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arXiv:2605.05917 · detector doi_compliance · incontrovertible · 2026-05-19 13:07:27.601353+00:00

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Yaguang Tao, Alan Both, Rodrigo I. Silveira, Kevin Buchin, Stef Sijben, Ross S. Purves, Patrick Laube, Dongliang Peng, Kevin Toohey, and Matt Duckham. A comparative analysis of trajectory similarity measures.GIScience & Remote Sensing, 58(5):643–669, 2021.doi:10.1080/15481603.2 021.1908927. A Proof of Proposition 9 Proposition 9. For every borderB the function B.apx computed by our algorithm is continuous. Moreover, allt, t ′ ∈dom(B)witht < t ′ satisfyB.apx(t) + opt ∥·∥(B(t),B(t ′))≥ B.apx(t ′). Proof. We start with the second property. The initialisations ofB.apx in lines 1–2 and lines 6–7 of Algorithm 1 yieldB.apx(t) + opt∥·∥(B(t),B (t′)) = B.apx(t′), as they use straight paths onB. InB.PropagateFromAdjoiningfrom Algorithm 2, whereA :=B.adj, we have the bound apxs∗(t) + opt∥·∥(B(t),B(t ′)) =A.apx(s ∗) + optA,B(s∗, t) + opt∥·∥(B(t),B(t ′)) ≥ A.apx(s∗) + optA,B(s∗, t′) = apxs∗(t′). This is because we can extend an optimal(A(s∗),B (t))-path onB to create an(A(s∗),B (t′))-path, whose cost is lower bounded by the cost of an optimal(A(s∗),B (t′))-path. For the same reason, we moreover haveapx≤s∗(t) +opt∥·∥(B(t),B (t′)) ≥apx ≤s∗(t′)in B.PropagateFromOpposing. The subroutines computeB.apx as the lower envelope of its initialisation withapxs∗ and apx≤s∗, and the lower envelope retains the desired property that is shared by these three functions. To show continuity, we proceed similarly within an inductive proof. The base case in lines 1–2 of Algorithm 1 gives continuous cost functio

Evidence payload

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  "reconstructed_doi": "10.1080/15481603.2021.1908927.A",
  "ref_index": 23,
  "resolved_title": null,
  "verdict_class": "incontrovertible"
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