ For a quotient map f , if the restriction of f to every simplexis injective, then f is called a gluing.

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Operations

Aggregate (gluing)

Aggregate (gluing)



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Operations

Aggregate (gluing)

Aggregate (gluing)



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Operations

Aggregate (gluing)

▶ Let G = (V,Σ, k, f) be a gridfield. Let g : (V,Σ) → (V ′,Σ′)be a gluing map.

▶ For each σ′ ∈ Σ′k, g

−1(σ′) ⊆ Σk.

▶ Define an aggregation function to be a function

ϕ : 2Σk × (Rn)Σk → (Rn)Σ′k .

▶ Thus, there is a grid field (V ′,Σ′, k, f ′) with

f ′ = ϕ(g−1(σ′), f).

32 / 41

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Operations

Aggregate (gluing)



−1(σ′) ⊆ Σk.


ϕ : 2Σk × (Rn)Σk → (Rn)Σ′k .


f ′ = ϕ(g−1(σ′), f).

32 / 41

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Operations

Aggregate (gluing)



−1(σ′) ⊆ Σk.


ϕ : 2Σk × (Rn)Σk → (Rn)Σ′k .


f ′ = ϕ(g−1(σ′), f).

32 / 41

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Operations

Aggregate (gluing)



−1(σ′) ⊆ Σk.


ϕ : 2Σk × (Rn)Σk → (Rn)Σ′k .


f ′ = ϕ(g−1(σ′), f).

32 / 41

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Operations

Subdivision

Subdivision

▶ Let K = (V,Σ) be a simplicial complex. The barycentricsubdivision SdK is defined as

(1) The vertex set of SdK is Σ.(2) The simplexes of SdK are chains (under the inclusion

relation) in Σ.

▶ A barycentric subdivision of a triangle.

33 / 41

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Operations

Subdivision

Subdivision


(1) The vertex set of SdK is Σ.

(2) The simplexes of SdK are chains (under the inclusionrelation) in Σ.


33 / 41

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Operations

Subdivision

Subdivision



relation) in Σ.


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Operations

Subdivision

Subdivision



relation) in Σ.


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Operations

Subdivision

▶ Let G = (V,Σ, k, f) be a gridfield. Let SdG = (V ′,Σ′) bethe barycentric subdivision of G = (V,Σ).

▶ To define a new gridfield SdG, we need a transition function

ϕ : Σ′k × (Rn)Σk → (Rn)Σ

′k .

▶ We could let SdG = (V ′,Σ′, k, f ′) where

f ′(σ′) = ϕ(σ′, f) ∀σ′ ∈ Σ′k.

34 / 41

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Operations

Subdivision




′k .


f ′(σ′) = ϕ(σ′, f) ∀σ′ ∈ Σ′k.

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Operations

Subdivision




′k .


f ′(σ′) = ϕ(σ′, f) ∀σ′ ∈ Σ′k.

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Operations

Product

Product

▶ A product of two simplicial complex.

a0

b0

c0a

b

c

a1b1

c1

0

1

▶ Let K = (V,Σ), L = (V ′,Σ′) be two simplicial complexes.(V,⩽1) and (V ′,⩽2) are two total order. The product K × Lis defined as follows.

▶ The vertex set of K × L is V × V ′.

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Operations

Product

Product


a0

b0

c0a

b

c

a1b1

c1

0

1



35 / 41

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Operations

Product

Product


a0

b0

c0a

b

c

a1b1

c1

0

1



35 / 41

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Operations

Product

▶ For each σ = {v0, · · · , vk} ∈ Σ and τ = {w0, · · · , wl} ∈ Σ′.v0 <1 · · · <1 vk and w0 <2 · · · <2 wl. Define σ × τ as a setof V × V ′.

σ × τ = {(vi0 , wj0), (vi1 , wj1), · · · , (vik+l, wjk+l

)}

such that

vi0 = v0, wi0 = w0, vik+l= vk, wik+l

= wl,

vi0 ⩽ vi1 ⩽ · · · ⩽ vik+l,

wi0 ⩽ wi1 ⩽ · · · ⩽ wik+l,

|is+1 − is|+ |js+1 − js| = 1.

