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ERROR: type should be string, got "https://doi.org/10.1007/s10796 021 10164 2\n\n# Speeding Up Reachability Queries in Public Transport Networks Using Graph Partitioning\n\n**Bezaye Tesfaye[1]** **· Nikolaus Augsten[1]** **· Mateusz Pawlik[1]** **· Michael H. B¨ohlen[2]** **· Christian S. Jensen[3]**\n\n\nAccepted: 17 June 2021\n© The Author(s) 2021\n\n\n/ Published online: 14 August 2021\n\n\n**Abstract**\nComputing path queries such as the shortest path in public transport networks is challenging because the path costs between\nnodes change over time. A reachability query from a node at a given start time on such a network retrieves all points of\ninterest (POIs) that are reachable within a given cost budget. Reachability queries are essential building blocks in many\napplications, for example, group recommendations, ranking spatial queries, or geomarketing. We propose an efficient\nsolution for reachability queries in public transport networks. Currently, there are two options to solve reachability queries.\n(1) Execute a modified version of Dijkstra’s algorithm that supports time-dependent edge traversal costs; this solution is\nslow since it must expand edge by edge and does not use an index. (2) Issue a separate path query for each single POI, i.e., a\nsingle reachability query requires answering many path queries. None of these solutions scales to large networks with many\nPOIs. We propose a novel and lightweight reachability index. The key idea is to partition the network into cells. Then, in\ncontrast to other approaches, we expand the network cell by cell. Empirical evaluations on synthetic and real-world networks\nconfirm the efficiency and the effectiveness of our index-based reachability query solution.\n\n**Keywords Reachability queries · Public transport networks · Temporal graphs · Spatial network databases**\n\n\n## 1 Introduction\n\nWe study the problem of scalable and efficient reachability querying in public transport networks. A reachability\nquery retrieves all points of interest (POIs) reachable from\na given query node at a specific start time within a given\n\n� Bezaye Tesfaye\n[bbelayneh@cs.sbg.ac.at](mailto: bbelayneh@cs.sbg.ac.at)\n\nNikolaus Augsten\n[augsten@cs.sbg.ac.at](mailto: augsten@cs.sbg.ac.at)\n\nMateusz Pawlik\n[mpawlik@cs.sbg.ac.at](mailto: mpawlik@cs.sbg.ac.at)\n\nMichael H. B¨ohlen\n[boehlen@ifi.uzh.ch](mailto: boehlen@ifi.uzh.ch)\n\nChristian S. Jensen\n[csj@cs.aau.dk](mailto: csj@cs.aau.dk)\n\n1 University of Salzburg, Salzburg, Austria\n\n2 University of Zurich, Z¨urich, Switzerland\n\n3 Aalborg University, Aalborg, Denmark\n\n\ntime budget. The start time is required since the reachability result changes over time. Interesting applications of\nreachability queries include group recommendations, ranking spatial queries, urban planning, and geomarketing. We\npresent two examples.\nConsider a platform that recommends events to a group\nof people such that the group members like to attend the\nevent together (Amer-Yahia et al., 2009; Jameson & Smyth,\n2007). Group members are query nodes and events are POIs.\nWhen the group is given, the events must be evaluated\nby various criteria to optimize the benefit to the group.\nOne important aspect is the location of the event relative\nto the group members. The start time and the travel time\nbudget to reach an event may differ for each member.\nEvents too far away are unlikely to be successful. A single\nrecommendation comprises multiple reachability queries,\none for each group member.\nAnother example is a real estate website that ranks\nproperties (query nodes) according to user preferences. The\nusers may customize reachability criteria for different POIs\n(e.g., school, working place, train station). Thereby, the time\nbudget for individual types of POIs may vary: a user may\nbe willing to commute to work for an hour, while a school\nmust be nearby. Ranking the results of a single user query\n\n\n-----\n\nrequires the computation of multiple reachability queries:\none for each property and parameter setting.\nTo support such applications, reachability queries must\nbe computed efficiently. Achieving this goal in public\ntransport networks is tricky since the shortest path between\ntwo nodes depends on the start time, and the time to\ntraverse a path may vary greatly across time. In a public\ntransport network, stations are nodes, and connections\nbetween stations are edges between nodes. An edge can\nonly be traversed at specific points in time as given by\na schedule. Therefore, computing an index for public\ntransport networks is more complex than for networks with\nconstant edge-traversal costs or networks in which an edge\ncan be traversed at any time (like pedestrian networks or\nroad networks).\n\n_Example 1 Consider, the public transport network in_\nFig. 1a. The nodes v1, v2, . . ., v12 represent stations, and the\ndirected edges represent connections between the stations.\nEach connection has a pair (td _, ta) of departure and arrival_\ntimes. For example, there is a connection leaving v4 at\ntime 10 and arriving at v3 at time 11. The traversal cost\nbetween nodes is expressed in terms of time units. The cost\nof traversing the edge (v4, v3) at time 9 is 2, since we have\na waiting time in addition to the edge traversal time. The\nshortest path from v10 to v11 at start time ts 9 has cost 2\n=\n(edge (v10, v11)), while at ts = 10, the cost of the shortest\npath is 3 (edges (v10, v12), (v12, v11)). At start time ts = 9,\n\n**Fig. 1 Temporal graph of public transport network and reachability index**\n\n\nthe nodes {v8, v9, v11} are reachable from v10 with budget\n_Δt = 2; at ts = 10 with the same budget, we can reach the_\nnodes {v9, v12}.\n\nThe state of the art in answering reachability queries\nin public transport networks includes two approaches.\nThe first is based on a temporal version of Dijkstra’s\nalgorithm (Dijkstra, 1959) that expands in the network until\nthe budget is exhausted. Algorithms following this approach\ncompute a so-called isochrone (the reachable region) and\nintersect it with the set of POIs (Bauer et al., 2008; Gamper\net al., 2011). Since all edges in the isochrone must be\nexpanded, these algorithms do not scale to large networks.\nThe second approach translates a single reachability query\ninto a set of path queries (e.g., shortest path or earliestarrival path (Seufert et al., 2013; Wang et al., 2015; Wu\net al., 2016)), one for each POI. Path queries require heavy\nindex structures and do not scale to large numbers of POIs.\nWe propose an index-based technique for reachability\nqueries in public transport networks. Instead of expanding\nedge by edge, in a precomputation step, we partition the\nnetwork into cells and construct a novel reachability index.\nAt query time, the index is used to expand cell by cell.\nEach cell covers a region of the network and all POIs in\nthat region. The precomputation effort for a specific cell\nis independent of the other cells such that the index scales\nto large networks. The index is small, even smaller than\nthe original graph for some inputs. This paper extends\n\n\n-----\n\nour preliminary results in Tesfaye et al. (2020) with an\nevaluation of different partitioning techniques. The idea of\npartitioning a public transport network to index reachability\nqueries was first published in Tesfaye and Augsten (2016),\nbut the proposed index is not exact and the quality of the\napproximation was not evaluated.\nThe rest of the paper is structured as follows. In Section 2,\nwe define the problem, and we give an overview of our\nsolution in Section 3. We introduce our reachability index\nin Section 4 and discuss query processing using the index\nin Section 5. In Section 6, we review related work. In\nSection 7 we investigate experimentally the performance of\nour solution. We conclude in Section 8.\n\n## 2 Preliminaries and Problem Definition\n\nIn a public transport network, stations are nodes and\nconnections are edges. A connection has a departure time\n_td and an arrival time ta, td < ta. We assume periodic_\nschedules as is typically the case in public transport\nnetworks, e.g., schedules repeat daily or weekly.\nA temporal graph G = (V, E, c) is a directed graph\nwith vertices V, edges E ⊆ _V × V, and a time-dependent_\ncost function c(e, t), c : E × R → R≥0 that captures\nthe time cost of traversing edge e starting at time t. We\nrepresent public transport networks as temporal graphs with\na specific cost function, which we derive from the schedule.