ABSTRACTThis paper introduces the Traveling Analyst Problem (TAP), anoriginal strongly NP-hard problem where an automated algo-rithm assists an analyst to explore a dataset, by suggesting themost interesting and coherent set of queries that are estimated tobe completed under a time constraint. We motivate the problem,study its complexity, propose a simple heuristic under simplify-ing assumptions for approximating it, and run preliminary teststo observe the behavior of this heuristic.
1 INTRODUCTIONInteractive data analysis (IDE) [10, 19] is an iterative process con-sisting in executing an action (e.g., a query or a pattern extractionalgorithm) over the data, receiving the result and deciding whatquery comes next. It is a challenging task that a number of pre-vious works aimed at facilitating (see e.g., [5, 19]). Automatingsuch a process raises a number of challenges [16, 25]: how todetermine the direction to follow in often very large and disori-enting datasets, how to decide what is the best query to apply,how to determine if a result is interesting, how to tell a storywith the data resulting from the analysis [8, 9], etc.
If we define a data story as a coherent sequence of queriesthat answer a user goal, we can express this problem as thecomputation of the most interesting and coherent data story thatcan be obtained within a reasonable time. Even with simplifyingassumptions, like restricting to exploratory OLAP queries overa multidimensional schema (e.g., a star-schema, which allowsnavigating hierarchically-structured data with a low formulationeffort) and giving a particular starting point, this problem remainsinherently highly combinatorial.
This paper presents a preliminary study of this problem, thatwe name the Traveling Analyst Problem (TAP). Similarly to auto-mated machine learning, which aims at finding the best modelon a dataset given a time budget (see e.g., [7]), TAP aims at (i)finding, from a very large set of candidate queries, a subset ofqueries that maximizes their interest within a limited time bud-get, and (ii) ordering them so that they narrate a coherent datastory. More formally, each query is associated to an interest scoreas well as to an execution cost. A distance between queries isused to order the queries so that the transition cost between twoconsecutive queries is minimized. Interestingly, a study of thestate-of-the-art reveals that TAP has not been studied in the Op-erations Research community, while being close to two classicaloptimization problems (the Traveling Salesman Problem and theKnapsack Problem) [11].
Practically, we envision TAP as being at the core of an IDEsystem being used by data analysts, like the one described in [16].Analysts would not directly enter TAP parameters, but woulduse storytelling mechanism instead (e.g., [8]) or patterns like “Iwant a quick short report around this query result” or “I want anin-depth analysis in this particular zone of the cube”.
The contributions of this preliminary effort include the for-malization of TAP as well as the proof of its strong NP-hardness(Section 2), an approximation algorithm (Section 3), and somepreliminary tests assessing the parameter sensitivity, executiontime, and closeness to the optimal of our heuristics (Section 4).Section 5 concludes the paper by discussing future research.
2 PROBLEM FORMULATION ANDCOMPLEXITY
In this section, we formalize TAP and show its NP-hardness.
2.1 Problem formulationTAP is formulated as:
Input: Given a set of n queries, a function interest estimat-ing an interestingness score for each query, a function costestimating the execution cost of each query, and a functiondist estimating a cognitive distance between queries,
Do: find a sequence ofm ≤ n queries (without repetition)S.T.: the sequence has the following properties:(1) it maximizes the overall interestingness score,(2) the sum of the costs does not exceed a user-specified
time budget t ,(3) it minimizes the overall cognitive distance between the
queries.
We assume that a storytelling mechanism (e.g., [8]) generatesa set of candidate queries towards producing a story. Thus, ourdeliberations start with a set of n candidate queries whose exe-cution has to be optimized. Formally, let Q be a set of n queries,each associated with a positive time cost cost(qi ) and a positiveinterestingness score interest(qi ). Each pair of queries is associ-ated with a metric dist(qi ,qj ) for their cognitive distance. Givena time budget t , the optimization problem consists in findinga sequence ⟨q1, . . . ,qm⟩ of queries, qi ∈ Q , without repetition,withm ≤ n, such that:
(1) max∑mi=1 interest(qi ),
(2)∑mi=1 cost(qi ) ≤ t
(3) min∑m−1i=1 dist(qi ,qi+1),
The decision problem associated with this optimization prob-lem is to decide if such a sequence exists.
