Time Algorithm For Finding Research Articles

Fixed-Parameter Tractability of Maximum Colored Path and Beyond

We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is as follows. We give a randomized algorithm, that given a colored \(n\) -vertex undirected graph, vertices \(s\) and \(t\) , and an integer \(k\) , finds an \((s,t)\) -path containing at least \(k\) different colors in time \(2^{k}n^{\mathcal{O}(1)}\) . This is the first FPT algorithm for this problem, and it generalizes the algorithm of Björklund, Husfeldt, and Taslaman on finding a path through \(k\) specified vertices. It also implies the first \(2^{k}n^{\mathcal{O}(1)}\) time algorithm for finding an \((s,t)\) -path of length at least \(k\) . Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an \(n\) -vertex undirected graph \(G\) , a matroid \(M\) whose elements correspond to the vertices of \(G\) and which is represented over a finite field of order \(q\) , a positive integer weight function on the vertices of \(G\) , two sets of vertices \(S,T\subseteq V(G)\) , and integers \(p,k,w\) , and the task is to find \(p\) vertex-disjoint paths from \(S\) to \(T\) so that the union of the vertices of these paths contains an independent set of \(M\) of cardinality \(k\) and weight \(w\) , while minimizing the sum of the lengths of the paths. We give a \(2^{p+\mathcal{O}(k^{2}\log(q+k))}n^{\mathcal{O}(1)}w\) time randomized algorithm for this problem.

A Fast Algorithm for the Product Structure of Planar Graphs

Dujmović et al. (FOCS2019) recently proved that every planar graph G is a subgraph of \(H\boxtimes P\), where \(\boxtimes\) denotes the strong graph product, H is a graph of treewidth 8 and P is a path. This result has found numerous applications to linear graph layouts, graph colouring, and graph labelling. The proof given by Dujmović et al. is based on a similar decomposition of Pilipczuk and Siebertz (SODA2019) which is constructive and leads to an \(O(n^2)\) time algorithm for finding H and the mapping from V(G) onto \(V(H\boxtimes P)\). In this note, we show that this algorithm can be made to run in \(O(n\log n)\) time.

Finding patterns and periods in Cartesian tree matching

We introduce a new metric of match, called Cartesian tree matching, which means that two strings match if they have the same Cartesian trees. Based on Cartesian tree matching, we define single pattern matching, multiple pattern matching, and indexing problems. We propose a linear time algorithm for single pattern matching, and randomized linear time algorithms for multiple pattern matching and indexing. We also define three types of periods, called full period, initial period, and general period. We propose O(n) time, O(nlog⁡log⁡n) time, and O(nlog⁡n) time algorithms for finding all the full periods, initial periods, and general periods of a string of length n, respectively. Our efficient algorithms use a representation of the Cartesian tree, called parent-distance representation.

A cubic time algorithm for finding the principal solution to Sylvester matrix equations over (max, +)

We propose an algorithm for finding the so-called principal solution of the Sylvester matrix equation over max-plus algebra. The derivation of our algorithm is based on the concept of tropical tensor product introduced by Butkovič and Fiedler. Our algorithm reduces the computational cost of finding the principal solution from quartic to cubic. It also reduces the space complexity from quartic to quadratic. Since matrix–matrix multiplication is the most important ingredient of our proposed technique, we show how to use column-oriented matrix multiplications in order to speed-up MATLAB implementation of our algorithm. Finally, we illustrate our results and discuss the connection with the residuation theory.

Maximum Flow in Directed Planar Graphs with Vertex Capacities

In this paper we present an O ( n log n ) time algorithm for finding a maximum flow in a directed planar graph, where the vertices are subject to capacity constraints, in addition to the arcs. If t...

An Optimal Algorithm for Untangling Binary Trees via Rotations

There are various ways to measure the shape difference between two n-node rooted binary trees (binary trees for short). A rotation on a binary tree is a local restructuring that changes the tree into another one preserving the in-order sequence. The rotation distance between two binary trees is the minimum number of rotations needed to transform one into another. Till now, no polynomial–time algorithm exists for computing the rotation distance between any two binary trees. Recently, Lucas (Comput. J., 47, 259–269, 2004) presented an O(n2)–time algorithm for finding the rotation distance between two binary trees, where the source tree is a degenerate tree and the destination tree is an angle tree. This paper improves the time-complexity to O(n) under this constraint.