\nEach station is a node in the graph, and there is an edge\nfrom node u to node v iff there is a direct connection (i.e.,\nthere are no intermediate stops) from the station of u to the\nstation of v. The cost function is periodic with period Π, i.e.,\n_c(e, t) = c(e, t + Π) and piecewise linear; all linear pieces_\nhave slope k = −1; the cost function is not continuous; all\ndiscontinuities are at departure times of some connections.\nThe time cost of taking a connection si = (td _, ta) to traverse_\nedge e in the period (td − _Π, td_ ] is σ(si, t) = ta − _t. If there_\nare multiple connections Se = {s1, s2, . . ., si} on edge e, the\ncost of traversing edge e departing at time t is the minimum\ncost over all the individual connections at time t, c(e, t) =\nmins∈Se {σ(s, t)}. Our cost function is consistent, i.e., for\nany edge e ∈ _E and all start times t1 ≤_ _t2: t1 + c(e, t1) ≤_\n_t2+c(e, t2). Intuitively, in a consistent cost function, it never_\npays off to skip a connection in order to wait for a later\nand faster connection. Consistency is required for the use\nof Dijkstra’s shortest-path algorithm (Kaufmann & Smith,\n1993).\n\n_Example 2 Consider an edge e with three connections_\n_s1 = (2, 5), s2 = (3, 4), s3 = (4, 6) and a period of_\n_Π = 6. Figure 2 shows the time costs (y-axis) of taking_\nthe individual connections to traverse edge e starting at\ntime t (x-axis): σ(s1, t) = 5 − _t, t ∈_ _(−4, 2] (blue),_\n\n\n**Fig. 2 Example of cost function c(e, t) of traversing edge e starting at**\ntime t derived from three connections s1, s2, and s3 (period Π = 6)\n\n_σ(s2, t) = 4 −_ _t, t ∈_ _(−3, 3] (green), σ(s3, t) = 6 −_ _t,_\n_t_ ∈ _(−2, 4] (red). The overall cost function c(e, t) =_\nmin{σ(s1, t), σ(s2, t), σ(s3, t)} is illustrated with black\ndashed lines. For example, c(e, 2) = 2 (take connection\n_s2 at cost σ(s2, 2) = 4 −_ 2 = 2: depart at time 2,\nwait for 1 unit and then travel for 1 unit), c(e, 4) = 2\n(depart at time 4, no waiting, take connection s3 with cost\n_σ(s3, 4) = 6 −_ 4 = 2). Note that the connection s1 does\nnot contribute to the cost function c because it departs\nearlier and arrives later than the connection s2. Thus, s1\nis obsolete and c(e, t) = min{σ(s2, t), σ(s3, t)} holds,\nmaking c consistent. In practice, connections like s1, if they\nexist, are removed from the schedule.\n\nA path p from u to v in a temporal graph G = (V, E, c)\nis a sequence of edges p = ⟨e1, e2, . . ., en⟩ such that\n_ei ∈_ _E, ei = (wi−1, wi), w0 = u, and wn = v; P (u, v)_\nis the set of all paths from node u to node v. The cost of a\npath is the fastest time to traverse the path at a given start\ntime. Due to the consistency property of our cost function,\nthe path cost is the sum of all edge costs. The cost of path\n_p = ⟨e1, e2, . . ., en⟩_ at time t, λ(p, t), is the cost sum of all\nedges in p: λ(p, t) = [�]1≤i≤n _[c(e][i][, t][i][)][, where][ t][1][ =][ t][ and]_\n_ti = ti−1 + c(ei−1, ti−1) for i > 1. The shortest-path cost_\nfrom node u to node v at time t is the minimum cost of any\npath from u to v, sp(u, v, t) = minp∈P (u,v){λ(p, t)}. A path\nwith the minimum cost is called the shortest path. A node\n_v is reachable from a node u at time t within budget Δt iff_\nthere is a path p ∈ _P (u, v) such that the cost of p at time t_\nis no larger than Δt, i.e., λ(p, t) ≤ _Δt._\n\n**Definition 1 Reachability query. Given a temporal graph**\n_G_ = _(V, E, c) with points of interst P OI_ ⊆ _V,_\n\n\n-----\n\nthe reachability query RQ(u, t, Δt) returns all points of\ninterest reachable from node u at time t within budget\n_Δt, i.e,_\n\n_RQ(u, t, Δt) = {v ∈_ _P OI | ∃p ∈_ _P (u, v), λ(p, t) ≤_ _Δt}_\n\n**Problem definition The goal of this work is to develop an**\nefficient index-based solution for reachability queries that\nscales to large temporal graphs.\n\n## 3 Solution Overview\n\nWe propose a novel index structure, the reachability\n_index, to answer reachability queries. We introduce a bulk_\nloading technique for our index, provide access methods for\nanswering reachability queries, and discuss the incremental\ninsertion and deletion of POIs in the index.\nThe reachability index is built in a precomputation step.\nTo construct the index, we partition the temporal graph into\ndisjoint cells. Any such partitioning yields correct results.\nThe choice of cells, however, affects the effectiveness of the\nindex. We define requirements for a good partitioning and\npropose a suitable partitioning technique.\nThe index is a temporal graph that contains only those\nnodes of the original graph that are POIs or directly connect\ndifferent cells, called border nodes. Each POI belongs to a\ncell. POIs can be inserted into and deleted from the index\nat any time; the update cost is low and depends on a single\ncell. The index consists of the original edges between border\nnodes of neighboring cells and new edges between the\nborder nodes within a cell. Further, an edge between each\nPOI and the border nodes in its cell is introduced. The edge\ncosts are the costs of shortest paths between the respective\nnodes in the original graph.\nA high number of border nodes per cell increases the\nindex size. Each POI adds as many edges to the index as\nthere are border nodes in its cell.\nA search query traverses the index cell by cell. The\nborder nodes are used to cross cells and to reach neighboring\ncells. For each border node, we verify if any of the POIs in\nthat cell is reachable.\n\n## 4 The Reachability Index\n\nThe reachability index R is a temporal graph that is\nconstructed from the original graph G as follows:\n\n1. _Graph partitioning. The nodes of graph G are split into_\ndisjoint cells. At query time, instead of expanding edge\nby edge in G, we expand cell by cell in the index.\n2. _Constructing the index core. Based on the graph_\npartitioning, we insert the so-called border nodes and\n\n\nedges into the initially empty index. This index core\nnever changes.\n3. _Computing the index cost function. The edge cost is_\ncomputed as a shortest-path cost for each departure time\nfrom a source node to a destination node.\n4. _Inserting POIs. Inserting a POI into a cell adds a new_\nnode and an edge to each border node of the cell. POIs\nthat are not border nodes can be inserted and deleted\ndynamically without modifying the rest of the index.\n\nWe detail each step of the index construction next.\nAdditionally, we discuss the factors that affect the size of\nthe reachability index and present a compaction technique\nto reduce the number of connections.\n\n### 4.1 Graph Partitioning\n\nWe partition the nodes of a temporal graph G = (V, E, c)\ninto a set of disjoint cells C = {C1, C2, . . ., Cn}, such that\neach node of G belongs to exactly one cell Ci, i.e., Ci∩Cj =\n∅ for any pair of cells with i ̸= j, and [�]1≤i≤n _[C][i][ =][ V][ .]_\nEach disconnected component of the graph that is supposed\nto be indexed[1] should be partitioned into at least two cells.\nThe nodes of a cell Ci that have an adjacent node in a cell\ndifferent from Ci are border nodes, Bi: A node v ∈ _Ci is a_\nborder node, v ∈ _Bi, if it has an edge to or from another cell,_\ni.e., there is a node w ∈ _V, w /∈_ _Ci, and an edge (v, w) ∈_ _E_\nor an edge (w, v) ∈ _E. For example, the temporal graph G_\nin Fig. 1b of our example public transport network (Fig. 1a)\nis partitioned into two cells (dashed boxes): C1 with border\nnodes v5 and v7, and C2 with border nodes v8 and v9.\nThe cells define the structure of the reachability index.\nThe index will be expanded cell by cell to answer\nreachability queries. A good partitioning should satisfy the\nfollowing properties:\n\n1. _Well connected inside. A cell comprises highly-linked_\nnodes with many edges and connections inside the cell.\n2. _Loosely connected outside. The number of border nodes_\nper cell is small.\n3. _Large distance between cells. Crossing cell borders is_\nexpensive: the number of connections between cells is\nsmall and their cost is high.\n\nFinding a good partitioning that satisfies our requirements is not straightforward. We identify two relevant\napproaches in literature: community detection (Blondel\net al., 2008; Newman & Girvan, 2004; Traag et al., 2019)\nand minimum edge-cut partitioning (Karypis & Kumar,\n1998a; 1998b). Both approaches strive to minimize the\n\n1Indexing very small disconnected components may not be useful\nsince a reachability query expands at most the edges within the\ndisconnected component regardless of the query parameters.