1 Introduction

Location-based services (LBS) attract great attention from both research and industry communities, and various queries have been studied. In Refs. [7, 26] discover useful information from trajectory data generated in daily life. In Refs. [17, 24] optimize query processing on location-based social networks. In Refs. [12, 18, 25] combine LBS with traditional keyword search. Travel planning has also been studied, and becomes an important issue in location-based services (LBS), which are to find tours among points of interest (POI), where POIs are with latitude and longitude in a two-dimensional space or in a road network. In Refs. [9, 26] study on how to find trajectories from an existing trajectory set. There are works that try to construct routes satisfying certain requirements. In [8] constructs the most popular routes between two given points. In [35] defines different queries as finding the earliest arrival, latest departure and shortest duration paths on the transportation networks. Some recent work study finding the shortest tour connecting two POIs [34, 38] and searching the optimal meeting point for a set of POIs, which are to minimize the sum of distances from these POIs to the meeting point [36, 37].

Fig. 1
figure 1

Different datasets. a TSPLIB (pcb3038). b Normal. c New York (POI). d Los Angeles (check-ins)

As an important issue in travel planning, traveling salesman problem (TSP) has been extensively studied, which finds a tour that traverses all the points exactly once with the minimum overall distance, for a given set of points, and is known as NP-hard problem. The hardness is mainly due to two reasons. First, given \(n\) points, there are \(n!\) possible routes to traverse, in order to find the one with minimum overall distance. Second, the local optimum property does not hold. The state-of-the-art exact TSP solution, Concorde, is based on linear programming (http://www.math.uwaterloo.ca/tsp/concorde.html). By randomization, Arora in [1] finds \((1 + \frac{1}{c})\)-approximate answer in \(O(n(\log n)^{O(c)})\) time, for every fixed \(c > 1\), which is known as the best theoretical result, but is difficult to implement. In the literature, numerous works have been proposed to study TSP [19]. A large number of heuristics are proposed to find a high-quality tour within reasonable time. In Refs. [5, 31] and the most recent [21] summarize and test many representative heuristics and compare them in both effectiveness and efficiency.

Given the hardness of TSP, TSP query is not well studied in database community. Recently, there are TSP-like problems being studied for LBS, which are with constraints to reduce the search space [6, 23, 33], and find a tour by adding nearest neighbors one by one in a manner of expanding the partial result found. In other words, the work reported [6, 23, 33] only use one heuristic, namely the nearest neighbor, among many possible heuristics. The questions that arise are as follows. Is the nearest neighbor the best in terms of accuracy? What are the other methods and which one should we use to process TSP if there are many? This issue is important, since it opens ways for us to explore different ways to deal TSP in LBS for real large datasets with different properties.

There are several attempts to study different heuristics. First, [21] studies heuristics for TSP queries that travel more than 1000 points. However, in many real applications, the number of points can vary in a large range. For LBS, the number of points can be much smaller than that number. Second, the TSPLIB benchmark (http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95) studies about 150 difficult cases, which is not sufficient to understand the heuristics in real datasets. Third, there are synthetic datasets [20, 21], but they do not reflect all real datasets. Figure 1 shows 4 datasets. Figure 1a shows a dataset with 3038 points in TSPLIB. Figure 1b shows a dataset containing 3000 points that follow normal distribution generated [21]. Figure 1c shows 3000 randomly sampled POIs in a real dataset in New York (NY). Figure 1d shows 3000 randomly sampled check-ins in Los Angeles (LA) from the location-based social network, Gowalla (https://en.wikipedia.org/wiki/Gowalla). Fourth, there are no performance studies to study all heuristics. In this work, we study 22 heuristics for TSP queries.

The main contributions of this work are summarized below. First, we study 22 TSP construction heuristics. The reason to study such heuristics is due to the efficiency requirement in LBS, since construction heuristics [21] are efficient to find TSP without any further refinement. Second, we propose new synthetic datasets to understand TSP in the real LBS setting. Third, we conduct extensive performance studies over the selected real datasets, TSP benchmarks, the existing synthetic datasets, and our new synthetic datasets. Fourth, we conclude that both the nearest-neighbor-based heuristics that are widely used in LBS and the best heuristics in TSPLIB for difficult setting are not the best to be used in LBS. We identify several that can achieve high accuracy efficiently. Among many issues, we also show that it is possible to get high-quality TSP by precomputing/indexing, even though it is hard to prove by theorem.

The rest of the paper is organized as follows. Section 2 discusses the preliminaries and gives the problem statement. We introduce all the 22 construction heuristics in Sect. 3. In Sect. 4, we discuss our new synthetic datasets generation in detail, and we report our finding over the 22 heuristics using real datasets, the selected 20 TSPLIB benchmarks, the existing synthetic datasets, and new synthetic datasets. We conclude this work in Sect. 5.

2 Preliminaries

Consider a set of points V in a two-dimensional space, where the distance between two points u and v in V is the Euclidean distance, denoted as \(d ({u},{v})\).

$$\begin{aligned} d ({u},{v}) = \sqrt{(u.x - v.x)^2 + (u.y - v.y)^2} \end{aligned}$$

We denote the two x and y coordinates of a point u as u.x and u.y.

An edge-weighted complete undirected graph \(G = (V, E)\) can be constructed for the set of given points. Here, V is the set of nodes for the same set of points, and E is a set of edges for every pair of nodes in V where an edge weight for an edge (uv) is the distance between u and \(v, d ({u},{v})\).

Let \(Q\) be a subset of nodes of size \(n = |Q |\) in V. A Hamilton path over \(Q\) is a simple path, \((v_1, v_2, v_3, \ldots , v_{n-1}, v_n)\), that visits every node exactly once, where \((v_i, v_{i+1})\) is an edge in the graph G. A Hamilton circuit over \(Q\) is a simple cycle over all nodes in G. Both Hamilton path and Hamilton circuit can be regarded as a permutation of nodes (or points) in \(Q\). Here, a permutation \(\pi\) over Q is a one-to-one mapping. In other words, a node can only appear at a specific position in a permutation. Below, we use \(\pi _i\) to indicate a specific node \(v\) in \(Q\) at the ith position. We indicate a permutation over \(Q\) as \(T = (\pi _1, \pi _2, \ldots , \pi _n)\). Given a permutation \(T\) over \(Q\), the distance of a Hamilton path by \(T\) is defined as \(d (T) = \sum _{1 \le i < n} d ({\pi _i},{\pi _{i+1}})\), and the distance of a Hamilton circuit by \(T\) is defined as \(d (T) = \sum _{1 \le i < n} d ({\pi _i},{\pi _{i+1}}) + d ({\pi _{n}},{\pi _1})\). Let \(\mathcal {T}\) be the set of all possible paths (circuits) for \(Q\). The size of \(\mathcal {T}\) is \(|\mathcal {T}| = n!\) for Hamilton paths, and the size is \(|\mathcal {T}| = \frac{(n- 1)!}{2}\) for Hamilton circuits.