2.2 ComplexityTAP can be related to two families of classical NP-hard opti-mization problems: (i) the Knapsack problem, which consistsin picking weighted items from a set S such that the sum oftheir values is maximum, without exceeding a given size [13],corresponding to constraints (1) and (2); and (ii) the travelingsalesman problem (TSP), which aims at finding the shortest routethat visits all cities and returns to the initial one [1] and is closeto constraint (3).
In our context, the former problem would find the most inter-esting queries given a time budget but, in its classical formulation,it would miss the ordering of queries. A variant [4] includes theposition of each object in the Knapsack via the definition of afunction (which is constant in the case of classical Knapsack).While this problem is closer to TAP, it involves a function thatonly relies on the position (or order) of an object in the Knapsack,and not on the positions of objects previously picked.
TAP could also be modeled as a TSP problem with particu-lar constraints: (i) the TSP cities are the queries; (ii) inter-citydistances correspond to the cognitive distance between queries,whose total must be minimized. However, differently from classi-cal TSP, TAP operates under strong constraints: (iii) it is possibleto visit only a subset of cities, each city has a weight correspond-ing to the action’s interest, whose total is to be maximized; and(iv) each city has a duration of visit corresponding to the cost ofthe query, whose sum must not go beyond a given time budget.
Interestingly, this optimization problem has not been studiedin the literature yet. The variant of the TSP called TSP with profit(TSPwP), described by Feillet & al. in [6], is closer to our problem,but still differs in two aspects: (i) it looks for circuits and doesnot reject any vertex in the solution, and (ii) it gives a limit interms of the travel distance (inter-queries distance in our case)while our limit is on the cost of queries (the duration of visit).
An in-depth study of the TAP complexity is beyond the scopeof this paper. However, we can easily show that our problem isstrongly NP-hard since the TSP is a particular case of it. Indeed, ifthe time budget t is high enough, i.e., all queries can be selected,then TAP is a TSP. This result means that, unless P=NP, the TAPproblem can only be solved to optimality by algorithms with aworst-case time complexity in O∗(cn ), with c a positive root andn the size of Q .
2.3 Size of the problem and our naiveapproach
We now discuss the size n of TAP (the number of queries inQ) since the size of an optimization problem usually impactsthe choice of resolution approaches. Theoretically, given a cubeschema, all non empty queries over this schema could be con-sidered. Practically, it is reasonable to consider that this set isgenerated from a given initial query q0.
In what follows, we restrict to star join queries over a starschema of the form q = (д, s,m) where д is the query group-byset, s is a set of selections, andm is a set of measures, all 3 setsbeing pairwise disjoint. Transforming a query into another witha simple OLAP operation means either changing the group-byset д, changing a selection in s , or changing a measure inm.
Even restricting Q to the set of queries that can be generatedby transforming an initial query q0 = (д0, s0,m0)with a sequenceof simple OLAP operations, the size ofQ is potentially very large,not allowing to look for an exact solution. A rough estimate ofthe number of queries that can be generated from q0 by applying
k OLAP operations, i.e., the size of Q , can be done by assumingfor simplicity that dimensions only have linear hierarchies (nobranches):
where hi is the number of levels in dimension i , D is the union ofthe active domains of all levels, andM is the set of all measuresin the cube. Changing the query group-by set means picking onegroup-by set among all the possible ones, excluding the currentoneд0. Changing the selected valuesmeans picking a set of valuesin the active domain, excluding the current one s0. Changing themeasure set means picking a set of measures among all possiblesets of measures excluding the current onem0.
In order to approach solutions of TAP for arbitrary large sets ofqueries, we adopt the following strategy. We first use a heuristicto solve the Knapsack problem and obtain a subset of queries,using estimated costs and interests, so that the estimated costssatisfy the time budget. Then, we order the queries by increasingestimated cost and evaluate them. We periodically check thedifference between the time budget constraint and the elapsedtime: if it is negative (too much time is taken) we redo a Knapsackto reduce the set of chosen queries; otherwise (we can benefitfrom additional time), we redo a Knapsack adding previously nottaken queries. Finally, we determine an order on the chosen set ofqueries so that cognitive distance is minimized, using a heuristicto solve the TSP.
3 APPROXIMATION ALGORITHMBefore presenting our approach we discuss its parameters, i.e.,the three functions for cost, interest, and distance, and the set Qof queries. Choosing the best three functions or defining the bestset of queries Q is outside the scope of this paper. Note that aframework for learning cell interestingness in a cube is the topicof a recent paper [17]. We give examples in this section, and weindicate precisely in the tests of Section 4 the parameters used.
