Space Efficient Algorithms for Breadth-Depth Search

Continuing the recent trend, in this article we design several space-efficient algorithms for two well-known graph search methods. Both these search methods share the same name {\it breadth-depth search} (henceforth {\sf BDS}), although they work entirely in different fashion. The classical implementation for these graph search methods takes $O(m+n)$ time and $O(n \lg n)$ bits of space in the standard word RAM model (with word size being $\Theta(\lg n)$ bits), where $m$ and $n$ denotes the number of edges and vertices of the input graph respectively. Our goal here is to beat the space bound of the classical implementations, and design $o(n \lg n)$ space algorithms for these search methods by paying little to no penalty in the running time. Note that our space bounds (i.e., with $o(n \lg n)$ bits of space) do not even allow us to explicitly store the required information to implement the classical algorithms, yet our algorithms visits and reports all the vertices of the input graph in correct order.Comment: 12 pages, This work will appear in FCT 201

FPT-space Graph Kernelizations

Let $n$ be the size of a parametrized problem and $k$ the parameter. We present a full kernel for Path Contraction and Cluster Editing/Deletion as well as a kernel for Feedback Vertex Set whose sizes are all polynomial in $k$, that are computable in polynomial time, and use $O(\rm{poly}(k) \log n)$ bits. By first executing the new kernelizations and subsequently the best known polynomial-time kernelizations for the problem under consideration, we obtain the best known kernels in polynomial time with $O(\rm{poly}(k) \log n)$ bits

Evaluating Regular Path Queries in GQL and SQL/PGQ: How Far Can The Classical Algorithms Take Us?

We tackle the problem of answering regular path queries over graph databases, while simultaneously returning the paths which witness our answers. We study this problem under the arbitrary, all-shortest, trail, and simple-path semantics, which are the common path matching semantics considered in the research literature, and are also prescribed by the upcoming ISO Graph Query Language (GQL) and SQL/PGQ standards. In the paper we present how the classical product construction from the theoretical literature on graph querying can be modified in order to allow returning paths. We then discuss how this approach can be implemented, both when the data resides in a classical B+ tree structure, and when it is assumed to be available in main memory via a compressed sparse row representation. Finally, we perform a detailed experimental study of different trade-offs these algorithms have over real world queries, and compare them with existing approaches