In this paper, we focus on Hamilton circuit, and may use “circuit,” “tour” and “route” interchangeably, since they are all used in reported studies. Among all possible permutations in \(\mathcal {T}\), the optimal Hamilton circuit over \(Q\) is the shortest Hamilton circuit, denoted as \(T ^*\), such that \(d (T ^*) = \min _{T \in \mathcal {T}}d (T)\). The problem of finding the optimal Hamilton circuit is known as traveling salesman problem (TSP), which is known to be NP-hard. The error-ratio for an approximate \(T\) is defined below.

$$\begin{aligned} {\mathsf {eratio}} (T) = \frac{d (T)-d (T ^*)}{d (T ^*)} \end{aligned}$$

It is worth mentioning that the TSP problem we study in this paper is the symmetric and metric TSP. Here, by symmetric it implies \(d ({u},{v}) = d ({v},{u})\), and by metric it implies \(\forall u, v, w \in V, d ({u},{w}) + d ({w},{v}) \ge d ({u},{v})\).

The Problem

In this paper, we study TSP query to find the shortest Hamilton circuit \(T^*\) for a given TSP query, \(Q\), which is a set of points, and explore the similarities and differences among 22 heuristics proposed for TSP using real datasets and new synthetic datasets in addition to the existing benchmarks and uniform/normal synthetic datasets.

3 The Heuristics

The TSP heuristics have been studied. In this work, we focus on tour construction heuristics [21]. By tour construction, it computes a tour (or a circuit) following some rules, and takes the resulting tour by the rules as the final result without further refinement. In [21], construction heuristics are divided into 3 categories: heuristics designed for speed, tour construction by pure augmentation and more complex tour construction. In this work, we cover more heuristics, and divide 22 construction heuristics into 3 new categories, namely (1) space-partitioning-based heuristics, (2) node-based heuristics and (3) edge-based heuristics. Table 2 lists all the 22 heuristics studied, where some are with guarantee of the approximate ratio. Below, we discuss 2 space-partitioning-based heuristics in Sect. 3.1, 4 edge-based heuristics in Sect. 3.3 and 16 node-based heuristics in Sect. 3.2.

3.1 Space-Partitioning-Based Heuristics

The space-partitioning-based methods compute TSP for a given set of points \(Q\) in three main steps: (1) partition nodes in \(Q\) into smaller subsets based on their pairwise Euclidean distances, (2) connect the nodes in the same subset into a Hamilton path and (3) determine the Hamilton circuit for \(Q\) by linking all Hamilton paths obtained for all subsets. We discuss two heuristics, namely \(\mathsf {Strip}\) and \(\mathsf {Hilbert}\).

First, \(\mathsf {Strip}\) computes the minimum bounding rectangle (MBR) in two-dimensional space that encloses all the query nodes \(Q\) of size \(n = |Q |\), and partitions the MBR into \(\sqrt{\frac{n}{3}}\) equal-width vertical strips. For each vertical strip, \(\mathsf {Strip}\) sequences all the inside nodes according to y-coordinate by alternately top to bottom and bottom to top. The final circuit is determined by connecting all the sequences computed for all strips. \(\mathsf {Strip}\) only involves sorting by x-coordinate and y-coordinate.

figure a

Second, the space filling curve is a widely used technique to map multidimensional data into one-dimensional data. The main idea behind the space filling curve is that it keeps the locality information of the original data after mapping such that two near nodes may still be close to each other after mapping. Therefore, visiting query nodes in \(Q\) in the order of their appearance along the space filling curve reduces the total length [29]. By space filling curve, it can recursively partition the whole plane into small units, where a unit is labeled with a string of binary digits based on where it is in the hierarchy. For instance, if the entire plane is divided into two units, one is labeled with “0,” and the other is labeled with “1.” Then it can get 4 units by further dividing each of the unit into another 2 smaller units in the similar manner. Such partitioning stops until there is at most one node at each unit. In this paper, we focus on the \(\mathsf {Hilbert}\) curve (or \(\mathsf {Hilbert}\) space filling curve) since it has better locality-preserving behavior.

Both \(\mathsf {Strip}\) and \(\mathsf {Hilbert}\) are easy to implement and are efficient. However, they only utilize pairwise distances to reduce the total route length, and neglect the overall distribution of all query nodes, which sacrifices accuracy. To improve the accuracy, in every step, there are several strategies that can be adopted to make the final circuit as short as possible. The state-of-the-art approximate algorithm [1] is based on partition. However, it utilizes dynamic programming to connect inner and inter nodes, which is beyond the scope of this work on simple construction heuristics.

3.2 Edge-Based Heuristics

The edge-based heuristics are based on the minimum spanning tree (MST). We discuss the greedy (\(\mathsf {Greedy}\)) which is known as multiple fragment heuristic, double-MST (\(\mathsf {DMST}\)), the Christofides algorithm (\(\mathsf {Chris}\)) and the savings algorithm (\(\mathsf {SV}\)).

First, \(\mathsf {Greedy}\) is designed based on the Kruskal’s algorithm [22] to find the minimum spanning tree for a undirected graph. As shown in Algorithm 1, it inserts every pair of u and v in \(Q\) as an edge into a min-heap \({\mathcal {H}}\) with \(d (u, v)\). In the while loop, it picks up the edge (uv) from \({\mathcal {H}}\), which is with the minimum distance, and checks if such an edge (uv) can connect two different subtrees as a larger subtree without a cycle. In addition, it further checks if the tree formed can end up a TSP by ensuring that the degree of u/v (\(\mathrm{deg}(u)\)/\(\mathrm{deg}(v)\)) is less than or equal to 2. At the end, it connects the two nodes in T with degree 1 to form a circuit.

Second, \(\mathsf {DMST}\) is an algorithm which traverses the minimum spanning tree T constructed for the edge-weighted undirected graph representation for \(Q\). To obtain the circuit, it keeps the traversal order and skips the nodes which are traversed before.

Third, the Christofides algorithm (\(\mathsf {Chris}\)) finds the circuit as follows. (1) Given the edge-weighted undirected graph representation \(G = (V, E)\) for \(Q\), it finds the minimum spanning tree \(T = (V_T, E_T)\). (2) It identifies a subset of \(V_T\), denoted as \(V_O\), which includes all those nodes in \(V_T\) that have an odd degree. (3) It then constructs an induced subgraph \(G_O(V_O, E_O)\) from G. (4) It finds a minimum weighted perfect matching M from \(G_O\), where a perfect matching M is a set of edges that do not have any common nodes. (5) It constructs a multigraph \(G_H = T \cup M\). (6) It then finds an Eulerian circuit in \(G_H\), because every node in \(G_H\) has an even degree. (7) Finally, it obtains the Hamilton circuit by removing the repeated nodes from the Eulerian circuit.

Fourth, the savings algorithm (\(\mathsf {SV}\)) takes a different approach, and does not build a circuit using a minimum spanning tree. \(\mathsf {SV}\) starts from a randomly selected node as the central node \(v_c\) and then builds a pseudo tour, \(T_P\), from the central node \(v_c\) to all other nodes in \(Q\). In order to make the tour short, \(\mathsf {SV}\) looks for shortcuts in the pseudo tour \(T_P\) constructed. In every iteration, \(\mathsf {SV}\) selects a pair of nodes u and v that connect with \(v_c\) in \(T_P\) based on Eq. (3), deletes the edge \((u, v_c), (v, v_c)\), and inserts a new edge (uv) as shortcut. In order to find the shortcut with maximum benefit, it defines a new cost function as:

$$\begin{aligned} (u, v) = {\mathop {\mathrm{argmin}}\limits _{u,v \in T_P \wedge u \ne v \ne v_c}} \{{c '(u, v)}\} \end{aligned}$$

where the cost function \(c '\) is given in Eq. (4).

$$\begin{aligned} c '(u, v) = d ({u},{v}) - d ({u},{v_c}) - d ({v},{v_c}) \end{aligned}$$

All the edge-based heuristics aim at finding the edge with the smallest distance directly into the circuit. \(\mathsf {DMST}\) and \(\mathsf {Chris}\) have a better approximate ratio than most heuristics studied in this work because a TSP circuit becomes a tree if any edge is removed from it, whose total length should be smaller than that of MST.

