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Destination-aware Task Assignment in SpatialCrowdsourcing: A Worker Decomposition

ApproachYan Zhao, Kai Zheng, Yang Li, Han Su, Jiajun Liu, and Xiaofang Zhou Fellow, IEEE

Abstract—With the proliferation of GPS-enabled smart devices and increased availability of wireless network, spatial crowdsourcing(SC) has been recently proposed as a framework to automatically request workers (i.e., smart device carriers) to performlocation-sensitive tasks (e.g., taking scenic photos, reporting events). In this paper we study a destination-aware task assignmentproblem that concerns the optimal strategy of assigning each task to proper worker such that the total number of completed tasks canbe maximized whilst all workers can reach their destinations before deadlines after performing assigned tasks. Finding the globaloptimal assignment turns out to be an intractable problem since it does not imply optimal assignment for individual worker. Observingthat the task assignment dependency only exists amongst subsets of workers, we utilize tree-decomposition technique to separateworkers into independent clusters and develop an efficient depth-first search algorithm with progressive bounds to prunenon-promising assignments. In order to make our proposed framework applicable to more scenarios, we further optimize the originalframework by proposing strategies to reduce the overall travel cost and allow each task to be assigned to multiple workers. Extensiveempirical studies verify that the proposed technique and optimization strategies perform effectively and settle the problem nicely.

Index Terms—Spatial Crowdsourcing, Spatial Task Assignment, Algorithm.

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1 INTRODUCTION

T HE increased popularity of GPS-equipped smart devices anddecreased cost of wireless mobile network (e.g., 4G network)

have enabled people to move as sensors and participate somelocation-based tasks. Spatial crowdsourcing is a recently proposedconcept and framework, which employs smart device carriersas workers to physically move to some specified locations andaccomplish these tasks.

One of the main research problems in spatial crowdsourcingis how to assign tasks to workers strategically. Existing worksfocused on assigning tasks to workers to maximize the totalnumber of completed tasks [13], the number of performed tasksfor a worker with an optimal schedule [7], or the reliabilityand diversity score of assignments [5]. An implicit assumptionshared by these work is that a worker can only or is willing toperform tasks that are close to her currently location (e.g., withina circle with given radius). While this is indeed realistic for manyapplications, we also observe some other scenarios where it isfeasible for workers to perform tasks beyond her spatial vicinity.For instance, a worker who is driving on road towards a certaindestination might not mind performing some tasks along the route

● Y. Zhao is with the Institute of Artificial Intelligence, School of ComputerScience and Technology, Soochow University, Suzhou, China and ZhejiangLab, China. Email: [email protected].

● Y. Li is with the School of Computer Science and Technology, SoochowUniversity, Suzhou, China. Email: [email protected].

● K. Zheng and H. Su are with University of Electronic Science and Technol-ogy of China, Chengdu, China. Email: {zhengkai, hansu}@uestc.edu.cn.*K. Zheng is the corresponding author of the paper.

● J. Liu is with Renmin University of China, Beijing, China. Email: [email protected].

● X. Zhou is with the University of Queensland, Brisbane, Australia andZhejiang Lab, China. Email: [email protected].

as long as the extra travel cost (e.g., detour cost) does not affecther scheduled deadline at the destination. Note that, these tasksare not necessarily close to her original location so a specific validrange cannot be defined for each worker.

In this paper, we investigate the task assignment of spatialcrowdsourcing under such a problem setting, namely Destination-aware Task Assignment (DATA). Specifically, given a user’s cur-rent location, destination and deadline, before which she needs toarrive at the destination, it aims at finding an optimal assignmentof tasks to workers such that the total number of task assignmentsis maximized. Note it actually consists of two sub-problems: 1)for each task, we need to assign it to the suitable workers; and2) for each worker, we need to schedule a sequence by whicha worker performs her assigned tasks. Compared to the previouswork, the hardness of our problem lies in that, once the travelcosts associated with moving to tasks’ locations and the expirationtime of tasks are taken into account, local optimal assignmentdoes not lead to global optimal result, that is assigning the mosttasks to each worker does not necessarily imply the maximumnumber of accomplished tasks by all workers. The only existingwork that considers task assignment and scheduling at the sametime is [8], in which an approximate approach is developed thatiteratively improves the assignment and scheduling to achievemore completed tasks. The second challenge is that the tasksreachable by each worker highly depend on the distance betweenorigin and destination as well as the tightness of deadline. Thismakes pruning infeasible tasks more difficult than the conventionalsettings, which specify a valid range for each worker [8], [13].

We propose an exact solution that finds the optimal assignmentresult in terms of the total number of task assignments. Themain idea of our approach is that, observing each worker onlyshares common tasks with a small portion of the entire workerset, we utilize a graph to represent the task dependency among



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workers (i.e., two workers sharing the same tasks have an edge in-between them) and apply a tree-decomposition procedure to divideall the workers into independent clusters. A top-down recursivesearch algorithm is then developed to traverse the tree in depth-first manner. In the meantime, dynamic upper and lower boundsare maintained during traversal in order to prune the tree nodesthat cannot lead to optimal results. Compared to the iterativeapproach [8], our method finds the final and optimal assignmentupon the completion of the search procedure, i.e., there is no re-matching and re-scheduling phase.

Although our previous work [27] has already achieved theoptimization goal of maximizing the overall task assignments, itfails to consider the travel cost (i.e., in time or distance) of theworkers during the assignment process, which is another criticalfactor since workers must physically go to the designated locationsin order to perform the assigned tasks on spatial crowdsourcingplatforms. However, the goals of maximizing the task assignmentand minimizing the travel cost are often conflicting, which meansoptimizing both simultaneously could be difficult. To address thisissue, in Section 4.1 of this extension, we incorporate a travelcost optimization strategy into the task assignment frameworkproposed in [27], which tries to minimize the overall travel cost ofworkers while keeping the number of task assignments unchangedby giving more priority to the performable task set with lowertravel cost for each worker.

The second limitation of our previous study [27] is that itcan only assign each task to a single worker. Nevertheless, someapplications require each task to be assigned to multiple workersdue to quality control purposes. Since allowing multiple workersto perform the same task (i.e., redundant task assignment mode)can affect the dependency relationship between workers duringthe independent worker partition phase, we carefully re-design ourprevious DATA solution in Section 4.2 to adapt it to the redundanttask assignment mode by introducing new algorithms for bothworker partition strategy and task assignment search strategy.

To summarize, our new technical contributions in this exten-sion are five folds.

1) We identify and study in depth two limitations in ourprevious DATA framework, which includes failing to considertravel cost factor and failing to support redundant task assignment.

2) We prove that the problems of Maximal Valid Task Setcalculation and DATA are both NP-hard.

3) We incorporate a travel cost optimization strategy into thetask assignment process, which tries to re-assign workers theperformable tasks with less travel cost whenever possible as longas the overall number of task assignment remains optimized.

4) We carefully re-design the worker partition algorithm andtask assignment algorithm to make the DATA framework applica-ble to scenarios where each task should be assigned to multipleworkers.

5) Extensive experiments are conducted to study the impactof the key parameters and effectiveness of our newly proposedalgorithms. In particular, compared with the original exact taskassignment approach, the travel cost optimization strategy canreduce the total travel cost by up to 24.35%, while the redundanttask assignment strategy can improve the overall task assignments,which guarantees at least 41.3% tasks can be assigned to multipleworkers in order to enhance the accuracy of task completion.

The remainder of this paper is organized as follows. Sec-tion 2 introduces the preliminary concepts and formulates thedestination-aware task assignment problem. The proposed algo-

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rithms and related techniques are presented in Section 3, followedby the extension in Section 4. We report the results from empiricalstudy in Section 5. Section 6 surveys the related work underdifferent problem settings and Section 7 concludes this paper.

2 PROBLEM DEFINITION

In this section, we define a set of preliminaries in the contextof self-incentivised single task assignment (i.e., a task can onlybe assigned to a worker) in spatial crowdsourcing with ServerAssigned Tasks (SAT) mode [13]. Table 1 lists the major notationsused throughout the paper.

