Debugging Embedded Multimedia Application Tracesthrough Periodic Pattern Mining

Patricia López Cueva*† Aurélie Bertaux* Alexandre Termier*

Jean François Méhaut* Miguel Santana†

†STMicroelectronicsCrolles, France

FirstName.LastName@st.com

*University of Grenoble, LIGGrenoble, France

FirstName.LastName@imag.fr

ABSTRACTIncreasing complexity in both the software and the underly-ing hardware, and ever tighter time-to-market pressures aresome of the key challenges faced when designing multime-dia embedded systems. Optimizing the debugging phase canhelp to reduce development time significantly. A powerfulapproach used extensively during this phase is the analy-sis of execution traces. However, huge trace volumes makemanual trace analysis unmanageable. In such situations,Data Mining can help by automatically discovering interest-ing patterns in large amounts of data. In this paper, we areinterested in discovering periodic behaviors in multimediaapplications. Therefore, we propose a new pattern miningapproach for automatically discovering all periodic patternsoccurring in a multimedia application execution trace.

Furthermore, gaps in the periodicity are of special inter-est since they can correspond to cracks or drop-outs in thestream. Existing periodic pattern definitions are too restric-tive regarding the size of the gaps in the periodicity. So, inthis paper, we specify a new definition of frequent periodicpatterns that removes this limitation. Moreover, in order tosimplify the analysis of the set of frequent periodic patternswe propose two complementary approaches: (a) a losslessrepresentation that reduces the size of the set and facilitatesits analysis, and (b) a tool to identify pairs of “competitors”where a pattern breaks the periodicity of another pattern.Several experiments were carried out on embedded videoand audio decoding application traces, demonstrating thatusing these new patterns it is possible to identify abnormalbehaviors.

Categories and Subject DescriptorsD.2.5 [Software Engineering]: Testing and Debugging,Tracing; H.2.8 [Database Management]: Database Ap-plications, Data Mining

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.EMSOFT’12, October 7-12, 2012, Tampere, Finland.Copyright 2012 ACM 978-1-4503-1425-1/12/09 ...$10.00.

KeywordsEmbedded Systems, Multimedia Applications, Periodic Pat-tern Mining, Debugging

1. INTRODUCTIONThe current trend in consumer electronics is to have pro-

ducts that support multimedia applications, such as set-topboxes, tablets, smartphones and MP4 players. All theseproducts are powered by highly integrated System-on-a-Chip(SoC) solutions, that contain, in a single die, several process-ing units, memory blocks and specialized units for audio andvideo decoding. Developing efficient and robust applicationsfor systems using these SoCs is a challenging issue. It is thuscritical for companies developing embedded software to havecomprehensive programming frameworks with advanced fea-tures for debugging and optimizing their applications.

Nowadays, the most widely used technique to give insightinto the behavior of an embedded application for debuggingand optimization purposes is trace analysis [19]. That is, tocollect events generated by the system and the application inorder to perform a post-mortem analysis of the execution.Many recent embedded systems directly integrate tracinghardware support in order to minimize intrusiveness, i.e.the act of tracing has minimal impact on the behavior ofthe application, allowing complex interactions to be shownin real-time applications such as video decoding.

Over the past decade, the software and hardware of em-bedded systems have faced an increase in complexity dueto the use of techniques such as multi-threading, power andmemory optimization, and so on. Consequently, the size ofthe execution traces of applications running on these sys-tems has also increased, and we can only expect an evenbigger increase with the introduction of many-core proces-sors in embedded systems. Therefore, the manual analysisof execution traces is becoming an unmanageable task.

To overcome this issue we intend to use data mining inorder to automatically extract pertinent information (calledpatterns) from traces. In this context, we consider a patternas a set of functions and system events that are found to-gether frequently in the trace. Multimedia applications havea periodic execution based on frame decoding, i.e. the sameoperations are performed on each frame or every n frames.Therefore, in this paper, we use frequent periodic pattern

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mining on execution traces to discover periodic behaviors ofmultimedia applications.