3.3 Node-Based Heuristics

The node-based heuristics construct a circuit by expanding the nodes in \(Q\) one by one until all of them are visited. There are three main issues in the heuristics, which are (a) how to initialize an initial node(s) and (b) in every iteration, how to select the next node to expand and where it is for the next node to be inserted. Among the node-based heuristics, we discuss the nearest-neighbor heuristics, the insertion heuristics, the convex hull-based insertion heuristics, the addition heuristics, and the augmented addition heuristics. Algorithm 2 shows the framework of node-based heuristics.

figure b

The Nearest-Neighbor Heuristics [3]

The nearest-neighbor heuristics do not spend time on finding an initial TSP by init(Q), and randomly picks one node from Q. In other words, T by init(Q) contains only one node randomly selected. Then, in every iteration in the while loop, it picks up a point from the nodes that have not been selected before, namely from \(Q \setminus T\), and inserts it into the end of the current partial path computed in the previous iteration. Assume \(T_i = (\pi _1, \pi _2, \ldots , \pi _i)\) is the partial path computed at the ith iteration. There are two ends in \(T_i\), namely \(\pi _1\) and \(\pi _i\). Consider the (i+1)th iteration to expand the path by adding one more node. The nearest-neighbor heuristic (\(\mathsf {NN}\)) selects the nearest-neighbor node to the node at the position \(\pi _i\) from \(Q{\setminus}T_i\), and inserts the node selected at \(\pi _{i+1}\). On the other hand, the double-ended nearest-neighbor heuristic (\(\mathsf {DENN}\)) considers the nearest-neighbor node to either of the two end points: \(\pi _1\) and \(\pi _i\). Assume the node selected is near to the node at \(\pi _1\), \(\mathsf {DENN}\) will insert the newly selected node at \(\pi _1\) in \(T_{i+1}\) and place the node at the jth position (\(\pi _j\)) in \(T_i\) at (j+1)th position \(\pi _{j+1}\) in \(T_{i+1}\). Otherwise, if the node selected is near to the node \(\pi _i\), \(\mathsf {DENN}\) behaves like \(\mathsf {NN}\). Both \(\mathsf {NN}\) and \(\mathsf {DENN}\) expand a path. After the while loop, both obtain the TSP by adding one edge from the last node to the first node in T. We omit such post-processing from Algorithm 2. In brief, comparing with \(\mathsf {NN}\), \(\mathsf {DENN}\) considers both ends of the current partial path when expanding and selects the one with shorter length. Consequently, \(\mathsf {DENN}\) consumes longer time than \(\mathsf {NN}\) to improve the accuracy.

Fig. 2
figure 2

A TSP Example

The Insertion Heuristics [31]

Like the nearest-neighbor heuristics, the insertion heuristics randomly pick one node from Q by init(Q). Unlike the nearest-neighbor heuristics which expand the current partial path in every iteration, the insertion heuristics enlarge the current partial circuit in every iteration. Let \({\mathsf {T}} _i\) be the partial circuit over nodes of size i such that \({\mathsf {T}} _i = (\pi _1, \pi _2, \ldots , \pi _i, \pi _1)\). In the (i+1)th iteration, the insertion heuristics attempt to add one node into the current circuit by minimizing the increment of the total distance of the circuit. There are two things. One is how to select a node, w, from \(Q \setminus {\mathsf {T}} _i\). The other is how to insert w into \({\mathsf {T}} _i\) to obtain \({\mathsf {T}} _{i+1}\). We first discuss how to insert a new node into \({\mathsf {T}} _i\), assuming the node to be inserted next is selected from \(Q \setminus {\mathsf {T}} _i\). We will discuss the node selection next (Table 1).

Consider an insertion of a node \(w\,(\notin {\mathsf {T}} _i)\) between u and v in \({\mathsf {T}} _i\). Here, for simplicity, we say to insert a node w into an edge (uv) in \({\mathsf {T}} _i\), where an edge (uv) implies that v is next to u in the permutation. In the new circuit to be, \({\mathsf {T}} _{i+1}\), the edge (uv) in \({\mathsf {T}} _i\) will be replaced by two edges (uw) and (wv). Among all edges in \({\mathsf {T}} _i\), the edge, (uv), selected for a given node w is to minimize the incremental cost by Eq. (5).

$$\begin{aligned} (u, v) = {\mathop {\mathrm{argmin}}\limits _{(u, v) \in {\mathsf {T}} _i}} \{c (u, v, w)\} \end{aligned}$$

where \(c (u, v, w)\) is a cost function to measure the incremental cost of inserting a node between two nodes (an edge) as given in Eq. (6).

$$\begin{aligned} c (u, v, w) = d (w, u) + d (w, v) - d (u, v) \end{aligned}$$

Next consider how to select the next node. There are 4 ways to select the next node w to be inserted into \({\mathsf {T}} _i\), namely the random insertion (\(\mathsf {RI}\)), the nearest insertion (\(\mathsf {NI}\)), the cheapest insertion (\(\mathsf {CI}\)) and the furthest insertion (\(\mathsf {FI}\)) [28]. Here, \(\mathsf {RI}\) randomly picks one as the next node w. \(\mathsf {NI}\) selects the next node w from \(Q \setminus {\mathsf {T}} _i\) that has the smallest distance to a node in \({\mathsf {T}} _i\) (Eq. 7)

$$\begin{aligned} w = {\mathop {\mathrm{argmin}}\limits _{w \notin {\mathsf {T}} _i}}\{d ({w},{v}), \forall v \in {\mathsf {T}} _i\} \end{aligned}$$

Note that \(\mathsf {NI}\) inserts the nearest node to the current circuit, instead of the end node as done by the nearest-neighbor heuristics (\(\mathsf {NN}\) or \(\mathsf {DENN}\)). \(\mathsf {CI}\) selects the next node based on the cost function.

$$\begin{aligned} w = {\mathop {\mathrm{argmin}}\limits _{w \notin {\mathsf {T}} _i}}\{c (u, v, w), \forall (u, v) \in {\mathsf {T}} _i\} \end{aligned}$$

We discuss the similarity between \(\mathsf {NI}\) and \(\mathsf {CI}\) in certain cases. Suppose the next node to be inserted is w. Assume \(v'\) is the node in \({\mathsf {T}} _i\) under which w is selected by Eq. (7), and assume (uv) is the edge under which the next node w is selected by \(\mathsf {CI}\) based on Eq. (8) As proved in [5], (uv) should satisfy at least one of the 3 conditions: (1) \(u = v'\) or \(v = v'\), which means one endpoint of this edge in \({\mathsf {T}} _i\) is the nearest neighbor to w; (2) given a circle C centered at w with the radius \(1.5 \times d (w, v')\), then either \(u \in C\) or \(v \in C\), which means one endpoint of this edge is inside a circle centered at w; and (3) for every pair of \((v_i, v_j)\) in \({\mathsf {T}} _i\), given a circle \(C_i\) centered at \(v _i\) with radius \(1.5 \times d ({v _i},{v _j})\), then either \(w \in C_{i}\) or \(w \in C_{j}\), which means w is inside the corresponding circle of u or v. Here, if the condition 1 is satisfied, \(\mathsf {CI}\) selects the same node as \(\mathsf {NI}\) does.