TABLE 1Summary of Notations

Notation Definition

s Spatial taskls Location of spatial task ses Expiration time of spatial task smaxWs Maximum acceptable workers for task sw Workerlw Current location of worker wdw Destination of worker wtw Deadline of worker wspeedw Movement speed of worker wR A task sequenceSw A task set for wV TS(w) A valid task set of wt(l) The arrival time of particular location lc(a, b) Travel distance from a to bA A spatial task assignment

Definition 1 (Spatial Task). A spatial task, denoted by s =<ls, es >, is a task to be answered at location ls, and will expireat es, where ls ∶ (x, y) is a point in the 2D space.

For simplicity and without loss of generality, we assume theprocessing time of each task is 0, which means that a worker willgo to the next task upon finishing the current task.

Definition 2 (Worker). A worker, w =< lw, dw, tw, speedw >,is a carrier of a mobile device who volunteers to performspatial tasks. A worker can be in an either online or offlinemode. A worker is offline when she is unable to performtasks and is online when she is ready to accept tasks. Anonline worker is associated with her current location lw, herdestination location dw, the deadline before when she mustarrive at destination tw, and her movement speed speedw.



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Figure 1 shows an example of several workers W ={w1,w2, ...,w8} and all tasks S (for simplicity, we just use theindex to denote a specific task, i.e., S = {1,2, ...,22}) along andnear the routes from lwi to wi.d. Each worker with her currentlocation, such as w1 located at (2,12), starts from time zero; eachtask is associated with a location and deadline: Task 1 located at(4,14) will expire after 3 time units. For the sake of simplicity,we set the movement speed of each worker to 1 in this runningexample.Definition 3 (Task Sequence). Given an online worker w and a set

of tasks assigned to her Sw, a task sequence on Sw, denotedas R(Sw), represents the order by which w visits each taskin Sw. The arrival time of w at task si ∈ Sw (the time ofcompleting task si) can be computed as follows:

tw,R(lsi) = {tw,R(lsi−1) + c(lsi−1 , lsi)/speedw if i ≠ 1c(lw, lsi)/speedw if i = 1,

(1)

where c(a, b) is the travel distance from location a to locationb. The arrival time at destination after completing all tasks inSw with the task sequence R is

tw,R(dw) = tw,R(ls∣Sw ∣) + c(ls∣Sw ∣ , dw)/speedw. (2)

When the context of w and R is clear, we use t(lsi) (t(dw))to denote tw,R(lsi) (tw,R(dw)).

Definition 4 (Valid Task Set (VTS)). A task set Sw is called a validtask set (VTS) for a worker w, if there exists a task sequenceR(Sw), such that,1) all the tasks of Sw can be completed before their respectiveexpiration time, i.e., ∀si ∈ Sw, t(lsi) ≤ esi , and2) the worker w can arrive at destination on time aftercompleting all tasks in Sw, i.e., t(dw) ≤ tw.

Definition 5 (Maximal Valid Task Set (MaxVTS)). A Valid TaskSet Sw is maximal if none of its super sets is still valid for aworker w.

Note that there may exist more than one maximal VTS for agiven worker w. In Figure 1, {4,13}, {14,13} and {4,14,13}are valid task sets for worker w2, but {2,4} is not a valid task setsince w2 cannot arrive at w2.d on time after finishing task 2 and4. Note that neither {4,13} nor {14,13} is a maximal VTS sinceit is contained by {4,14,13}.

The MaxVTS calculation problem can be proved to be NP-hard by reduction from a Destination-aware Traveling SalesmanProblem (DTSP). In the following, we give the definition of DTSPand prove it as NP-Complete.Definition 6 (Destination-aware Traveling Salesman Problem

(DTSP)). Given a complete graph G(V,E) with weight func-tion c: V × V → Z , a source vertex a, a destination vertex band cost k ∈ Z , where k ≥ c(a,x) + c(x, b) for any x ≠ a andx ≠ b, the DTSP problem < G, c, a, b, k > is to determinewhether there exists a tour which visits each vertex exactlyonce, starting from the source vertex a and finishing at thedestination vertex b with the cost of at most k.

Lemma 1. The DTSP problem is NP-Complete.

Proof 1. The proof is shown in Appendix A.

Lemma 2. Given a worker w (with her current location lw,destination location dw and deadline tw), s set of n tasksS and number m, deciding whether there exists a valid tasksequence R (by which worker w has to start from lw and end

at dw before tw), st. ∣R∣ = m, is NP-Complete. That is, thedecision problem of MaxVTS calculation is NP-Complete.

Proof 2. The proof is shown in Appendix B.

Since we have proved that the decision version of MaxVTScalculation problem is NP-Complete, we can conclude that theMaxVTS calculation problem is NP-hard.

Definition 7 (Spatial Task Assignment). Given a set of workersW and a set of tasks S, a spatial task assignment, denotedby A, consists of a set of < worker, V TS > pairs inthe form of < w1, V TS(w1) >, < w2, V TS(w2) >,...,< w∣W ∣, V TS(w∣W ∣) >, where V TS(w1) ∩ V TS(w2)... ∩V TS(w∣W ∣) = ∅.

Let A.S denote the set of tasks that are assigned to allworkers, i.e., A.S = ∪w∈WSw and A denote all possible waysof assignments. The problem investigated in our paper can beformally stated as follows.

Problem Statement: Given a set of workers W and a setof tasks S, the Destination-aware Task Assignment (DATA)problem aims to find the global optimal assignment Aopti, suchthat ∀ Ai ∈ A, ∣Ai.S∣ ≤ ∣Aopti.S∣.Lemma 3. The DATA problem is NP-hard.

Proof 3. The proof is shown in Appendix C.

3 ALGORITHM

Since the DATA problem is NP-hard, a simple greedy algorithmis to use the maximum valid task set for each worker as theassignment result. This can hardly be a satisfying result sincemultiple workers may be assigned the same set of tasks that mayleave more tasks unassigned. In this paper, we develop an exactsolution with three steps. First, we devise a dynamic programmingalgorithm to find the set of maximal valid task sequences for eachworker. It can be shown that the global optimal result is the unionof one possible valid task sequence of all workers. Second, toavoid exhaustive search through all the possible combination ofvalid task sequences, we utilize a tree-decomposition technique toseparate all workers into independent clusters and organize theminto a tree structure, such that the workers in sibling nodes of thetree do not share the same valid tasks. In the final step, the treeis traversed in depth-first manner to find the optimal assignment.During the traversal, a lower bound that indicates the minimalnumber of required tasks for each sub-tree is dynamically main-tained and compared against its upper bound (i.e., the maximumnumber of tasks that can be assigned to the sub-tree). If thelower bound is greater than the upper bound, the sub-tree canbe eliminated without further exploration.

3.1 Valid Task Set Generation

3.1.1 Finding Reachable TasksDue to the constraint of workers’ deadlines and tasks’ expirationtime, each worker can only complete a small subset of tasks.Therefore, we firstly find the set of tasks that can be reachedby each worker without violating the constraints. The reachabletask subset for a worker w, denoted as RSw, should satisfy thefollowing two conditions: ∀s ∈ RSw,

1) c(lw, ls) ≤ es and2) c(lw, ls) + c(ls, dw) ≤ tw.



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The above two conditions guarantee that a worker can travelfrom her origin to the location of task s directly before it expiresand still have sufficient time to arrive her destination beforedeadline. From computational perspective, the reachable tasks(satisfying the condition 1) fall inside an ellipse with the worker’sorigin and destination as focus and the maximum travel distance(i.e., tw × speedw) as the length of major axis. It is easy to seethe time complexity is O(∣W ∣ ⋅ ∣S∣), where ∣W ∣ and ∣S∣ are thenumbers of workers and tasks respectively. In Figure 1, the bluenumbered circles denote all the reachable tasks and the grey onesrepresent the unreachable tasks.

3.1.2 Finding Maximal Valid Task Set

Given the reachable task set for each worker, we next find the set ofMaxVTS, which is shown to be an NP-hard problem in Lemma 2.However, the reachable task set for each worker is usually notlarge, which means this problem can still be solved by an efficientalgorithm in practice. Moreover, finding the MaxVTSs for eachworker is completely independent and can be easily parallelized.

