Efficient Enumeration of Induced Subtrees in a K-Degenerate Graph

In this paper, we address the problem of enumerating all induced subtrees in an input k-degenerate graph, where an induced subtree is an acyclic and connected induced subgraph. A graph G = (V, E) is a k-degenerate graph if for any its induced subgraph has a vertex whose degree is less than or equal to k, and many real-world graphs have small degeneracies, or very close to small degeneracies. Although, the studies are on subgraphs enumeration, such as trees, paths, and matchings, but the problem addresses the subgraph enumeration, such as enumeration of subgraphs that are trees. Their induced subgraph versions have not been studied well. One of few example is for chordless paths and cycles. Our motivation is to reduce the time complexity close to O(1) for each solution. This type of optimal algorithms are proposed many subgraph classes such as trees, and spanning trees. Induced subtrees are fundamental object thus it should be studied deeply and there possibly exist some efficient algorithms. Our algorithm utilizes nice properties of k-degeneracy to state an effective amortized analysis. As a result, the time complexity is reduced to O(k) time per induced subtree. The problem is solved in constant time for each in planar graphs, as a corollary

Amortized Rotation Cost in AVL Trees

An AVL tree is the original type of balanced binary search tree. An insertion in an $n$-node AVL tree takes at most two rotations, but a deletion in an $n$-node AVL tree can take $\Theta(\log n)$. A natural question is whether deletions can take many rotations not only in the worst case but in the amortized case as well. A sequence of $n$ successive deletions in an $n$-node tree takes $O(n)$ rotations, but what happens when insertions are intermixed with deletions? Heaupler, Sen, and Tarjan conjectured that alternating insertions and deletions in an $n$-node AVL tree can cause each deletion to do $\Omega(\log n)$ rotations, but they provided no construction to justify their claim. We provide such a construction: we show that, for infinitely many $n$, there is a set $E$ of {\it expensive} $n$-node AVL trees with the property that, given any tree in $E$, deleting a certain leaf and then reinserting it produces a tree in $E$, with the deletion having done $\Theta(\log n)$ rotations. One can do an arbitrary number of such expensive deletion-insertion pairs. The difficulty in obtaining such a construction is that in general the tree produced by an expensive deletion-insertion pair is not the original tree. Indeed, if the trees in $E$ have even height $k$, $2^{k/2}$ deletion-insertion pairs are required to reproduce the original tree

New Variants of Pattern Matching with Constants and Variables

Given a text and a pattern over two types of symbols called constants and variables, the parameterized pattern matching problem is to find all occurrences of substrings of the text that the pattern matches by substituting a variable in the text for each variable in the pattern, where the substitution should be injective. The function matching problem is a variant of it that lifts the injection constraint. In this paper, we discuss variants of those problems, where one can substitute a constant or a variable for each variable of the pattern. We give two kinds of algorithms for both problems, a convolution-based method and an extended KMP-based method, and analyze their complexity.Comment: 15 pages, 2 figure

A simple and optimal ancestry labeling scheme for trees

We present a $\lg n + 2 \lg \lg n+3$ ancestry labeling scheme for trees. The problem was first presented by Kannan et al. [STOC 88'] along with a simple $2 \lg n$ solution. Motivated by applications to XML files, the label size was improved incrementally over the course of more than 20 years by a series of papers. The last, due to Fraigniaud and Korman [STOC 10'], presented an asymptotically optimal $\lg n + 4 \lg \lg n+O(1)$ labeling scheme using non-trivial tree-decomposition techniques. By providing a framework generalizing interval based labeling schemes, we obtain a simple, yet asymptotically optimal solution to the problem. Furthermore, our labeling scheme is attained by a small modification of the original $2 \lg n$ solution.Comment: 12 pages, 1 figure. To appear at ICALP'1

Cache-Oblivious Persistence

Partial persistence is a general transformation that takes a data structure and allows queries to be executed on any past state of the structure. The cache-oblivious model is the leading model of a modern multi-level memory hierarchy.We present the first general transformation for making cache-oblivious model data structures partially persistent