3.1 CostThe cost of a query is related to its execution time. Classically,this cost can be estimated by a query optimizer before the exe-cution of the query. We therefore consider that we can measurea query cost in two ways, to obtain an a priori cost (e.g., usingthe RDBMS optimizer) and an a posteriori cost (the actual queryexecution time). The a priori cost is used to decide if a query canbe included or not in the solution, while the a posteriori cost isused to compute the elapsed time.
3.2 Interestingness measureA crucial part of TAP lies in the definition of an interestingnessmeasure to determine the optimal subset of queries. To quicklydecide if a query is interesting, it is preferable that this measurecan be computed before the actual evaluation of the query bythe DBMS, therefore that it relies on the text of the query. In thissense, we propose to follow the idea of subjective interestingnessmeasure of a pattern as developed by De Bie in the context of Ex-ploratory Data Mining (EDM) [2] and to extend it to measure thesubjective interestingness of a query as expressed by a coherentset of query parts.
In the information-theoretic formalism proposed by De Bie, in-terestingness is conditioned by the prior knowledge belie f (p) ona pattern p of the data space, which is expressed as a probability
distribution over the set of patterns P . The interestingness mea-sure IM is derived by normalizing the belief by the complexityof the pattern as follows:
IM(p) =− log(belie f (p))complexity(p)
(1)
In the context of BI, [3] introduces an innovative approachto learn the belief distribution associated to query parts. Theapproach considers a connected graph of query parts based onthe schema of the cube and the past usage of query parts, and usesa random walk on this graph to produce the expected long-termdistribution over query parts. Interestingly, by construction, theprobabilities obtained for each query part are independent, whichallows to propose a simple formulation for the interestingnessmeasure of a query q = (д, s,m) based on its query parts p ∈
(д ∪ s ∪m):
IM(q) =−∑p∈(д∪s∪m) log(belie f (p))
|д | + |s | + |m |(2)
3.3 Cognitive distanceTo order a set of queries we draw inspiration from Hullman et al.[9] who estimate the cognitive cost of transiting from the resultof one query to the result of another. We use this cost as a proxyfor cognitive distance. Interestingly, this cost can be estimatedwithout having evaluated the query. Using controlled studies ofpeoples’ preferences for different types of single visualization-to-visualization transitions, Hullman et al. [9] proposed a transi-tion cost model that approximates the cognitive cost of movingfrom one visualization to the next in a sequence of static views.The transition cost is defined as the number of transformationsrequired to convert the data shown in the first view to the sec-ond. A single transformation is defined as a change to one ofthe data fields shown from the first view to the second. We usethis cost model to define a simple distance between queries asthe Jaccard distance between sets of query parts. Formally, letq1 = (д1, s1,m1) and q2 = (д2, s2,m2) be two queries; we define:
dist(q1,q2) = 1 −|(д1 ∪ s1 ∪m1) ∩ (д2 ∪ s2 ∪m2)|
|д1 ∪ s1 ∪m1 ∪ д2 ∪ s2 ∪m2 |(3)
3.4 Set of queriesTAP is defined for a given setQ of queries. Practically, we considerthat this set is generated from a given initial query, q0. This initialquery can be interpreted as a particular moment in an interactiveanalysis where the user considers what is retrieved is importantand worth exploring “around” it, but has too many directionsto explore. In what follows, we consider this initial query as aparameter of our approach.
Defining the set of queries worth exploring after q0 shouldbe rooted in the earlier OLAP literature, especially automaticreporting [8], automatic cube exploration [12], discovery drivenanalysis [22, 23], and more generally realistic OLAP workloads[21]. In the Cinecubes approach [8], the authors consider twotypes of queries: (i) queries for values similar to those definingthe selection filters of the initial query (i.e., siblings of ancestors),and (ii) direct drill-downs into the dimensions of the initial re-sult, one dimension at a time. In the DICE approach [12], theauthors consider direct roll-up queries, direct drill-down queries,sibling queries (a change of a dimension value, i.e., coordinate,in a dimension), and pivot queries (a change in the inspected
dimension). In the DIFF and RELAX operators [22, 23], the au-thors consider direct or distant drill-down (resp. roll-up) queriesthat detail (resp. aggregate) two particular cells of a query result.Finally, the CubeLoad OLAP workload generator [21] is basedon patterns modeling realistic OLAP sessions, that could be usedto generate the queries of Q .