In the sequel, we present a dynamic programming algorithmthat iteratively expands the sets of tasks in the ascending order ofset size and find all MaxVTSs in each iteration. For each taskin one set, we consider the scenario that it is finished in theend, and find all completed task sequences. Specifically, givena worker w, and a set of tasks Q ⊆ RSw. We define opt(Q,sj)as the maximum number of tasks completed by scheduling all thetasks in Q with constraints starting from lw and ending at lsj ,and R as the corresponding task sequence on Q to achieve thisoptimum value. We also use si to denote the second-to-last taskbefore arriving at sj inR, andR′ to denote the corresponding tasksequence for opt(Q−{sj}, si). Then opt(Q,sj) can be calculatedby Equation 3.

opt(Q,sj) =⎧⎪⎪⎨⎪⎪⎩

1 if ∣Q∣ = 1max

si ∈Q,si ≠sjopt(Q − {sj}, si) + δij otherwise, (3)

δij = { 1 if t(lsj ) ≤ esj , and t(lsj ) + c(lsj , dw) ≤ tw0 otherwise,

where δij is an indication function, in which δij = 1 means sj canbe finished after appending sj to R′ and the worker can arrive thedestination before her deadline.

When Q contains only one task si, the problem is trivialand opt({si}, si) is set to 1. When ∣Q∣ > 1, we need to searchthrough Q to examine all possibilities of valid task sets and findthe particular si that achieves the optimum value of opt(Q,sj).Algorithm 1 outlines the structure of this procedure. Note thatwe use pre(Q,sj) to record the last-to-second task si beforeachieving opt(Q,sj) to facilitate the reconstruction of the optimalvalid task sequence R. After initialization, the algorithm generatesand processes sets in the increasing order of their size from 2to n (lines 7-8). For each task sj ∈ Q, it computes opt(Q,sj)and pre(Q,sj) according to Equation 3 (lines 10-11). Finally,whenever Q can be added to Qw, we remove its proper subsetsthat already exist inQw (lines 14-16). To save space, the procedureof constructing R∗ from tables opt and pre is omitted here.

Algorithm 1 correctly computes MaxV TS set withO(2∣RSw ∣ ⋅ ∣RSw ∣3) time complexity, where ∣RSw ∣ is the numberof reachable tasks for worker w. Table 2 shows the MaxVTSsof workers based on Equation 3 and their maximum number ofcompletable tasks ∣maxS∣.

Algorithm 1: MaxVTS

1 Input: w,RSw

Output: Qw

2 Qw ← null;3 for each task si in RSw = {s1, s2, ..., sn} do4 opt({si}, si)← 1;5 Qw ← Qw ∪ {{si}};6 pre({si}, si)← null;

7 for len ← 2 to n do8 for each subset Q ⊆ RSw of size len do9 for each sj ∈ Q do

10 opt(Q,sj)← maxsi∈Q,si≠sj

opt(Q − {sj}, si) + δij ;

11 pre(Q,sj)← arg maxsi∈Q,si≠sj

opt(Q − {sj}, si) + δij ;

12 if δij = 1 then13 Qw ← Qw ∪ {Q};14 for each Q′ ∈ Qw do15 if Q′ ⊂ Q then16 Remove Q′;

17 compute R∗ based on opt and pre;18 return Qw

TABLE 2Maximal Valid Task Sets

W Maximal Valid Task Sets ∣maxS∣w1 {s1, s2},{s2, s3},{s3, s4} 2w2 {s2},{s4, s13, s14} 3w3 {s14},{s6, s8},{s5, s6},{s4, s6},

{s4, s5},{s3, s4}2

w4 {s10, s17},{s11, s17},{s13, s14, s17} 3w5 {s6, s8},{s7, s9},{s8, s9},{s8, s10},{s9, s10} 2w6 {s10, s16},{s9, s10, s18},{s9, s12, s16},

{s12, s16, s18}3

w7 {s10},{s11},{s12} 1w8 {s18, s19} 2

3.2 Worker Partition

The main computational challenge lies in huge search space whenenumerating all possible combinations of the valid task sets ofeach worker, which increases exponentially with respect to thenumber of workers. However, in practice a worker shares thesame tasks with only a few other workers who have similar orintersected travel routes.

Definition 8 (Worker Dependency). Given two workers wi, wj ,and their respective reachable task sets, RSwi ,RSwj , they areindependent with each other if RSwi ∩RSwj = ∅. Otherwise,they are dependent with each other.

For instance, in Figure 1, w1 has dependency with w2 andw3, but is independent with the rest of workers. In our work, weaim to leverage the independency amongst workers and partitionthe worker set into independent groups, such that the optimalassignment can be found more efficiently in each groups.

3.2.1 Worker Dependency GraphGiven a worker set W and task set S, we can construct a WorkerDependency Graph (WDG) G(V,E), where each node v ∈ Vrepresents a worker wv ∈W . An edge e(u, v) ∈ E exists betweenu and v if the two workers wu and wv are dependent with eachother. The time complexity of WDG construction is O(∣W ∣2 ⋅∣RS∣), where ∣RS∣ is the average number of reachable takes foreach worker. Figure 2(a) illustrates the WDG for worker set shownin Figure 1.



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3.2.2 Graph PartitionIn this part, we need to decompose the dependency relationshipby partitioning WDG. To this end, we utilize the notion of tree-decomposition [16], which transforms a graph into a tree structure.Definition 9 (Tree Decomposition). Given an undirected graph

G = (V,E) composed of a set V of vertices and a set Eof edges. A tree-decomposition of G is a pair (X, T ), whereX = {X1, ...,Xn} is a family of subsets of V , and T is atree whose nodes are the subsets Xi, satisfying the followingproperties [16]:1) ∪i ∈nXi = V , and2) ∀(v,w) ∈ E,∃Xi ∈ X containing both v and w, and3) if Xi, Xj and Xk are nodes, and Xk is on the path fromXi to Xj , then Xi ∩Xj ⊆Xk.

The tree decomposition of a graph is far from unique. Next,we briefly introduce some related concepts and then describe themaximum cardinality search (MCS) algorithm [20] to find thetree-decomposition.Definition 10 (Chordal Graph). A graph is a chordal graph if

every cycle of length > 3 has a chord, i.e., edge joining twonon-consecutive vertices of a cycle [1].

Definition 11 (Maximal Clique). Every maximal clique of achordal graph G = (V,E) is of the form {v}∪C(v), for somevertex v ∈ V where C(v) = {w∣(v,w) ∈ E, δ(v) < δ(w)}.δ is a perfect elimination ordering of vertices in a graph suchthat, for each vertex v, v and the neighbors of v that occurlater than v in the order form a clique [17].

Property 1. A tree-decomposition of a chordal graph consists ofthe set of its maximal cliques [20].

The MCS algorithm consists of following steps.1) Given a WDG, construct the corresponding chordal graph

by adding suitable new edges.2) Find δ on the derived chordal graph.3) Identify the maximal cliques in the chordal graph. For each

vertex v of δ, the maximal clique containing v is a graph with thenodes {v} ∪C(v).

4) The maximal cliques will be the nodes (i.e., X) of the tree-decomposition result.

The time complexity for the above algorithm is O(∣V ∣+ ∣E′∣),where E′ is the number of edges in the chordal graph obtained.Take the WDG shown in Figure 2(a) as an example. Since it isalready a chordal graph according to its definition, we do notneed to take any actions in the first step. In the second step, wefind the perfect elimination ordering of graph G and sort all thenodes by δ in ascending order: {w8,w1,w2,w7,w3,w5,w6,w4}.Lastly, the maximal cliques of the graph can be found andoutput as the nodes of tree-decomposition: X = {{w1,w2,w3},{w2,w3,w4}, {w3,w4,w5}, {w4,w5,w6,w7}, {w6,w8}}, asshown in Figure 2(b).

3.2.3 Tree ConstructionAccording to the definition of tree-decomposition, if two nodesdo not share the same vertexes, the workers belonging to the twonodes are independent with each other. In this step, our goal isto organize the subsets of workers in a tree structure such thatthe sibling nodes are independent with each other. Facilitated bysuch a tree structure, we can solve the optimal assignment sub-problem on each sibling node independently. Since the search costis largely affected by the number of workers, we would like tomake the tree as balanced as possible, i.e., to avoid any node withsignificantly more workers than the others. To this end, we devisethe following Recursive Tree Construction (RTC) algorithm:

1) Try to remove the vertices in each node Xi ∈ X (output inthe graph partition step) from the WDG G. G will be separatedinto a few components, of which the largest one is recorded asGmax. ∣Gmax∣ is the number of vertices in Gmax.

