By Guo L.-T.

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Lemma 3 The time complexity of the SP-TAG algorithm is O(m(log T + log n)) where T is the number of time instants, n is the number of nodes and m is the number of edges in the time aggregated graph. Proof The cost model analysis assumes an adjacency list representation of the graph with two significant modifications. The edge time series is stored in the sorted order. Attached to every adjacent node in the linked list are the edge time series and the travel time series. For every node extracted from the priority queue Q, there is one edge time series look up and a priority queue update for each of its adjacent nodes.

Min min S Pmin = pq∈P d min pq ≤ kl∈P ∗ dkl and dkl ≤ dkl (t). min S Pmin ≤ ≤ kl∈P ∗ dkl kl∈P ∗ dkl (t) = S P(t). The heuristic function is admissible. Lemma 5 The heuristic function h(n) is monotone. Proof A heuristic function h(n) is monotone if h(i) ≤ di j + h( j)∀i j ∈ E. Here, min ≤ d (t) + S P min . S P min ≤ d min + S P min ; else, it is a contradiction to the S Pid ij jd id ij jd ( min . Since d min ≤ d t), S P min ≤ d (t) + S P min . optimality of S Pid ij ij id jd ij 34 3 Shortest Path Algorithms for a Fixed Start Time Since the heuristic function is admissible and monotone, the A∗ algorithm finds an optimal solution and performs and optimal search [24, 37].

Lemma 8 The NF-SP-TAG algorithm is correct. Proof The algorithm runs on the transformed TAG where the edge costs are the arrival times at the end node of the edge. Here we prove that the algorithm computes the shortest path using the greedy strategy. 4 Shortest Path Algorithm for a Given Start Time in a Non-FIFO Network (NF-SP-TAG) 37 Algorithm 3 Shortest Path (NF-SP-TAG) Algorithm Input: 1) G(N , E): a graph G with a set of nodes N and a set of edges E; Each node n ∈ N has a property: Node Presence Time Series : series of positive integers; Each edge e ∈ E has two properties: Edge Presence Time Series, Arrival_time series : series of positive integers; au,v (t) - arrival time at v for a start time t at u.