Parallel Complexity of Depth-First-Search and Maximal path in restricted graph classes

TL;DR

This work advances the deterministic parallel complexity of DFS and maximal-path problems by establishing tight bounds for restricted graph families. It extends prior planar results to graphs of bounded genus and single-crossing-minor-free graphs, placing DFS in $\AC^1(\text{UL}\cap\text{co-UL})$ for bounded genus, and in $\NC$ for single-crossing-minor-free undirected graphs, while achieving $\mathsf{L}$ for bounded-treewidth graphs. For maximal paths, it obtains a $\mathsf{L}$-time algorithm in planar graphs by leveraging triconnected-component decompositions and outerplanar layering. Collectively, the results use a rich mix of path/cycle separators, gadget constructions, and Courcelle-type MSO techniques to push deterministic parallel bounds closer to the limits for increasingly general graph classes.

Abstract

Constructing a Depth First Search (DFS) tree is a fundamental graph problem, whose parallel complexity is still not settled. Reif showed parallel intractability of lex-first DFS. In contrast, randomized parallel algorithms (and more recently, deterministic quasipolynomial parallel algorithms) are known for constructing a DFS tree in general (di)graphs. However a deterministic parallel algorithm for DFS in general graphs remains an elusive goal. Working towards this, a series of works gave deterministic NC algorithms for DFS in planar graphs and digraphs. We further extend these results to more general graph classes, by providing NC algorithms for (di)graphs of bounded genus, and for undirected H-minor-free graphs where H is a fixed graph with at most one crossing. For the case of (di)graphs of bounded tree-width, we further improve the complexity to a Logspace bound. Constructing a maximal path is a simpler problem (that reduces to DFS) for which no deterministic parallel bounds are known for general graphs. For planar graphs a bound of O(log n) parallel time on a CREW PRAM (thus in NC2) is known. We improve this bound to Logspace.

Figures (7)

• Figure 1: An example of a graph $G$. We ignore directions here.
• Figure 2: A clique sum decomposition of $G$. Red nodes are the clique nodes and blue node the piece nodes. Dashed edges denote virtual edges.
• Figure 3: $R$ is the central node with children subgraphs $G_1,G_2,\ldots G_l$ attached to it via cliques $c_1,c_2,\ldots c_l$ respectively. $G_1$ consists of subgraphs $G_{11},G_{12}\ldots G_{1k}$ glued at $c_1$. $G_2$ is same as $G_{21}$. Dashed edges denote virtual edges in $R$. In the case when $R$ is a planar piece, it is useful to think of it as a sphere, with separating triplets forming empty triangular faces in $R$, on which $G_1,G_2,\ldots G_l$ are attached. The cycle in green is an example of interior-exterior-cycle separator $\widetilde{C}$ (though gadget replacements are not shown in the figure). Of the attached subgraphs shown, $G_2,G_l$ are interior components with respect to $\tilde{C}$ whereas $G_1$ is an exterior component with respect to $\tilde{C}$.
• Figure 4: Figure (a) shows the gadget for graph $G_{2}$ for Case 2, (b) for Case 3 and (c) for Case 4. The faces formed by cycles of the gadget corresponding to the subgraphs $H_0,H_1,H_2,H_3$ of $G_2$ are assigned the weight functions as shown. The total weight in each gadgets sums up to $w(G_{2})$. The subgraph $H_0$ of $G_2$ has no cut vertex in each case. In the gadget for Case 4, it is possible that the block $H_0$ is empty, i.e. there exists a vertex $(x=y=z)$ in $G_2$ which on removal disconnects all of $v_1,v_2,v_3$ from each other. We don't draw that gadget separately since its structure is clear.
• Figure 5: Both cases for reduction of maximal path problem to the triconnected components of $G$. Dashed edges denote virtual edges in $R$. The edge $(r_0,r_1)$ may be real or virtual. In these figures it is virtual. Figure (a) shows the case when there is a leaf node $L$ that is a $3$-connected graph. From $r$ we find a path in $G'$ to reach $r_0$ as shown in figure (a). This reduces the problem to finding a maximal path in $L$ that does not end at $r_1$ and does not use the virtual edge $(r_0,r_1)$. Figure (b) illustrates the case when all the leaf nodes (and hence $L$) are cycles. We find a path from $r$ touching the boundary of $G'$ (which must be a simple cycle) for the first time at $v$. From $v$ we encircle the boundary of $G'$ first reaching $r_0$ and finally $r_1$. From $r_1$ we encircle the cycle $L$ till $x$. Since both the neighbours of $x$ are visited, this path is maximal.

Theorems & Definitions (24)

• Theorem 1
• Definition 2
• Theorem 3
• Theorem 4
• Theorem 5
• Corollary 6
• Corollary 7
• Proposition 8
• Definition 9
• Theorem 10