\n\n\n-----\n\nedges connecting different partitions. An additional objective of the minimum edge-cut partitioning is to produce\npartitions of balanced size. Community detection algorithms automatically adjust the number of partitions such\nthat the resulting partitions are densely connected inside.\nFor the minimum edge-cut partitioning the number k of partitions is a user-defined parameter. Both approaches allow\nfor edge weights that increase the cost for cutting an edge;\nwe choose the number of connections on an edge to be its\nweight.\nWe assess how well do community detection and\nminimum edge-cut partitioning suit our scenario. In this\nsection, we revisit the commonly used algorithms and\nthoroughly evaluate their impact on our solution in\nSection 7.\n\n**4.1.1 Community Detection**\n\nIn contrast to fixing the number of partitions upfront, community detection aims at identifying naturally occurring\ndensely connected subgraphs. One of the best-known objective functions for community detection is modularity (Newman & Girvan, 2004). Modularity is the difference between\nthe edges in the identified communities and the expected\nnumber of edges in communities of an equivalent network\nwith edges distributed at random. Optimizing modularity,\ni.e., finding communities that maximize modularity, is NPhard. We describe two approximations: Louvain (Blondel\net al., 2008), and its recent improvement Leiden (Traag\net al., 2019).\n\n**Louvain by Blondel et al. (2008) is a two-phase approach.**\nInitially, each node forms its own community. In the first\nphase, nodes are moved to neighboring communities such\nthat the modularity increases. The result is the input to the\nsecond phase, where an aggregated network is created. Each\nnode in the aggregated network represents a community\ndetected in the first phase. The two phases are repeated until\nmodularity cannot be increased any further.\nLouvain produces partitions with the following guarantees: no communities can be merged and no nodes can be\nmoved between communities to further improve modularity (Traag et al., 2019).\n\n**Leiden by Traag et al. (2019) is an improvement of**\nLouvain. Traag et al. observe that Louvain may result\nin poorly connected or even disconnected communities,\nwhich is not desired. Leiden adds an additional phase to\nLouvain. Initially, each node is in its community. In the\nfirst phase, similarly to Louvain, Leiden moves nodes to\nneighboring communities. However, unlike Louvain that\nvisits each node, Leiden uses an improvement, called fast\n_local move, to visit only those nodes whose neighborhood_\n\n\nhas changed. In the second phase, called refinement, Leiden\nfinds sub-communities within each community resulted\nfrom the first phase. Each of the sub-communities will\nform their own community. The refinement phase is used\nto avoid poorly connected and disconnected communities.\nIn the third phase, unlike Louvain, Leiden uses the refined\ncommunities to aggregate the network.\nLeiden provides two new guarantees in addition to the\nguarantees provided by Louvain: communities are well\nconnected, and individual nodes are well connected to their\ncommunities.\n\n**4.1.2 Minimum Edge-Cut Partitioning**\n\nThe minimum edge-cut partitioning computes k partitions\nsuch that: (a) the sizes of the partitions are balanced,\n(b) the number of edges that connect nodes from\ndifferent partitions (the so-called edge-cut) is minimized.\nFinding the minimum edge-cut is NP-hard. Efficient\napproximate methods resulting in high-quality partitions\nfollow a multilevel partitioning paradigm. For our purpose\nwe chose the multilevel k-way partitioning (Karypis\n& Kumar, 1998b) from the commonly used METIS\nframework (Karypis & Kumar, 1998a).\nThe multilevel k-way partitioning (Karypis & Kumar,\n\n1998b) is a three-phase approach, consisting of coarsening,\n_partitioning, and uncoarsening. In coarsening phase an_\ninput graph is reduced by collapsing nodes and edges\niteratively. In each step, the graph from the previous step\nis reduced further. The coarsening phase stops when the\ncoarsest graph has a small number of nodes, or the reduction\nbetween two consecutive coarser graphs is too small. The\n_partitioning phase splits the coarsest graph into k partitions._\nThe uncoarsening phase projects the partitioned graph back\nto the original input by unfolding nodes and edges. To do so,\nit follows the backward sequence of coarsened graphs until\nall nodes and edges are unfolded. During uncoarsening, a\nrefinement technique is applied to decrease the edge-cut.\nVarious algorithms have been proposed for each of the\nphases; Karypis and Kumar (1998b) thoroughly evaluate the\noptions.\nIn our scenario, the number k of desired partitions is\nnot known up front. This parameter is hard to guess. Since\nLeiden and Louvain do not require this input parameter, the\nnumber of partitions produced by these algorithms could\nserve as a guideline. This is the approach that we take in our\nempirical evaluation.\n\n### 4.2 Constructing the Index Core\n\nGiven a temporal graph G = (V, E, c) and a partitioning C\nof G, we construct the core of our reachability index. The\nindex core is independent of POIs and never changes for\n\n\n-----\n\na given partitioning. The reachability index is a temporal\ngraph R = (VR, ER, cR) with nodes VR ⊂ _V, edges_\n_ER ⊆_ _VR × VR, and cost function cR(e, t) on the edges_\n_e ∈_ _ER. For an edge e = (u, v) ∈_ _ER, cR is defined_\nas the shortest-path cost from u to v at time t in G, i.e.,\n_cR(e, t) = sp(u, v, t)._\n_Index nodes. For each cell Ci ∈_ _C, we insert all its border_\n_nodes Bi into the node set VR of the index. Thus, the nodes_\nof the index VR = [�]1≤i≤|C| _[B][i][. Figure][ 1][c shows the index]_\ncore of the temporal graph (Fig. 1b) with cells C1 = {v5, v7}\nand C2 = {v8, v9}.\n\n**Index edges The edges of the index core are ER = BB ∪**\n_BC. BB is the set of all edges between border nodes of_\nneighboring cells. For each edge (u, v) ∈ _E between two_\nborder nodes of different cells in C, u ∈ _Ci, v ∈_ _Cj_ _, i ̸= j_,\nwe insert a new edge between the respective nodes into the\nindex, ER = ER ∪{(u, v)}. BC is the set of all edges\nbetween pairs of border nodes within a cell. For each pair\n_u, v ∈_ _Bi, we insert two new edges (u, v) and (v, u) into_\nthe index, ER = ER ∪{(u, v), (v, u)}. For example, BB =\n{α3, α4, α7} and BC = {α1, α2, α5, α6} in Fig. 1c.\n\n### 4.3 Points of Interest\n\nPOIs can be inserted and deleted at any time, also after index\nconstruction. This is beneficial because POIs may change\nover time. A POI v ∈ _V may be any node in the original_\ntemporal graph. If v is a border node, no action is required\nbecause such a node is in the index core already. Otherwise,\nsimilarly to border nodes, inserting v into the index involves\nthree steps. (1) We add v to the index nodes (VR = VR∪{v}).\n(2) We add a new edge from each border node of v[′]s cell to\n_v (we call such edges BP edges). (3) The cost function based_\non shortest paths (same procedure as for all other edges)\nis computed. Deleting a POI from the index removes the\nPOI node and all its incoming edges. For example, consider\ninserting two POIs, v2, v10, into the index in Fig. 1d. We add\nedges BP = {α8, α9, α10, α11}.\n\n### 4.4 Computing the Index Cost Function\n\nThe cost function cR of an edge e = (u, v) ∈ _ER in_\nindex R is defined as the shortest-path cost from u to\n_v at time t in graph G, i.e., cR(e, t) = sp(u, v, t). For_\ncomputing the values of the cost function cR, we execute\nDijkstra’s single-source shortest-path algorithm once for\nevery border node b ∈ _Bi and every departure time at b._\nThe expansion stops when all other border nodes in the\ncell and all direct neighbors of b (i.e., nodes reachable\nfrom b via a BB edge) are visited. Since the cells are\nsmall compared to the overall graph, typically only a small\nnumber of nodes needs to be considered for each execution\n\n\nof Dijkstra’s algorithm. BC and BP edges may connect\nnodes that are not reachable in the original temporal graph.\nIf a node is not reached during one of the shortest-path\ncomputations, we assign infinite cost to the respective\nedges.\n\n_Example 3 Cost examples for the index core in Fig. 1c are:_\n_cR(α3, 14) = 3, cR(α1, 9) = 2, cR(α5, 9) = 2. In Fig. 1d,_\nthe costs for the edges of type BP are cR(α8, 9) = 3,\n_cR(α9, 8) = 7, cR(α10, 9) = 1, and cR(α11, 11) = 1._\n\n### 4.5 Index Size\n\nThe index consists of border nodes and POIs. Thus, the\nnumber of index nodes is at most the number of nodes in\nthe temporal graph. We introduce three types of edges into\nthe index. BB edges connect border nodes between different\ncells, and they are a subset of the temporal graph edges.