arXiv subjects

Shahbaz Khan

Publications and source records attributed to Shahbaz Khan.

22 records · Page 2Linked to original sources

arXiv2018

Dynamic DFS Tree in Undirected Graphs: breaking the $O(m)$ barrier

Depth first search (DFS) tree is a fundamental data structure for solving various problems in graphs. It is well known that it takes $O(m+n)$ time to build a DFS tree for a given undirected graph $G=(V,E)$ on $n$ vertices and $m$ edges. We address the problem of maintaining a DFS tree when the graph is undergoing {\em updates} (insertion and deletion of vertices or edges). We present the following results for this problem. (a) Fault tolerant DFS tree: There exists a data structure of size ${O}(m ~polylog~ n)$ such that given any set ${\cal F}$ of failed vertices or edges, a DFS tree of the graph $G\setminus {\cal F}$ can be reported in ${O}(n|{\cal F}| ~polylog~ n)$ time. (b) Fully dynamic DFS tree: There exists a fully dynamic algorithm for maintaining a DFS tree that takes worst case ${O}(\sqrt{mn} ~polylog~ n)$ time per update for any arbitrary online sequence of updates. (c) Incremental DFS tree: Given any arbitrary online sequence of edge insertions, we can maintain a DFS tree in ${O}(n ~polylog~ n)$ worst case time per edge insertion. These are the first $o(m)$ worst case time results for maintaining a DFS tree in a dynamic environment. Moreover, our fully dynamic algorithm provides, in a seamless manner, the first deterministic algorithm with $O(1)$ query time and $o(m)$ worst case update time for the dynamic subgraph connectivity, biconnectivity, and 2-edge connectivity.

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arXiv2017

Near Optimal Parallel Algorithms for Dynamic DFS in Undirected Graphs

Depth first search (DFS) tree is a fundamental data structure for solving graph problems. The classical algorithm [SiComp74] for building a DFS tree requires $O(m+n)$ time for a given graph $G$ having $n$ vertices and $m$ edges. Recently, Baswana et al. [SODA16] presented a simple algorithm for updating DFS tree of an undirected graph after an edge/vertex update in $\tilde{O}(n)$ time. However, their algorithm is strictly sequential. We present an algorithm achieving similar bounds, that can be adopted easily to the parallel environment. In the parallel model, a DFS tree can be computed from scratch using $m$ processors in expected $\tilde{O}(1)$ time [SiComp90] on an EREW PRAM, whereas the best deterministic algorithm takes $\tilde{O}(\sqrt{n})$ time [SiComp90,JAlg93] on a CRCW PRAM. Our algorithm can be used to develop optimal (upto polylog n factors deterministic algorithms for maintaining fully dynamic DFS and fault tolerant DFS, of an undirected graph. 1- Parallel Fully Dynamic DFS: Given an arbitrary online sequence of vertex/edge updates, we can maintain a DFS tree of an undirected graph in $\tilde{O}(1)$ time per update using $m$ processors on an EREW PRAM. 2- Parallel Fault tolerant DFS: An undirected graph can be preprocessed to build a data structure of size O(m) such that for a set of $k$ updates (where $k$ is constant) in the graph, the updated DFS tree can be computed in $\tilde{O}(1)$ time using $n$ processors on an EREW PRAM. Moreover, our fully dynamic DFS algorithm provides, in a seamless manner, nearly optimal (upto polylog n factors) algorithms for maintaining a DFS tree in semi-streaming model and a restricted distributed model. These are the first parallel, semi-streaming and distributed algorithms for maintaining a DFS tree in the dynamic setting.

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arXiv2017

Incremental DFS algorithms: a theoretical and experimental study

Depth First Search (DFS) tree is a fundamental data structure for solving graph problems. The DFS tree of a graph $G$ with $n$ vertices and $m$ edges can be built in $O(m+n)$ time. Till date, only a few algorithms have been designed for maintaining incremental DFS. For undirected graphs, the two algorithms, namely, ADFS1 and ADFS2 [ICALP14] achieve total $O(n^{3/2}\sqrt{m})$ and $O(n^2)$ time respectively. For DAGs, the only non-trivial algorithm, namely, FDFS [IPL97] requires total $O(mn)$ time. In this paper, we carry out extensive experimental and theoretical evaluation of existing incremental DFS algorithms in random and real graphs, and derive the following results. 1- For insertion of uniformly random sequence of $n \choose 2$ edges, ADFS1, ADFS2 and FDFS perform equally well and are found to take $Θ(n^2)$ time experimentally. This is quite surprising because the worst case bounds of ADFS1 and FDFS are greater than $Θ(n^2)$ by a factor of $\sqrt{m/n}$ and $m/n$ respectively. We complement this result by probabilistic analysis of these algorithms proving $\tilde{O}(n^2)$ bound on the update time. Here, we derive results about the structure of a DFS tree in random graphs, which are of independent interest. 2- These insights led us to design an extremely simple incremental DFS algorithm for both undirected and directed graphs. This algorithm theoretically matches and experimentally outperforms the state-of-the-art in dense random graphs. It can also be used as a single-pass semi-streaming algorithm for incremental DFS and strong connectivity in random graphs. 3- Even for real graphs, both ADFS1 and FDFS perform much better than their theoretical bounds. Here again, we present two simple algorithms for incremental DFS for directed and undirected real graphs. In fact, our algorithm for directed graphs almost always matches the performance of FDFS.

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arXiv2017

Multiple Source Dual Fault Tolerant BFS Trees

Let $G=(V,E)$ be a graph with $n$ vertices and $m$ edges, with a designated set of $σ$ sources $S\subseteq V$. The fault tolerant subgraph for any graph problem maintains a sparse subgraph $H$ of $G$, such that for any set $F$ of $k$ failures, the solution for the graph problem on $G\setminus F$ is maintained in $H\setminus F$. We address the problem of maintaining a fault tolerant subgraph for Breath First Search tree (BFS) of the graph from a single source $s\in V$ (referred as $k$ FT-BFS) or multiple sources $S\subseteq V$ (referred as $k$ FT-MBFS). The problem of $k$ FT-BFS was first studied by Parter and Peleg [ESA13]. They designed an algorithm to compute FT-BFS subgraph of size $O(n^{3/2})$. Further, they showed how their algorithm can be easily extended to FT-MBFS requiring $O(σ^{1/2}n^{3/2})$ space. They also presented matching lower bounds for these results. The result was later extended to solve dual FT-BFS by Parter [PODC15] requiring $O(n^{5/3})$ space, again with matching lower bounds. However, their result was limited to only edge failures in undirected graphs and involved very complex analysis. Moreover, their solution doesn't seems to be directly extendible for dual FT-MBFS problem. We present a similar algorithm to solve dual FT-BFS problem with a much simpler analysis. Moreover, our algorithm also works for vertex failures and directed graphs, and can be easily extended to handle dual FT-MBFS problem, matching the lower bound of $O(σ^{1/3}n^{5/3})$ space described by Parter [PODC15].The key difference in our approach is a much simpler classification of path interactions which formed the basis of the analysis by Parter [PODC15]. Our dual FT-MBFS structure also seamlessly gives a dual fault tolerant spanner with additive stretch of +2 having size $O(n^{7/8})$.

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