An Optimal Rotation Distance Set

A rotation in a binary tree is a local restructuring that changes the tree into another and preserves the inorder sequence. The rotation distance between two binary trees is the minimum number of rotations needed to transform one into another. However, a polynomial-time algorithm for computing rotation distances between any two binary trees has still not been found. Lucas (The Computer Journal, 47, 259–269, 2004) recently presented an O(n2)-time algorithm for finding the exact rotation distance between two binary trees that are of a restricted form (each node has at most one child in the source tree and there is at most one zig-zag pair in the destination tree), where n is the number of nodes in each binary tree. In this paper, by using the coding technique of left weight sequences, which was proposed by Pallo (The Computer Journal, 9, 171–175, 1986), we find another restricted set of binary trees in which any two of them can be transformed with the exact rotation distance. Our algorithm can be performed in linear time for finding the rotation distance between any two trees in the restricted set. Moreover, the actual sequence of transforming rotations can also be built.

FINDING ALL DOOR LOCATIONS THAT MAKE A ROOM SEARCHABLE

A room is a simple polygon with a prespecified point, called the door, on its boundary. Search starts at the door, in order to detect any intruder who was in the room at the time it started. Search may be conducted by two guards on the boundary who can detect any intruder that crosses the line connecting them, or by a single searcher with a flashlight, moving on the boundary. A room may or may not be searchable, depending on where the door is placed or no matter where the door is placed. The main contribution of this paper is an O(n log n) time algorithm for finding all intervals on the polygon boundary, if any, where the door can be placed for the resulting room to be searchable, where n is the number of sides of the given polygon.

Minimum face-spanning subgraphs of plane graphs

A plane graph is a planar graph with a fixed embedding. Let G = (V, E) be an edge weighted connected plane graph, where V and E are the set of vertices and edges, respectively. Let F be the set of faces of G. For each edge e ∊ E, let w(e) ≥ 0 be the weight of the edge e of G. A face-spanning subgraph of G is a connected subgraph H induced by a set of edges S ⊆ E such that the vertex set of H contains at least one vertex from the boundary of each face f ∊ F of G. A minimum face-spanning subgraph H of G is a face-spanning subgraph of G, where is minimum. In this paper we consider the problem of finding a minimum face-spanning subgraph of a plane graph and deal with the following problem which we call “the face-spanning subgraph problem”: “Is there any face-spanning subgraph H of G such that , for a positive real number b?”. We prove that the face-spanning subgraph problem of a plane graph is NP-complete, which implies that it is unlikely to have a polynomial time algorithm for finding a minimum face-spanning subgraph of a plane graph. In this paper, we consider a variation of the face-spanning subgraph problem called “minimum-vertex face-spanning subgraph problem” where the objective is to minimize the number of vertices instead of edge cost and we prove that the minimum-vertex face-spanning subgraph problem is also NP-complete. We also present approximation algorithms for both the problems. We calculate the approximation ratios and time complexities for both the algorithms. In this paper, we also present a linear time algorithm for finding a minimal face-spanning subgraph of a plane graph. We calculate the upper bound and lower bound on the number of vertices for a minimal face-spanning subgraph. We show that the upper bound and the lower bound are tight. Note that the graphs being dealt with in the paper are all undirected.

SEMI-BALANCED PARTITIONS OF TWO SETS OF POINTS AND EMBEDDINGS OF ROOTED FORESTS

Let m be a positive integer and let R1, R2 and B be three disjoint sets of points in the plane such that no three points of R1 ∪ R2 ∪ B lie on the same line and |B| = (m-1)|R1|+m|R2|. Put g = |R1∪R2|. Then there exists a subdivision X1∪X2∪⋯∪Xg of the plane into g disjoint convex polygons such that (i) |Xi ∩ (R1 ∪ R2)| = 1 for all 1 ≤ i ≤ g; and (ii) |Xi∩B| = m-1 if |Xi∩R1| = 1, and |Xi∩B| = m if |Xi∩R2| = 1. This partition is called a semi-balanced partition, and our proof gives an O(n4) time algorithm for finding the above semi-balanced partition, where n = |R1| + |R2| + |B|. We next apply the above result to the following theorem: Let T1,…,Tg be g disjoint rooted trees such that |Ti| ∈ {m,m+1} and vi is the root of Ti for all 1 ≤ i ≤ g. Let P be a set of |T1|+⋯+|Tg| points in the plane in general position that contains g specified points p1,…,pg. Then the rooted forest T1 ∪ ⋯ ∪ Tg can be straight-line embedded onto P so that each vi corresponds to pi for every 1 ≤ i ≤ g.