\n_BC edges connect border nodes in a single cell, and their_\ncardinality is at most quadratic in the number of border\nnodes. Each POI adds as many BP edges as border nodes in\na cell. The numbers of BC and BP edges depend only on the\nsubset of temporal graph nodes that are in a single cell. The\nnumbers do not depend on the graph size. In sparse graphs,\nwhere many nodes have only a few edges, the reachability\nindex may grow larger than the temporal graph: we can\nremove only a small number of original edges but need to\ninsert new BC and BP edges.\nEach edge has as many edge cost values as there\nare departure times from a node. The edge costs are\ncomputed for each single cell in isolation, making parallel\ncomputation possible. In particular, the edge cost of\na specific border node at a specific departure time is\nindependent of all other edge costs.\n\n### 4.6 Index Compaction\n\nThe index size, as well as the size of the temporal graph,\nis dominated by the size of the schedule, i.e., the number\nof edge connections. After computing the edge costs in the\nindex, we observe that many different departure times have\nthe same arrival time at the destination. It is enough to keep\nonly one connection per arrival time, namely the one with\nthe maximum departure time. We leverage that and compact\nthe index by reducing the number of connections as follows.\nConsider an edge e(u, v) ∈ _ER and set S of departure–_\narrival connection pairs (d, a) on that edge. We compact S\nto S[′] ⊆ _S, such that S[′]_ = {(d, a) ∈ _S : ∄(di_ _,ai_ _)∈Sai = a ∧_\n_di > d}. Experiments show that this compaction technique_\nis highly effective and reduces the index size by up to 74%\n(cf. Section 7). For example, the set of all connections on\nedge α8 in Fig. 1d, {(8, 12), (9, 12), (11, 15)}, is compacted\ninto {(9, 12), (11, 15)}.\n\n\n-----\n\n## 5 Answering Reachability Queries\n\nThe core idea of our reachability algorithm is to expand cell\nby cell rather than edge by edge. The BB edges between\nborder nodes of different cells allow us to expand to the\nneighboring cells; the BC edges between border nodes of\nthe same cell reflect the time to cross a cell; the direct\n_BP edges from border nodes to POIs allow for a quick_\nevaluation of which POIs can be reached. In addition, we\ndiscuss a heuristic to avoid unnecessary edge expansions\nand processing of query nodes that are non-border nodes.\n\n**The reachability algorithm Algorithm 1 takes as an input**\nthe reachability index R = _(VR, ER, cR), query node_\n_q, start time ts, and the cost budget Δt. The expansion_\nproceeds like in Dijkstra’s algorithm and returns the set N of\nreachable POIs in R. Nodes and their costs from q are stored\nin a min-heap M initialized to M[q] = 0, and M[v] = ∞\nfor all other nodes v (line 1). The closest node v to q is\npopped from the min-heap (line 4), and the costs for nodes\nadjacent to v are updated if smaller (lines 9–11). To retrieve\nthe correct edge cost, we do a binary search in the list of\nedge costs sorted by departure time (line 9). Each node\nis traversed only once. The algorithm terminates when no\nmore nodes with cost lower than the budget are in the heap\n(line 5). Consider the reachability index in Fig. 1d. Here,\n_RQ(R, v5, 8, 6) = {v2, v10} because sp(v5, v2, t) = 4_\n(through α8) and sp(v5, v10, t) = 5 (through α4 and α10).\n_RQ(R, v5, 6, 6) = {v2} because sp(v5, v2, t) = 6 (through_\n_α8) but sp(v5, v10, t) = 7 (through α4 and α10)._\n\n**Avoiding unnecessary expansions Regarding the edges**\nwithin a cell, we observe the following. Consider Algorithm 1 processing a border node b of a cell Ci. Then, the\n\n\ncosts of the other nodes, vj ∈ _Ci, are updated w.r.t. the cost_\nof reaching them from b. When we pop a node vj in a later\nround, and if vj was last updated by b, there is no point in\nfollowing the edges from vj to the other nodes in the cell.\nThe cost of accessing the other nodes in the cell through\n_vj cannot be smaller than the cost of accessing these nodes_\ndirectly from b since all edge costs are shortest paths. If,\nhowever, vj was updated through an edge from a neighboring cell, the edges to the other nodes in the cell need to\nbe followed. We exploit this observation to avoid following\nedges inside a cell that cannot lead to an update and thus\ndo not affect the solution. We flag the nodes whenever their\ncost was updated by processing a node from within a cell,\nand we remove the flag, otherwise (line 12). The outgoing\nedges that must be expanded are selected based on the flag\n(line 7).\nNote that the number of edges within a cell is quadratic in\nthe number of border nodes of that cell. Thanks to the use of\nflags we avoid unnecessary expansions. In particular, if the\ncheapest way to reach all nodes in a cell is through k border\nnodes, we only expand k(w − 1) edges per cell, where w\nis the number of all border nodes and POIs in a cell. The\nvalue of k is expected to be small and will often be 1 (i.e.,\nthe shortest path from a query node q to all nodes in the cell\ncrosses the border node that is closest to q).\n\n**Non-border query nodes The reachability index does not**\ncontain all nodes of the original graph. If the query node\n_q in cell Ci is not a border node, the algorithm starts the_\nexpansion from q in the temporal graph. All POIs reached\nin cell Ci are part of the result. Once a border node b[′] ∈ _Bi_\nis reached, the expansion continues in the index at time\n_ts + sp(q, b[′], ts)._\n\n**Correctness We show that the shortest-path costs in the**\nindex and the original temporal graph are identical.\nLet u, w ∈ _VR_ be two index nodes and p =\n⟨(v0, v1), (v1, v2), . . ., (vn−1, vn)⟩ be the corresponding\nshortest path in the temporal graph, i.e., u = v0, w =\n_vn. If there is a direct edge between u and w in the_\nindex, the shortest-path cost is the cost of that edge: this\ncost is precomputed using Dijkstra’s algorithm for each\ndeparture time in the original temporal graph; since our\ncost function is consistent (cf. Section 2), the edge cost is\ncorrect (Kaufmann & Smith, 1993). Otherwise, u and w\nare not in the same cell (all nodes in a cell are connected\nwith an edge). So, there must be a path along index nodes\n_u1, u2, . . . uk ⊆_ _v1, . . . vn−1 that are all on path p since cells_\ncan be exited only through border nodes. We show that the\ncost of the index path is indeed the shortest path. Assume\na node ui exists such that sp(v0, vn, t) < sp(u, ui, t) +\n_sp(ui, w, t) + sp(u, ui, t). On a path of length two, the_\n\n\n-----\n\ncosts of edges (u, u1) and (u1, w) are precomputed shortestpath costs, and they are therefore correct. The assumption,\nhowever, implies that one of the edge costs could be\ndecreased, i.e., the assumption is incorrect. This argument\ncan be extended edge by edge to paths of arbitrary length.\n\n## 6 Related Work\n\nShortest-path and reachability queries on road networks,\ni.e., graphs with constant edge cost, have been studied\nextensively. Unfortunately, these works cannot be applied\nreadily to public transport networks (Bast, 2009). An\nevaluation by Bast et al. (Bast et al., 2016) shows a large\nperformance gap between the two types of networks. This\nis due to the time-dependent edge costs of public transport\nnetworks, which makes the precomputation efforts of many\nalgorithms infeasible.\nCurrent solutions for public transport networks either\nrely on Dijkstra’s algorithm (Dijkstra, 1959) or require\nheavy precomputations. Dijkstra’s algorithm follows an\nexpansion technique that visits edges in all possible direction until the target is reached. Dijkstra-based\napproaches include isochrone algorithms for multimodal\nnetworks (Bauer et al., 2008; Gamper et al., 2011). They\nexpand from a query point using Dijkstra’s algorithm and\ncompute a so-called isochrone, which is the reachable portion of the network at a given point in time. Since all edges\nin the isochrone must be expanded, this approach does not\nscale to large networks.\nMany works fall into the category of labeling approaches.