However, sometimes this periodic behavior is not perfect:cracks in the sound or drop-outs in the video stream areoften the consequence of a disruption in the periodicity of apattern, showing up as a gap. Therefore, these gaps shouldbe investigated as a priority for debugging and optimizingmultimedia applications. The state of the art (see Section7) considers only regular sized gaps, i.e. a multiple of theperiod, while in multimedia applications it is not possible toknow the size of the gaps in advance. Therefore, we proposea new definition of periodic patterns that allows to havearbitrary sized gaps.

Frequent pattern mining generally generates a largeamount of patterns which can make their analysis difficult.Therefore, we propose a reduced representation of the set offrequent periodic patterns, based on ternary relation theory[22] and the minimal generator concept [18], while keepingthe same amount of information.

In this paper, we explore the current status of trace de-bugging on embedded systems as well as explain how to pre-process execution traces to analyze them with our patternmining algorithm in Section 2. We propose a new definitionof frequent periodic pattern in Section 3, and introduce ourapproach to reduce the set of outputted patterns mentionedabove in Section 4. In Section 5, we present a new algorithmto mine the reduced set of frequent periodic patterns, as wellas introduce an analysis tool that helps to identify pairs ofcompetitor periodic patterns. By competitors we mean thattheir executions are in competition, i.e. a pattern breakingthe periodicity of another pattern, which helps to identifypossible conflicts between different parts of the system. InSection 6, we present the insights given by analyzing applica-tion traces. Next, we explore the state-of-the-art regardingpattern mining in system analysis, periodic pattern miningand pattern mining of ternary relations in Section 7. Finally,we present our conclusions and future work in Section 8.

2. MULTIMEDIA APPLICATIONS ANDEMBEDDED PLATFORMS

In this section, we introduce the context in which tracedebugging is used on embedded systems. We also explainhow to pre-process the traces so that they can be analyzedby our pattern mining algorithm.

2.1 Trace Debugging on Embedded SystemsDuring the development of applications for embedded sys-

tems, development and evaluation boards are used by devel-opers to test their applications. An example is the STi7200-MBoard platform [21] which contains an STi7200 SoC, who-se architecture is shown in Figure 1. This SoC is used inhigh-definition set-top boxes produced by STMicroelectron-ics. Apart from all the necessary connectivity, this system-on-chip contains a ST40 core and two ST231 cores. TheST40 core is dedicated to application execution and devicecontrol, while the ST231 cores are in charged of the audioand video decoding.

These development boards offer a way of collecting tracesthat can vary from a dedicated trace port, to a networkconnexion that allows the developer to retrieve the tracesfrom the board. Execution traces provide a full view of the

Figure 1: STi7200 SoC architecture

running system (operating system and application) whileminimizing the intrusiveness.

In some cases, software tracing solutions are provided bythe operating system. For instance, on an ST40 core, ap-plications run on a Linux distribution for STMicroelectron-ics products. This operating system provides a tracing toolbased on KProbes [8], which registers system and applica-tion events: interrupts, context switches, function calls, sys-tem calls, etc. In this paper, we focus on the analysis ofsoftware execution traces. Moreover, the execution tracesused for the experiments were produced by the tracing toolmentioned above in this paragraph.

In other cases, hardware tracing solutions provide very ac-curate information about the execution of the system withno intrusion. The main problem with hardware-based trac-ing techniques is the extremely large amount of data gen-erated. Moreover, transferring the trace out of the chip re-quires a large I/O bandwidth with a significant cost increase.

Embedded systems are complex entities composed of sev-eral processing units, accelerators, GPUs, DSPs and so on.Moreover, multimedia applications often consist of filter pi-pelines, in which certain steps are carried out by acceleratorsor GPUs instead of the main core. Therefore, in order tohave an insight into what is really happening in the system,it is necessary to trace every component of the SoC. Com-panies such as Intel [16] or STMicroelectronics are currentlyworking on system tracing solutions that tackle this neces-sity. Certainly traces obtained by this new approach willbe a lot richer in terms of the amount of information, butalso a lot bigger than current traces. In consequence, theneed for automatic trace analysis tools will be even moreimportant in the near future. Currently, we focus on theanalysis of software execution traces, but our approach isgeneric and would be easily extended to be able to analyzethis new generation of traces.