2) Pick the node Xmin that leads to the least ∣Gmax∣ upon thecompletion of the previous loop (pick the smallest Xi as Xmin

when there is a tie on ∣Gmax∣ and randomly pick a Xi as Xmin

when their cardinalities are same). Set Xmin as the parent nodefor each output of the recursive procedure in step 3.

3) Apply the MCS algorithm on each sub graph by removingworkers of Xmin and recursively perform this algorithm on theoutput of MCS algorithm.

4) Return N =Xmin as the root node of this sub-tree.We can derive from the RTC algorithm that the balanced tree

(constructed from G(V,E)), denoted by T (with a set of nodesNT = {n1, n2, ..., n∣NT ∣}), satisfies the following properties:

1) ∪i∈∣NT ∣ni = V , and2) for each node ni ∈ NT , removing ni from GTi leads to the

least ∣GmaxTi ∣, where Ti is the subtree rooted with node ni, GTi is

the WDG for workers in the subtree Ti and GmaxTi is the largest

subgraph by removing ni from GTi , and3) workers in the subtrees rooted with sibling nodes are

independent with each other.The time complexity of RTC in the ith recursion is O(∣Xi∣ +

∣Gisub∣ ⋅ (∣V i∣+ ∣E′i∣)) (including finding node Xi

min from Xi andapplying MCS algorithm in each subgraph), where Gi

sub is thesubgraph set by performing step 1 in the ith recursion. Thus thetotal time complexity of RTC is O(∑m

i=1(∣Xi∣ + ∣Gisub∣ ⋅ (∣V i∣ +

∣E′i∣))), where m is the number of recursions. Constructing thetree with nodes in Figure 2(b) by RTC algorithm, we get the finaltree structure, which is illustrated in Figure 2(c).

3.3 SearchIn this section, we present our search algorithm framework forusing a tree-decompositon to solve the DATA problem.

Once the worker dependency graph has been transformed to atree structure, the optimal assignment can be found by a depth-firstsearch through the tree. First of all, we give an overview of wholeprocess including the previous steps in Algorithm 2.



6

Given the worker set W and task set S, the reachable task setRSw and maximal valid task setsQw are computed for each work-erw (line 2-4), and the corresponding worker dependency graphGis constructed (line 5). Then for each connected component g ∈ G,we decompose g into a set of vertex clusters with MCS algorithm(line 7) and organize them into a tree with RTC algorithm (line 8).Lastly the depth-first search algorithm (DFSearch) is invokedon each tree to find the optimal assignment in each sub problem.Since each component of G is independent with each other, thefinal result is to simply sum up the optimal assignment of eachcomponent g (line 9).

Algorithm 2: Solution OverviewInput: W,SOutput: Opt

1 Opt← 0; Q′ ← ∅; S′ ← S;2 for each worker w ∈W do3 RSw ← compute the reachable tasks for w;4 Qw ←MaxV TS(w,RSw);5 G← construct worker dependency graph;6 for each connected component g ∈ G do7 Xg ← decompose g into vertex clusters;8 Ng ← organize Xg into a tree;9 Opt← Opt +DFSearch(Ng , S,WNg , LB(Ng));

10 return Opt;

Next we elaborate the details of DFSearch procedure inAlgorithm 3. The procedure takes four parameters: the root nodeN of the sub-tree to be traversed, the remaining unassigned taskset S, the remaining available workers WN in node N and aheuristic h indicating the minimum required number of tasks yetto be assigned in order to beat the current optimal assignment.

The algorithm starts with computing an upper bound UB(N)of the number of tasks that can be assigned to the workerscontained in the sub-tree rooted with N (line 2), and compares itagainst with the heuristic h that represents a lower bound LB(N)of the number of tasks that need to be assigned to this sub-treein order to beat the optimal assignment Opt found so far (line 3).Obviously, if UB < LB, this sub-tree can be safely pruned sinceit cannot lead to a better assignment. The ways to derive UB(N)and LB(N) will be discussed in Section 3.3.1 and Section 3.3.2.

Then the algorithm branches depending onWN , the worker setcontained by current nodeN . If there are still workers to be probed(line 5), we will sequentially examine each available worker inWN (line 6), get a new maximal VTS (Qw) by eliminating theassigned tasks (Q′) from the the existing maximal VTS (line 7),and then recursively call the DFSearch procedure by passingin the updated remaining task set (S − Q), updated worker set(WN − w) and the updated heuristic (h − ∣Q∣). The optimalassignment is updated if the returned assignment plus ∣Q∣ (i.e., thenumber of tasks assigned to current examined worker) is greater(line 10). The heuristic value h is updated accordingly since abetter assignment is just found (line 11). On the other hand, if allthe workers have been enumerated (line 13), the algorithm willinvoke DFSearch procedure on each child node of N . Sinceeach child node (and their sub-tree) is independent with each otheras guaranteed by our tree-decomposition algorithm in previousphase, the problem of finding optimal assignment for each sub-tree can be solved independently, and then summed up to obtainthe global optimal assignment (line 14-15). The time complexityof Algorithm 3 is O(∑r

i=1(∣W iN ∣ ⋅ ∣Qi

w ∣ + ∣N ichild∣)), where r is

the number of recursions, ∣W iN ∣ is the number of workers in the

node W iN in the ith recursion, ∣Qi

w ∣ is the number of MaxVTSs ofworker w in the ith recursion, and ∣N i

child∣ is the number of childnodes of N in the ith recursion.

Accuracy. In the worker partition phase, we separate allworkers into independent clusters and organize them into a treestructure, where workers in sibling nodes of the tree do not sharethe same valid tasks. Algorithm 3 illustrates that, given any non-empty nodeN of the tree, we check all the task assignments for allthe workers (contained in the subtree whose root is N ) and theirvalid task sets, thus the optimal task assignment with maximalnumber of assigned tasks can be found in this subtree rooted withN (line 5-12). Since workers in sibling nodes do not share thesame valid tasks, we can simply sum up the task assignment resultsin the subtrees rooted with sibling nodes in order to get the globaloptimal task assignment (line 14-15).

Algorithm 3: DFSearchInput: N,S,WN , hOutput: Opt

1 Opt← 0;2 UB(N)← compute the upper bound of assigned tasks for the sub-tree

rooted with N ;3 if UB(N) < h then4 return 0;

5 if WN ≠ ∅ then6 for each worker w ∈WN do7 Qw ← Qw −Q′;8 for each maximal valid task set Q ∈ Qw do9 Q′ = Q′ ∪Q;

10 Opt←max{DFSearch(N,S −Q,WN −w,h −∣Q∣) + ∣Q∣,Opt};

11 h← Opt;12 Q′ ← ∅;

13 else14 for each child node Ni of N do15 Opt+← DFSearch(Ni, S,WNi

, LB(Ni));

16 return Opt;

3.3.1 Upper Bound Estimation

The upper bound of a node N , denoted as UB(N), representsthe maximum number of tasks that can be finished by the sub-treerooted at N . A simple estimation of UB(N) is to sum up thecardinality of the maximum valid task set of each worker in thissub-tree, i.e.,

UB(N) =∣W ∣

∑i=1(∣maxSwi ∣), (4)

where W denotes all the workers in the current sub-tree, andmaxSwi denotes the maximum valid task set that can be finishedby wi. maxSwi can be obtained by choosing the maximal validtask set of wi with the greatest cardinality, i.e., maxSwi =max{Q∣Q ∈MaxV TS(wi, S)}.

For example, when the search algorithm reaches N3 in Fig-ure 2(c), UB(N3) can be estimated as follows:

UB(N3) = ∣maxSw6 ∣ + ∣maxSw7 ∣ + ∣maxSw8 ∣ = 3 + 1 + 2 = 6,

where the ∣maxS∣ can be looked up from Table 2.

Lemma 4. UB(N) upper bounds the optimal assignment of thesub-tree rooted at N , and the bound is tight.



7

Proof 4. Since in any task assignment A, including the optimalone, the number of tasks completed by each worker w cannotexceed ∣maxSw ∣, the following inequality always holds:

∣A.S∣ = ∣ ∪w∈W Sw ∣ ≤ ∑w∈W

∣Sw ∣ ≤ ∑w∈W

∣maxSw ∣ = UB(N).

When all the workers in N are independent with each other,i.e., they do not share any task, the optimal assignment will beequal to UB(N) since the above inequality becomes

∣ ∪w∈W Sw ∣ = ∑w∈W

∣Sw ∣ = ∑w∈W

∣maxSw ∣.

