Characterizing Matrices with Consecutive Ones Property N.S. Narayanaswamy, IIT Madras (Joint work...
-
Upload
miranda-mclaughlin -
Category
Documents
-
view
214 -
download
0
Transcript of Characterizing Matrices with Consecutive Ones Property N.S. Narayanaswamy, IIT Madras (Joint work...
Characterizing Matrices with Consecutive Ones Property
N.S. Narayanaswamy, IIT Madras(Joint work with R. Subashini, NITC)
The Problem Does a given 0-1 Matrix have the
Consecutive Ones Property (COP)?
Consecutive Ones Property Permute the rows such that the ones in
each column occur consecutively
DNA Physical Mapping[9]m2 m1 m5 m4 m3
Applications
Maximal Clique-Vertex incidence matrix Interval graph characterization
Characterizing cubic Hamiltonian graphs Databases Computational Biology
Previous Work Poly time solvable
Fulkerson and Gross Forbidden matrix configurations – Tucker
View the matrix as a maximal clique-vertex incidence matrix
Asteroidal triple Induced cycles larger than K3
Linear time algorithm Booth and Lueker Running time of O(m+n+#non-zero entries)
Consecutive Ones Testing (COT) Trees PQ-trees
L-R order yields one permutation Leaves are the rows Internal nodes are P and Q nodes P node - all permutations of its children yields a
valid permutation Q node - exactly two permutations are permitted
Algorithm outputs a PQ-tree only if the matrix has the COP Addressed by PQR trees
Permutations and Intervals A feasible permutation of rows yields an interval
assignment to the columns Length of the interval is the number of ones in the
column Intersection cardinality of a pair of intervals is the
number of rows in which a 1 occurs in both the corresponding columns.
Does such an assignment imply a feasible permutation?
Feasible Interval Assignments Each Permutation gives a interval
assignment Is it sufficient to find an interval
assignment to the sets to preserve intersection cardinalities
If yes, can we get a permutation from an interval assignment?
Preserving intersection cardinalities - Sufficiency
Sort the intervals in increasing order of left end point and break ties using the right end points Discard identical columns
Consider (P1,Q1) Pi row indices in i-th column Qi is the interval assigned to the i-th column Encodes all permutations in which Pi is mapped to Qi
Refining the set of permutations Iteratively filter the current set of
permutations Using strictly intersecting pairs Pair of intersecting intervals, neither
contained in the other
Algorithm 1 - Permutations from an ICPIA
Let 0 = {(Ai, Bi) | 1 ≤ i ≤ m}
j = 1;
while there is (P1,Q1), (P2,Q2) Є j-1 with Q1 and Q2 strictly intersecting do
j = j-1\{(P1,Q1), (P2, Q2)}
j = j {(P1 P2 ,Q1 Q2),(P1\ P2, Q1\ Q2),(P2\ P1 ,Q2\ Q1)}
j= j+1;
end while
= j
Return ;
Proof Helly property for intervals
For any 3 mutually intersecting intervals one is contained in the union of the other two.
Intersection cardinality preserved
Invariants Q is an interval for each (P,Q). |P|=|Q| for each (P,Q) For any two (P',Q'), (P'',Q''),
|P'P''|=|Q' Q''|. At the end no interval is strictly
intersecting with another interval Either disjoint or contained.
Completing the refinement The set of (P,Q) yields a natural
containment tree
Algorithm 2 – Permutations from Algorithm 1function Post-order-traversal(T, root-node, )if (root-node is a leaf) then
return
endif
while (root-node has unexplored children) do
Next-root-node = an unexplored child of root node
Post-order-traversal(T, next-root-node, П)end while
if (root-node has no unexplored children) then
let (P,Q) denote the element of П associated with the root-node
let (P1,Q1)…(Pk,Qk) be the pairs associated with the children of root-node
П = П\{(P,Q)}
П = П U {(P\(P1U …Pk), Q\(Q1U…. Qk))}
Return
endif
Consequence
Given an interval assignment We have a data structure that encodes all
permutations which yield this interval assignment
Finding good interval assignments
For a set of proper intervals and its flipping the intersection graph are isomorphic- [1,8],[5,10],[2,7] is isomorphic to [1,6],[3,10],[4,9]
Feasible interval assignments
Intuition To assign intervals to a set system, there
are only two choices and these will be decided at the first step.
An ordering of the sets First set
A set such that all those sets which intersect it have a pair-wise non-empty intersection - candidate for the leftmost interval
Next Set (iteratively) One that has a strict intersection with one
of the chosen sets.
Assigning the Intervals First Set-Left most interval Second set - has strict intersection with first
set. So two interval choices Next set (iteratively)-has strict intersection
with some interval Exactly one choice of interval, given intersection
cardinality constraints Failure implies no feasible interval assignment
Linear time in the number of sets, but computing intersection is costly
Sets left out Do not have a strict overlap with the
sets considered Disjoint Contained
Two distinct sets are related if they have a strict overlap Consider connected components in this
undirected graph
On the components Each component is a sub-matrix formed by
the columns Two components are either
Disjoint Or all the sets in one are contained in a single set
of the other. An interval assignment to each component
implies an interval assignment to the whole set system
Putting the interval assignments together Given that an interval assignment to each of
the components is feasible. Containment tree/forest on the components
An arc between vertices corresponding to two components if the sets of one are all contained in one set of the other
Construct the interval assignment in a BFS fashion starting from the root of each tree
Applications
Can test if rows can be permuted so that columns are sorted 1s occur in a circular fashion
Further Research Solves an isomorphism problem to a target
class of matrices in which 1s in each column are consecutive
NP-hard when 1s are in at most 3 consecutive regions.
References1. N.S. Narayanaswamy , R. Subashini , “A new characterization of
matirces with Consecutive Ones Propery”, Discrete Applied Mathematics, August 2009.
2. K.S. Booth, G.S. Lueker, “Tesing for the consecutive ones property,
interval graph and graph planarity using PQ-tree algorithms”, Journal of
computer System Science,1976.
3. M.C. Golumbic, “Algorithmic Graph Theory and Perfect Graphs”,
Academic Press, 1980.
4. R.Wang, F.C.M Lau, Y.C. Zhao, “Hamiltonicity of regular graphs and
blocks of consecutive ones in symmetric matrices”, Discrete Applied
Mathematics, 2007.
5. S. Ghosh, “File organization: The consecutive retrieval property”,
Communications of the ACM,1979.
6. D. Fulkerson, O.A Gross, “Incidence matrices and interval graphs”,
Pacific Journal of Mathematics,1965.
7. A. Tucker, “A structure theorem for the consecutive ones property”,
Journal of Combinatorial Theory, 1972.
8. J. Meidanis, E. Munuera, “A theory for the consecutive ones property”,
Proceedings of the III South American Workshop on String Processing, 1996.
9. J. Meidanis, Oscar Porto, Guilherme P. Telles, “On the Consecutive
Ones Property”, Discrete Applied Mathematics, 1998.