ARCHEOLOGICALSERIATION AND INTERVALGRAPHSARCHEOLOGICALSERIATION AND INTERVALGRAPHS
                             By
                        Pranathi Reddy Tetali
OutlineOutline
Problem statement
Graph Construction
Relation to graph problem
NP-Hard problem
Special Properties
Depicting graph solution
Comments
Problem StatementProblem Statement
Archeology seriation is the attempt to place aset of items in their proper chronologicalorder.
The problem also called sequence dating isto figure out the time relationships betweenset of artifacts, found in graves and the timeintervals during which they were in use.
Problem statementProblem statement
It involves mapping all the artifacts foundin each grave to the corresponding timeduration.
This problem has much in common withinterval graphs and consecutive 1sproperty of incidence matrices.
Graph ConstructionGraph Construction
Assumptions:
If two different artifacts occurred together inthe same grave, then their time periods musthave overlapped.
Since number of graves was large, if timeperiods overlapped then the artifacts appearedtogether in some graves.
Graph ConstructionGraph Construction
Consider 6 artifacts: a,b,c,d,e,f The adjacency matrix tells which pairs of artifacts occurred together in graves. 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑎 0 1 0 0 0 0 𝑏 1 0 1 1 1 0 𝑐 0 1 0 1 1 0 𝑑 0 1 1 0 1 0 𝑒 0 1 1 1 0 1 𝑓 0 0 0 0 1 0
Graph ConstructionGraph Construction
The problem now is to represent them inchronological order. This can be done by permutingadjacency matrix to incidence matrix withconsecutive 1’s property.
However, this method produces many correctpermutations.
To limit the number of correct orders we will usegraph theory of interval graphs.
Graph ConstructionGraph Construction
Let G be a graph whose vertices representartifacts and edges correspond to pairs ofartifacts that appear together in same grave.
Relation to a graph problemRelation to a graph problem
This Real world problem is converted tointerval graph problem.
The problem in graphical terms can bedescribed as-
“To obtain an interval model with all theadjacent vertices intersecting while the nonadjacent vertices are apart.
Relation to a graph problemRelation to a graph problem
From the graph we construct a set of intervalson the real line corresponding to time periodsduring which the artefacts were in use.
Artefacts correspond to overlapping intervalsand sets of artifacts correspond to overlappingintervals.
Relation to a graph problemRelation to a graph problem
The interval model obtained from thegraph:
An NP- Hard problemAn NP- Hard problem
It takes many years to determine all possiblepermutations and obtain correct order.
The problem is solvable in polynomial timeon interval graphs that is NP-complete whileit is NP-Hard in general case.
Special PropertiesSpecial Properties
The clique matrix of an undirected graph isan incidence matrix having maximal cliquesas rows and vertices as columns.
Corollary: An undirected graph G is aninterval graph if and only if the clique matrixof G has the consecutive ones property forcolumns.
Special PropertiesSpecial 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. X is the set of maximal cliques of G. F = {S (v)│v ∈ V}, S(v) is set of maximal cliques of G.
Special PropertiesSpecial Properties
Algorithm calculates 𝜋 (F): 1: procedure consecutive (X ,F, 𝜋) 2: let 𝜋 be the set of all permutations of X 3: for all S ∈ F do 4: remove from 𝜋 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 PropertiesSpecial Properties
Theorem:
Interval graphs can be recognized in O(n+m)time. Moreover, if G is an interval graph, thenthere is an algorithm taking O(n+m) time toconstruct a proper PQ-tree T such thatconsistent(T) is the set of orderings of themaximal cliques of G in which, for everyvertex v of G, the maximal cliques containingvertex v occur consecutively.
Special PropertiesSpecial Properties
Some other properties that define interval graphs: It is chordal and its complement G is a comparability graph. It contains no induced 𝐶 4 and G is transitively orientable. It is chordal and contains no asteroidal triple (AT).
Depicting Graph SolutionDepicting Graph Solution
The interval model directly displays thechronological order.
From the interval graph we get, thefollowing intervals
CommentsComments
The interval graph is used to optimize theseriation process.
It is not simple in practice as few differentarrangements of intervals are possible.
Additional information is required to exactlydetermine one order from the fewpermutations.
ReferencesReferences
Kendall, D. (1969). INCIDENCEMATRICES, INTERVAL GRAPHS ANDSERIATION IN ARCHAEOLOGY. PACIFICJOURNAL OF MATHEMATICS, 28(3), 565-570. Retrieved October 7, 2014, fromhttp://projecteuclid.org/download/pdf_1/euclid.pjm/1102983306
ReferencesReferences
Interval Graph Isomorphism. (n.d.).Retrieved October 7, 2014, fromhttp://www.lsi.upc.edu/~valiente/graph-00-01-d.pdf
Mertzios, G. (2008).A matrix characterization of interval andproper interval graphs. AppliedMathematics Letters, 21, 332-337. Retrieved October 7, 2014, fromhttps://community.dur.ac.uk/george.mertzios/papers/Jour/Jour_NIR_SNIR.pdf
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