Graph Traversal Techniques and the Maximum Flow Problem in Distributed Computation

Research on space efficient graph algorithms, particularly for st-connectivity, has a long history including the celebrated polynomial time, O(lg n) bits1 algorithm in undirected graphs by Reingold J. JACM. 55(4) (2008), and polynomial time, \(n/2^{\Theta (\sqrt {\lg n})}\) bits algorithm in directed graphs by Barnes et al. SICOMP. 27(5), 1273–1282 (1998). Recent works by Asano et al. ISAAC (2014) and Elmasry et al. STACS (2015), reconsidered classical fundamental graph algorithms focusing on improving the space complexity. Elmasry et al. gave, among others, an implementation of breadth first search (BFS) in a graph G with n vertices and m edges, taking the optimal O(m + n) time using O(n) bits improving the naive O(n lg n) bits implementation. Similarly, Asano et al. provided space efficient implementations for performing depth first search (DFS) in a graph G. We continue this line of work focusing on improving the space requirement for implementing a few classical graph algorithms. Our first result is a simple data structure that can maintain any subset S of a universe of u elements using just u + o(u) bits and supports in constant time, apart from the standard insert, delete and membership queries, the operation findany that finds and returns any element of the set (or outputs that the set is empty). It can also enumerate all elements present currently in the set in no particular order in O(k + 1) time where k is the number of elements currently belonging to the set. While this data structure supports a weaker set of operations than that of Elmasry et al. STACS (2015), it is simple, more space efficient and is sufficient to support a BFS implementation optimally in O(m + n) time using at most 2n + o(n) bits. Later, we further improve the space requirement of BFS to at most n lg 3 + o(n) bits albeit with a slight increase in running time to O(m lg nf(n)) time where f(n) is any extremely slow growing function of n, and the o term in the space is a function of f(n). We also discuss a similar time-space tradeoff result for finding a minimum weight spanning tree of a weighted (bounded by polynomial in n) undirected graph using n + O(n/f(n)) bits and O(m lg nf(n)) time, for any function f(n) such that 1 ≤ f(n) ≤ n. For DFS in a graph G, we provide an implementation taking O(m + n) time and O(n lg m/n) bits. This partially answers at least for sparse graphs, a question asked by Asano et al. ISAAC (2014) whether DFS can be performed in O (m + n) time and using O(n) bits, and also simultaneously improves the DFS result of Elmasry et al. STACS (2015). Using our DFS algorithm and other careful implementations, we show how one can also test for biconnectivity, 2-edge connectivity, and find cut vertices and bridges of a given undirected graph within the same time and space bounds; earlier classical linear time algorithms for these problems used Ω(n lg n) bits of space.

Graph searching is one of the simplest and most widely used tools in graph algorithms. Every graph search method is defined using some partic-ular selection rule, and the analysis of the corre-sponding vertex orderings can aid greatly in de-vising algorithms, writing proofs of correctness, or recognition of various graph families. We study graphs where the sets of vertex order-ings produced by two di˙erent search methods coincide. We characterise such graph families for ten pairs from the best-known set of graph searches: Breadth First Search (BFS), Depth First Search (DFS), Lexicographic Breadth First Search (LexBFS) and Lexicographic Depth First Search (LexDFS), and Maximal Neighborhood Search (MNS).

Graph Algorithms

Even though a large number of I/O-efficient graph algorithms have been developed, a number of fundamental problems still remain open. For example, no space- and I/O-efficient algorithms are known for depth-first search or breadth-first search in sparse graphs. In this paper we present two new results on I/O-efficient depth-first search in an important class of sparse graphs, namely undirected embedded planar graphs. We develop a new efficient depth-first search algorithm and show how planar depth-first search in general can be reduced to planar breadth-first search. As part of the first result we develop the first I/O-efficient algorithm for finding a simple cycle separator of a biconnected planar graph. Together with other recent reducibility results, the second result provides further evidence that external memory breadth-first search is among the hardest problems on planar graphs.

Even though a large number of I/O-efficient graph algorithms have been developed, a number of fundamental problems still remain open. For example, no space- and I/O-efficient algorithms are known for depth-first search or breadth-first search in sparse graphs. In this paper we present two new results on I/O-efficient depth-first search in an important class of sparse graphs, namely undirected embedded planar graphs. We develop a new efficient depth-first search algorithm and show how planar depth-first search in general can be reduced to planar breadth-first search. As part of the first result we develop the first I/O-efficient algorithm for finding a simple cycle separator of a biconnected planar graph. Together with other recent reducibility results, the second result provides further evidence that external memory breadth-firstsearch is among the hardest problems on planar graphs.KeywordsPlanar GraphExternal MemorySimple CycleOuterplanar GraphBoundary CycleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Breadth-first search (BFS) is a cornerstone of algorithmic algorithms for complex graphs. In the worst case, BFS's linear complexity is determined by the number of edges and vertices, yet a recently discovered bottom-up method can reduce this complexity to the number of vertices in the best-case scenario, which is usually much less than the number of edges. The conventional top-down approach, however, always takes the same amount of time as the lowest case. To determine the direction, directional algorithms combine top-down and bottom-up methods, adapting between them as the boundary expands. Although bottom-up methods are not always the best option, their use in combination with top-down methods produces efficient algorithms. The main focus of this research is to improve search efficiency and reduce complexity by implementing a version of top-down algorithm optimisation. A scalable distributed-memory parallelisation technique is presented in this study, which has achieved speedups that are ten times faster than the previous purely toplevel approach. In previous studies, it has been demonstrated that this technique is superior to a 1D decomposition method.