2.2 Execution Traces in Pattern MiningAn execution trace is a sequence of events and their times-

tamp, registered during the execution of an application, asshown in Figure 2. However, most pattern mining algo-rithms search for patterns common to a sequence of sets[1]. This makes it necessary to split the execution trace intosets of events, called transactions, in order to apply patternmining algorithms.

We propose two methods to split the trace:

(a) using a time interval [11], where a transaction containsall events of the trace registered in the same time win-dow, or

(b) using a function name, where a transaction containsall events of the trace that occurred between two oc-currences of the given function.

As an example, we can observe the result of splitting atrace using a time interval of 0.1 ms in Figure 2. The times-tamp of the events is not included in the transactions, so

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68.770630 getFrame68.770697 displayFrame68.770741 int1668.770768 swint1668.770869 getFrame68.770913 displayFrame68.770959 write1668.770982 cpu clock68.771032 getFrame68.771099 displayFrame68.771150 read1668.771235 fork68.771324 get pid68.771346 getFrame68.771372 displayFrame68.771402 prink68.771456 sem up68.771487 sem down68.771540 getFrame68.771586 displayFrame

12345678910

Preprocessing

getFrame, displayFrameint16, swint16getFrame, displayFramewrite16, cpu clockgetFrame, displayFrameread16fork, get pidgetFrame, displayFrame, prinksem up, sem downgetFrame, displayFrame

Figure 2: Preprocessing of an execution trace bysplitting it into time intervals of 0.1 ms. Timestamp:seconds.microseconds

the information about when exactly an event was executedis not considered in the analysis. In this paper, we are in-terested in discovering sets of events that occur periodicallyin the execution trace, but the order in which those eventsare executed is not taken into account. Considering thatthe executing environment is multi-threaded, the order ofthe events might change according to the decision taken bythe scheduler. Therefore, even if the order is not taken intoaccount, our approach is able to discover interesting rela-tionships between the events in the execution trace.

For a more exhaustive analysis where the order of eventexecution is considered important, there exist pattern min-ing techniques that can discover ordered sets of events, calledsequences, that occur frequently in the trace. Nevertheless,these techniques are more complex and computationally ex-pensive. The results obtained by these techniques might beinteresting but these techniques are out of the scope of thispaper.

Unless stated otherwise, in this paper we use the functionthat starts the decoding of a frame to split the trace in orderto discover events that happen while decoding frames. Beingable to identify the events that occur when decoding a frameallows us to more easily identify sets of events that mightaffect the application’s behavior.

3. DEFINITIONSIn this section, we give basic definitions that allow us to

present our definition of frequent periodic patterns.As we have seen in Section 2, an execution trace is split

into sets of events which, in data mining, are called transac-tions. We define our dataset D as the ordered set of transac-tions {t1, t2, ..., tn} obtained by splitting an execution trace,and the set of items I as the set of all possible events foundin the trace. An itemset is a set of events belonging to I, e.g.{sem up, int16, cpu clock}. For instance, Table 1 presentsthe dataset obtained after preprocessing the execution traceshown in Figure 2 by splitting the trace using a time intervalof 0.1 ms.

Definition 1. Given a set of items I = {i1, i2, ..., ir},a dataset D is an ordered set of transactions {t1, t2, ..., tn}where each transaction is a subset of I, i.e. tk ⊆ I for1 ≤ k ≤ n, and where the order is defined by ti < tj if andonly if i < j. The length of the dataset D is the number oftransactions that form part of the dataset and is denoted by|D|.

Table 1: A dataset in the context of system traceanalysis.

tk Itemsett1 getFrame, displayFramet2 int16, swint16t3 getFrame, displayFramet4 write16, cpu clockt5 getFrame, displayFramet6 read16t7 fork, get pidt8 getFrame, displayFrame, printkt9 sem up, sem downt10 getFrame, displayFrame

An itemset X is denoted by {x1, x2, ..., xj} where xt is anitem, i.e. xt ∈ I. Considering X ⊆ I, we say that an itemsetX occurs in the transaction tk if and only if X ⊆ tk.