Therefore the upper bound is tight.

3.3.2 Lower Bound Estimation

In order to prune the unpromising branch as early as possible, wealso calculate an lower bound heuristic LB(N) and pass it as aparameter h to the recursive procedure on the child node. LB(N)implies the minimal number of tasks that must be completed bythe workers in the subtree rooted at N in order to find a betterassignment than the current best one. Obviously, whenever theupper bound of one node is less than the lower bound, its sub-treecan be discarded safely since none of the possible assignment onthe workers in this sub-tree can lead to better global assignment.

Now let us describe how to estimate the lower bound ofNi as the child node of N . Suppose N has m child nodes,i.e., N1,N2, ...,Nm. The DFSearch algorithm will invoke theprocedure on each child node and try to get a better assignmentOpt for each node before returning to its parent node N (line 14).We can estimate the lower bound of Ni by the following formula,

LB(Ni) = h −i−1∑j=1

Opt(Nj) −m

∑j=i+1

UB(Nj), (5)

where (1) h is the minimal number of tasks required for theworkers in all child nodes of N (i.e., N1,N2, ...,Nm); (2)∑i−1

j=1Opt(Nj) represents the optimal assignment of the sub-trees that have already been traversed by the procedure; (3)∑m

j=i+1UB(Nj) represents the upper bound of the optimal as-signment of the sub-trees that are yet to be probed.

Lemma 5. LB(Ni) is the minimal number of tasks to be com-pleted by workers in the sub-tree rooted at Ni.

Proof 5. If the optimal number of tasks assigned to Ni is lessthan LB(Ni), then the maximum number of tasks that can becompleted by workers in all child nodes of N will be less thanh:m

∑j=1

Opt(Nj) = Opt(Ni) +i−1∑j=1

Opt(Nj) +m

∑j=i+1

Opt(Nj)

< LB(Ni) +i−1∑j=1

Opt(Nj) +m

∑j=i+1

UB(Nj)

= h.

Therefore in order to complete at least h tasks, the optimalassignment of Ni must satisfy Opt(Ni) ≥ LB(Ni).

When invoking the DFSearch procedure for the first time(line 10 of Algorithm 2), the lower bound of the entire tree can beestimated by the assignment of a greedy algorithm, i.e., the unionof maximal valid task sets of all workers.

TABLE 3Time Complexity of the DATA Algorithm

Operation Complexity

Valid Task Set Generation O(∣W ∣ ⋅ ∣S∣ + ∣W ∣ ⋅ 2∣RS∣ ⋅ ∣RS∣3))Worker Partition O(∣W ∣2 ⋅ ∣RS∣ +∑m

i (∣Xi∣ + ∣Gisub∣ ⋅ (∣V i∣ + ∣E′i∣)))

Search O(∑ri (∣W i

N ∣ ⋅ ∣Qiw ∣ + ∣N i

child∣))

3.3.3 OptimizationIn this part we briefly discuss some optimization schemes tofurther reduce the search cost.

1) Re-ordering sub-tree traversal: before invoking theDFSearch procedure on the sub-trees of a node, we firstly sortall the sub-trees in the ascending order of the number of theirassociated workers. The rationality behind this optimization isthat, the impact of loose pruning bounds in the very beginning tosearch performance can be reduced when applying on a sub-treewith less workers, while the tighter bounds can make the search onlarger sub-trees more efficient. We update DFSearch algorithmby sorting all the sub-trees in the ascending order of the numberof their associated workers before line 14 of Algorithm 3.

2) Optimization of UB: we can further improve the up-per bound of each sub-tree by applying a modified version ofDFSearch on the whole tree in bottom-up manner. The onlydifference lies in that, when searching the sub-trees, the inputtask set S is always the entire task set. Essentially this processfinds the optimal assignment for workers in each sub-tree byassuming they can access all tasks (while they cannot in factdue to existence of other dependent workers). It is worth notingthat the extra overhead incurred by this optimization is minimal,since the “optimal result” of a sub-tree is now independent withits parent node. Therefore when invoking DFSearch from leafnodes all the way up to the root, we can record the return valueas UB on each node, and its parent node can simply use thisrecord to derive its upper bound without recursively applying thisprocedure again. We can update DFSearch algorithm by using“Opt+ ← DFSearch(Ni, S

′,WNi, LB(Ni))” to replace line

15 of Algorithm 3, where S′ is the entire task set initialized inline 1 of Algorithm 2. The return value is the upper bound ofassigned tasks for the sub-tree rooted with N . We compute theupper bound for each node N from bottom up.

3.4 Limitation of DATA

Our DATA problem requires each worker to specify her destinationand deadline when she is ready to perform tasks. As its name(i.e., Destination-aware Task Assignment) suggests, this problemcan only be applied in the destination-aware scenarios. In thiswork, we assume the processing time of each task is 0, whichis a common assumption in spatial crowdsourcing studies [7],[8] due to the fact that most existing spatial tasks (e.g., takingphotos/videos) are simple enough to be completed instantaneously.However, our proposed algorithms can be extended to handle morecomplex spatial tasks. To this end, we can modify the arrival timein Equation 1 and 2 by adding the processing time of each task inthe corresponding task sequence.

Though our algorithm can provide an exact solution, thecalculation is relatively inefficient. Table 3 summarizes the costof each operation of our algorithms. Clearly the cost is dominatedby the MaxVTS generation phase with an exponential time com-plexity, which is computationally expensive when ∣RS∣ is large.



8

Therefore, our algorithm is not suitable for a task-dense area, i.e.,each worker has a large number of reachable tasks. However, inpractice, the algorithm is still efficient because the number (i.e.,∣RS∣) of reachable tasks for each worker is a relatively smallvalue. Moreover, since the MaxVTS generation of each workeris independent with each other, we can calculate the MaxVTSs foreach worker in parallel to improve efficiency.

4 EXTENSION

As the extension of our previous work [27], we will present in thissection two optimizations that will reduce the overall travel costand support redundant task assignment respectively.

4.1 Travel Cost Optimization StrategyThe problem with the original framework is that it only maximizesthe number of task assignments, without considering the travelcost (e.g., in time or distance) of the workers during the assign-ment process. In the problem settings of spatial crowdsourcing,travel cost is also critical issue since workers must physicallygo to the location of the spatial task in order to perform it. Toaddress this issue, we propose a strategy, referred to as Travel CostOptimization Strategy, to improve the overall task assignment bygiving higher priority to the valid task sets with lower travel cost.

We first introduce the notion of valid route, by which a workercan travel from her original location to her destination passing aset of valid tasks. More formally,Definition 12 (Valid Route). Given a worker w and a valid task

set assigned to her, V TS, there may exist more than one tasksequences in V TS for w. Let Ri(V TS) denote a task se-quence on V TS, representing the order by whichw visits eachtask in V TS, and R(V TS) = {R1(V TS),R2(V TS), ...}denote all possible task sequences on V TS. A route from theworker’s original location to her destination passing all thetasks of V TS, lw → V TS → dw, is called a valid route forw, if there exists a task sequence Ri(V TS), such that,1) all the tasks of VTS can be completed before their respectiveexpiration time, e.g., ∀si ∈ V TS, tw,Ri(V TS)(lsi) ≤ esi , and2) the workerw can arrive destination on time after completingall tasks in VTS, e.g., tw,Ri(V TS)(dw) ≤ tw, and3) the arrival time at destination by following the task se-quence Ri(V TS) is minimal, e.g., ∀Rj(V TS) ∈ R(V TS),tw,Ri(V TS)(dw) ≤ tw,Rj(V TS)(dw).Intuitively, tasks which are closer to a worker have smaller

travel costs. Therefore, we define the travel cost from a worker’slocation lw to her destination dw passing a valid task set V TS,in terms of the Euclidean distance of the corresponding validroute, denoted by c(lw, dw, V TS). Consequently, by computingthe distance among every worker, her performable spatial tasks(i.e., those in a valid task set), and her destination, we can associatehigher priorities to the closer valid task sets. Moreover, given a setof workers W and a set of tasks S, we define the aggregate travelcost, denoted by ac(W,S), as the sum of the Euclidean distancesof the valid routes for all workers inW while satisfying the spatio-temporal constraints of workers and tasks.

