A Review of Graph Traversal Algorithms: Techniques and Applications in Network Analysis

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).

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.

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.

Nowadays, Graph Traversal (GT) is a fundamental procedure for developing a measure in computing systems and algorithms. Graph traversal algorithms usually iterate input datasets of a graph to provide a logical convergence in every iteration. Graphic Processing Units (GPUs) have been utilized broadly as accelerators of GT due to their massive parallelism and accessibility of high-transfer speed memory. Furthermore, graph-based computation is pervasive and challenging for memory systems implementation in a specific time period. This paper provides quadratic and single GPUs, allowing developers to extend graph-based GPU computation effortlessly. The paper concentrates on the creditability and validation of sparse graphs in GPUs. Since the benchmark characteristic of Breadth-First Search (BFS) plays a critical role in graphs, this paper considers a primary characteristic benchmark for BFS computation usage. This paper explores the effectiveness and applicability of BFS for analyzing graph systems by utilizing the mentioned characteristics in BFS computation. This paper also utilizes the penalization of GPU BFS during the analysis and simulation results. In addition, using a comparative simulation result, this paper concludes that improved traversal rates speed up the computation and reduce memory usage for the single and quadratic GPU structures.

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

The subject of the research is one of the ways of updating modern training systems for solving problems of graph theory, namely, automatic generation of graphs. This approach will reduce the load on the training system database and generate tasks for the user in real-time without updating the bank of tasks. In the course of the work, the advantages and disadvantages of this approach were identified. The most suitable method for the implementation of the research was chosen to represent graphs in electronic computers. The requirements for generated graphs and possible ways of implementing these requirements are identified and substantiated. Namely: in the implemented program, simple connected undirected graphs will be generated. We considered an important detail in working with graphs — graph traversal using the “Depth (width) search” algorithm, which in this task is used to check the graph for connectivity. The result of the work is presented — a software implementation of the graph generation algorithm in the C# programming language. In it, graphs are represented by an adjacency list, generated randomly, and checked for connectivity using the DFS (Depth First Search) function. DFS is a software implementation of the Depth First Search algorithm.

While most of the research efforts on network reliability analysis have focused on networks with imperfect links but perfectly reliable nodes, binary decision diagram–based methods have recently been developed to determine reliability of networks with imperfect links and imperfect nodes. The smaller the binary decision diagram size is, the better the performance of the binary decision diagram–based algorithm will be. The binary decision diagram size heavily depends on the chosen variable ordering. As finding the optimal variable ordering is a nondeterministic polynomial time-complete problem, a heuristic is typically used. Various heuristics based on depth-first search, breadth-first search, and network-driven search have been developed for network reliability analysis. However, they often assume the networks have perfectly reliable nodes. This article advances the state-of-the-art by proposing a new ordering heuristic for the binary decision diagram–based reliability analysis of networks with imperfect links/edges and nodes. Empirical studies show that the proposed new ordering heuristic can generate significantly smaller binary decision diagram size for a wide variety of network structures than the traditional depth-first search–based, breadth-first search–based, network-driven search–based heuristics for most cases, enabling efficient binary decision diagram–based reliability analysis of large-scale networks considering both link and node failures. Robustness of the proposed heuristic is also investigated through empirical studies.

This paper presents two new methods for network analysis. Ant colony optimization is a nature inspired algorithm succesfull in graph traversal and network path finding whereas network reduction based on stability introduces two new properties of network vertices based on their long-term behavior, their role in the network and the understanding of how memory works. We illustrate the algorithms on applications in social network analysis and information retrieval using the DBLP dataset and a small network of hyperlinked documents.

Graph search, the process of visiting vertices in a graph in a specific order, has demonstrated magical powers in many important algorithms. But a systematic study was only initiated by Corneil et al. a decade ago, and only by then we started to realize how little we understand it. Even the apparently naive question vertex can be the last visited by a graph search algorithm, known as the end vertex problem, turns out to be quite elusive. We give a full picture of all maximum cardinality searches on chordal graphs, which implies a polynomial-time algorithm for the end vertex problem of maximum cardinality search. It is complemented by a proof of NP-completeness of the same problem on weakly chordal graphs. We also show linear-time algorithms for deciding end vertices of breadth-first searches on interval graphs, and end vertices of lexicographic depth-first searches on chordal graphs. Finally, we present 2^n * n^O(1)-time algorithms for deciding the end vertices of breadth-first searches, depth-first searches, and maximum cardinality searches on general graphs.

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.

There is an ever-increasing need for exploring large-scale graph data sets in computational sciences, social networks, and business analytics. However, due to irregular and memory-intensive nature, graph applications are notoriously known for their poor performance on parallel computer systems. In this paper we propose a new hybrid MPI/Pthreads breadth-first search (BFS) algorithm featuring with (i) overlapping computation and communication by separating them into multiple threads, (ii) maximizing multi-threading parallelism on multi-cores with massive threads to improve throughputs, and (iii) exploiting pipeline parallelism using lock-free queues for asynchronous communication. By comparing it with traditional MPI-only BFS algorithm, we learned several valuable lessons that would help to understand and exploit parallelism in graph traversal applications. Experiments show our algorithm is 1.9� faster than the MPI-only version, capable of processing 1.45 billion edges per second on a 32-node SMP cluster. At a large scale, our algorithm is 1.49� than the MPI-only BFS algorithm in Combinatorial BLAS Library with 6,144 cores.