Given a transaction tk, its transaction identifier, denotedas tid(tk), is its position in the dataset, i.e. k. We alsodefine the distance d between two transactions ti and tjas the difference between their transaction identifiers, i.e.d = j − i with i ≤ j.

When an itemset occurs over a set of transactions andthe distance between any two consecutive transactions isconstant, this set of transactions forms a cycle.

Definition 2. Given an itemset X and a period p, a cy-cle of X, denoted cycle(o, p, l,X), is a maximal set of ltransactions in D containing X, starting at transaction toand separated by equal distance p: cycle(o, p, l,X) = {tk ∈D|X ⊆ tk, k = o + p ∗ i, 0 ≤ i < l,X * to−p, X * to+p∗l}where 0 ≤ o < n, 2 ≤ l ≤ n and n is the number of transac-tions in D.

Example 1. In the dataset in Table 1, the itemset X ={getFrame, displayFrame} is found at a period p = 2 ontransactions from t1 (o = 1) to t5, therefore it forms a cycleof length l = 3, denotedcycle(1,2, 3, {getFrame, displayFrame}) = {t1, t3, t5}1.

Figure 3: Examples of cycles and non-cycles

Figure 3 offers a graphical representation of a set of trans-actions where cycle1(1,2, 3, {getFrame, displayFrame}) ismaximal. cycle2(3,2, 2, {getFrame, displayFrame}) andcycle3(1,2, 2, {getFrame, displayFrame}) are not maxi-mal since they can be extended with transactions t1 andt5 respectively, and therefore they are not valid cycles inour context.

A set of consecutive cycles (the end of a cycle happensbefore the beginning of the following cycle) over the sameitemset and the same period forms a periodic pattern.

1Periods are presented in bold for clarity.

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Figure 4: Periodic pattern formation

Definition 3. An itemset X together with a set of cyclesC and a period p form a periodic pattern if the set of cyclesC = {(o1, l1), ..., (ok, lk)}2, with 1 ≤ k ≤ m and m being themaximum number of cycles of period p in the dataset D, isa set of cycles of X such that:

1. All cycles have the same period p.

2. All cycles are consecutive: ∀(oi, li), (oj , lj) ∈ C suchthat 1 ≤ i < j ≤ k, we have oi < oj.

3. Cycles do not overlap: ∀(oi, li), (oj , lj) ∈ C such thati < j, we have oi + (p ∗ (li − 1)) < oj.

We denote this periodic pattern P (X, p, s, C).

The support of a periodic pattern, denoted s, is the sumof all cycle lengths in C, i.e. given C = {(o1, l1), ..., (ok, lk)}with 1 ≤ k ≤ m then s =

∑ki=1 li.

Example 2. In Figure 4 we can observe that the cyclescycle1(1,2, 3, {getFrame, displayFrame}) and cycle2(8,2,2, {getFrame, displayFrame}) form a periodic patternP1 = ({getFrame, displayFrame},2, 5, {(1, 3)(8, 2)}) withperiod 2 and a support of 5 transactions.

We now introduce the notion of frequent periodic patterns.

Definition 4. Given a minimum support thresholdmin sup, a periodic pattern P is frequent if its support isgreater than min sup, i.e. P (X, p, s, C) is frequent if andonly if s ≥ min sup.

Example 3. Given the dataset shown in Table 1 and aminimum support of two transactions, the set of frequentperiodic patterns is presented in Table 23.

As we can see in Table 2, the set of frequent periodic pat-terns is highly redundant. On one hand, all combinations oflarge itemsets are consider as patterns. For example, P1, P2

and P3 are present in exactly the same transactions and theitemset of P3 contains the itemsets of P1 and P2. ThereforeP1 and P2 do not give any more information than P3. Onthe other hand, combinations of small periods by additionor multiplication generate redundant patterns. For example,P9 is redundant with respect to P3 since its period is multi-ple of P3’s period while the transactions in P9 can be foundin P3. Therefore P9 does not give any more informationthan P3.