Taking the worker w8 and her valid task set {s18, s19} inFigure 1 as a case, worker w8 can perform task s18 and s19 inturn, or she can successively perform task s19 and s18 beforearriving at her destination. However, the valid route for w8 basedon the valid task set {s18, s19} is the route from the location

of w8 to her destination passing task s19 and s18 in turn, sincetw8,(s19,s18)(dw8) is less than tw8,(s18,s19)(dw8). Correspond-ingly, the travel cost for {s18, s19} is the arrival time at thedestination of w8 by following the task sequence (s19, s18), i.e.,c(lw8 , dw8 ,{s18, s19}) = tw8,(s19,s18)(dw8).

With the knowledge of the travel cost of valid routes for allthe workers, we incorporate the travel cost in the search process tomaximize the task assignments while minimizing the travel cost ofthe workers whenever possible (line 13, 19 and 22 of Algorithm 4).

4.2 Redundant Task Assignment StrategyAnother problem with the proposed DATA solution is that itcan just be applied for the single task assignment in spatialcrowdsourcing, in which each spatial task is only assigned toone worker. The assumption here is that all the workers aretrusted, and thus they complete the spatial tasks correctly withoutany malicious intentions [6], [12]. In practice, however, thereinevitably exist some workers who either intentionally (e.g., mali-cious workers) or unintentionally (i.e., making mistakes) performthe tasks incorrectly (i.e., being dishonest about physically goingto the locations of the spatial tasks), which cannot guarantee thequality of task completion. To tackle this problem, we improvethe DATA solution by changing the single task assignment to theredundant task assignment [13], in which each spatial task can becompleted by a few available workers in proximity of the task,such that majority voting can be applied to improve the quality oftask completion. The intuitive assumption shared by the redundanttask assignment is based on the idea of the wisdom of crowds [19]that the majority of the workers can be trusted and thus the validityof task results provided by a group of workers can be verified bymajority voting.

In detail, a task s, associated with its maximum capacity (e.g.,maximum acceptance workers, maxWs), can be performed by atmost maxWs workers instead of being completed by a particularworker, where maxWs is specified by the requester who issuesthe task s. Clearly, the higher the maxWs value is, the morechance that the task is completed correctly. To apply the DATAsolution to the redundant task assignment problem, for each tasks, we first calculate the workers who are allowed to performit, namely Available Worker Set (AWS), and correspondingly∣AWS∣ denotes the number of available workers for s. Then were-define the worker dependence based on AWS.

To obtain the available workers for each task, we employ theinverted file to improve the retrieval speed. The Available WorkerSet of each task in our running example is illustrated in Table 2.

TABLE 4Available Worker Set

S Available Worker Set S Available Worker Sets1 w1 s10 w4, w5, w6, w7

s2 w1, w2 s11 w4, w7

s3 w1, w3 s12 w6, w7

s4 w1, w2, w3 s13 w2, w4

s5 w3 s14 w2, w3, w4

s6 w3, w6 s16 w5

s7 w5 s17 w6

s8 w3, w5 s18 w6, w8

s9 w5, w6 s19 w8

Based on the AWS for each task, we now re-define the notionof worker dependency.Definition 13 (Worker Dependency). Given two workers wi, wj ,

and their respective reachable task sets, RSwi ,RSwj , they



9

are independent with each other if either of the followingconditions is satisfied:1) RSwi ∩RSwj = ∅, or2) ∀s ∈ {RSwi ∩RSwj}, ∣AWS(s)∣ ≤maxWs.Otherwise, they are dependent with each other.

Consider the running example in Figure 1 and set the maxi-mum number of acceptance workers of s18 to 3, i.e., maxWs18 =3. w8 is independent with w6 since the available workers’ numberof their shared task (i.e., s18) is 2, which is less than maxWs18 .w8 also has independency with the rest of workers because w8

shares no reachable tasks with them.After constructing the balanced tree by MCS algorithm (in

Section 3.2.2) and RTC algorithm (in Section 3.2.3) based on thenew worker dependency relationship, the search algorithm has tobe modified to achieve the goal of maximizing the overall taskassignments when dealing with the redundant task assignmentproblem. Algorithm 4 depicts the improved DFSearch process.Once the search algorithm assigns each maximal valid task set foreach available worker in WN (line 9 and 11), it finds the task setQ′′ from the current maximal valid task set Q, in which the taskshave already used their capacity, i.e., for any task s in Q′′, it hasalready been assigned to maxWs workers (line 14-18). Then thealgorithm recursively calls the Improved DFSearch procedureby passing in the updated remaining task set (S −Q′′), updatedworker set (WN − w) and the updated heuristic (h − ∣Q′∣) (line20).

Algorithm 4: Improved DFSearchInput: N,S,WN , hOutput: Opt

1 Opt← 0;2 S.n← 0;3 t← 0;4 ac← +∞;5 UB(N)← compute the upper bound of assigned tasks for the sub-tree

rooted with N ;6 if UB(N) < h then7 return 0;

8 if WN ≠ ∅ then9 for each worker w ∈WN do

10 Qw = Qw −Q′;11 for each maximal valid task set Q ∈ Qw do12 Q′ ← Q′ ∪Q;13 c(lw, dw,Q)← compute the travel cost from lw to dw

passing Q with the corresponding valid route;14 Q′′ ← ∅;15 for each task s ∈ Q do16 s.n+← 1;17 if s.n =maxWs then18 Q′′+← s;

19 t+← c(lw, dw,Q);20 Opt←max{Improved DFSearch(N,S −

Q′′,WN −w,h − ∣Q∣) + ∣Q∣,Opt};21 h← Opt;22 ac←min{t, ac};23 Q′ ← ∅;

24 else25 for each child node Ni of N do26 Opt+← Improved DFSearch(Ni, S,WNi

, LB(Ni));

27 return Opt;

5 EXPERIMENT

5.1 Experiment Setup

Due to the lack of benchmark for spatial crowdsourcing algo-rithms, we use a real trajectory dataset generated by taxis in abig city to simulate the travel behaviors of workers, in which thelengths (i.e., travel distances) of these trajectories vary from 1kmto 30km and the travel times vary from 105s to 2723s. Theaverage speed of each worker can be easily computed based onthe travel distance and time of each trajectory. For each test werandomly choose 1000 trajectories from the dataset with similar(Euclidean) travel distance controlled by travel distance coefficienttc, which is defined as the ratio between the origin-destinationdistance and the maximum travel distance in the dataset. Eachworker’s deadline is set by multiplying her actual travel time witha deadline coefficient dc. Then we uniformly generate ∣S∣/∣W ∣tasks inside the workers’ elliptic reachable region and set eachtask’s expiration time as ec ∗ c(lw, ls)/speedw, where c(lw, ls)is the travel distance from origin to the location of task andspeedw is the worker’s average speed. The default values of allthe parameters used in our experiments are summarized in Table 5.For each experiment, we run 50 test cases and report the averageresults. All the algorithms are implemented on an Intel Core i5-2400 CPU @ 3.10G HZ with 8 GB RAM.

TABLE 5Experiment Parameters

Parameter Default value

Number of tasks ∣S∣ 4000Worker travel distance coefficient tc 0.1Worker deadline coefficient dc 1.5Task expiration time coefficient ec 2.5Maximum acceptable workers maxW 1

5.2 Experiment Results

5.2.1 Performance of Worker PartitionIn this part we evaluate the performance of worker partition phaseand its impact to subsequent search. While applying the samegraph partitioning algorithm (Section 3.2.2), we introduce a base-line algorithm for tree construction, Random Tree Constructionalgorithm (RTA), which randomly selects a worker cluster as theroot node of sub-tree. Two metrics are compared between RTAand our proposed balanced tree-construction algorithm (BTA): 1)search depth: the maximum number of workers enumerated whensearching from the root node to leaf nodes within one depth-firsttraversal; 2) CPU time: the CPU time cost for finding the optimalassignment with the resulting tree.

Effect of ∣S∣. First, we investigate how the number of tasksaffects the resulting trees. As shown in Figure 3(a), though thesearch depths of both tree construction algorithms increase with∣S∣, BTA can generate a much more balanced tree, which inturn leads to more efficient search than RTA as confirmed inFigure 3(b).