▶ The simplexes in K × L is the union of all such σ × τ .

36 / 41

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Operations

Product

▶ For each σ = {v0, · · · , vk} ∈ Σ and τ = {w0, · · · , wl} ∈ Σ′.v0 <1 · · · <1 vk and w0 <2 · · · <2 wl. Define σ × τ as a setof V × V ′.

σ × τ = {(vi0 , wj0), (vi1 , wj1), · · · , (vik+l, wjk+l

)}

such that

vi0 = v0, wi0 = w0, vik+l= vk, wik+l

= wl,

vi0 ⩽ vi1 ⩽ · · · ⩽ vik+l,

wi0 ⩽ wi1 ⩽ · · · ⩽ wik+l,

|is+1 − is|+ |js+1 − js| = 1.

▶ The simplexes in K × L is the union of all such σ × τ .

36 / 41

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Operations

Product

▶ Let G = (V,Σ, k, f),H = (V ′,Σ′, l, g) be two gridfields andk = dimK, l = dimL. Then the product G×H of G and His a gridfield

(V × V ′,Σ× Σ′, k + l, f × g).

▶ Here we can only consider the product of top dimensionalcells. The product of tables may lead contradictions if weconsider the product of non-top dimensional cells.

37 / 41

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Operations

Product

▶ Let G = (V,Σ, k, f),H = (V ′,Σ′, l, g) be two gridfields andk = dimK, l = dimL. Then the product G×H of G and His a gridfield

(V × V ′,Σ× Σ′, k + l, f × g).

▶ Here we can only consider the product of top dimensionalcells. The product of tables may lead contradictions if weconsider the product of non-top dimensional cells.

37 / 41

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Operations

Intersection, union

Intersection, union

▶ If for all σ ∈ K, τ ∈ L with nonempty intersection we haveσ ⩽ τ or τ ⩽ σ, then we say K,L are compatible.

v0

v2

v1

w0 w1

A non-compatible case, K = [v0, v1, v2], L = [w0, w1]

▶ Given two compatible simplicial complexes K,L, we define theintersection and union as

K ∩ L = {σ|σ ∈ K, σ ∈ L},

K ∪ L = {σ|σ ∈ K or σ ∈ L}.

38 / 41

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Operations

Intersection, union

Intersection, union


v0

v2

v1

w0 w1



K ∩ L = {σ|σ ∈ K, σ ∈ L},

K ∪ L = {σ|σ ∈ K or σ ∈ L}.

38 / 41

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Operations

Intersection, union

Intersection, union


v0

v2

v1

w0 w1



K ∩ L = {σ|σ ∈ K, σ ∈ L},

K ∪ L = {σ|σ ∈ K or σ ∈ L}.

38 / 41

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Operations

Intersection, union

Intersection, union (non-compatible case)

▶ If K,L are not compatible, we need to construct subdivisionsK ′, L′ of K,L such that K ′, L′ are compatible.

v0

v2

v1

w0 w1

|K| ∩ |L| = ∅v0

v2

v1K ′

w0 w1

L′v0

v2

v1K ′ ∩ L′

▶ Then we define K ∩ L = K ′ ∩ L′,K ∪ L = K ′ ∪ L′.

39 / 41

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Operations

Intersection, union



v0

v2

v1

w0 w1

|K| ∩ |L| = ∅v0

v2

v1K ′

w0 w1

L′v0

v2

v1K ′ ∩ L′


39 / 41

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Operations

Intersection, union



v0

v2

v1

w0 w1

|K| ∩ |L| = ∅v0

v2

v1K ′

w0 w1

L′v0

v2

v1K ′ ∩ L′


39 / 41

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Summary

Summary

▶ We proposed a simplicial complex based model for scientificdata.

▶ The model could manipulate the topological data.

▶ We investigate the operations of related gridfield.

40 / 41

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Summary

Summary




40 / 41

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Summary

Summary




40 / 41

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Thank You!

41 / 41

Documents

A Simplicial Complex Model for Large Shared Scienti c Data ... · A Simplicial Complex Model for Large Shared Scienti c Data Repositories Introduction Motivation Motivation We want