Unlike the 3 heuristics to select the next node discussed above, \(\mathsf {FI}\) picks up the next node which is far away from \(T_i\) with a higher priority. It chooses the next node following Eq. (9).

$$\begin{aligned} w = {\mathop {\mathrm{argmax}}\limits _{w \notin {\mathsf {T}} _i}}\{d (w, {\mathsf {T}} _i)\} \end{aligned}$$

where \(d (w, {\mathsf {T}} _i)\) is the smallest distance from a node \(w \notin {\mathsf {T}} _i\) to a node in the current \({\mathsf {T}} _i\) such that \(d (w, {\mathsf {T}} _i) = \min _{v \in T _p}d (v, w)\).

Table 1 Expansion order of the node-based heuristics
Table 2 Twenty-two heuristics (default start node: \(v _1\))

Convex Hull-Based Insertion Heuristics

The insertion heuristics pick the start node randomly, which may affect the quality of the result. As pointed out in [27], it is an effective approach to construct an initial circuit for the points \(Q\) and then insert the remaining nodes into the initial circuit. It is proved that for the optimal circuit, the nodes lying on the boundary of the convex hull will be visited in their cyclic order [14]. This suggests that the convex hull can serve as a sketch to guide future insertions. The convex hull-based insertions are proposed to find the convex hull of all nodes in \(Q\) first using \(T \leftarrow init(Q)\) and then compute the circuit using one of the insertion heuristics for the remaining nodes in \(Q \setminus T\). Here, we investigate 4 heuristics: convex hull cheapest insertion (\(\mathsf {CHCI}\)), convex hull nearest insertion (\(\mathsf {CHNI}\)), convex hull random insertion (\(\mathsf {CHRI}\)) and convex hull furthest insertion (\(\mathsf {CHFI}\)). Note that [21] only shows the testing results for \(\mathsf {CHCI}\).

The Addition Heuristics [5]

The insertion heuristics determine an edge for a node to be inserted among all the edges in \({\mathsf {T}} _i\). To further reduce the computational cost, for a node w selected from \(Q \setminus {\mathsf {T}} _i\), the addition heuristics pick a node v at \(\pi _j\) on the current circuit \({\mathsf {T}} _i\) and only consider the insertion of w either between two nodes at \((\pi _{j-1}, \pi _j)\) or between two nodes at \((\pi _j, \pi _{j+1})\). The edge can be selected with the smallest cost by Eq. (6). We study 3 ways to select the next node w, namely random addition (\(\mathsf {RA}\)), nearest addition (\(\mathsf {NA}\)) and furthest addition (\(\mathsf {FA}\)). We do not study cheapest addition, since the insert position by the cheapest heuristic is decided once the next node is determined. Here, \(\mathsf {RA}\) selects the next node w from \(Q \setminus {\mathsf {T}} _i\) randomly, and identifies \(v \in {\mathsf {T}} _i\) based on Eq. (10).

$$\begin{aligned} v = {\mathop {\mathrm{argmin}}\limits _{v \in {\mathsf {T}} _i}}\{d (w, v)\} \end{aligned}$$

\(\mathsf {NA}\) selects a pair of nodes, w and v such that \(w \notin {\mathsf {T}} _i\) and \(v \in {\mathsf {T}} _i\) in a similar way like the nearest insertion, based on Eq. (11).

$$\begin{aligned} (v, w) = {\mathop {\mathrm{argmin}}\limits _{v \in {\mathsf {T}} _i, w \notin {\mathsf {T}} _i}}\{d (v, w)\} \end{aligned}$$

Here, w is the next node to be selected, and v is the node at \(\pi _j\) position in the current \({\mathsf {T}} _i\). \(\mathsf {FA}\) selects the next node w based on Eq. (9), and identifies the insertion position \(\pi _j\) by Eq. (10), and inserts w either between \(\pi _{j-1}\) and \(\pi _{j}\) or between \(\pi _{j}\) and \(\pi _{j+1}\) following the minimal incremental cost (Eq. 6).

Augmented Addition Heuristics

Consider the insertion heuristics and the addition heuristics. On the one hand, the insertion heuristics explore all edges in \({\mathsf {T}} _i\) to insert a node w between a pair of nodes, u and v, as an edge (uv). On the other hand, the addition heuristics only consider 2 edges incident to the node v at \(\pi _j\), when inserting w. Different from insertion/addition heuristics, the augmented additions attempt to explore more than 2 edges up to some extent. Here, like the addition heuristics, they select the node \(w\,(\notin {\mathsf {T}} _i)\) and the insertion position v at \(\pi _j\) in \({\mathsf {T}} _i\) using either \(\mathsf {RA}\), \(\mathsf {NA}\), and \(\mathsf {FA}\). Then, the augmented addition heuristics select an edge with the minimum cost from all edges in a circle centered at w with the radius of \(\alpha \cdot r\), where \(\alpha \ge 1\) and \(r = d (w, v)\) in the two-dimensional space. Here, an edge (uv) is in the circle if u or v appears in the circle. We denote such augmented addition heuristics as random augmented addition (\(\mathsf {RA+}\)), nearest augmented addition (\(\mathsf {NA+}\)) and furthest augmented addition (\(\mathsf {FA+}\)).

3.4 An Example

Figure 2 shows the optimal TSP, T, for a TSP query \(Q\) with 20 points, \(v_i\), for \(1 \le i \le 20\) sampled from NY. The positions of the 20 points are also given in Fig. 2, which forms 3 clusters, one with 10 points, \(\{v_i\}\), for \(8 \le i \le 17\), one with 2 points, \(\{v_7, v_8\}\) and one with 5 points, \(\{v_1, v_2, v_3, v_4, v_{20}\}\). There are some points which are at a distance from any of the clusters, such as \(v_5, v_{18}\) and \(v_{19}\). Table 2 shows the 22 heuristics in the three categories: space-partitioning-based, edge-based, and node-based. For each heuristic, the approximate ratio is given, if any, which is for a TSP, \(Q\), with n points. Also, in Table 2, the 5th column shows the \(\mathsf {eratio}\), for the TSP of 20 points shown in Fig. 2. Here, as default, we select \(v_1\) as the first node to start except for the 4 convex hull-based insertion heuristics, which identify a convex hull with nodes, \(v_8, v_{10}, v_{19}\), and \(v_4\), to start. In particular, we also show how the node-based heuristics select the next node in every iteration in Table 1. In terms of the accuracy, among the 22 heuristics, there are 3 heuristics that get the optimal TSP.

Table 3 Four real datasets
Table 4 Twenty Selected TSPLIB Benchmarks
Table 5 Parameters of new synthetic datasets

4 Performance Studies

We study all 22 heuristics covering a large range of datasets: 4 real datasets, 20 datasets from TSPLIB benchmark [30], 2 existing synthetic datasets [21] and new synthetic datasets.

Datasets: The 4 real datasets used to test are shown in Table 3. Here, NY (New York) and BJ (Beijing) are real POIs of the two cities. LA (Los Angeles) and HK (Hong Kong) are real check-in data we crawled from the location-based social network in the two cities from Twitter (https://twitter.com/) and Gowalla, respectively.

