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Path-Value Functions for Which Dijkstra’s Algorithm Returns Optimal Mapping

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Abstract

Dijkstra’s algorithm (DA) is one of the most useful and efficient graph-search algorithms, which can be modified to solve many different problems. It is usually presented as a tool for finding a mapping which, for every vertex v, returns a shortest-length path to v from a fixed single source vertex. However, it is well known that DA returns also a correct optimal mapping when multiple sources are considered and for path-value functions more general than the standard path-length. The use of DA in such general setting can reduce many image processing operations to the computation of an optimum-path forest with path-cost function defined in terms of local image attributes. In this paper, we describe the general properties of a path-value function defined on an arbitrary finite graph which, provably, ensure that Dijkstra’s algorithm indeed returns an optimal mapping. We also provide the examples showing that the properties presented in a 2004 TPAMI paper on the image foresting transform, which were supposed to imply proper behavior of DA, are actually insufficient. Finally, we describe the properties of the path-value function of a graph that are provably necessary for the algorithm to return an optimal mapping.

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Notes

  1. Apparently, there was a typo in the version of DA from [21], since they defined a set \(\mathcal{F}\), never used, to avoid reprocessing the vertices in the inner loop of the algorithm. A proper use of \(\mathcal{F}\) would make [21, Lemma 2] valid.

  2. Actually, it is enough to assume only that every hereditarily \(\psi \)-optimal path is monotone.

  3. Notice that in case of the algorithm DA, the value of \(\pi [\mathsf {w}_k]\) can still further change, as shown in Example 6. But, in the presented argument, \(\pi _k\) remains fixed.

  4. Note that if we weaken the assumptions by replacing (R*) with the property (R\(^+\)) obtained by replacing in (R*) symbols \(\preceq \) with the equation =, then the implication does not hold any more: \(\psi _\mathrm{dif}\) from Example 4 satisfies (M) and, for the example from Fig. 2, also (R\(^+\)), but fails the conclusion of Proposition 2.

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Authors and Affiliations

  1. Department of Mathematics, West Virginia University, Morgantown, WV, 26506-6310, USA

    Krzysztof Chris Ciesielski

  2. MIPG, Department of Radiology, University of Pennsylvania, Philadelphia, PA, 19104-6021, USA

    Krzysztof Chris Ciesielski

  3. Institute of Computing, University of Campinas, Campinas, SP, 13083-852, Brazil

    Alexandre Xavier Falcão

  4. Institute of Mathematics and Statistics, University of São Paulo, São Paulo, SP, 05508-090, Brazil

    Paulo A. V. Miranda

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  1. Krzysztof Chris Ciesielski
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Correspondence to Krzysztof Chris Ciesielski.

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Ciesielski, K.C., Falcão, A.X. & Miranda, P.A.V. Path-Value Functions for Which Dijkstra’s Algorithm Returns Optimal Mapping. J Math Imaging Vis 60, 1025–1036 (2018). https://doi.org/10.1007/s10851-018-0793-1

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