In real datasets tens of thousands of frequent periodic pat-terns might be found, which are impractical to analyze by anapplication developer. In order to produce a reduced repre-sentation of the set of frequent periodic patterns we adopt atriadic approach by introducing the periods into the dataset.

2For simplicity of notation, a cycle is represented here by its origin

o and its length l since all cycles in the set share the same itemset Xand period p.

3Only periods not greater that the length of the dataset divided

by min sup are considered.

Table 2: Set of frequent periodic patternsFrequent Periodic Patterns

P1({getFrame},2, 5, {(1, 3)(8, 2)})P2({displayFrame},2, 5, {(1, 3)(8, 2)})P3({getFrame, displayFrame},2, 5, {(1, 3)(8, 2)})P4({getFrame},3, 2, {(5, 2)})P5({displayFrame},3, 2, {(5, 2)})P6({getFrame, displayFrame},3, 2, {(5, 2)})P7({getFrame},4, 2, {(1, 2)})P8({displayFrame},4, 2, {(1, 2)})P9({getFrame, displayFrame},4, 2, {(1, 2)})P10({getFrame},5, 2, {(3, 2)})P11({displayFrame},5, 2, {(3, 2)})P12({getFrame, diaplayFrame},5, 2, {(3, 2)})P13({getFrame},5, 2, {(5, 2)})P14({displayFrame},5, 2, {(5, 2)})P15({getFrame, displayFrame},5, 2, {(5, 2)})

4. NON-REDUNDANT PATTERNSIn this section, we present the application of ternary rela-

tion theory to periodic pattern mining and the definition ofminimal periodic generators that will allow us to generatea reduced set of periodic patterns that are easier to analyzeby the developer.

As we have seen in Section 3, the set of frequent periodicpatterns is highly redundant. This is a common drawbackwhen using pattern mining techniques which complicates theanalysis of the results. A major breakthrough has been thestudy of closed operators applied to patterns such as item-sets [18], gradual patterns [2] and so on. The set of closedpatterns, mined by closed operators, is a reduced represen-tation of the set of frequent patterns but contains the sameinformation.

No previous study has been carried out regarding closedperiodic patterns. This might be because the theory behindclosed patterns, called Galois connection, is based on binaryrelations, such as the relation items×transactions for item-set mining. Analyzing the structure of periodic patterns, wecan observe that they are based on ternary relations involv-ing the items, the transactions and the periods. In ternaryrelations, it is not possible to make use of a Galois connectionto generate closed patterns [3], therefore it is not possible toapply the Galois theory to periodic pattern mining.

Instead, we are going to use a triadic approach to FormalConcept Analysis [10] to mine a reduced representation ofthe set of frequent periodic patterns, which in this contextis called the set of triadic concepts. Nevertheless, sinceall possible periods are considered, the set of triadic con-cepts might contain redundant periods, such as multiples ofsmaller periods, periods formed by the concatenation of twosmaller periods, and so on. So, to reduce this redundancy,we define the concept of minimal periodic generators whichallows us to extract the minimal set of periodic patterns

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Table 3: Representation of the relation YI/P − T 2 3

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10getFrame × × × × × × ×displayFrame × × × × × × ×...

I/P − T 4 5t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10

getFrame × × × × × ×displayFrame × × × × × ×...

that explains the behavior of the application, without losinginformation.

4.1 Triadic Approach to Periodic PatternMining

Our dataset, introduced in Definition 1, corresponds to arelation between two attributes, items × transactions, i.e.R ⊆ I × D and each r ∈ R can be represented by a coupler = {(i, t)|i ∈ I, t ∈ D}, denoting that the item i occurs ontransaction t.

In order to mine periodic patterns, the period should beincluded in the dataset, but in order to do so, the binaryrelation R ⊆ I × D has to be transformed into a ternaryrelation Y ⊆ I×P×D, with P the set of all possible periodsthat we limit to the range [1..|D|/min sup].

To formalize this ternary relation we are going to intro-duce a triadic approach to formal concept analysis first intro-duced by Lehmann et al. [10] in 1995. Specifically, the con-cepts of triadic context and triadic concepts are presentedhere including some modifications needed in order to adaptthem to periodic pattern mining.