The 20 datasets selected from the TSPLIB benchmark [30] cover 3 major types of TSPLIB data: ATT, EUD_2D, and GEO, and are summarized in Table 4, where the 1st and 4th columns are the short names, the 2nd and 5th columns are the names used in the benchmark, and the 3rd and 6th columns are the size of points used in the dataset. There are 10 datasets where \(n = |Q |\) is selected between 100 and 1000, denoted as TB_H, and there are 10 datasets where \(n = |Q |\) is selected between 1000 and 10000, denoted as TB_T.

Table 6 Average inter-cluster distance (normalized)

We discuss our new synthetic datasets generation in this work. Note that the 2 existing synthetic datasets by uniform distribution and normal distribution [21] are not the best for LBS, since there is a gap between uniform/normal and the real datasets. In this work, we simulate that people visit n POIs in a real dataset of N POIs, where a small number of hot POIs are visited by many people. In brief, we synthesize large datasets of size N followed by randomly selecting \(n = |Q |\) from the large synthetic datasets. A synthetic dataset is generated with \(N = 100,000\) points using 3 parameters, namely the number of clusters (\(K\)), the inter-cluster distance (\(l\)) and the distribution of points in clusters (\(\alpha\)). First, the total number of points \(N = 100,000\) is used because it is similar to the sizes of the real datasets we use (Table 3). Second, we randomly generate \(K\) points, as centers of the clusters, following uniform distribution in a square with size \(l \times l\). Let \(C_i\) and \(k_i\) be the ith cluster and the center of \(C_i\), for \(1 \le i \le K\). Third, we randomly generate \(N_i = \frac{100,000}{K}\) points for each of the \(K\) clusters, \(C_i\). The points in a cluster follow Gaussian distribution, which is widely used to model spatial data/events [13, 32] and user mobility [10]. Let \(\sigma _i\) be the variance for \(C_i\) with \(N_i\) points centered at \(k_i\). The covariance matrix of the Gaussian distribution will be in the form of \([[\sigma _i^2, 0], [0, \sigma _i^2]]\). All \(\sigma _i\) for all clusters \(C_i\) (\(1 \le i \le K\)) follow Pareto distribution (\(\alpha\)). The Pareto distribution is used because it is proved in [15] that human mobility patterns follow a power law distribution, which means a small number of places are visited by most people. The corresponding Gaussian distribution will be compact for a small \(\sigma\), i.e., the small region has high visiting frequency. Table 5 shows the parameters with the default values.

Fig. 3
figure 3

Pareto distribution

With our synthetic datasets, we can study different settings including uniform and normal. For the normal distribution, it is by setting \(K = 1\). where there is only one cluster. For the uniform distribution, it is by setting \(K\) as a large value, e.g., \(K = N\). We discuss \(l\) and \(\alpha\). Recall that \(l\) is a parameter to decide the size of plane. When \(l\) is large, the average inter-cluster distance will be large, and the overlap between clusters will be small, as shown in Table 6. We show the average inter-cluster distances by varying three parameters. It is nearly in proportionate to \(l\), and the varying of \(K\) and \(\alpha\) will have no influence on it. For Pareto distribution, the parameter \(\alpha\) decides the skewness. As shown Fig. 3, the larger \(\alpha\) is, the more skew the distribution will be.

Figure 4 shows the data distributions with different parameters. Each figure contains 3000 points similar to Fig. 1. Figure 4a shows the distribution with default settings, which simulates LA (Fig. 1d). Figure 4b shows the distribution when \(K = 1\). It is a normal distribution, similar to the benchmark shown in Fig. 1b. Comparing Fig. 4a, d shows the case when there are more clusters with a smaller variance (larger \(\alpha\)). Consequently, it is more skewed. Figure 4d shows how to simulate NY (Fig. 1c). Figure 4c has a larger \(l\).

Fig. 4
figure 4

Data distribution with different parameters. a \(K =64, l =10, \alpha =1\). b \(K =1, l =10, \alpha =1\). c \(K =64, l =50, \alpha =1\). d \(K =64, l =10, \alpha =4\)

TSP Queries

For real datasets, the TSP query size is \(n = |Q |\), where the default is 60. For TSPLIB datasets, we test it using the same number of points as given in the benchmark. For the 2 existing synthetic datasets, uniform and normal, used in [21], we generate a set of n points for a given size to test. For both real datasets and our synthetic datasets generated using the parameters with \(N = 100,000\) points, we conduct testing 100 times for a TSP query with n points randomly selected, and report the average.

The heuristics

We study the 22 heuristics listed in Table 2, which are implemented in C++ following [5, 21], where KD tree [4] is used for efficient search. The convex hull for a TSP query is implemented by Graham scan [16]. The implementation details can be found in [5]. We have conducted extensive experiments on a PC with two Intel Xeon X5550@2.67GHz CPU and 48GB main memory.

The Measures

We measure the heuristics by accuracy and efficiency. The accuracy is based on the error-ratio \(\mathsf {eratio}\) (Eq. 2), and the efficiency is based on CPU time. We focus on the accuracy, since all the heuristics are fast as reported in [5].

Below, we give an overview for the 22 heuristics using real datasets and the existing synthetic datasets in Sect. 4.1. We give details in terms of accuracy in Sect. 4.2, and discuss the issue whether we can find a TSP using indexing in Sect. 4.3. Finally, we discuss the heuristics using our new synthetic datasets in Sect. 4.4.

Fig. 5
figure 5

Accuracy versus Runtime (\(|Q | = 60\)). a BJ. b NY. c LA. d HK

4.1 Accuracy Versus Efficiency

Figure 5 shows CPU time and error-ratio for TSP queries of size \(n = 60\), for the 4 real datasets. The results shown in Fig. 5 for real datasets highlight the difference from the results conducted in [21] for 9 heuristics with \(n = 10{,}000\) using uniform distributed datasets. On the one hand, as shown in [21], \(\mathsf {SV}\) has the highest accuracy followed by \(\mathsf {FI}\), \(\mathsf {Greedy}\) is better than \(\mathsf {CHCI}\), and \(\mathsf {NN}\) is better than \(\mathsf {NI}\). On the other hand, for these 4 real datasets, as shown in Fig. 5, the convex hull-based insertion heuristics achieve near-optimal accuracy, especially for NY, LA, and HK, since BJ is close to the normal distribution due to its urban planning. In other words, the results of BJ share the similarity with those reported in [21].

CPU Time

The results of using real datasets are similar to the finding given in [21]. Both \(\mathsf {Strip}\) and \(\mathsf {Hilbert}\) are the fastest, as they only need to sort \(n\) values. \(\mathsf {NN}\) and \(\mathsf {DENN}\) are fast, because the nearest neighbors can be found efficiently using KD tree. The convex hull-based insertion heuristics spend time to select the initial convex hull as a sketch in \(O(n\lg n)\) time, which is cost-effective since it reduces the number of nodes for insertion. Therefore, the convex hull-based insertion heuristics can run even faster than the other insertion heuristics which randomly select an initial node to start. For the node-based heuristics, the main CPU cost is to select a node in iterations, and there are 4 expanding orders of nodes in iterations: the random order, the nearest order, the cheapest and the furthest. The random is the fastest, as it picks the next node for insertion randomly. The CPU cost for the nearest and the furthest has only marginal difference, since both can be done using KD tree. The cheapest takes the longest time, since it needs to calculate the possible insertion cost for every node and every edge in iterations. For the edge-based heuristics, \(\mathsf {Greedy}\) is the fastest. Its CPU time is comparable to that by the insertion heuristics. \(\mathsf {DMST}\) and \(\mathsf {Chris}\) build an minimum spanning tree before generating the route, and consume more time.