On Even Triangulations of 2-Connected Embedded Graphs

Recently, Hoffmann and Kriegel proved an important combinatorial theorem [SIAM J. Discrete Math., 9 (1996), pp. 210--224]: Every 2-connected bipartite plane multigraph G without 2-cycle faces has a triangulation in which all vertices have even degree (this is called an even triangulation). Combined with the classical Whitney's theorem, this result implies that every such graph has a 3-colorable plane triangulation. Using this theorem, Hoffmann and Kriegel significantly improved the upper bounds of several art gallery and prison guard problems. A complicated O(n2) time algorithm was obtained in [SIAM J. Discrete Math., 9 (1996), pp. 210--224] for constructing an even triangulation of G. Hoffmann and Kriegel conjectured that there is an O(n3/2) time algorithm for solving this problem. In this paper, we develop a simple proof of the above theorem. Our proof reveals and relies on a natural correspondence between even triangulations of G and certain orientations of G. Based on this new proof, we obtain a very simple O(n) time algorithm for finding an even triangulation of G. We also extend our proof to show the existence of even triangulations for similar graphs on high genus surface.

1-restricted Simple 2-matchings

Abstract A simple 2-matching in a graph is a subgraph all of whose nodes have degree ≤ 2. A simple 2-matching is called k-restricted if every connected component has > k edges. We consider the problem of finding a k-restricted simple 2-matching with a maximum number of edges, which is a relaxation of the problem of finding a Hamilton cycle in a graph. Our main result is a new polynomial time algorithm for finding a 1-restricted simple 2-matching with a maximum number of edges. We also present a Tutte-type existence theorem and a Tutte-Berge-type formula for this problem.

Optimal algorithms for recombination distance problem

Recombination of sequences is an important combinatorial optimization problem due to its applications in computational biology. It is related to the alignment with recombination which is an edit distance problem involved recombination, and reconstructing a history of recombination from a set of sequences, etc. Recently, Wu and Gu discussed a distance problem involved recombination consisting of single crossover. [S. Wu and X. Gu (2001). A Greedy algorithm for optimal recombination. Proceeding of COCOON2001, Lecture Notes in Computer Science 2108, pp. 86–90.] 𝒜 and 𝒮 denote two collections of sequences where |𝒮| = 2. The goal is to generate 𝒜 from 𝒮 by a series of recombinations in minimum number of steps. They defined a special class 𝒜 of sequences called ‘tree’, and presented a greedy algorithm which was claimed optimal for finding the recombination distance from 𝒮 to 𝒜. In this paper, we revisit this result. We show that the greedy algorithm is not always optimal, and propose a revised algorithm which can solve the problem optimally. We further propose a new class 𝒜 of sequences called ‘chain’ and provide a polynomial time algorithm for finding the optimal recombination evolutionary history from 𝒮 to 𝒜. †chenting@math.zju.edu.cn

FINDING AN OPTIMAL BRIDGE BETWEEN TWO POLYGONS

Let π(a,b) denote the shortest path between two points a, b inside a simple polygon P, which totally lies in P. The geodesic distance between a and b in P is defined as the length of π(a,b), denoted by gd(a,b), in contrast with the Euclidean distance between a and b in the plane, denoted by d(a,b). Given two disjoint polygons P and Q in the plane, the bridge problem asks for a line segment (optimal bridge) that connects a point p on the boundary of P and a point q on the boundary of Q such that the sum of three distances gd(p′,p), d(p,q) and gd(q,q′), with any p′ ∈ P and any q′ ∈ Q, is minimized. We present an O(n log 3 n) time algorithm for finding an optimal bridge between two simple polygons. This significantly improves upon the previous O(n2) time bound. Our result is obtained by making substantial use of a hierarchical structure that consists of segment trees, range trees and persistent search trees, and a structure that supports dynamic ray shooting and shortest path queries as well.

Optimal algorithm for tree scheduling with unit time communication delays

A program is usually designed to act as a set of interacting tasks that can be used to construct a graph. These tasks can be assigned to processing elements in different ways. The scheduling problem is applied to defining an assignment of tasks in order to process elements that will optimise certain performance indices. One of the most important reasons for scheduling indices is to try to optimise the finishing time. The finishing time of a program is the result of two components: the computing time and communication delays. An optimal schedule is defined as the one that assigns tasks to processing elements in such a way as to finish at the earliest possible time. Completion of the optimum solution for arbitrary directed acyclic task graphs (DAGs) has been shown to be NP-complete. However, for some special classes of DAGs (such as fork, join, coarse-grain trees and some fine grain trees) many algorithms have been introduced to find optimal schedules. Note that most of these algorithms do not take into consideration the delays due to message passing among the processors. In the paper, the authors study the increase in time complexity due to communication delays. The particular problem that is addressed is the scheduling problem of the in-forest/out-forest. This paper introduces the first known linear time algorithm for finding the optimal schedule, for the special case of tree-like graphs, with unit time computation and unit time communication delays.