Ideally, the set Q should include all these potentially relevantfollow-up queries to q0. For the present work we will considerdifferent sets varying the generation of roll-up, drill-down, andsibling queries (see Section 4).
3.5 A simple heuristic for approximating TAPAlgorithm 1 presents our heuristic, named Reopt , since it is basedon a naive re-optimization principle. Note that the generation ofthe setQ from an initial queryq0 is considered as a pre-processingstep. Algorithm 1 takes advantage of the fact that functions in-terest, cost, and distance can be computed solely on the basis ofthe query expression (and not on the result of the queries).
First, Algorithm 1 splits the time budget t in tk (time for theglobal query execution) and to (time for other tasks, like solvingthe Knapsack, solving the TSP, etc.). Then the Knapsack is solved(line 2), and the queries selected are ordered by their estimatedevaluation cost (line 3). Queries are then executed (line 6) and,after each execution, the elapsed time is checked. If it is esti-mated that the time budget tk will not be respected (line 9), thenanother Knapsack is triggered with the remaining time. If it isestimated that the time budget will not be completely spent (line13), then another Knapsack is triggered with all the remainingqueries. Finally, once all the queries of the chosen set of queriesare executed, the TSP is solved. It is easy to verify that Algorithm1 converges: the set K is at worst Q and, at each iteration of thefor loop (line 5-16), the set E is augmented with one more queryof K while such query is removed from K .
Note that, when actually executing the queries, Reopt attachesa particular importance to the estimated cost (see for instanceLine 3) compared to interest or distance. The main cause behindthis decision is due to the time budget that has to be respected:had we given priority to distance from q0 or interest, we mighthave executed first costly queries or we might have many ties(since many queries would be at the same distance from q0).Although alternative formulations are also possible, due to thesensitivity of time efficiency in user interactions (keep in mindthat we operate in the context of exploratory data analysis), it isimperative that the other factors do not guide query executionto take up too much time.
Implementation-wise, the Knapsack problem is solved usinga Fully Polynomial Time Approximation that gives a bound onthe divergence from the optimum [14], and for the TSP using theheuristic of [15].
4 PRELIMINARY EXPERIMENTSOur preliminary experiments aim at understanding the behaviorof our naive algorithm Reopt . To this end, we have comparedit to two other algorithms: a brute force one that looks for theoptimal solutions (named optimal ), which obviously works onlyon small instances of TAP, and a simplistic one that consists ofsolving the Knapsack, then solving the TSP, and then executingthe queries (namedK +TSP ). All the algorithms are implementedin Java 12 and run on a Core i7 with 16GB RAM under LinuxFedora. The code is available via Github (https://github.com/OLAP3/CubeExplorer). Note that, in all tests, costs are estimated
Algorithm 1: Reopt: simple re-optimization heuristic forTAP approximationData: An instance (Q, interest(), dist(), cost(), t) of TAPResult: A sequence of queries
1 Split t = tk + to2 K = knapsack(Q, tk )
3 sort K by increasing cost4 E = ∅
5 for each query q ∈ K do6 execute q7 K = K \ {q}
8 E = E ∪ {q}
9 if elapsedTime +∑q∈K cost(q) > tk then
10 //we may have less time than expected11 K = knapsack(K, tk − elapsedTime)
12 sort K by increasing cost13 else if elapsedTime +
∑q∈K cost(q) < tk then
14 //we may have more time than expected15 K = knapsack(Q \ E, tk − elapsedTime)
16 sort K by increasing cost
17 S = TSP(E); // construct sequence18 return S
Figure 1: Optimization time
using not the estimation coming from the query optimizer of theDBMS but a linear regressive model based on the query lengthin ASCII, the number of tables, the number of projections, thenumber of selections, and the number of aggregations of the SQLstar join query. This is because our empirical tests have shownthat this model was more accurate than the raw estimates givenby the query optimizer.
4.1 Optimization timeIn this first test, we use a simple synthetic cube under the schemaof the SSB benchmark [20], with an instance of 1GB under Mari-aDB, produced with the TPC-H data generator. The cube hasonly one fact table and 5 dimension tables, which enables to keepthe number of queries in Q under control for comparing Reoptand K +TSP with optimal . Over this schema, we have used theCubeLoad generator [21] to generate 40 q0 queries. These queriesare used to generate Q with direct roll-up or drill down queries,i.e., Q can vary between 5 and 10.
