Graph Reconstruction via Distance Oracles

Mathieu, Claire; Zhou, Hang

doi:10.1007/978-3-642-39206-1_62

Cited by 9 publications

(17 citation statements)

References 23 publications

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“…To bound OPT and get the first statement in Theorem 1, it is enough to prove the desired bound for a different verification algorithm. This algorithm is a more sophisticated recursive version of the algorithm in [11]. Recursion is a challenge because, when we query the pair (u, v) in a recursive subgraph, the oracle returns the distance between u and v in the entire graph, not just within the subgraph.…”

Section: Our Resultsmentioning

confidence: 99%

“…Yet, this topology can be extremely difficult to find, due to the dynamic structure of the network and to the lack of centralized control. The network reconstruction problem has been studied extensively [1,2,6,7,11,16]. Sometimes we have some idea of what the network should be like, based perhaps on its state at some past time, and we want to check whether our image of the network is correct.…”

Section: Introductionmentioning

confidence: 99%

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Near-Linear Query Complexity for Graph Inference

Kannan

Mathieu

Zhou

2015

Automata, Languages, and Programming

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Abstract. How efficiently can we find an unknown graph using distance or shortest path queries between its vertices? Let G = (V, E) be an unweighted, connected graph of bounded degree. The edge set E is initially unknown, and the graph can be accessed using a distance oracle, which receives a pair of vertices (u, v) and returns the distance between u and v. In the verification problem, we are given a hypothetical graphĜ = (V,Ê) and want to check whether G is equal toĜ. We analyze a natural greedy algorithm and prove that it uses n 1+o(1) distance queries. In the more difficult reconstruction problem,Ĝ is not given, and the goal is to find the graph G. If the graph can be accessed using a shortest path oracle, which returns not just the distance but an actual shortest path between u and v, we show that extending the idea of greedy gives a reconstruction algorithm that uses n 1+o(1) shortest path queries. When the graph has bounded treewidth, we further bound the query complexity of the greedy algorithms for both problems byÕ(n). When the graph is chordal, we provide a randomized algorithm for reconstruction usingÕ(n) distance queries.

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Section: Our Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Near-Linear Query Complexity for Graph Inference

Kannan

Mathieu

Zhou

2015

Automata, Languages, and Programming

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“…A lower bound of Ω(n 2 ) queries is shown by Reyzin and Srivastava [32] for general graphs. Mathieu and Zhou [24] generalize this lower bound to allow approximate distance oracles, provide an upper bound of  O(n 3/2 ) for constant-degree graphs, and  O(n) for outerplanar graphs.…”

Section: Related Workmentioning

confidence: 96%

“…The problem of reconstructing an unknown graph from oracle queries has been studied in many different contexts, and most notably using edge detection queries [16,2,1,5,6], edge counting queries [17,7,25], or distance queries [20,21,32,24].…”

Section: Related Workmentioning

confidence: 99%

Reconstructing Markov processes from independent and anonymous experiments

Micali

Zhu

2016

Discrete Applied Mathematics

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a b s t r a c tWe investigate the problem of exactly reconstructing, with high confidence and up to isomorphism, the ball of radius r centered at the starting state of a Markov process from independent and anonymous experiments. In an anonymous experiment, the states are visited according to the underlying transition probabilities, but no global state names are known: one can only recognize whether two states, reached within the same experiment, are the same.We prove quite tight bounds for such exact reconstruction in terms of both the number of experiments and their lengths.

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“…These differ from our work in that (1) particular types of graphs or betweenness relations are often studied, whereas we consider general graphs, and (2) arbitrary rather than unique shortest paths are usually considered. This line of theory has applications to algorithms for computing the betweenness centrality of a graph [11,5,6,7,1] and to algorithms for reconstructing the hidden edges of a graph by querying its distances or its betweenness relation [17,26,24,2,23].…”

Section: Future Directions and Related Workmentioning

confidence: 99%

On the Structure of Unique Shortest Paths in Graphs

Bodwin¹

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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This paper develops a structural theory of unique shortest paths in real-weighted graphs. Our main goal is to characterize exactly which sets of node sequences, which we call path systems, can be realized as unique shortest paths in a graph with arbitrary real edge weights. We say that such a path system is strongly metrizable.An easy fact implicit in the literature is that a strongly metrizable path system must be consistent, meaning that no two of its paths may intersect, split apart, and then intersect again. Our main result characterizes strong metrizability via some new forbidden intersection patterns along these lines. In other words, we describe a family of forbidden patterns beyond consistency, and we prove that a path system is strongly metrizable if and only if it is consistent and it avoids all of the patterns in this family. We offer separate (but closely related) characterizations in this way for the settings of directed, undirected, and directed acyclic graphs.Our characterizations are based on a new connection between shortest paths and topology; in particular, our new forbidden patterns are in natural correspondence with two-colored topological 2-manifolds, which we visualize as polyhedra. We believe that this connection may be of independent interest, and we further show that it implies some additional structural corollaries that seem to suggest new and possibly deep-rooted connections between these areas. *

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Graph Reconstruction via Distance Oracles

Cited by 9 publications

References 23 publications

Near-Linear Query Complexity for Graph Inference

Near-Linear Query Complexity for Graph Inference

Reconstructing Markov processes from independent and anonymous experiments

On the Structure of Unique Shortest Paths in Graphs

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