\nThe earliest work, 2-hop labeling (Cohen et al., 2003),\nis designed for weighted graphs and is based on 2-hop\ncovers of shortest paths. Recent works strive to decrease the\nindex size and construction time (Cheng et al., 2013; Jin &\nWang, 2013), which are bottlenecks of 2-hop labeling and\nprevent application to large graphs. Time Table Labeling\n(TTL) (Wang et al., 2015) and Top Chain (Wu et al.,\n2016) adapt 2-hop labeling to public transport networks;\nthey support shortest-path and point-to-point reachability\nqueries. In TTL, the main idea is to precompute label sets for\neach node v containing reachable nodes from and to v. Top\nChain creates a directed acyclic graph (DAG), where each\nnode represents a departure time, and decomposes the DAG\nto create the label sets. Creating label sets in both techniques\nrequires high precomputation costs and large index sizes.\nTo decrease the index size, Top Chain only stores K label\nsets, called chains. The index size of Top Chain for small K\nvalues is smaller than that of TTL, but there is no guarantee\nthat the query results can be found using the index.\nNon-labeling techniques include Scalable Transfer Patterns (Bast et al., 2016), Connection Scan Algorithm\n(CSA) (Strasser, 2016), and Contraction Hierarchy for\n\n\nTimetables (CHT) (Geisberger, 2010). Transfer Patterns\nrequire an expensive profile search from each node to find\nthe optimal paths to all other nodes. CSA organizes a\nschedule as two sequences of edges. The first sequence contains sorted edges based on arrival times, and the second\nsequence sorts edges based on departure times. At query\ntime, CSA scans the sequences in linear time to answer\nearliest arrival path queries. CHT organizes vertices in hierarchies and applies a contraction technique to reduce the\ngraph size for query processing. SPs are precomputed by\nadding new edges to the graph, which are leveraged at query\ntime. These approaches involve expensive precomputations\nor large index sizes, which limits their scalability.\nTo compute reachability queries as defined in this paper,\nall techniques based on point-to-point queries require the\ncomputation of shortest paths from a given query node to\nevery POI, which does not scale to large number of POIs.\n\n## 7 Experiments\n\nWe experimentally evaluate our solution, RQ, and compare\nit to two competitors, a no-index solution, NI, and a fullyindexed solution, SP . We report on the index size and\nefficiency of the algorithms w.r.t. the number of expanded\nedges, which is the work that an algorithm has to do to find\nreachable nodes.\nOur algorithm, RQ, partitions the input graph in order to\nbuild the index. We study the effect of different partitioning\ntechniques discussed in Section 4.1 on the index structure\nand the performance of our RQ algorithm. We identify one\npartitioning technique to be used in conjunction with RQ .\n\n### 7.1 Implementation details\n\nThe algorithms are implemented in Python 3 and executed\non a virtual machine with 32 cores and 117GB RAM,\nrunning Debian 10(buster). For the partitioning of the input\ngraphs we use the following Python libraries: for Louvain\npython-louvain v0.13[2], for Leiden leidenalg\nv0.8.2[3], for multilevel k-way partitioning metis v2020.1[4]\n\nwith the underlying C METIS library v5.1.0[5].\n\n### 7.2 Competitors\n\n**No-Index (NI) The no-index solution, NI, operates on the**\noriginal temporal graph and does not build an index. The\nreachability query is computed with a modified version\n\n[2https://github.com/taynaud/python-louvain](https://github.com/taynaud/python-louvain)\n[3https://github.com/vtraag/leidenalg](https://github.com/vtraag/leidenalg)\n[4https://github.com/inducer/pymetis](https://github.com/inducer/pymetis)\n[5http://glaros.dtc.umn.edu/gkhome/metis/metis/overview](http://glaros.dtc.umn.edu/gkhome/metis/metis/overview)\n\n\n-----\n\nof Dijkstra’s algorithm that supports our cost function\n(cf. Section 5).\n\n**Shortest-Path (SP) The fully-indexed solution, SP, precom-**\nputes and stores all shortest paths from every node in the\ntemporal graph to each POI at every departure time. SP\nrepresents the collection of works that index the shortest\npaths between pairs of nodes (cf. Section 6). In terms of\nlookups per shortest-path, SP is optimal since only a single\nlookup is required per shortest-path query. Other solutions\nfor reachability queries based on indices for shortest-path\nqueries cannot outperform SP in terms of lookups per query.\nInstead, these solutions trade in lookup performance to\nachieve a smaller memory footprint (which is high for SP).\nTherefore, the number of edge expansions performed by SP\nfor reachability queries cannot improve by substituting SP\nby another shortest-path index.\n\n### 7.3 Datasets\n\nWe use two real-world public transport networks represented as temporal graphs, Zurich and Berlin (2020),\nand one synthetic graph, Synthetic. Zurich and Berlin are\nobtained in GTFS format that is further processed. For these\ngraphs, we chose all transport modes and all connections\noperating on Mondays. Synthetic is a 6 6 grid of equally×\nsized spider-web subgraphs. Each spider-web subgraph has\none edge to every neighboring subgraph (to its left, right,\ntop, and bottom). This graph simulates loosely connected\ncities that are densely connected inside. In such a case we\nwould expect a good partitioning to assign nodes of each\nspider-web subgraph into a separate partition. Table 1 shows\nthe statistics, where #Conn is the number of all connections\n(departure-arrival pairs) that can be used to cross an edge.\nFigure 3 visualizes the structure of our public transport\nnetworks.\n\n### 7.4 Effect of Partitioning\n\nOur solution, RQ, will work with any partitioning of\nthe input graph and compute correct results. However, we\nobserve that the index structure and performance of query\nanswering varies depending on the specific partitioning used\nto build the index (cf. Section 4.1).\n\n**Table 1 Dataset statistics**\n\nDataset #Nodes #Edges #Conn\n\nZurich 2,508 5,630 555,713\nBerlin 12,984 34,791 1,348,070\nSynthetic 145,188 433,272 31,042,468\n\n\nWe investigate the effect of different partitioning\ntechniques on our solution RQ . In our analysis, we include\ntwo community detection algorithms, Louvain (Blondel\net al., 2008) and Leiden (Traag et al., 2019), and the\nmultilevel k-way partitioning (Karypis & Kumar, 1998b)\nfrom the METIS framework (Karypis & Kumar, 1998a)\n(denoted as METIS). In Louvain and Leiden, the so-called\nmodularity of the partitioning is optimized to find good\ncommunities. To compute the modularity, a resolution and\na weight between pairs of nodes needs to be specified. We\nuse the default value 1 for the resolution and the number of\nconnections as the edge weight: the more connections exist\nbetween two nodes, the better they are connected.\nWhile Louvain and Leiden auto-detect the number of\npartitions, METIS requires the number of partitions as an\ninput parameter. Defining a good number of partitions is not\nstraightforward since this parameter inherently depends on\nthe network structure. We evaluate METIS using the mean\nof the partition numbers detected by Louvain and Leiden\nwhen we compare to these algorithms in terms of index size\nand query performance.\n\n**7.4.1 Structure of the Partitioning**\n\nTable 2 shows the statistics of partitions resulting from\napplying different techniques to our input graphs: number\nof partitions, partition sizes (average/min/max), number of\nborder nodes (total sum, average/min/max per partition).\nWe visualize the partitions for Zurich, Berlin and Synthetic\ngraphs in Figs. 4, 5, and 6, respectively.\n\n**Number of Partitions Louvain and Leiden auto-detect the**\nnumber of partitions. The number of detected partitions\nis similar for both algorithms: Louvain detects 4% fewer\npartitions than Leiden for Zurich and 15% more partitoins\nfor Synthetic; both algorithms produce the same number\nof partitions for Berlin. METIS requires the number of\npartitions as an input parameter. We choose the mean\nvalue of the partition numbers detected by Louvain and\nLeiden for METIS. For Synthetic, in addition, we provide\nMETIS with the number of spider webs (6 6 36)\n× =\nthat we used to construct the graph. The numbers in\nTable 2 show that METIS correctly identifies all spider\nwebs when the number of partitions is well chosen, resulting\nin a low number of border nodes. With the number of\npartitions that we automatically detect using Louvain and\nLeiden (44 partitions), however, the performance of METIS\nsignificantly drops. The visualization of the partitions in\nFig. 6 confirms this result.