Bipartite Perfect Matching in Pseudo-Deterministic NC

We present a pseudo-deterministic NC algorithm for finding perfect matchings in bipartite graphs. Specifically, our algorithm is a randomized parallel algorithm which uses poly(n) processors, poly(log n) depth, poly(log n) random bits, and outputs for each bipartite input graph a unique perfect matching with high probability. That is, on the same graph it returns the same matching for almost all choices of randomness. As an immediate consequence we also find a pseudo-deterministic NC algorithm for constructing a depth first search (DFS) tree. We introduce a method for computing the union of all min-weight perfect matchings of a weighted graph in RNC and a novel set of weight assignments which in combination enable isolating a unique matching in a graph. We then show a way to use pseudo-deterministic algorithms to reduce the number of random bits used by general randomized algorithms. The main idea is that random bits can be reused by successive invocations of pseudo-deterministic randomized algorithms. We use the technique to show an RNC algorithm for constructing a depth first search (DFS) tree using only O(log^2 n) bits whereas the previous best randomized algorithm used O(log^7 n), and a new sequential randomized algorithm for the set-maxima problem which uses fewer random bits than the previous state of the art. Furthermore, we prove that resolving the decision question NC = RNC, would imply an NC algorithm for finding a bipartite perfect matching and finding a DFS tree in NC. This is not implied by previous randomized NC search algorithms for finding bipartite perfect matching, but is implied by the existence of a pseudo-deterministic NC search algorithm

Biconnectivity, Chain Decomposition and st-Numbering Using O(n) Bits

Recent work by Elmasry et al. (STACS 2015) and Asano et al. (ISAAC 2014) reconsidered classical fundamental graph algorithms focusing on improving the space complexity. Elmasry et al. gave, among others, an implementation of depth first search (DFS) of a graph on n vertices and m edges, taking O(m lg lg n) time using O(n) bits of space improving on the time bound of O(m lg n) due to Asano et al. Subsequently Banerjee et al. (COCOON 2016) gave an O(m + n) time implementation using O(m+n) bits, for DFS and its classical applications (including testing for biconnectivity, and finding cut vertices and cut edges). Recently, Kammer et al. (MFCS 2016) gave an algorithm for testing biconnectivity using O(n + min{m, n lg lg n}) bits in linear time. In this paper, we consider O(n) bits implementations of the classical applications of DFS. These include the problem of finding cut vertices, and biconnected components, chain decomposition and st-numbering. Classical algorithms for them typically use DFS and some Omega(lg n) bits of information at each node. Our O(n)-bit implementations for these problems take O(m lg^c n lg lg n) time for some small constant c (c leq 3). Central to our implementation is a succinct representation of the DFS tree and a space efficient partitioning of the DFS tree into connected subtrees, which maybe of independent interest for space efficient graph algorithms

Fast and Compact Distributed Verification and Self-Stabilization of a DFS Tree

We present algorithms for distributed verification and silent-stabilization of a DFS(Depth First Search) spanning tree of a connected network. Computing and maintaining such a DFS tree is an important task, e.g., for constructing efficient routing schemes. Our algorithm improves upon previous work in various ways. Comparable previous work has space and time complexities of $O(n\log \Delta)$ bits per node and $O(nD)$ respectively, where $\Delta$ is the highest degree of a node, $n$ is the number of nodes and $D$ is the diameter of the network. In contrast, our algorithm has a space complexity of $O(\log n)$ bits per node, which is optimal for silent-stabilizing spanning trees and runs in $O(n)$ time. In addition, our solution is modular since it utilizes the distributed verification algorithm as an independent subtask of the overall solution. It is possible to use the verification algorithm as a stand alone task or as a subtask in another algorithm. To demonstrate the simplicity of constructing efficient DFS algorithms using the modular approach, We also present a (non-sielnt) self-stabilizing DFS token circulation algorithm for general networks based on our silent-stabilizing DFS tree. The complexities of this token circulation algorithm are comparable to the known ones

Indexing Graph Search Trees and Applications

We consider the problem of compactly representing the Depth First Search (DFS) tree of a given undirected or directed graph having n vertices and m edges while supporting various DFS related queries efficiently in the RAM with logarithmic word size. We study this problem in two well-known models: indexing and encoding models. While most of these queries can be supported easily in constant time using O(n lg n) bits of extra space, our goal here is, more specifically, to beat this trivial O(n lg n) bit space bound, yet not compromise too much on the running time of these queries. In the indexing model, the space bound of our solution involves the quantity m, hence, we obtain different bounds for sparse and dense graphs respectively. In the encoding model, we first give a space lower bound, followed by an almost optimal data structure with extremely fast query time. Central to our algorithm is a partitioning of the DFS tree into connected subtrees, and a compact way to store these connections. Finally, we also apply these techniques to compactly index the shortest path structure, biconnectivity structures among others

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