A Simplified Õ(nm) Time Edge-Splitting Algorithm in Undirected Graphs

Let G=(V,E) be a multigraph which has a designated vertex s ∈ V with an even degree. For two edges e 1 = (s,u 1 ) and e 2 = (s,u 2 ) , we say that a multigraph G' is obtained from G by splitting e 1 and e 2 at s if two edges e 1 and e 2 are replaced with a single edge (u 1 ,u 2 ) . It is known that all edges incident to s can be split without losing the edge-connectivity of G in V-s . This complete splitting plays an important role in solving many graph connectivity problems. The currently fastest algorithm for a complete splitting [14] runs in O(n(m+n log n) log n) time, where n = |V| and m is the number of pairs of vertices between which G has an edge. Their algorithm is first designed for Eulerian multigraphs, and then extended for general multigraphs. Although the part for Eulerian multigraphs is simple, the rest for general multigraphs is considerably complicated. This paper proposes a much simpler O(n(m+n log n) log n) time algorithm for finding a complete splitting. A new edge-splitting theorem derived from our algorithm is also presented.

Partial and perfect path covers of cographs

A set P of disjoint paths in a graph G is called a (complete) path cover of G if every vertex of G belongs to one of the paths in P . A path cover of any subgraph of G is called a partial path cover of G. For fixed k>0, a k-blanket of graph G is a partial path cover P of G, consisting of exactly k paths, that maximizes the size of the subgraph covered by P . A k-core of graph G is a partial path cover P of G, consisting of exactly k paths, that minimizes the sum, over all vertices v of G, of the distance of v to its closest path in P . The problems of finding a k-blanket or a k-core (for fixed k) of an arbitrary graph G as well as the dual minimum-path-cover problem (find a path cover of minimum size) are all NP-hard. A linear-time algorithm is known (C.J. Chang and D. Kuo, SIAM J. Discrete Math. 9 (1996) 309–316) for the minimum-path-cover problem on cographs (graphs that can be constructed from a collection of isolated vertices by union and complement operations). However, prior to this paper, polynomial-time algorithms for the k -core problem were known only for trees — and even then for k = 1, 2 only (Becker and Perl, Discrete Appl. Math. 11 (1985) 103–113; Morgan and Slater, SIAM J. Appl. Math. 37 (1979) 539–560). In this paper, we introduce a variant of a minimum path cover, called a perfect path cover. We show that every cograph has a perfect path cover, and we exploit this to obtain an O( m + n log n)-time algorithm for finding, for any arbitrary k, a k-blanket or a k-core of a arbitrary cograph on n vertices and m edges.

A New Subgraph of Minimum Weight Triangulations.

In this paper, two sufficient conditions for identifying a subgraph ofminimum weight triangulation of a planar point set are presented. Theseconditions are based on local geometric properties of an edge to beidentified. Unlike the previous known sufficient conditions for identifyingsubgraphs, such as Keil‘sβ-skeleton and Yang and Xu‘s double circles, The localgeometric requirement in our conditions is not necessary symmetric withrespect to the edge to be identified. The identified subgraph is differentfrom all the known subgraphs including the newly discovered subgraph:so-called the intersection of local-optimal triangulations by Dickerson etal. An O(n3) time algorithm for finding this subgraph from a set ofn points is presented.

Covering a string

We consider the problem of finding the repetitive structures of a given stringx. The periodu of the stringx grasps the repetitiveness ofx, sincex is a prefix of a string constructed by concatenations ofu. We generalize the concept of repetitiveness as follows: A stringw covers a stringx if there is a superstring ofx which is constructed by concatenations and superpositions ofw. A substringw ofx is called aseed ofx ifw coversx. We present anO(n logn)-time algorithm for finding all the seeds of a given string of lengthn.

Finding a maximum matching in a permutation graph

We present anO(n log logn) time algorithm for finding a maximum matching in a permutation graph withn vertices, assuming that the input graph is represented by a permutation.