Effect of tc. As illustrated in Figure 4, the performances ofboth algorithms deteriorate as the worker travel distance coeffi-cient increases. This is because the dependency among workersincreases when there are more reachable tasks for each worker.Another observation is that, the performance gap of both ap-proaches in terms of search cost is also increasing. This is dueto the fact, when the tree is unbalanced, the search cost is moresensitive to the average number of valid task sets of each worker



10

0

10

20

30

40

2 3 4 5 6 7

Sea

rch

dept

h

Number of tasks (k)

RTABTA

(a) Search depth

1

10

102

103

104

105

106

2 3 4 5 6 7

CP

U ti

me

(mill

isec

onds

)

Number of tasks (k)

RTABTA

(b) CPU time

Fig. 3. Performance of Worker Partition: Effect of ∣S∣

5

10

15

20

25

30

35

0.06 0.08 0.1 0.12 0.14

Sea

rch

dept

h

Travel distance

RTABTA

(a) Search depth

102

103

104

105

106

0.06 0.08 0.1 0.12 0.14

CP

U ti

me

(mill

isec

onds

)

Travel distance

RTABTA

(b) CPU time

Fig. 4. Performance of Worker Partition: Effect of tc

that increases with tc. In such circumstances, the benefits of amore balanced tree become more significant.

Effect of dc. Evidently, the effect of workers’ travel distancecoefficient tc and deadline coefficient dc are strongly correlated,which explains why the impact of dc shares the similar trendwith that of tc (see Figure 5(a)). While the search cost of bothalgorithms increases quickly with dc, RTA deteriorates muchfaster and cannot even return a result within tolerated time whendc > 1.6, which demonstrates the importance of tree structure tothe search performance.

Effect of ec. As shown in Figure 6(a), the search depth is notaffected by the expiration time. This is because the dependencyamong workers does not change much as long as the reachabletask set remains stable for each worker. However, as noted inFigure 6(b), the search cost of RTA increases much faster thanBTA since a greater ec results in more valid task sets hencemore VTS enumeration during the search. This again confirms thesuperiority of a tree with more balanced sub-trees.

5.2.2 Performance of Task Assignment Algorithm

In this part, we compare the efficiency (i.e., CPU time) of follow-ing algorithms:

5

10

15

20

25

30

35

40

1.1 1.3 1.5 1.7 1.9

Sea

rch

dept

h

Worker deadline

RTABTA

(a) Search depth

102

103

104

105

106

1.1 1.3 1.5 1.7 1.9

CP

U ti

me

(mill

isec

onds

)

Worker deadline

RTABTA

(b) CPU time

Fig. 5. Performance of Worker Partition: Effect of dc

10

12

14

16

18

20

22

24

1.5 2 2.5 3 3.5

Sea

rch

dept

h

Task expiration

RTABTA

(a) Search depth

0

2000

4000

6000

8000

1.5 2 2.5 3 3.5

CP

U ti

me

(mill

isec

onds

)

Task expiration

RTABTA

(b) CPU time

Fig. 6. Performance of Worker Partition: Effect of ec

1) DFS: our proposed DFSearch algorithm based on thebalanced tree.

2) DFS+W : DFS with optimization of re-ordering sub-treetraversal based on the number of workers.

3) DFS + W&U : DFS + W with optimization of UBcomputation.

4) DFS +W&U + TCopt: DFS +W&U with travel costoptimization.

5) GALS: Global Assignment and Local Scheduling algorith-m that iteratively assigns tasks for workers based on maximumflow method and schedules the suitable tasks for each worker [8],where the capacity of worker w is set to the number (i.e., ∣RSw ∣)of her reachable tasks. When scheduling tasks, GALS has toensure the worker can arrive her destination before deadline aftercompleting all the scheduled tasks.

For effectiveness of task assignment, we first compare thenumber of task assignments in the following methods:

1) DFS +W&U .2) DFS +W&U + TCopt.3) GALS.4) GA: Greedy Algorithm that assigns each worker with the

maximal valid task sets from the unassigned tasks, until all thetasks are assigned or all the workers are exhausted.

5) IGA: Iteratively Greedy Algorithm that repeats GA pro-cedure multiple times with every worker as the first one to beassigned and choose the best assignment as the final result.

Moreover, we compare the travel cost among DFS +W&U ,DFS+W&U+TCopt andGALS algorithms. The effectivenessis measured as the aggregate travel cost, which is the sum ofthe Euclidean distances of the valid routes for all workers whilesatisfying the spatio-temporal constraints of workers and tasks.

Effect of ∣S∣. In this set of experiments, we evaluate thescalability of all the approaches by varying the number ∣S∣ oftasks from 2k to 7k. As we can see from Figure 7(a), all ourproposed algorithms have similar performance when the numberof tasks is low, which means there are not many benefits gainedfrom the optimizations. However, the benefits of re-ordering sub-tree traversal and tighter upper bound become more obviouswhen ∣S∣ > 3000. Another observation is that, the CPU costof DFS + W&U + TCopt is a little bit higher than that ofDFS +W&U , but it saves huge travel cost (see Figure 7(c)),which demonstrates the benefits of our proposed travel costoptimization strategy. Although GALS is fastest among all themethods, it assigns less tasks than our proposed methods (i.e.,DFS+W&U andDFS+W&U+TCopt), shown in Figure 7(b).Figure 7(b) also depicts that the greedy algorithms are moredisadvantaged than others with the growth of task numbers. FromFigure 7(c) we can see GALS obtains lower travel cost than bothDFS +W&U and DFS +W&U + TCopt. This is because,DFS +W&U and DFS +W&U + TCopt have more assignedtasks and workers need to take more travel cost to perform suchassigned tasks.

Effect of tc. Figure 8 illustrates the effect of tc on the per-formance of all algorithms. As expected in Figure 8(a), increasingthe travel distance will incur more CPU time for all algorithms.This may be due to the fact that, more valid tasks need to besearched when worker’s travel distance is getting longer. Withthe increase of travel distance, search space increases explosivelythat explains the more benefits of our optimized approaches. Fig-ure 8(b) shows that the optimality of greedy algorithms deteriorateas each worker has more valid task sets. Furthermore, we notice



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that DFS +W&U +TCopt outperforms DFS +W&U a largemargin (up to 24.3%) in terms of travel cost, which shows theeffectiveness of the proposed travel cost optimization strategyagain (see Figure 8(c)).

Effect of dc. In this set of experiments, we study the effect ofworker’s deadline. Not surprisingly, as we can see in Figure 9(a),the performance gaps among the search algorithms (i.e., DFS,DFS +W and DFS +W&U ) become larger when the dead-lines are more relaxed. Figure 9(b) demonstrates that the greedyalgorithms’ utilization of workers’ increased capability to performmore tasks is marginal compared to our exact algorithm. Theworker deadline has similar effect on the aggregate travel costwith worker travel distance, as demonstrated in Figure 9(c). Theintrinsic reason lies in the more valid task sets generated as theincreasing deadline.

Effect of ec. Figure 10 illustrates the effect of task expirationtime. As expected, longer expiration time means on average eachworker has more freedom to schedule the tasks, which results ingreater search space. However, this performance deterioration canbe greatly relieved by the proposed optimizations. On the otherhand, as shown in Figure 10(b), the number of assigned tasks isnot heavily affected since the reachable task set of each workeris unchanged. When comes to the travel time, we notice that theincrease of aggregate travel cost of our proposed algorithms (i.e.,DFS+W&U andDFS+W&U+TCopt) becomes slower whenec ≥ 2.5 since with longer task expiration time there is increasingchance that most of the tasks have already been added into thevalid task sets and thus few tasks need to be added into the validtask sets.

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5.2.3 Performance of Redundant Task Assignment Strate-gy

Finally, we test the performance of the redundant task assignmentstrategy proposed in Section 4.2. In particular, we evaluate bothefficiency and effectiveness of the three strategies, i.e., a task canonly be assigned to 1 worker (maxW = 1), a task can be assignedto at most 2 workers (maxW = 2) and a task can be assigned toat most 3 workers (maxW = 3). Each set of experiments measurethe CPU time and the number of task assignments.