Figure 2: Total interestingness
Figure 1 shows, on a logarithmic scale, the average time takenby the 3 algorithms to optimize (not counting the query execu-tion time) by different sizes of Q with time budget varying from1 second to 10 seconds for Reopt and K + TSP . Results are asexpected, with K +TSP outperforming its two competitors, sinceit only runs once Knapsack solving. Notably, the time taken byReopt remains under control.
To assess the benefit of our re-optimization step (line 9-16of Algorithm 1), we counted the number of times each branchof the if statement in line 9 of Reopt is taken, i.e., the numberof times for negative (lines 10-12) and for positive (lines 14-16)re-optimizations. We have observed that both branches are used,with positive re-optimizations being rarely done compared to neg-ative ones; precisely, over 399 total runs, positive re-optimizationsare done 116 times while negative re-optimizations are done 851times. This demonstrates that Reopt can adaptively respond toinaccurate estimates.
4.2 Distance and interestingness vs timebudget
For the next series of tests, we use a cube issued by a Frenchproject on energy vulnerability. It is organized as a star schemawith 19 dimensions, 68 (non-top) levels, and 24 measures, and itcontains 37,149 facts recorded in the fact table. The cube is storedin SQL Server. We use 19 real user explorations over this cube(navigation traces generated by master students when analysingdata during a course) and pick as q0 the first query of the explo-rations. For each of these queries, we generateQ by rolling-up q0in a dimension, drilling down q0 in a dimension, or computinga sibling query in a dimension. We use this cube and set Q toobtain more realistic instances of TAP. Precisely, |Q | ranges from29 to 149 queries for these tests. Over these sets Q we run Reoptand K +TSP and observe how total interestingness and averagedistance between queries change when increasing time budget tfrom 500 milliseconds to 5 seconds.
Figures 2 and 3 show the results. It can be seen that Reoptoutperforms the simplistic K +TSP in terms of interestingness,which illustrates the benefit of our re-optimization heuristic,while both algorithms achieve comparable average distance.
4.3 Unused timeIn this last test, using the same cube and protocol as in the previ-ous subsection, we want to understand if the budget time is usedproperly.
Figure 3: Average distance
Figure 4: Unused time and candidate queries left
Figure 4 plots against different time budgets the proportion ofunused time versus the number of candidate queries left in Q toexecute. As we can see, regardless of the budget, our algorithmmanages to take advantage of all the time available unless allqueries of the Q have been explored, in which case the ratio ofunused time increases. On the contrary, the K +TSP approach,which is unaware of the possible gain in executing the otherqueries, has a larger ratio of unused time and does not manageto explore completely Q . The absence of idle time clearly provesthe advantages of our adaptive re-optimization heuristic over thestatic K +TSP method, which cannot compensate for the errorsin the prediction of the cost, nor ensure that the execution timeis near to the time budget.
5 CONCLUSIONThis paper introduced the Traveling Analyst Problem (TAP), theproblem of computing the most interesting and coherent datastory that can be obtained within a reasonable time. We formalizethe problem, show its strong NP-hardness and propose a heuristicfor finding approximate solutions to it. Our preliminary experi-ments show that a heuristic based on simple re-optimization is apromising direction to obtain acceptable solutions.
We believe that TAP opens many interesting research direc-tions. Obviously, the first step is an in-depth theoretical studyof TAP, to understand which types of optimization algorithmsare more appropriate. Importantly, TAP should be investigated
in the context of data exploration, which means that optimiza-tion algorithms should take advantage of classical data manage-ment optimizations, like re-optimization (e.g., [18]), and that TAPshould be declaratively formulated, for instance by having start-ing points expressed in an intentional fashion (e.g., [24]). Usertests should be conducted for further evaluating the approach.Finally, we also note that the definition of TAP is general, leavingroom for variants, e.g., changing the definition of queries (e.g.,from non-OLAP SQL queries to more complex actions involvingqueries and pattern mining or statistical tests), as well as chang-ing cost (e.g., using self-adjusting cost model), interest (e.g., usingstatistics or data sampling), and distance functions.
Acknowledgment. The authors would like to thank VincentT’Kindt for his insights on TAP complexity.
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