Enumerating Cyclic Orientations of a Graph

Acyclic and cyclic orientations of an undirected graph have been widely studied for their importance: an orientation is acyclic if it assigns a direction to each edge so as to obtain a directed acyclic graph (DAG) with the same vertex set; it is cyclic otherwise. As far as we know, only the enumeration of acyclic orientations has been addressed in the literature. In this paper, we pose the problem of efficiently enumerating all the \emph{cyclic} orientations of an undirected connected graph with $n$ vertices and $m$ edges, observing that it cannot be solved using algorithmic techniques previously employed for enumerating acyclic orientations.We show that the problem is of independent interest from both combinatorial and algorithmic points of view, and that each cyclic orientation can be listed with $\tilde{O}(m)$ delay time. Space usage is $O(m)$ with an additional setup cost of $O(n^2)$ time before the enumeration begins, or $O(mn)$ with a setup cost of $\tilde{O}(m)$ time

On the Number of Synchronizing Colorings of Digraphs

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have different colors. Such a coloring corresponds naturally to an automaton. The road coloring theorem states that every primitive digraph has a synchronizing coloring. In the present paper we study how many synchronizing colorings can exist for a digraph with $n$ vertices. We performed an extensive experimental investigation of digraphs with small number of vertices. This was done by using our dedicated algorithm exhaustively enumerating all small digraphs. We also present a series of digraphs whose fraction of synchronizing colorings is equal to $1-1/k^d$, for every $d \ge 1$ and the number of vertices large enough. On the basis of our results we state several conjectures and open problems. In particular, we conjecture that $1-1/k$ is the smallest possible fraction of synchronizing colorings, except for a single exceptional example on 6 vertices for $k=2$.Comment: CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1

Maximum Independent Sets in Subcubic Graphs: New Results

The maximum independent set problem is known to be NP-hard in the class of subcubic graphs, i.e. graphs of vertex degree at most 3. We present a polynomial-time solution in a subclass of subcubic graphs generalizing several previously known results

On the (non-)existence of polynomial kernels for Pl-free edge modification problems

Given a graph G = (V,E) and an integer k, an edge modification problem for a graph property P consists in deciding whether there exists a set of edges F of size at most k such that the graph H = (V,E \vartriangle F) satisfies the property P. In the P edge-completion problem, the set F of edges is constrained to be disjoint from E; in the P edge-deletion problem, F is a subset of E; no constraint is imposed on F in the P edge-edition problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size k of the edge set F, it has been proved that if P is an hereditary property characterized by a finite set of forbidden induced subgraphs, then the three P edge-modification problems are FPT. It was then natural to ask whether these problems also admit a polynomial size kernel. Using recent lower bound techniques, Kratsch and Wahlstrom answered this question negatively. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. Kratsch and Wahlstrom asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P4-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that parameterized cograph edge modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the Pl-free and Cl-free edge-deletion problems for large enough l

Complexity of Coloring Graphs without Paths and Cycles

Let $P_t$ and $C_\ell$ denote a path on $t$ vertices and a cycle on $\ell$ vertices, respectively. In this paper we study the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of $P_5$-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle have shown that $k$-colorability of $P_5$-free graphs for $k \geq 4$ does not. These authors have also shown, aided by a computer search, that 4-colorability of $(P_5,C_5)$-free graphs does have a finite forbidden induced subgraph characterization. We prove that for any $k$, the $k$-colorability of $(P_6,C_4)$-free graphs has a finite forbidden induced subgraph characterization. We provide the full lists of forbidden induced subgraphs for $k=3$ and $k=4$. As an application, we obtain certifying polynomial time algorithms for 3-coloring and 4-coloring $(P_6,C_4)$-free graphs. (Polynomial time algorithms have been previously obtained by Golovach, Paulusma, and Song, but those algorithms are not certifying); To complement these results we show that in most other cases the $k$-coloring problem for $(P_t,C_\ell)$-free graphs is NP-complete. Specifically, for $\ell=5$ we show that $k$-coloring is NP-complete for $(P_t,C_5)$-free graphs when $k \ge 4$ and $t \ge 7$; for $\ell \ge 6$ we show that $k$-coloring is NP-complete for $(P_t,C_\ell)$-free graphs when $k \ge 5$, $t \ge 6$; and additionally, for $\ell=7$, we show that $k$-coloring is also NP-complete for $(P_t,C_7)$-free graphs if $k = 4$ and $t\ge 9$. This is the first systematic study of the complexity of the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. We almost completely classify the complexity for the cases when $k \geq 4, \ell \geq 4$, and identify the last three open cases