In this section, we will provide further explanation on the propositions and proofs in the main text for the readers' reference. For the sake of clarity, the descriptions of the propositions here differ slightly from those in the main text, but this does not affect the content.
PropositionS1. Given a network-constrained path segment 𝑠 𝑖𝑗 𝑞 with speed 𝑣 𝑖𝑗 𝑞 and a network-constrained path segment 𝑠 𝑖𝑗 𝑏 with speed 𝑣 𝑖𝑗 𝑏 , their synchronous shortest-path distance varies linearly with time during their shared period [𝑡 𝑖 , 𝑡 𝑗 ]. Proof. Consider a pair of network-constrained path segments 𝑠 𝑖𝑗 𝑞 ∈ 𝑃 𝑞 and 𝑠 𝑖𝑗 𝑏 ∈ 𝑃 𝑏 with speeds 𝑣 𝑖𝑗 𝑞 and 𝑣 𝑖𝑗 𝑏 , respectively. hhe path segment 𝑠 𝑖𝑗 𝑞 connects two consecutive control points 𝑃 𝑞 (𝑡 𝑖 ) = 〈𝑙 𝑢 , 𝑚 𝑢,𝑖 , 𝑡 𝑖 〉 and 𝑃 𝑞 (𝑡 𝑗 ) = 〈𝑙 𝑢 , 𝑚 𝑢,𝑗 , 𝑡 𝑗 〉 on the same link 𝑎 𝑢 . Here, 𝑙 𝑢 represents the identifier of the link 𝑎 𝑢 on which point 𝑃 𝑞 (𝑡 𝑖 ) is located, 𝑚 𝑢,𝑖 denotes the relative position of point 𝑃 𝑞 (𝑡 𝑖 ) on the link, and 𝑡 𝑖 is the timestamp of point 𝑃 𝑞 (𝑡 𝑖 ). Similarly, the path segment 𝑠 𝑖𝑗 𝑏 connects two consecutive control points 𝑃 𝑏 (𝑡 𝑖 ) = 〈𝑙 𝑣 , 𝑚 𝑣,𝑖 , 𝑡 𝑖 〉 and 𝑃 𝑏 (𝑡 𝑗 ) = 〈𝑙 𝑣 , 𝑚 𝑣,𝑗 , 𝑡 𝑗 〉 on the same link 𝑎 𝑣 . At time 𝑡 𝑘 , the location on the path segment 𝑠 𝑖𝑗 𝑞 is 〈𝑙 𝑢 , 𝑚 𝑢,𝑘 〉, and the location on the path segment