Definition 5. A periodic triadic context is defined asa quadruple (I,P,D,Y) where I is the set of items, P isthe set of periods, D is the set of transactions, and Y is aternary relation between I, P and D, i.e. Y ⊆ I × P ×D.

An element of the relation y ⊆ Y is denoted by the tripley = {(i, p, t)|i ∈ I, p ∈ P, t ∈ D} and is read: the transac-tion t forms part of a cycle of period p of the item i.

Example 4. Given the dataset shown in Table 1, the cor-responding triadic context is shown in Table 34. Each crossin the table represents an element of the ternary relation Y.For example, P1({getFrame},2, 5, {(1, 3)(8, 2}) in Table 2is transformed into the triples (getFrame, 2, t1),(getFrame, 2, t3), (getFrame, 2, t5), (getFrame, 2, t8),(getFrame, 2, t10) shown by the corresponding crosses in Ta-ble 3.

Definition 6. Given a minimum support thresholdmin sup, a triple (I, P, T ), with I ⊆ I, P ⊆ P, T ⊆ D andI × P × T ⊆ Y, is frequent if and only if I 6= ∅, P 6= ∅ and|T | ≥ min sup.

Example 5. In Table 3, given a min sup of 2, we canobserve several frequent triples such as ({getFrame}, {2, 3},{t5, t8}) or ({getFrame, displayFrame}, {5}, {t3, t5, t8,t10}), since the number of transactions forming those triplesis greater or equal to 2.

4All items, excluding getFrame and displayFrame, do not form

any cycle of any possible period and, for clarity, they are not includedin the table.

The set of frequent triples is highly redundant since it in-cludes all possible combinations between items, periods andtransactions included in the ternary relation Y. A losslessreduced representation of this set was introduced by Wille[22] and named triadic concepts. Saying the representationis lossless means that it is possible to reconstruct the set offrequent triples from the set of triadic concepts without anyextra information.

Definition 7. A triadic concept of a triadic context(I,P,D,Y) is a triple (I, P, T ) with I ⊆ I, P ⊆ P andT ⊆ D, such that none of its three components can be en-larged without violating the condition I × P × T ⊆ Y.

Example 6. In Table 4, we can observe the set of triadicconcepts extracted from the set of frequent triples obtainedfrom the dataset shown in Table 3. The triples forming thisset are triadic concepts since it is not possible to extend anyof the attributes of the triple without violating the relationY.

Table 4: Set of triadic conceptsTriadic Concepts

T1({getFrame, displayFrame}, {2}, {t1, t3, t5, t8, t10})T2({getFrame, displayFrame}, {2, 4}, {t1, t5})T3({getFrame, displayFrame}, {2, 5}, {t3, t5, t8, t10})T4({getFrame, displayFrame}, {2, 3, 5}, {t5, t8})T5({getFrame, displayFrame}, {2, 3, 4, 5}, {t5})

It can be observed that the set of triadic concepts is a loss-less representation of the set of frequent periodic patternseven if they are presented using a different notation. More-over, it is important to note that the set of triadic conceptspresented on Table 4 is considerably smaller than the set offrequent periodic patterns presented in Table 2.

This set can be translated into a set of frequent peri-odic patterns by calculating the cycles included in the set oftransactions of each triadic concept. For instance,T2({getFrame, displayFrame}, {2, 4}, {t1, t5}) containsonly one cycle of period 4 with transactions t1 and t5 whichwould give us the periodic pattern ({getFrame,displayFrame}, 4, 2, {(1, 2)}) (period=4, support=2, cycleoffset=1 and cycle length=2) which is P9 from Table 2, be-ing able to deduce as well P7 and P8 since their itemsets aresubsets of {getFrame, displayFrame}.

Nevertheless, this set still contains redundant informationin terms of redundant periods. For example, if we considerthe triadic concepts T1 and T2 from the set of triadic con-cepts shown in Table 4, we can see that T2 is “included” inT1, i.e. they have the same itemset and the transactionsbelonging to T2 are a subset of the transactions belongingto T1 and therefore, period 4 of T2 can be “deduced” fromthe set of transactions of T1. As a result, T2 can be removed

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without losing information. The same logic can be appliedto T3, T4 and T5, reducing the set to pattern T1.

