Simultaneous Time-Space Upper Bounds for Red-Blue Path Problem in Planar DAGs

Abstract

In this paper, we show that given a weighted, directed planar graph G, and any > 0, there exists a polynomial time and O(n 1 2 + ) space algorithm that computes the shortest path between two fixed vertices in G.We also consider the RedBluePath problem, which states that given a graph G whose edges are colored either red or blue and two fixed vertices s and t in G, is there a path from s to t in G that alternates between red and blue edges. The RedBluePath problem in planar DAGs is NL-complete. We exhibit a polynomial time and O(n 1 2 + ) space algorithm (for any > 0) for the RedBluePath problem in planar DAG.In the last part of this paper, we consider the problem of deciding and constructing the perfect matching present in a planar bipartite graph and also a similar problem which is to find a Hall-obstacle in a planar bipartite graph. We show the time-space bound of these two problems are same as the bound of shortest path problem in a directed planar graph.

State Complexity Characterizations of Parameterized Degree-Bounded Graph Connectivity, Sub-Linear Space Computation, and the Linear Space Hypothesis

“…Determine whether or not the replacement of 2qlinN by 2N or even 2N/poly is possible. 5. At this moment, we cannot assert that the failure of LSH derives Statements (1)-(2) of Theorem 1.…”

“…Such a bound is informally called "sub-linear" in a strong sense. It has been conjectured that, for every constant ε ∈ [0, 1), no polynomial-time O(n ε )-space algorithm solves DSTCON with n vertices (see references in, e.g., [1,5]). For convenience, we denote by PsubLIN the collection of all parameterized decision problems (L, m) solvable deterministically in time polynomial in |x| using space at most m(x) ε ℓ(|x|) for certain constants ε ∈ [0, 1) and certain polylogarithmic (or polylog, for short) functions ℓ [16].…”

“…it moves only to the right), and making nondeterministic choices only at the right endmarker. 5 For a positive integer c, a c-branching 2nfa makes only at most c nondeterministic choices at every step and a family of 2nfa's is called c-branching if every 2nfa in this family is c-branching. A computation of an 2afa is normally expressed as a tree whose nodes are labeled by (surface) configurations; however, it is more convenient and more concise to view the computation as a "graph," in which the same configurations at the same depth are all treated as the same vertex.…”

“…A 2nfa with a tape head that sweeps a circular tape is called "rotating" in[10,13] 5. This requirement is known in[6,13] as "outer nondeterminism."…”

Sublinear-Space Approximation Algorithms for Max r-SAT

“…Proposition 6 (Chakraborty and Tewari [6], Theorem 1). There is an algorithm which takes as input a planar graph on n vertices and computes a BFS sequence for G in time n O (1) using O( √ n log n) bits of space.…”