Fig. 6
figure 6

Accuracy of 22 Heuristics under 26 Datasets (\(n = 20, 40, 60, 80, 100\)). a BJ. b NY. c LA. d HK. e Uniform. f Normal. g TB_H. h TB_T

The Error-Ratio

In addition to the efficiency (x axis) and the accuracy (y axis) shown in Fig. 5, we further conduct testing for all 22 heuristics over 26 datasets: 4 real datasets (BJ, NY, LA, and HK), 2 synthetic datasets (uniform and normal) and 20 TSPLIB benchmarks (10 TB_H and 10 TB_T) for \(n = |Q |\) to be selected over 20, 40, 60, 80, and 100. Figure 6 shows the results using candlesticks for the TSP queries tested. The differences among the 22 heuristics in terms of accuracy are more obvious than the differences in terms of efficiency, for the datasets tested. Note that the accuracy is related to the distribution, whereas the efficiency is related to the heuristics. As shown in Fig. 6, in terms of accuracy, the range of \(\mathsf {eratio}\) is between 0.0001 and 10 (y axis). In a short summary, in general, the error-ratio of the 22 heuristics over the real datasets (BJ, NY, LA, HK) is lower than that of the synthetic datasets by uniform and normal as well as TB, especially for the best cases. Among the 4 real datasets, BJ follows the normal distribution, the error-ratio for the heuristics over BJ is relatively higher. Over NY, LA and HK, most heuristics can generate results with an error-ratio below 0.1 (below 10%). As a comparison, for normal and TB, the error-ratio is larger than 0.1 (10%). Among the 22 heuristics, the convex hull-based insertion heuristics are the best for most cases, whereas the space-partitioning-based heuristics are the worst. The insertion heuristics are better than the augmented addition heuristics which are better than the addition heuristics. For the edge-based heuristics, \(\mathsf {SV}\) and \(\mathsf {Chris}\) can be used to obtain accurate answers in some circumstances.

Fig. 7
figure 7

Nearest-neighbor-based heuristics. a NY. b LA. c Normal. d TB_H

4.2 The Accuracy

In this section, we focus on the accuracy over 2 real datasets, NY and LA, a synthetic dataset by normal, and TB_H benchmarks. The main purpose is to show that the heuristics behave differently in real datasets comparing to normal and TB_H benchmarks. We focus on the node-based and edge-based heuristics, and do not discuss the space-partitioning heuristics since they do not show their advantages in terms of accuracy for LBS.

The Nearest-Neighbor Heuristics

Figure 7 shows how the accuracy of \(\mathsf {NN}\)/\(\mathsf {DENN}\) changes while increasing query size \(n\). For the real datasets, the error-ratios increase monotonically. However, it does not hold for non-real datasets, especially for the 10 datasets in TB_H. In TB_H, the first with 101 points has the largest error. This suggests that the accuracy is related to data distribution. In terms of accuracy, \(\mathsf {DENN}\) is a little better than \(\mathsf {NN}\). When \(n\) is small, for \(n = 20\), the error-ratios for the two over NY and LA are less than \(10\%\), whereas the error-ratio is around 20% for normal, and is even higher than 30% for TB_H. When \(n\) becomes larger, the error-ratio for NY and LA increases noticeably, as shown in Fig. 7a, b, which is different from the error-ratios observed from normal.

Fig. 8
figure 8

insertion Heuristics. a NY. b LA. c Normal. d TB_H

Fig. 9
figure 9

Convex hull-based insertion heuristics. a NY. b LA. c Normal. d TB_H

Fig. 10
figure 10

Addition Heuristics. a NY. b LA. c Normal. d TB_H

The Insertion Heuristics

Figure 8 shows the error-ratio for the insertion heuristics. Like Fig. 7, the error-ratio increases with query size for the real datasets (NY and LA). The error-ratios for real datasets are much better than that of non-real datasets. The error-ratios range from 0 to 10% for NY and LA, from 0 to 25% for normal, and can be up to 35% for TB_H. Among the 4 insertion heuristics, \(\mathsf {CI}\) and \(\mathsf {NI}\) perform in a similar way. It is surprised to notice that \(\mathsf {RI}\) can perform better than \(\mathsf {CI}\) and \(\mathsf {NI}\) in terms of error-ratio. \(\mathsf {FI}\) performs the best. Both \(\mathsf {RI}\) and \(\mathsf {FI}\) have an error-ratio less than 5% for NY, is even less than 3% for LA for all queries tested on average.

The Convex Hull-Based Insertion Heuristics

They are the optimal choices for most cases, as shown in Fig. 6. Different from the insertion-based heuristics (\(\mathsf {RI}\), \(\mathsf {NI}\), \(\mathsf {CI}\), and \(\mathsf {FI}\)) which randomly pick up a node to start, the convex hull-based heuristics find the convex hull as a sketch for a TSP query first, and expands the remaining nodes under the guide of the sketch in a similar way as the insertion-based (\(\mathsf {RI}\), \(\mathsf {NI}\), \(\mathsf {CI}\) and \(\mathsf {FI}\)). There are \(\mathsf {CHRI}\), \(\mathsf {CHNI}\), \(\mathsf {CHCI}\), and \(\mathsf {CHFI}\). First, comparing Fig. 9 with Fig. 8, the convex hull-based heuristics outperform the corresponding insertion-based heuristics. The error-ratios are noticeably reduced. Second, the reduction on error-ratio by the convex hull changes the order of the heuristics in terms of the accuracy. Consider Figs. 8a and 9a for NY. Without the convex hull, as shown in Fig. 8a, \(\mathsf {CI}\) performs the worst. Both \(\mathsf {RI}\) and \(\mathsf {FI}\) perform in a similar way, and when \(n = 100\), \(\mathsf {RI}\) even outperforms \(\mathsf {FI}\). On the other hand, with the convex hull, as shown in Fig. 9a, \(\mathsf {CHNI}\) (or convex hull plus \(\mathsf {NI}\)) performs the worst, and \(\mathsf {CHFI}\) (or convex hull plus \(\mathsf {FI}\)) outperforms others.

Fig. 11
figure 11

Augmented addition heuristics. a NY. b LA. c Normal. d TB_H

Fig. 12
figure 12

Edge-based heuristics. a NY. b LA. c Normal. d TB_H

Fig. 13
figure 13

Effectiveness of Start Nodes (NY). a \(\mathsf {NN}\)/\(\mathsf {DENN}\). b Insertion. c Addition. d Edge-Based

Fig. 14
figure 14

Error-Ratio in the ith Iteration. a Insertion. b Convex Hull-based Insertion. c Addition. d Augmented Addition

Fig. 15
figure 15

Reoptimization. a Insertion. b Convex hull-based insertion. c Addition. d Augmented addition

The Addition Heuristics

The addition heuristics pick the next node to insert in the same way as the corresponding insertion heuristics. As shown in Table 1 for the TSP example in Fig. 2, both \(\mathsf {FI}\) and \(\mathsf {FA}\), and both \(\mathsf {NI}\) and \(\mathsf {NA}\) have the same expanding order to select the next node in iterations, respectively. On the other hand, the addition heuristics do not consider all the insertion positions in the current circuit, instead consider only between the two edges that are incident to a node v in the current circuit, where v is the nearest neighbor to the next node selected to insert. As expected, the addition heuristics cannot outperform the corresponding insertion-based heuristics. This is also observed by comparing Fig. 10 with Fig. 8. The advantage of the addition heuristics is the efficiency, since they check much less number of insertion positions on the current circuit in iterations.