\n\n**Size of Partititions One of the optimization goals of METIS**\nis to produce partitions well balanced in size, whereas\nLouvain and Leiden detect communities regardless of their\n\n\n-----\n\n**Fig. 3 Structure of the different public transport networks**\n\nsize. This is confirmed by our results. The partitions of\nMETIS are well balanced, while the community detection\nalgorithms produce some very small and some very large\n\n**Table 2 Statistics for different partitioning techniques**\n\n\npartitions for the real world datasets. Figure 7 shows the\ndistribution of nodes per partition for the three algorithms\non Zurich, Berlin and Synthetic.\n\n\nDataset Partitioning Border nodes\n\nAlgo. #Part avg min max sum avg min max\n\nZurich Louvain 44 57 2 163 314 7 0 20\nLeiden 46 54.5 2 158 298 6.5 0 17\nMETIS 45 55.7 54 57 439 9.8 1 20\nBerlin Louvain 51 254.6 2 924 1250 24.5 0 52\nLeiden 51 254.6 2 747 1138 22.3 0 43\nMETIS 51 254.6 247 262 1439 28.2 9 53\nSynthetic Louvain 48 3024.8 1042 4037 1979 41.2 2 90\nLeiden 41 3541.2 1463 4036 904 22 2 84\nMETIS 44 3299.7 3204 3398 4395 99.9 51 227\n36 4033 4031 4035 120 3.3 2 4\n\n\n-----\n\n**Fig. 4 Partitioning of Zurich**\n\n**Number of Border Nodes As discussed in Section 4.1, we**\nstrive for partitions that are well connected inside and\nloosely connected outside. A low number of border nodes\nindicates loose connections between partitions since edges\nbetween partitions can exist only between border nodes. For\nZurich and Berlin, Leiden exhibits the lowest values for\n\n**Fig. 5 Partitioning of Berlin**\n\n\nborder nodes. We observe that for these datasets the good\nbalance of partition sizes of METIS comes at the cost of\nmore border nodes.\nFor the Synthetic graph, if METIS is given the optimal\nnumber of 36 partitions, all spider webs are detected and\noptimal partitions are produced, resulting in two border\n\n\n-----\n\n**Fig. 6 Partitioning of Synthetic**\n(each partition is colored\ndifferent)\n\n**Fig. 7 Distribution of nodes per**\npartition\n\n\n-----\n\n**Table 3 Index core size for different partitionings**\n\nDataset Algorithm #Nodes #Edges #Conn #Total\n\nZurich RQ LO 314 3,333 405,898 409,545\nRQ LE 298 2,953 377,818 381,069\nRQ ME45 439 5,131 436,281 441,851\n\nBerlin RQ LO 1,250 38,286 2,229,584 2,269,120\nRQ LE 1,138 30,124 1,936,377 1,967,639\nRQ ME51 1,439 45,664 1, 976, 750 2,023,853\n\nSynthetic RQ LO 1,979 150,840 13,263,713 13,416,532\nRQ LE 904 63,622 5,833,430 5,897,956\nRQ ME36 120 416 22,025 22,561\nRQ ME44 4,395 598,896 35,192,991 35,796,282\n\n\nnodes for the spider webs in the four corners, three border\nnodes for the spider webs on the boundaries of the graph,\nand four border nodes for all other spider webs. With the\nauto-detected value of 44 partitions, METIS produces the\nlargest number of border nodes for Synthetic. The reason is\nthat METIS strives to balance the partition sizes also at the\ncost of more border nodes and border edges.\nZurich and Berlin include small disconnected components. Louvain and Leiden detect the small disconnected\ncomponents and do not further partition them. This explains\nthe zero values for the minimum number of border nodes in\nTable 2. METIS includes the disconnected components into\nlarger partitions to balance their sizes.\n\n**7.4.2 Index Size and Structure**\n\nThe border nodes resulting from a partitioning define the\nstructure of our RQ index. Table 3 presents the index\ncore size for different partitionings. The suffixes LO,\n\n\nLE, and MEk in the Algorithm column denote Louvain,\nLeiden, and METIS with k partitions, respectively. The\nvalues that increase the index size are the number of\nnodes (#Nodes) and edges (#Edges), and the number of\nconnections (#Conn), which is the dominating factor. The\ntotal sum of these three values (#Total) is used to compare\nindex sizes among the different partitionings.\nTable 3 suggests that the most promising partitioning is\nLeiden, which results in the smallest overall index core size\n(#Total). An exception is Synthetic if we provide METIS\nwith the optimal number of 36 partitions: the resulting\nindex core is much smaller than the index core for the\nother partitionings. With the auto-detected number of 44\npartitions, however, METIS produces the largest index core.\nTo evaluate our RQ index size after additions of POIs,\nwe randomly pick 5% of the input graph nodes to be POIs\nand we built our index for different partitionings across\nfive different POI samples. Table 4 shows average (avg)\nand standard deviation (σ, rounded values) of the counts\n\n\n**Table 4 Index size for different partitionings given multiple samples of POIs**\n\nDataset Algorithm #Nodes #Edges #Conn\n\nAlgorithm avg _σ_ avg _σ_ avg _σ_\n\nZurich RQ LO 421 4 4,173 47 456,913 7,206\nRQ LE 406 2 3,733 46 425,621 11,207\nRQ ME45 542 2 6,102 35 494,138 8,437\n\nBerlin RQ LO 1,834 8 54,988 283 2,385,623 7,103\nRQ LE 1,732 5 44,643 161 2,717,979 13,747\nRQ ME51 2,022 10 61,893 199 2,539,917 14,743\n\nSynthetic RQ LO 9,135 9 350,070 2,811 28,873,351 259,401\nRQ LE 8,122 7 163,449 1,982 13,646,302 173,456\nRQ ME36 7,375 2 24,594 61 1,420,407 7,379\nRQ ME44 11,423 18 1,281,986 5,555 78,219,714 205,325\n\n\n-----\n\nthat affect the index size (number of nodes, edges, and\nconnections). In Zurich and Berlin, the average number\nof nodes and edges using Leiden is the smallest. On the\nSynthetic dataset, METIS with 36 partitions outperforms all\nthe other partitioning algorithms for all values. The number\nof nodes varies across the samples due to a varying number\nof POIs that are border nodes. The standard deviation is\nbelow 2% for any of the values, indicating that the particular\nchoice of POIs does not have a large effect on the index\nsize.\n\n**7.4.3 Query Performance.**\n\nTo measure the effect of different partitioning techniques on\nthe querying performance, we count the number of edges\nthat RQ must expand for answering a reachability query.\nWe compare the indices constructed over the partitioning of\nLouvain, Leiden, and METIS.\nIn this experiment, we use the RQ index constructed for\none of the POI samples discussed in Section 7.4.2. All query\nnodes are border nodes. Since the different partitionings\nproduce different border nodes, we pick all border nodes\nthat exist in all partitionings as query nodes. For each query\nnode, we executed a total of 10 different reachability queries\nvarying the starting time (8:00, 12:00, 16:00, 18:00, 22:00)\nand time budget (60 and 120 minutes).\nThe results for our three input graphs are shown in Fig. 8.\nOn the y-axis we measure the relative difference of RQ\n\n**Fig. 8 Relative number of**\nexpanded edges w.r.t. NI\n(100%)\n\n\ncompared to no-index baseline NI (horizontal red line,\n100%). The data points are ordered along the x-axis by\nthe number of expanded edges, separately for each data\nseries. We evaluate Louvain (RQ LO), Leiden (RQ LE),\nand METIS (RQ MEk). The number k of partitions for\nMETIS is different for each dataset to approximately match\nthe number of partitions that Louvain and Leiden produce\nfor the respective setting (cf. Table 2).\nIn Zurich and Synthetic, all the three partitioning\ntechniques perfom similarly for most of the data points.\nLeiden (RQ LE) and Louvain (RQ LO) perform slightly\nworse than METIS (RQ MEk) for some of the data\npoints on the Berlin dataset. Independently of the chosen\npartitioning, RQ clearly outperforms the baseline NI on all\ndatesets: Except for a small number of outliers, the index\nconstructed using Leiden reduces the number of expanded\nedges by at least 55% for Zurich, 70% for Berlin, and 90%\nfor Synthetic.\n\n**7.4.4 Which Partitioning to Choose?**\n\nWe summarize our experimental findings and evaluate the\ndifferent partitioning techniques.\nLeiden outperforms Louvain with respect to the index\nsize. Given a similar number of partitions as Leiden and\nLouvain, (1) METIS produces more balanced partition\nsizes, but the partitions have more border nodes on average,\nand also (2) the index size (in particular the number of\n\n\n-----\n\n**Table 5 Index (RQ using Leiden partitioning)**\n\nDataset Algorithm #Nodes #Edges #Conn\n\nZurich RQ 406 3,733 425,621\n_SP_ 2,508 313,500 69,464,152\n_NI_ 2,508 5,630 555,713\n\nBerlin RQ 1,732 44,643 2,717,979\n_SP_ 12,984 8,426,616 874,897,430\n_NI_ 12,984 34,791 1,348,070\n\nSynthetic RQ 8,122 163,449 13,646,302\n_SP_ 145,188 1,053,919,692 225,337,275,212\n_NI_ 145,188 433,272 31,042,468\n\n\nconnections that dominates the index size) is smaller for\nLeiden than for METIS.