공간 효율적인 그래프 알고리즘의 성능 분석

학위논문 (석사)-- 서울대학교 대학원 : 공과대학 컴퓨터공학부, 2019. 2. Satti, Srinivasa Rao.Various graphs from social networks or big data may contain gigantic data. Searching such graph requires memory scaling with graph. Asano et al. ISAAC (2014) initiated the study of space efficient graph algorithms, and proposed algorithms for DFS and some applications using sub-linear space which take slightly more than linear time. Banerjee et al. ToCS 62(8), 1736-1762 (2018) proposed space efficient graph algorithms based on read-only memory(ROM) model. Given a graph G with n vertices and m edges, their BFS algorithm spends O(m + n) time using 2n + o(n) bits. The space usage is further improved to nlg3 + o(n) bits with O(mlgn f(n)) time, where f(n) is extremely slow growing function of n. For DFS, their algorithm takes O(m + n) time using O(mlg(m/n)). Chakraborty et al. ESA (2018) introduced in-place model. The notion of in-place model is to relax the read-only restriction of ROM model to improve the space usage of ROM model. Algorithms based on in-place model improve space usage exponentially, to O(lgn) bits, at the expense of slower runtime. In this thesis, we focus on exploring proposed space efficient graph algorithms of ROM model and in-place model in detail and evaluate performance of those algorithms. We implemented almost all the best-known space-efficient algorithms for BFS and DFS, and evaluated their performance. Along the way, we also implemented several space-efficient data structures for representing bit vectors, strings, dictionaries etc.소셜 네트워크나 빅 데이터로부터 생성된 다양한 그래프들은 방대한 양의 데이터를 포함하고 있다. 이러한 그래프를 탐색하기 위해서는 그래프의 크기에 비례하여 필요한 메모리의 용량이 늘어난다. Asano 등(ISAAC (2014))은 공간 효율적 그래프 알고리즘 연구를 개시했다. 이 연구를 통해 선형적 시간보다 약간 더 걸리는 대신 저선형적 공간을 사용하는 DFS 알고리즘과 활용 방안들이 제안됐다. Banerjee 등(ToCS 62(8), 1736-1762 (2018))은 ROM 모델을 기반으로 하는 공간 효율적인 그래프 알고리즘들을 제안했다. 그래프 G의 n개의 정점과 m개의 간선이 주어졌을 때, O(m + n)의 시간과 2n + o(n) 의 비트를 사용하는 BFS가 제안됐고, f(n)을 n에 비례해서 매우 느리게 커지는 함수라고 했을 때, O(mlgnf(n))의 시간과 nlg3 + o(n)의 비트를 사용하는 알고리즘이 제안됐다. DFS의 경우, O(m + n)의 시간과 O(mlg m n )의 비트를 사용하는 알고리즘이 제안됐다. Chakraborty 등(ESA (2018))은 ROM 모델이 가지고 있는 한계점을 넘기 위해 ROM 모델의 제한점을 완화시키는 in-place 모델을 소개했다. In-place 모델을 기반으로 한 알고리즘들은 n + O(lgn)의 비트를 사용하여 BFS와 DFS를 수행할 수 있고, 추가적으로 더 긴 시간을 소요하여 O(lgn) 비트의 공간만으로 알고리즘을 수행할 수 있다. 이 논문에서는 ROM 모델과 in-place 모델에서 제안된 다양한 알고리즘들을 연구 및 구현하고 실험을 통하여 이들 알고리즘의 수행 결과를 평가한다.Abstract i Contents iii List of Figures v List of Tables vi Chapter 1 Introduction 1 1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Organization of the Paper . . . . . . . . . . . . . . . . . . . . . . 2 Chapter 2 Preliminaries 4 2.1 ROM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 In-place Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Succinct Data Structure . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Changing Base Without Losing Space . . . . . . . . . . . . . . . 6 2.5 Dictionaries With Findany Operation . . . . . . . . . . . . . . . 7 Chapter 3 Breadth First Search 9 3.1 ROM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Rotate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Implicit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 iii Chapter 4 Depth First Search 14 4.1 ROM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Rotate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 Implicit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Chapter 5 Experimental Results 22 5.1 BFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 DFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter 6 Conclusion 40 요약 46 Acknowledgements 47Maste

Shortest Distances as Enumeration Problem

We investigate the single source shortest distance (SSSD) and all pairs shortest distance (APSD) problems as enumeration problems (on unweighted and integer weighted graphs), meaning that the elements $(u, v, d(u, v))$ -- where $u$ and $v$ are vertices with shortest distance $d(u, v)$ -- are produced and listed one by one without repetition. The performance is measured in the RAM model of computation with respect to preprocessing time and delay, i.e., the maximum time that elapses between two consecutive outputs. This point of view reveals that specific types of output (e.g., excluding the non-reachable pairs $(u, v, \infty)$, or excluding the self-distances $(u, u, 0)$) and the order of enumeration (e.g., sorted by distance, sorted row-wise with respect to the distance matrix) have a huge impact on the complexity of APSD while they appear to have no effect on SSSD. In particular, we show for APSD that enumeration without output restrictions is possible with delay in the order of the average degree. Excluding non-reachable pairs, or requesting the output to be sorted by distance, increases this delay to the order of the maximum degree. Further, for weighted graphs, a delay in the order of the average degree is also not possible without preprocessing or considering self-distances as output. In contrast, for SSSD we find that a delay in the order of the maximum degree without preprocessing is attainable and unavoidable for any of these requirements.Comment: Updated version adds the study of space complexit