It is worth noticing that there are two main things in iterations. One is to select the next node to insert, and the other is to find an insertion position to insert. The following heuristics, \(\mathsf {RI}\), \(\mathsf {CHRI}\) and \(\mathsf {RA}\) are to select the next node randomly in iterations. As shown in Figs. 8 and 9, \(\mathsf {RI}\) and \(\mathsf {CHRI}\) perform well. However, Fig. 10 shows that \(\mathsf {RA}\) does not perform well, and performs the worst for NY. This suggests that such random heuristics need to explore a certain number of insertion positions on the current circuit, in order to achieve a better accuracy. The same occurs to the furthest heuristics. Both \(\mathsf {FI}\) and \(\mathsf {CHFI}\) perform well, but \(\mathsf {FA}\) does not perform well on the other hand due to the limited number of exploring insertion positions.

The Augmented Addition Heuristics

Such heuristics are positioned between the insertion heuristics and the addition heuristics, due to the ways of exploring insertion positions on the current circuit. Figure 11 shows the results for \(\mathsf {NA+}\), \(\mathsf {RA+}\) and \(\mathsf {FA+}\). Comparing Fig. 10 and Figs. 8, 11 shows that, by the limited additional number of insertion positions, \(\mathsf {RA+}\) and \(\mathsf {FA+}\) outperform \(\mathsf {NA+}\), in a similar way as the corresponding \(\mathsf {RI}\) and \(\mathsf {FI}\) outperform \(\mathsf {NI}\) and the corresponding \(\mathsf {CHRI}\) and \(\mathsf {CHFI}\) outperform \(\mathsf {CHNI}\).

The Edge-Based Heuristics

The performance studies done in [21] using \(n = 10{,}000\) points under the uniform distribution conclude that \(\mathsf {SV}\) is the best among all 9 the heuristics tested. \(\mathsf {FI}\) outperforms \(\mathsf {Greedy}\) which in turn outperforms \(\mathsf {CHCI}\). Different from the results reported in [21] using \(n = 10,000\) points under the uniform distribution, as shown in Fig. 6, with many different datasets, \(\mathsf {SV}\) is not the best, even though \(\mathsf {SV}\) like other edge-based heuristics can get better accuracy. Figure 12 shows the details for the edge-based heuristics. \(\mathsf {SV}\) performs the best. For LA, the error-ratio for \(\mathsf {SV}\) ranges from 0.2 to 2.3%, which is marginally higher than the error-ratio for \(\mathsf {CHFI}\) that ranges from 0.1 to 1.2%. Both \(\mathsf {Chris}\) and \(\mathsf {DMST}\) are based on MST, and \(\mathsf {Chris}\) outperforms \(\mathsf {DMST}\). Note that the approximate ratio is 1.5 for \(\mathsf {Chris}\), and is 2 for \(\mathsf {DMST}\). As shown in Fig. 5, \(\mathsf {Chris}\) takes longer CPU time since it needs to find a perfect matching and Euler circuit after generating the minimum spanning tree. \(\mathsf {Greedy}\) performs well as reported in [21]. However, in our testing settings, \(\mathsf {Greedy}\) is inferior of \(\mathsf {Chris}\) in the two synthetic datasets NY and LA, generates similar results with \(\mathsf {DMST}\) in NY, and is the worst for LA.

Fig. 16
figure 16

Impact of parameters: An overview. a Query size. b Cluster number. c Inter-cluster distance. d Cluster distribution

Fig. 17
figure 17

Impact of Parameters: The Details. a Query size. b Cluster number. c Inter-cluster distance. d Cluster distribution

The Effectiveness of Start Nodes

All heuristics except for the convex hull-based need to select a start node to start expanding randomly, which may have a great impact on the resulting circuit. We study the selection of start nodes using the real dataset NY by varying the number of points \(n = 20, 40, 60, 80, 100\). For each n value, for example, \(n = 60\), we randomly select 100 sets of points of size n from NY. Assume \(Q\) is one of the 100 sets for a given n. We test a certain heuristics by selecting every node in \(Q\) as a start node. Given a query size n, we record the difference between the shortest and longest routes for every start node in every of the 100 randomly selected sets, and show the average difference in Fig. 13. It shows how the choice of start node affects the quality. As observed, the error-ratios increase monotonically while the query size increases. This suggests that the random selection of the first node to start is more sensitive when there are more node. As shown in Fig. 13, among the 2 nearest-neighbor heuristics, \(\mathsf {DENN}\) is more stable than \(\mathsf {NN}\), given that \(\mathsf {DENN}\) is only difference from \(\mathsf {NN}\) by looking at the two ends of the current path in iterations, instead of only one end. For the insertion heuristics, \(\mathsf {CI}\), \(\mathsf {NI}\), and \(\mathsf {FI}\) perform in a similar way, whereas \(\mathsf {RI}\) is more sensitive to the first node selected to start (Fig. 13b). The sensitive of the random is also shown in Fig. 13c where \(\mathsf {RA}\) is the worst, and \(\mathsf {NA}\) and \(\mathsf {FA}\) outperform \(\mathsf {RA}\) significantly. For the edge-based heuristics, Fig. 13d shows that it is less sensitive for the edge-based heuristics to select the start node. \(\mathsf {Chris}\) is the best among all heuristics, in terms of random selection of the start node.

4.3 More on Accuracy

We further analyze the error-ratios for the insertion, the convex hull-based insertion, the addition, and the augmented addition heuristics, and we focus on three issues: (a) the error-ratios in iterations, (b) the error correlation between the intermediate error-ratios and the final error-ratio and (c) the possibility of reoptimization to obtain the (i+1)th expansion by heuristics given the optimal TSP for the first i nodes selected. We conduct testing over the real dataset NY, for \(n = 60\). We randomly select 100 sets of points of size \(n = 60\) from NY, and report the average. Let \({\mathsf {T}} _i\) be a circuit with i nodes, and let \({\mathsf {T}} ^*_i\) be the optimal circuit over the same set of nodes.

The Error-Ratio in the i th Iteration

The error-ratio \({\mathsf {eratio}} ({\mathsf {T}} _i)\) is computed by Eq. (2) for \({\mathsf {T}} _i\) in the ith iteration. Figure 14 shows the results. Several observations are made. First, for the random methods (\(\mathsf {RI}\), \(\mathsf {CHRI}\), \(\mathsf {RA}\) and \(\mathsf {RA+}\)), the error-ratios increase in the ith iteration when i becomes larger. The error-ratio in the ith iteration becomes comparatively smaller, if it tries to find an insertion position among more choices, i.e., \(\mathsf {RI}\) is better than \(\mathsf {RA+}\), which is better than \(\mathsf {RA}\). Among all the random-based methods, \(\mathsf {CHRI}\) is best given the convex hull computed. Second, the error-ratios for the addition heuristics are high due to the limited insertion positions in iterations. Third, for the nearest-neighbor methods (\(\mathsf {NI}\), \(\mathsf {CHNI}\), \(\mathsf {CHCI}\)), the error-ratio increases in the first iterations and then drop in the late iterations. The reason is that in the first iterations, it takes near-to-far approach, whereas in the late iterations it may insert the next node between two nodes to refine the circuit. For \(\mathsf {CHCI}\), given the convex hull computed, it increases, and terminates before it finds the position to drop. Fourth, for the furthest methods (\(\mathsf {FI}\), \(\mathsf {CHFI}\), \(\mathsf {FA+}\), and \(\mathsf {FA}\)), the error-ratios are small and grow slowly. Among all heuristics, \(\mathsf {CHFI}\) performs the best.

