ARCHEOLOGICAL SERIATION AND INTERVAL GRAPHS

By Pranathi Reddy Tetali

Outline

Problem statement

Graph Construction

Relation to graph problem

NP-Hard problem

Special Properties

Depicting graph solution

Comments

Problem Statement

Archeology seriation is the attempt to place a

set of items in their proper chronological

order.

The problem also called sequence dating is

to figure out the time relationships between

set of artifacts, found in graves and the time

intervals during which they were in use.

Problem statement

It involves mapping all the

artifacts found in each grave to

the corresponding time duration.

This problem has much in

common with interval graphs and

consecutive 1s property of

incidence matrices.

Graph Construction

Assumptions:

If two different artifacts occurred

together in the same grave, then their

time periods must have overlapped.

Since number of graves was large, if time

periods overlapped then the artifacts

appeared together in some graves.

Graph Construction

Consider 6 artifacts:

a,b,c,d,e,f

The adjacency matrix tells

which pairs of artifacts

occurred together in graves.

Graph Construction

The problem now is to represent them in

chronological order. This can be done by

permuting adjacency matrix to incidence

matrix with consecutive 1’s property.

However, this method produces many

correct permutations.

To limit the number of correct orders we will

use graph theory of interval graphs.

Graph ConstructionLet G be a graph whose vertices

represent artifacts and edges

correspond to pairs of artifacts that

appear together in same grave.

a

b

c

e

d

f

Relation to a graph problemThis Real world problem is converted to

interval graph problem.

The problem in graphical terms can be

described as-

“To obtain an interval model with all

the adjacent vertices intersecting while

the non adjacent vertices are apart.

Relation to a graph problem From the graph we construct a set of

intervals on the real line

corresponding to time periods during

which the artefacts were in use.

Artefacts correspond to overlapping

intervals and sets of artifacts

correspond to overlapping intervals.

Relation to a graph problemThe interval model obtained from the graph: c

a d f

e

b

-4000 -3800 -3600 -3400 -3200 -3000 -2800 -2600 -2400

An NP- Hard problem

It takes many years to determine all

possible permutations and obtain

correct order.

The problem is solvable in

polynomial time on interval graphs

that is NP-complete while it is NP-

Hard in general case.

Special Properties

The clique matrix of an undirected graph

is an incidence matrix having maximal

cliques as rows and vertices as columns.

Corollary: An undirected graph G is an

interval graph if and only if the clique

matrix of G has the consecutive ones

property for columns.

Special Properties

Given a finite set X and a collection F of subsets

of X, the consecutive arrangement problem is to

determine whether or not there exists a

permutation π of X in which the elements of

each subset S F appear as a consecutive

subsequence of π.

I. X is the set of maximal cliques of G.

II. F = {S (v)│v V}, S(v) is set of maximal

cliques of G.

Special PropertiesAlgorithm calculates (F):1: procedure consecutive (X ,F, ) 2: let be the set of all permutations of X3: for all S F do4: remove from those permutations in which the elements of S do not occur as a subsequence 5: end procedure

Alternatively we can use PQ-Tree representation.

Special PropertiesTheorem:

Interval graphs can be recognized in O(n+m) time. Moreover, if G is an interval graph, then there is an algorithm taking O(n+m) time to construct a proper PQ-tree T such that consistent(T) is the set of orderings of the maximal cliques of G in which, for every vertex v of G, the maximal cliques containing vertex v occur consecutively.

Special Properties

Some other properties that define interval graphs:It is chordal and its complement

G is a comparability graph.It contains no induced and G is

transitively orientable.It is chordal and contains no

asteroidal triple (AT).

Depicting Graph Solution

The interval model directly displays the chronological order.From the interval graph we get, the following intervals

a : (-4000,-3000) b : (-3800,-3200) c : (-3600,-3000) d : (-3400,-3200) e : (-3400,-2600) f : (-2800,-2400)

Comments

The interval graph is used to optimize

the seriation process.

It is not simple in practice as few

different arrangements of intervals are

possible.

Additional information is required to

exactly determine one order from the

few permutations.

ReferencesKendall, D. (1969). INCIDENCE

MATRICES, INTERVAL GRAPHS AND SERIATION IN ARCHAEOLOGY. PACIFIC JOURNAL OF MATHEMATICS, 28(3), 565-570. Retrieved October 7, 2014, from http://projecteuclid.org/download/pdf_1/euclid.pjm/1102983306

ReferencesInterval Graph Isomorphism. (n.d.).

Retrieved October 7, 2014, from http://www.lsi.upc.edu/~valiente/graph-00-01-d.pdf

Mertzios, G. (2008). A matrix characterization of interval and proper interval graphs. Applied Mathematics Letters, 21, 332-337. Retrieved October 7, 2014, from https://community.dur.ac.uk/george.mertzios/papers/Jour/Jour_NIR_SNIR.pdf

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