ARCHEOLOGICAL SERIATION AND INTERVAL GRAPHS By Pranathi Reddy Tetali.
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Transcript of ARCHEOLOGICAL SERIATION AND INTERVAL GRAPHS By Pranathi Reddy Tetali.
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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