A Spectral Algorithm for Seriation and the Consecutive Ones Problem

Abstract

Abstract. In applications ranging from DNA sequencing through archeological dating to sparse matrix reordering, a recurrent problem is the sequencing of elements in such a way that highly correlated pairs of elements are near each other. That is, given a correlation function f reflecting the desire for each pair of elements to be near each other, find all permutations π with the property that ifThis seriation problem is a generalization of the well-studied consecutive ones problem. We present a spectral algorithm for this problem that has a number of interesting features. Whereas most previous applications of spectral techniques provide only bounds or heuristics, our result is an algorithm that correctly solves a nontrivial combinatorial problem. In addition, spectral methods are being successfully applied as heuristics to a variety of sequencing problems, and our result helps explain and justify these applications.Key words. seriation, consecutive ones property, eigenvector, Fiedler vector, analysis of algorithms AMS subject classifications. 05C15, 15A18, 15A48, 68E15PII. S00975397952857711. Introduction. Many applied computational problems involve ordering a set so that closely coupled elements are placed near each other. This is the underlying problem in such diverse applications as genomic sequencing, sparse matrix envelope reduction, and graph linear arrangement as well as less familiar settings such as archeological dating. In this paper we present a spectral algorithm for this class of problems. Unlike traditional combinatorial methods, our approach uses an eigenvector of a matrix to order the elements. Our main result is that this approach correctly solves an important ordering problem we call the seriation problem which includes the wellknown consecutive ones problem (C1P) [5] as a special case.More formally, we are given a set of n elements to sequence; that is, we wish to bijectively map the elements to the integers 1, . . . , n. We also have a symmetric, real valued correlation function (sometimes called a similarity function) that reflects the desire for elements i and j to be near each other in the sequence. We now wish to find all ways to sequence the elements so that the correlations are consistent; that is, if π is our permutation of elements andAlthough there may be an exponential number of such orderings, they can all be described in a compact data structure known as a PQ-tree [5], which we review in the next section. Not all correlation functions allow for a consistent sequencing. If a consistent ordering is possible we will say the problem is well posed.

Minimising the number of gap-zeros in binary matrices

“…A heuristic algorithm for the Gap0 problem is suggested by Greenberg and Istrail [14]. They experiment with the so-called spectral algorithm initially developed for the C1P problem in [3], evaluating its performance mainly in biological terms in the context of genome reconstruction without assessing the quality of heuristic solutions from the combinatorial optimisation point of view.…”

Graph matching and clustering using spectral partitions

“…However, recently a graph-spectral solution has been found to the problem. Atkins et al [14] have shown how to use the Fiedler eigenvector of the Laplacian matrix to sequence relational data. The method has been successfully applied to the consecutive ones problem and a number of DNA sequencing tasks.…”

A graph spectral approach to shape-from-shading

“…The aim is to recover the integration path that maximises the sum of weights across the field of surface normals. This can be viewed as the problem of ordering the set of nodes in a graph in a sequence such that strongly correlated elements are placed next to one another, and this is often referred to as graph-seriation [24]. The seriation problem can be approached in a number of ways.…”

“…However, recently a graph-spectral solution has been found to the problem. Atkins, Boman and Hendrikson [24] have shown how to use the leading eigenvector of the Laplacian matrix to sequence relational data. The method has been successfully applied to the consecutive ones problem and a number of DNA sequencing tasks.…”

“…However, in the case of a random walk the path is not guaranteed to encourage edge connectivity. The spectral seriation method on the other hand does impose edge connectivity constraints on the recovered path, Unfortunately, the analysis of the seriation problem presented in the paper by Atkins, Bowman and Hendrikson [24] is not directly applicable to our surface reconstruction problem. We provide an analysis which establishes the relationship between the seriation path and the leading eigenvector of the weight matrix.…”