\nIn terms of query performance, METIS is competitive\nwith the other approaches and in some cases slightly\noutperforms them. The catch is that the performance of\nMETIS depends on the parameter k, which is hard to choose\nwithout running experiments similar to the ones in our tests.\nThe parameters k so determined may not be valid for other\nnetworks or query loads.\nOverall we suggest to use the Leiden partitioning in\nconjunction with our RQ index.\n\n### 7.5 RQ vs. Competitors\n\nIn this section we compare RQ with its competitors\n_SP_ and NI . We use RQ with Leiden partitioning (as\nwe suggest in Section 7.4.4) for all experiments in this\nsection.\n\n**7.5.1 Index Size**\n\nWe compare the index size of RQ (with Leiden partitioning)\nto the index sizes of its competitors. Although NI does not\nrequire precomputation, the input graph has to be kept in\nmemory. In Table 5, we compare the index sizes (RQ, SP )\nto the input graph size (NI ).\n_RQ and SP precompute certain shortest paths and build_\nan index structure that is sufficient to answer reachability\nqueries. If the index of SP is stored as a graph, the\nindex nodes are identical to the input graph nodes (POIs\nare a subset of the input graph nodes), the number of\nindex edges is computed as #Nodes #POIs from the\n×\nnumber of input graph nodes and POIs (an edge is inserted\nbetween every node and every POI), and the number of\nindex connections results from computing a shortest path at\nevery departure time to every POI. For RQ the index size\ndepends on the position of the border nodes. We randomly\npick 5% of the input graph nodes as POIs and show the\n\n\naverage values for #Nodes, #Edges, and #Conn over five\nsamples.\nThe index size of RQ is always smaller than that of\n_SP_ (up to four orders of magnitude in the Synthetic\ngraph). RQ is also significantly smaller than the original\n_Zurich and Synthetic graphs (NI ). Although RQ_ has\nsignificantly fewer nodes and edges than the original graph\nfor Berlin, the number of connections is higher. This is\ncaused by the sparsity of Berlin (cf. Section 4.5). Finally,\n#Connections is the number of edge connections stored.\nFor RQ, we list the absolute number of connections after\nthe compaction (cf. Section 4.6). The reduction rate of\ncompaction is high: 74% for Zurich and Berlin, and 73% for\n_Synthetic._\nTable 6 shows the runtime to build our RQ index. The\nshortest path computations are executed in parallel on 20\ncores. For Zurich, the index builds in less than 2 min for\nLouvain, Leiden, and METIS. All runtimes for Berlin are\nbelow 30 min. For Synthetic, the index construction time\nfor METIS is only 7 min with 36 partitions, but 12 hrs\nwith 44 partitions. The high construction time for METIS\nwith 44 partitions is due to the poor partitioning and the\n\n**Table 6 Index construction time**\n\nDataset Algorithm Time (min)\n\nZurich RQ LO 0.7\nRQ LE 0.7\nRQ ME45 1.1\n\nBerlin RQ LO 26.3\nRQ LE 20.2\nRQ ME51 29.1\n\nSynthetic RQ LO 48.4\nRQ LE 25.6\nRQ ME36 7.2\nRQ ME44 720.4\n\n\n-----\n\n**Fig. 9 Number of expanded**\nedges (y-axis in log scale)\n\n**Fig. 10 Query time in ms**\n(y-axis in log scale)\n\n\n-----\n\nresulting high number of connections that require many\nshortest path computations (before compaction, there are\n292,883,908 connections, each resulting from one shortest\npath computation).\n\n**7.5.2 Performance of Query Answering**\n\nTo evaluate the efficiency, we compare the number of edges\nthat each algorithm has to process in order to find all\nreachable POIs (Fig. 9).\nIn this experiment, we use our RQ index constructed\nusing one of the samples discussed in Section 7.4.2. One\ndata point in the figure (scatter plot) is a single reachability\nquery. Data points are sorted along the x-axis by the number\nof expanded edges. We execute one reachability query\nstarting at every border node in our index. We do so at\nfive different start times (8:00, 12:00, 16:00, 18:00, 22:00)\nand for two time budgets (60 and 120 minutes). Thus, the\nnumber of data points is 10 #Border nodes. The budgets\n×\nare large enough to force RQ to traverse multiple edges.\nSince the edge costs of large cells in the RQ index are often\nabove 15 minutes (and above 30 minutes in about half of the\ncases), budgets near these values provide little insight. Since\nSP precomputes the path to each POI, it always evaluates\none edge per POI. This is a lower bound on the cost of any\npoint-to-point index. Although the index of SP is orders of\nmagnitude larger, RQ expands significantly fewer edges for\nmany of the data points. We observe the largest differences\nfor the budget of 120 min. On Synthetic, the number of\nedges expanded by RQ is up to three orders of magnitude\nlower than that of SP, and it is up to one order of magnitude\nlower than that of NI. RQ always expands fewer edges than\nNI. Values equal to zero indicate that an algorithm cannot\nexpand due to high connection costs.\nIn Fig. 10 we compare the query times of RQ and NI . We\nmeasure at the granularity of milliseconds, which results in\nruntime zero for very short query executions. As expected\nfrom the lower number of expanded edges, RQ outperforms\n_NI on almost all data points across the three datasets._\nOverall, our experiments show that, despite its small\nsize, RQ substantially reduces the number of expanded\nedges, and therefore speeds up reachability queries in public\ntransport networks.\n\n## 8 Conclusion\n\nThe paper offers improved support for reachability queries\nin temporal graphs that retrieve all reachable points of\ninterest (POIs) from a given query node at a specific start\ntime within a given time budget. We observe that current\nsolutions do not scale to large networks (solutions based\non Dijkstra’s algorithm without a pre-computed index) or\n\n\nto networks with many POIs (solutions based on an index\nfor single-path queries that must be executed for each\nPOI separately). We propose a solution based on a novel\naccess structure, the reachability index. This index partitions\nthe original temporal graph into cells, thus enabling us to\nexpand the graph cell by cell rather than edge by edge.\nWe evaluate different graph partitioning techniques and\nstudy the effect on index size and query performance. Our\nempirical evaluations suggest that our technique is both\neffective and efficient.\n\n**Acknowledgements This work was supported by Austrian Science**\nFund (FWF): W1237. We wish to thank Christine Gfrerer for her\nvaluable suggestions, as well as Alfred Egger and Manuel Kocher for\ntheir help with the experiments.\n\n**Funding Open access funding provided by Paris Lodron University**\nof Salzburg. Open access funding provided by Austrian Science Fund\n(FWF).\n\n**Open Access This article is licensed under a Creative Commons**\nAttribution 4.0 International License, which permits use, sharing,\nadaptation, distribution and reproduction in any medium or format, as\nlong as you give appropriate credit to the original author(s) and the\nsource, provide a link to the Creative Commons licence, and indicate\nif changes were made. The images or other third party material in this\narticle are included in the article’s Creative Commons licence, unless\nindicated otherwise in a credit line to the material. If material is not\nincluded in the article’s Creative Commons licence and your intended\nuse is not permitted by statutory regulation or exceeds the permitted\nuse, you will need to obtain permission directly from the copyright\n[holder. To view a copy of this licence, visit http://creativecommons.](http://creativecommons.org/licenses/by/4.0/)\n[org/licenses/by/4.0/.](http://creativecommons.org/licenses/by/4.0/)\n\n## References\n\nAmer-Yahia, S., Roy, S. B., Chawlat, A., Das, G., & Yu, C. (2009).\nGroup recommendation: Semantics and efficiency. Proceedings of\n_the VLDB Endowment, 2(1), 754–765._\nBast, H. (2009). Car or public transport - two worlds. In Efficient\n_algorithms: Essays dedicated to kurt mehlhorn on the occasion of_\n_his 60th birthday (pp. 355–367)._\nBast, H., Delling, D., Goldberg, A. V., M¨uller-Hannemann, M., Pajor,\nT., Sanders, P., Wagner, D., & Werneck, R.F. (2016). Route\nplanning in transportation networks. In Algorithm engineering:\n_Selected results and surveys (pp. 19–80)._\nBast, H., Hertel, M., & Storandt, S. (2016). Scalable transfer patterns.