Logic synthesis with reversible circuits has received considerable interest in the light of advances recently made in quantum computation. In this paper, we propose an improved technique for synthesizing reversible circuits based on a combined depth-first search (DFS) and breadth-first search (BFS) algorithm. A method based on DFS alone may often take a long time to converge, whereas, a BFS based method requires a large amount of memory for designing a circuit of moderate complexity. To strike a balance between these two approaches, we propose a hybrid DFS-BFS based synthesis algorithm that reduces the computation time compared to the DFS method and requires less space compared to the BFS method, while optimizing the cost of the circuit. Synthesis results on several reversible benchmark circuits have been reported.

Symbolic Execution is a widely used technique for program testing and analysis. When a program executes a trace symbolically, it simulates all possible paths. This results in an exponential growth of the number of states within the problem, which is commonly referred to as path explosion. We therefore propose novel strategies that only require limited resources to give priority to more valuable paths. In this paper, we utilize a method based on the Monte Carlo tree search (MCTS) strategy to resolve the problem. We then compare the proposed MCTS-based strategy with other methods such as depth-first search (DFS) and breadth-first search (BFS). We also perform different scales of experiments based on time and space resource constraints. For smaller test cases, we found that MCTS performs on average 1.4 times better than BFS and DFS in terms of the block discovery rate. In addition, for larger test cases, MCTS performs on average 2.8 times better than DFS and BFS in terms of the block discovery rate.

Efficient access to timely information is critical in today's digital era. Web crawlers, automated programs that navigate the Internet, play an important role in collecting data from websites such as Kumparan, a leading news site in Indonesia. This research shows the effectiveness of the Breadth-First Search (BFS) and Depth-First Search (DFS) algorithms in indexing Kumparan content. The results of the research show that BFS consistently indexes more files comprehensively but with longer execution times compared to DFS, which provides faster initial results but with fewer files. For example, at depth 4 BFS indexed 949 files in 886.94 seconds, while DFS indexed 470 files in 233.02 seconds. These findings highlight the balance between precision and speed when selecting a crawling algorithm tailored to the needs of a particular website. This research provides insights into optimizing web crawler technology for complex websites such as Coil and suggests avenues for further research to improve permission efficiency and adaptability across a variety of crawling scenarios.

Mining maximal hyperclique pattern: A hybrid search strategy

Graph mining is an emerging application of great importance to big data analytic. Graph mining algorithms are bottle-necked by both computation complexity and memory access, hence necessitating specialized hardware accelerators to improve the processing efficiency. Current accelerators have extensively exploited task-level and fine-grained parallelism in these algorithms. However, their task scheduling still has room for optimization. They use either breadth-first search, depth-first search or a combination of both, leading to either poor intermediate data locality, low parallelism or inter-depth barriers.

NeuroGraph is a simulation environment for design, construction and execution of neural networks and genetic algorithms in a distributed computing environment. The simulator parts either run on single computers or as distributed applications on UNIX/X-based networks consisting of personal computers, workstations or multi-processors. The parallelization component offers the possibility to divide computational tasks into concurrently executable modules, according to restrictions due to the neural net topology and computer net capabilities, i.e. NeuroGraph tries to select the best configuration out of the available distributed hardware environment to fit performance requirements.

With Moore's law in effect, as the complexity of digital electronic circuits increases, the amount of time spent by the electronic design automation (EDA) tools to design such circuits also increases. It brings us to the point, where we need to improve the performance of EDA algorithms to fulfil the present and the future requirements of the EDA industry. Out of many algorithms used by these tools, Breadth First Traversal (BFT) is one of the most commonly used algorithms to traverse the gates of electronic circuits. In this paper, we present a new simple, fast and parallelizable BFT algorithm for sparse graphs, named H-BFT. We show that the CPU implementation of H-BFT is about 75× faster than the CPU implementation of the state of the art, the Sparse Matrix-Vector Product (SMVP) based BFT. Further, with the new features introduced by NVIDIA in their GPUs, we have accelerated both the state of the art SMVP based BFT implementation and our new H-BFT implementation. The best speedups we achieved via these accelerations are 180× and 25× for the SMVP-BFT and H-BFT respectively.

We present linear-I/O algorithms for fundamental graph problems on embedded outerplanar graphs. We show that breadth-first search, depth-first search, single-source shortest paths, triangulation, and computing an � separator of size O(1/� )t akeO(scan(N)) I/Os on embedded outerplanar graphs. We also show that it takes O(sort(N)) I/Os to test whether a given graph is outerplanar and to compute an outerplanar embedding of an outerplanar graph, thereby providing O(sort(N))-I/O algorithms for the above problems if no embedding of the graph is given. As all these problems have linear-time algorithms in internal memory, a simple simulation technique can be used to improve the I/O-complexity of our algorithms from O(sort(N)) to O(perm(N)). We prove matching lower bounds for embedding, breadth-first search, depth-first search, and singlesource shortest paths if no embedding is given. Our algorithms for the above problems use a simple linear-I/O time-forward processing algorithm for rooted trees whose vertices are stored in preorder.

Legacy applications based on the Distributed Computing Environment (DCE) are subject to several significant limitations. As the development and support of DCE wanes, object‐oriented development becomes more desirable, and transmission over HTTP is established as the preferred protocol over the Internet, DCE application managers and developers are pressed to find extensions and alternatives to DCE. This paper briefly discusses several alternative targets for migration of DCE systems, then proceeds to detail, compare, and contrast two preferred candidates: CORBA and SOAP. Although we have found that developing a general migration solution for legacy DCE applications to CORBA or SOAP to be a non‐trivial, long‐term project, developing specific solutions that are based on a general architecture is feasible. Given a short list of reasonable premises, many DCE applications may be ported to technologies such as CORBA and SOAP. Copyright © 2002 John Wiley & Sons, Ltd.