Effect of ∣S∣. As demonstrated in Figure 11(a), all the methodsbecome more time consuming when ∣S∣ increases since more validtask sets that need to be traversed are generated. It is worth notingthat the strategy of maxW = 1 is most time consuming. Thisis because that workers tend to be dependent with fewer otherworkers when the maximum number of acceptable workers fortasks gets larger, which generates a simpler tree and thus makesthe search procedure simpler and more effectual. Regarding theireffectiveness (see Figure 11(b)), naturally the number of taskassignments generated from all strategies increases when moretasks are involved. In addition, the strategy of maxW = 3performs the best followed by that of maxW = 2 and thenmaxW = 1. It is interesting to see that at least 71.4% tasks canbe assigned to 2 workers when maxW = 2 and at least 41.3%tasks can be assigned to 3 workers when maxW = 3, which canimprove the accuracy of task results.

Effect of tc. Figure 12 shows the performance of travelcost optimization by changing over the length of workers’ traveldistance. A longer travel distance is more likely to make moretasks valid for a worker, which takes more search time andprovides more task assignments, which is indicated in Figure 12(a)and 12(b).

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Effect of dc. We also study the effects of the worker deadlinecoefficient by varying it from 1.1 to 1.9. As expressed in Fig-ure 13(a), all the methods have the trend of a growth in both theCPU time and the number of task assignments, whose reason issimilar with the effect of worker travel cost, i.e., as dc gets larger,the search procedure needs to check and assign more valid tasksfor workers.

Effect of ec. In this set of experiment, we change the task expi-ration time coefficient ec from 1.5 to 3.5. Obviously, as illustratedin Figure 14(a), the CPU time of all the methods are growingwhen ec is enlarged. Another observation is that the strategy ismost efficient when maxW = 3, which is approximately 33%slower than that of maxW = 2 and 55% slower than that ofmaxW = 1. The reason behind it is that the larger maxW inredundant task assignment mode leads to a simpler tree to besearched with simpler dependence among workers. In terms of thenumber of task assignments, more task assignments are generatedas maxW grows, which is clearly demonstrated in Figure 14(b).

6 RELATED WORK

Spatial Crowdsourcing (SC) is a new class of crowdsourcing,which employs smart device carriers as workers to physicallymove to some specified locations and perform spatial tasks [11],[18]. Based on the task publish mode, SC can be classified intoServer Assigned Tasks (SAT) mode and Worker Selected Tasks(WST) mode [13]. In SAT mode, the server assigns each task tonearby workers based on the system optimization goals such asmaximizing the number of assigned tasks after collecting all thelocations of workers [4], [8], [13], [14], [23], [25], maximizingthe total payoff from assigned tasks [2], maximizing the expectedtotal utility achieved by all workers [3], [21], [24], maximizingtask reliability for dynamic task assignment [10], maximizingthe expected quality of results from workers by a real-timebudget-aware task package allocation [26], or maximizing thespatial/tempral coverage where/when workers perform tasks [11].For instance, [8] considers task assignment and scheduling at thesame time, in which an approximate approach is developed thatiteratively improves the assignment and scheduling to achievemore completed tasks. However, their paper assumes that eachworker can only perform tasks in a specific spatial region, so thesearch space in their problem settings is much smaller than ours.Moreover, their work proposes an approximate algorithm while

we offer an exact solution. In WST mode, the server publishesvarious spatial tasks online, and workers can select any taskswithout the coordination with the server [7]. For example, Denget al. [7] formulate SC as a scheduling problem by reducing itinto a specialized Traveling Salesman Problem. The exact andapproximation algorithms are proposed to find a schedule tomaximize the number of tasks that can be completed by a workerwhen both travel cost of workers and expiration time of tasks aretaken into consideration.

Moveover, with spatial crowdsourcing, tasks can be assignedin two different modes: Single Task Assignment (STA) mode andRedundant Task Assignment (RTA) mode [13]. STA mode assumesthat all workers are trusted and can perform the tasks correctlywithout any malicious intentions, so that each task is only assignedto one worker in STA mode. However, there inevitably existsome malicious workers that might intentionally complete tasksincorrectly (i.e., being dishonest about physically moving to thelocations of tasks). Therefore, RTA mode is proposed to improvethe validity of task completion by assigning each task to severalnearby workers. In RTA mode, the task completion result with themajority vote is regarded as correct.

Among the above studies in SC, travel cost plays a crucialrole, due to the fact that SC workers have to physically moveto the locations of spatial tasks in order to perform them [9],[15], [22]. For instance, considering task localness, which refersto workers’ preferences based on their travel cost (i.e., workersare more likely to accept nearby tasks), [9] proposes an effectivetask assignment framework by modelling task acceptance rate as adecreasing function of travel distance. Cheung et al. [15] formulatethe interactions among users as a non-cooperative Task SelectionGame (TSG), and propose an Asynchronous and Distributed TaskSelection (ADTS) algorithm, which balances the rewards andtravel costs of the workers for completing tasks.

7 CONCLUSION

In this paper we study the problem of finding the optimal task as-signment for destination-aware spatial crowdsourcing, where eachworker can complete all the assigned tasks before their expirationtime and reach her destination before a given deadline. To settlethe intractable complexity of this problem, we propose a graphpartitioning based approach to decompose the complex workerdependency graph into smaller independent worker clusters, and atree construction algorithm to organize the clusters into balancedtree structure. Then a depth-first search strategy is devised witheffective lower and upper bounds to avoid unpromising traversals.Finally, we further optimizing the original algorithm by proposingstrategies to reduce the overall travel cost and allow each taskto be assigned to multiple workers in order to generalize theapplicability of the proposed framework. Extensive empiricalstudy demonstrates our proposed solution is efficient enough todeliver the maximum number of assignment within reasonablysmall amount of time, and the optimization strategies can alsoresult in less overall travel cost and increased total task assignmentwhen redundant task assignment is requested.

ACKNOWLEDGMENT

This work is partially supported by Natural Science Foundationof China (No. 61532018, 61836007 and 61832017), AustralianResearch Council (Grants No. DP170101172), and Major Projectof Zhejiang Lab (No. 2019DH0ZX01).



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Yan Zhao received the Master degree in Ge-ographic Information System from University ofChinese Academy of Sciences, in 2015. She iscurrently a PHD student in Soochow University.Her research interests include spatial databaseand trajectory computing.

Kai Zheng is a Professor of Computer Sci-ence with University of Electronic Science andTechnology of China. He received his PhD de-gree in Computer Science from The Universi-ty of Queensland in 2012. He has been work-ing in the area of spatial-temporal databases,uncertain databases, social-media analysis, in-memory computing and blockchain technolo-gies. He has published over 100 papers in pres-tigious journals and conferences in data man-agement field such as SIGMOD, ICDE, VLDB

Journal, ACM Transactions and IEEE Transactions. He is a member ofIEEE.

Yang Li received a Bachelor degree in Comput-er Science and Technology at Soochow Univer-sity, in 2015. He is currently a Master studentat Soochow University. His research interestsinclude data mining and spatial crowdsourcing.

Han Su received the BS degree in software en-gineering from Nanjing University, in 2011 andthe PhD degree in computer science from theUniversity of Queensland, in 2015. She is cur-rently an associate professor in the Big Data Re-search Center, University of Electronic Scienceand Technology of China. Her research interestsinclude trajectory querying and mining.

Jiajun Liu is an associate professor at RenminUniversity of China. He received his PhD andBEng from The University of Queensland, Aus-tralia and from Nanjing University, China in 2012and 2006 respectively. Before joining RenminUniversity he has been a Postdoctoral Fellowat the CSIRO of Australia from 2012 to 2015.From 2006 to 2008 he also worked as a Re-searcher/Software Engineer for IBM China Re-search/Development Labs. His main researchinterests are in multimedia and spatio-temporal

data management and mining. He serves as a reviewer for multiplejournals such as VLDBJ, TKDE, TMM, and as a PC member for ACMMM and CCF Big Data.

Xiaofang Zhou received the bachelors andmasters degrees in computer science from Nan-jing University, in 1984 and 1987, respectively,and the PhD degree in computer science fromthe University of Queensland in 1994. He is aprofessor of computer science with the Univer-sity of Queensland. He is the head of the Dataand Knowledge Engineering Research Division,School of Information Technology and ElectricalEngineering. He is also a specially appointed ad-junct professor with Soochow University, China.

His research is focused on finding effective and efficient solutions tomanaging integrating, and analyzing very large amounts of complex da-ta for business and scientific applications. His research interests includespatial and multimedia databases, high performance query processing,web information systems, data mining, and data quality management.He is a fellow of IEEE.

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