\nIn Proceedings of the meeting on algorithm engineering and\n_experiments (ALENEX) (pp. 15–29)._\nBauer, V., Gamper, J., Loperfido, R., Profanter, S., Putzer, S., &\nTimko, I. (2008). Computing isochrones in multi-modal, schedulebased transport networks. In Proceedings of the ACM SIGSPATIAL\n_international conference on advances in geographic information_\n_systems._\nBlondel, V. D., Guillaume, J.-L., Lambiotte, R., & Lefebvre, E. (2008).\nFast unfolding of communities in large networks. Journal of\n_Statistical Mechanics: Theory and Experiment, 2008(10)._\nCheng, J., Huang, S., Wu, H., & Fu, A.W. (2013). TF-Label: A\ntopological-folding labeling scheme for reachability querying in\na large graph. In Proceedings of the ACM SIGMOD international\n_conference on management of data (pp. 193–204)._\n\n\n-----\n\nCohen, E., Halperin, E., Kaplan, H., & Zwick, U. (2003). Reachability\nand distance queries via 2-hop labels. SIAM Journal on\n_Computing, 32(5), 1338–1355._\nDijkstra, E. W. (1959). A note on two problems in connexion with\ngraphs. Numerische Mathematik, 1, 269–271.\nGamper, J., B¨ohlen, M., Cometti, W., & Innerebner, M. (2011). Defining isochrones in multimodal spatial networks. In Proceedings of\n_the ACM international conference on information and knowledge_\n_management (CIKM) (pp. 2381–2384)._\nGeisberger, R. (2010). Contraction of timetable networks with realistic\ntransfers. In Proceedings of the international symposium on\n_experimental algorithms (pp. 71–82)._\nJameson, A., & Smyth, B. (2007). Recommendation to groups. In\n_The adaptive web: Methods and strategies of web personalization_\n(pp. 596–627).\nJin, R., & Wang, G. (2013). Simple, fast, and scalable reachability\noracle. Proceedings of the VLDB Endowment, 6(14), 1978–1989.\nKarypis, G., & Kumar, V. (1998). A fast and high quality multilevel\nscheme for partitioning irregular graphs. SIAM Journal on\n_Scientific Computing, 20(1), 359–392._\nKarypis, G., & Kumar, V. (1998). Multilevel k-way partitioning\nscheme for irregular graphs. Journal of Parallel and Distributed\n_Computing, 48(1), 96–129._\nKaufmann, D. E., & Smith, R. L. (1993). Fastest paths in timedependent networks for intelligent vehicle-highway systems\napplication. Journal of Intelligent Transportation Systems, 1(1),\n1–11.\nNewman, M. E. J., & Girvan, M. (2004). Finding and evaluating\ncommunity structure in networks. Physical Review E, 69, 026113.\nSeufert, S., Anand, A., Bedathur, S. J., & Weikum, G. (2013).\nFERRARI: Flexible and efficient reachability range assignment\nfor graph indexing. In Proceedings of the IEEE international\n_conference on data engineering (ICDE) (pp. 1009–1020)._\nStrasser, B. (2016). Intriguingly simple and efficient time-dependent\n[routing in road networks. arXiv:1606.06636.](http://arxiv.org/abs/1606.06636)\nTesfaye, B., & Augsten, N. (2016). Reachability queries in public\ntransport networks. In Proceedings of the 28th GI-Workshop\n_Grundlagen von Datenbanken, N¨orten Hardenberg, Germany,_\n_May 24-27, 2016, (Vol. 1594 pp. 109–114)._\nTesfaye, B., Augsten, N., Pawlik, M., B¨ohlen, M. H., & Jensen,\nC.S. (2020). An efficient index for reachability queries in public\ntransport networks. In Advances in databases and information\n_systems - 24th european conference, ADBIS 2020, lyon, france,_\n_august 25-27, 2020, proceedings, (Vol. 12245 pp. 34–48)._\nTraag, V. A., Waltman, L., & van Eck, N.J. (2019). From louvain\nto leiden: guaranteeing well-connected communities. Nature\n_scientific reports 9._\n\n\nWang, S., Lin, W., Yang, Y., Xiao, X., & Zhou, S. (2015). Efficient\nroute planning on public transportation networks A labelling\napproach. In Proceedings of the ACM SIGMOD international\n_conference on management of data (pp. 967–982)._\nWu, H., Huang, Y., Cheng, J., Li, J., & Ke, Y. (2016). Reachability\nand time-based path queries in temporal graphs. In Proceedings\n_of the IEEE international conference on data engineering (ICDE)_\n(pp. 145–156).\n[Zurich, & Berlin (2020). GTFS. https://data.stadt-zuerich.ch/dataset/](https://data.stadt-zuerich.ch/dataset/vbz_fahrplandaten_gtfs)\n[vbz fahrplandaten gtfs,](https://data.stadt-zuerich.ch/dataset/vbz_fahrplandaten_gtfs) [https://daten.berlin.de/datensaetze/](https://daten.berlin.de/datensaetze/vbb-fahrplandaten-gtfs)\n[vbb-fahrplandaten-gtfs, Accessed January 31, 2020.](https://daten.berlin.de/datensaetze/vbb-fahrplandaten-gtfs)\n\n**Publisher’s Note Springer Nature remains neutral with regard to**\njurisdictional claims in published maps and institutional affiliations.\n\n**Bezaye Tesfaye is a PhD candidate within the Database research**\ngroup at the University of Salzburg. She received her MSc degree\nin Geoinformatics from University of Twente in 2013. Her research\ninterests include efficient path queries in graphs and extension of\nspatio-temporal databases. She is part of the European Software\nSustainability Infrastructure (EuSSI) consortium, and is a member of\ndeRSE (German Research Software Engineers association). She is also\na software carpentry instructor.\n\n**Nikolaus Augsten is a full professor in the Department of Computer**\nScience at the University of Salzburg, where he heads the Database\nGroup. He received his PhD degree in Computer Science from Aalborg\nUniversity, Denmark, in 2008, and holds a MSc degree (Dipl.-Ing.)\nfrom Graz University of Technology, Austria. Prior to joining the\nUniversity of Salzburg in 2013, he was an assistant professor at\nthe Free University of Bozen-Bolzano, Italy. He was on leave at\nTU M¨uchen, Germany, in 2010/2011 and visited Washington State\nUniversity in 2005/2006. In his research, Augsten deals with technical\nissues of efficiently storing, updating, and querying large amounts\nof data. He is particularly interested in similarity queries, which\ndo not require an exact match but allow some degree of fuzziness.\nHis research is motivated by concrete applications and results in\nnew algorithms that are implemented and tested on the motivating\napplication. The results of his research were published in the most\nprestigious outlets of the database field. For his work on top-k\nqueries over tree data he received the Best Paper Award at the\nIEEE International Conference on Data Engineering in 2010. Augsten\nregularly serves on the program committee of the most important\ninternational database conferences and was an associate editor of the\nVLDB Journal.\n\n\n-----\n\n**Mateusz Pawlik is a senior scientist at the University of Salzburg and a**\nmember of the Database Research Group and the Centre for Cognitive\nNeuroscience. He received his PhD degree in Computer Science in\n2014 from the University of Bozen/Bolzano (Italy), and his MSc\ndegree from Adam Mickiewicz University in Pozna´n (Poland). His\nresearch interests span similarity of tree-structured data, reachability\nin public transport networks, and recently efficient management of\nneuroscientific data. His research results appeared in the top outlets\nof the databases community. Pawlik is the main developer of the Tree\n[Similarity Library (http://tree-edit-distance.dbresearch.uni-salzburg.](http://tree-edit-distance.dbresearch.uni-salzburg.at/)\n[at/).](http://tree-edit-distance.dbresearch.uni-salzburg.at/)\n\n**Michael H. B¨ohlen is a professor of computer science at the University**\nof Z¨urich where he heads the database technology group. His research\nfocuses on temporal data management and analytics, including query\nprocessing and indexing, statistical data management, and machine\nlearning. He received his Ph.D. degree from ETH Z¨urich and was\na faculty member at Aalborg University and the Free University of\nBozen-Bolzano. He served as Program Chair for VLDB 2013, EDBT\n2018 and SSDBM 2018, was a member of the VLDB Endowment’s\nBoard of Trustees, and served as associate editor for ACM TODS and\nthe VLDB Journal.\n\n**Christian S. Jensen is Professor of Computer Science at Aalborg**\nUniversity, Denmark. His research focuses on temporal and spatiotemporal data management and analytics, including machine learning,\ndata mining, and query processing. He is an ACM and IEEE Fellow,\nand he is a member of Academia Europaea, the Royal Danish\nAcademy of Sciences and Letters, and the Danish Academy of\nTechnical Sciences. He has received several awards for his research,\nmost recently the 2019 IEEE TCDE Impact Award. He is President of\nthe steering committee of the Swiss National Research Program on Big\nData. In Germany, he serves on the scientific advisory board the Max\nPlanck Institute for Informatics; and in Norway, he chairs the Scientific\nAdvisory Board of the Norwegian Research Center for AI Innovation.\n\n\n-----\n\n"