arXiv subjects

Keerti Choudhary

Publications and source records attributed to Keerti Choudhary.

23 records · Page 2Linked to original sources

arXiv2019

Graph Realizations: Maximum and Minimum Degree in Vertex Neighborhoods

The classical problem of degree sequence realizability asks whether or not a given sequence of $n$ positive integers is equal to the degree sequence of some $n$-vertex undirected simple graph. While the realizability problem of degree sequences has been well studied for different classes of graphs, there has been relatively little work concerning the realizability of other types of information profiles, such as the vertex neighborhood profiles. In this paper, we initiate the study of neighborhood degree profiles. We focus on the natural problem of realizing maximum and minimum neighborhood degrees. More specifically, we ask the following question: Given a sequence $D$ of $n$ non-negative integers $0\leq d_1\leq \cdots \leq d_n$, does there exist a simple graph with vertices $v_1,\ldots, v_n$ such that for every $1\le i \le n$, the maximum (resp. minimum) degree in the neighborhood of $v_i$ is exactly $d_i$? We provide in this work various results for both maximum as well as minimum neighborhood degree for general $n$ vertex graphs. Our results are first of its kind that studies extremal neighborhood degree profiles. For maximum neighborhood degree profiles, we provide a {\em complete realizability criteria}. In comparison, we observe that the minimum neighborhood profiles are not so well-behaved, for these our necessary and sufficient conditions for realizability {\em differ by a factor of at most two}.

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arXiv2019

Efficiently Realizing Interval Sequences

We consider the problem of realizable interval-sequences. An interval sequence comprises of $n$ integer intervals $[a_i,b_i]$ such that $0\leq a_i \leq b_i \leq n-1$, and is said to be graphic/realizable if there exists a graph with degree sequence, say, $D=(d_1,\ldots,d_n)$ satisfying the condition $a_i \leq d_i \leq b_i$, for each $i \in [1,n]$. There is a characterisation (also implying an $O(n)$ verifying algorithm) known for realizability of interval-sequences, which is a generalization of the Erdos-Gallai characterisation for graphic sequences. However, given any realizable interval-sequence, there is no known algorithm for computing a corresponding graphic certificate in $o(n^2)$ time. In this paper, we provide an $O(n \log n)$ time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is non-realizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence, in the same time. Finally, we consider variants of the problem such as computing the most regular graphic sequence, and computing a minimum extension of a length $p$ non-graphic sequence to a graphic one.

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arXiv2019

Diameter Spanners, Eccentricity Spanners, and Approximating Extremal Distances

The diameter of a graph is one if its most important parameters, being used in many real-word applications. In particular, the diameter dictates how fast information can spread throughout data and communication networks. Thus, it is a natural question to ask how much can we sparsify a graph and still guarantee that its diameter remains preserved within an approximation $t$. This property is captured by the notion of extremal-distance spanners. Given a graph $G=(V,E)$, a subgraph $H=(V,E_H)$ is defined to be a $t$-diameter spanner if the diameter of $H$ is at most $t$ times the diameter of $G$. We show that for any $n$-vertex and $m$-edges directed graph $G$, we can compute a sparse subgraph $H$ that is a $(1.5)$-diameter spanner of $G$, such that $H$ contains at most $\tilde O(n^{1.5})$ edges. We also show that the stretch factor cannot be improved to $(1.5-ε)$. For a graph whose diameter is bounded by some constant, we show the existence of $\frac{5}{3}$-diameter spanner that contains at most $\tilde O(n^\frac{4}{3})$ edges. We also show that this bound is tight. Additionally, we present other types of extremal-distance spanners, such as $2$-eccentricity spanners and $2$-radius spanners, both contain only $\tilde O(n)$ edges and are computable in $\tilde O(m)$ time. Finally, we study extremal-distance spanners in the dynamic and fault-tolerant settings. An interesting implication of our work is the first $\tilde O(m)$-time algorithm for computing $2$-approximation of vertex eccentricities in general directed weighted graphs. Backurs et al. [STOC 2018] gave an $\tilde O(m\sqrt{n})$ time algorithm for this problem, and also showed that no $O(n^{2-o(1)})$ time algorithm can achieve an approximation factor better than $2$ for graph eccentricities, unless SETH fails; this shows that our approximation factor is essentially tight.

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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

An efficient strongly connected components algorithm in the fault tolerant model

In this paper we study the problem of maintaining the strongly connected components of a graph in the presence of failures. In particular, we show that given a directed graph $G=(V,E)$ with $n=|V|$ and $m=|E|$, and an integer value $k\geq 1$, there is an algorithm that computes in $O(2^{k}n\log^2 n)$ time for any set $F$ of size at most $k$ the strongly connected components of the graph $G\setminus F$. The running time of our algorithm is almost optimal since the time for outputting the SCCs of $G\setminus F$ is at least $Ω(n)$. The algorithm uses a data structure that is computed in a preprocessing phase in polynomial time and is of size $O(2^{k} n^2)$. Our result is obtained using a new observation on the relation between strongly connected components (SCCs) and reachability. More specifically, one of the main building blocks in our result is a restricted variant of the problem in which we only compute strongly connected components that intersect a certain path. Restricting our attention to a path allows us to implicitly compute reachability between the path vertices and the rest of the graph in time that depends logarithmically rather than linearly in the size of the path. This new observation alone, however, is not enough, since we need to find an efficient way to represent the strongly connected components using paths. For this purpose we use a mixture of old and classical techniques such as the heavy path decomposition of Sleator and Tarjan and the classical Depth-First-Search algorithm. Although, these are by now standard techniques, we are not aware of any usage of them in the context of dynamic maintenance of SCCs. Therefore, we expect that our new insights and mixture of new and old techniques will be of independent interest.

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