The Reoptimization

For reoptimization, the rationale behind is whether there is any possibility to answer a TSP query based on an indexing to maintain certain length optimal circuit, even though it is known that the local optimal property does not hold. For a given heuristic method to obtain \({\mathsf {T}} _{i+1}\) by expanding a new node from \({\mathsf {T}} _i\), we consider expanding the new node into the optimal \({\mathsf {T}} ^*_i\) over the same set of nodes in \({\mathsf {T}} _i\) generated by the same method. We denote such a circuit obtained as \({\mathsf {T}} '_{i+1}\), and the error-ratio by \({\mathsf {eratio}} ({\mathsf {T}} ')\) is the error-ratio introduced in the last iteration only. Figure 15 shows the introduced error-ratio for different heuristics. The introduced errors are very small, which indicates that the reoptimization can generate high-quality solutions in most cases. For the addition heuristics, the error-ratio is higher than the others (Fig. 15c). For the insertion heuristics, \(\mathsf {FI}\) outperforms the others, and \(\mathsf {RI}\) also performs well. It is worth noting that when \({\mathsf {T}} _i\) becomes longer, the error-ratios become smaller. When \(i > 30\), the error-ratio is below 0.001, and when \(i > 40\), the error-ratio is close to zero. In general, the convex hull-based insertion better than the insertion.

4.4 The New Synthetic Datasets

In order to better understand the heuristics in real applications for LBS, we study the 22 heuristics in terms of accuracy using the new synthetic datasets proposed in this work.

Figure 16 gives an overview by candlesticks, where each figure is presented by varying the parameter in concern while fixing the other parameters by their default value. In this study, we use \(n = 100\) as the default query size. Figure 16a shows that most node-based heuristics are sensitive to the query size, especially of \(\mathsf {CHCI}\). The edge-based heuristics are stable with the change of \(n\). The difference between the best case and worst case of \(\mathsf {Chris}\) is just 1.5%. On the other hand, the edge base heuristics are sensitive to the cluster number.

We focus on 6 heuristics in Fig. 17, namely \(\mathsf {CHFI}\), \(\mathsf {CHRI}\), \(\mathsf {SV}\), \(\mathsf {CHCI}\), \(\mathsf {Greedy}\) and \(\mathsf {DENN}\), which are selected for the following reasons. \(\mathsf {CHFI}\), \(\mathsf {CHCI}\) and \(\mathsf {CHRI}\) are the better choices for real datasets tested, \(\mathsf {SV}\), \(\mathsf {CHCI}\), and \(\mathsf {Greedy}\) are the better choices as concluded in [21], and \(\mathsf {DENN}\) is selected as the nearest-neighbor heuristics which is used in LBS. Note that \(\mathsf {DENN}\) outperforms \(\mathsf {NN}\) in terms of accuracy.

Query Size (\(n\)): The error-ratios for all heuristics increase while increasing \(n\). As shown in Fig. 16a, it has higher influence on node-based heuristics (\(\mathsf {CHCI}\), \(\mathsf {CHRI}\) and \(\mathsf {CHFI}\)). When \(n\) is small, \(\mathsf {CHCI}\), \(\mathsf {CHRI}\) and \(\mathsf {CHFI}\) outperform \(\mathsf {SV}\). However, as \(\mathsf {SV}\) increases slower, \(\mathsf {SV}\) becomes the best when \(n \ge 400\). For \(\mathsf {Greedy}\), it looks constant when \(n \ge 80\), and will perform well when \(n\) is very large. This explains that \(\mathsf {Greedy}\) outperforms \(\mathsf {CHCI}\) in [21].

Cluster Number (\(K\)): As shown in Fig. 17b, the cluster number has little influence on the heuristics. The dataset follows a normal distribution when \(K = 1\) and gradually changes to uniform distribution when \(K\) becomes large. The same can be observed in Fig. 6e and f.

Inter-Cluster Distance (\(l\))

The inter-cluster distance controls the distance between clusters. Note that the overlap between clusters becomes small when \(l\) is large. The error-ratios for \(\mathsf {CHFI}\) and \(\mathsf {SV}\) increase slightly while increasing \(l\). For \(\mathsf {DENN}\), \(\mathsf {CHRI}\), \(\mathsf {CHCI}\), and \(\mathsf {Greedy}\), the error-ratio increases first and then decrease. \(\mathsf {SV}\) increases faster than \(\mathsf {CHFI}\). When \(l = 1\), \(\mathsf {SV}\) outperforms \(\mathsf {CHFI}\). However, \(\mathsf {CHFI}\) outperforms \(\mathsf {SV}\) when \(l \ge 10\) when the boundaries between clusters are more clear.

Cluster Distribution (\(\alpha\))

It is to control the variances in clusters. When \(\alpha\) is small, the data distribution is highly skewed, and most clusters have a small variance. On the other hand, when \(\alpha\) is large, more clusters have a large variance and the distribution over all clusters in terms of variance is more uniform. As shown in Fig. 17d, the error-ratio for \(\mathsf {CHFI}\) increases while increasing \(\alpha\) at the beginning and then decreases. The error-ratio for \(\mathsf {SV}\) increases monotonously. \(\mathsf {CHFI}\) is more suitable for the skewed case than \(\mathsf {SV}\).

As a summary, we conclude the following. First, \(\mathsf {CHFI}\) works well for real LBS applications, in particular, when the query size is relatively small and the query is highly skewed. Second, \(\mathsf {SV}\) is a better choice when the query size is large and query distribution is uniform as also observed in the existing work. Third, the nearest-neighbor heuristics is currently used in LBS for the efficiency, and is not the best for accuracy. Fourth, \(\mathsf {CHRI}\) can generate the near-optimal answers in many cases by randomly selecting next nodes to insert, and shows that the random heuristics is deserved to be investigated.

5 Conclusion

In this work, we investigate 22 construction heuristics for TSP in LBS by extensive performance studies over 4 real datasets, 20 datasets from TSPLIB benchmark, and 2 existing synthetic datasets. In addition, in order to understand real LBS setting, we also conduct extensive testing over the new synthetic datasets proposed in this work to simulate that a small number of hot POIs are visited by many people. Different from the existing work, we find that \(\mathsf {CHFI}\) works well for real LBS applications, whereas \(\mathsf {CHCI}\) get a good answer when the query size is small. Also, \(\mathsf {CHRI}\) can generate the near-optimal answers in many cases by randomly selecting next points to insert. In addition, for the issue of precomputing/indexing, we find that the quality of the circuit \({\mathsf {T}} _{i+1}\) by expanding a new point by heuristics from the optimal \({\mathsf {T}} ^*_i\) is high, which shows that it is deserved to study precomputing/indexing to support TSP queries.