The way of deducing T2 from T1 is by calculating the pos-sible periods between all transactions in T1, and then gen-erating the cycles belonging to each period, this way, fromT1 we can obtain periods 2 ({t1,t3,t5,t8,t10}), 3 ({t5,t8}), 4({t1,t5}) and 5 ({t3,t5,t8,t10}). The last step is to calcu-late the maximal subsets involving several periods on thesame set of transactions as is the case with t1 and t5 whichare found in periods 2 and 4 generating the triadic concept({getFrame, displayFrame}, {2, 4}, {t1, t5}).

In order to obtain a reduced representation of the set oftriadic concepts we propose removing triadic concepts withredundant periods. For this, we present here the definitionof minimal periodic generator which allows us to extractthe set of triadic concepts that does not contain redundantperiods.

4.2 Minimal Periodic GeneratorsIn general terms, a minimal generator is the smallest pat-

tern that will determine a closed pattern using the closureoperator [18]. In this context, a minimal periodic generatoris the triadic concept with the smallest set of periods andthe biggest set of transactions that can determine anothertriadic concept using the method introduced above. For itto determine another triadic concept they have to share thesame itemset, it has to include all transactions of the othertriadic concept and have a smaller set of periods than theother triadic concept. Indeed, the set of periods of the mini-mal periodic generator is a subset of the set of periods of theother triadic concept. This is because the set of transactionsof the triadic concept are included in the set of transactionsof the minimal periodic generator, and therefore all thosetransactions have associated the periods of the minimal pe-riodic generator and a few more.

Definition 8. A triadic concept (I, P, T ) is a minimalperiodic generator if there does not exist any other triadicconcept (I ′, P ′, T ′) such that I = I ′, P ′ ⊂ P and T ′ ⊃ T .

Example 7. In Table 5, we can observe the set of mini-mal periodic generators extracted from the set of triadic con-cepts shown in Table 4. For instance, T2({getFrame,displayFrame}, {2, 4}, {t1, t5}) is not a minimal periodic ge-nerator since there exists T1({getFrame, displayFrame},{2}, {t1, t3, t5, t8, t10}) with the same itemset {getFrame,displayFrame}, a smaller set of periods {2} ⊂ {2, 4} and abigger set of transactions {t1, t3, t5, t8, t10} ⊃ {t1, t5}.

Table 5: Set of minimal periodic generatorMinimal Periodic Generators

M1({getFrame, displayFrame}, {2}, {t1, t3, t5, t8, t10})

It is important to note that the set of minimal periodicgenerators shown in Table 5 is considerably smaller thanthe set of triadic concepts shown in Table 4, and thereforesmaller than the set of frequent periodic patterns shown inTable 2, and that it does not contain redundant periods.

In order to fit the periodic pattern notation introduced inDefinition 3, the set of minimal periodic generators shouldbe post-processed. For each minimal periodic generator theset of periods is extracted. Then, for each period in the setof periods, the set of cycles corresponding to that period isgenerated by reading the set of transactions, generating a

new periodic pattern containing the period and the set ofcycles.

In the example, from Table 5 which contains only one mi-nimal periodic generator M1({getFrame, displayFrame},{2}, {t1,t3,t5,t8,t10}), we obtain the frequent periodic pat-tern P ({getFrame, displayFrame}, 2, 5, {(1, 3)(8, 2)}),which corresponds to P3 from Table 2. This is a reducedrepresentation of the set of frequent periodic patterns con-taining enough information to deduce all other frequent pe-riodic patterns. As explained in the previous section, the setof triadic concepts can be deduced from the set of minimalperiodic generators. Then, the whole set of frequent periodicpatterns can be deduced from the set of triadic concepts bygenerating all triples contained in the set and then groupingthem by item and period (taking into account all possiblecombinations between the items to form the itemsets).

5. MINING PERIODIC PATTERNSIn this section, we introduce an algorithm to mine minimal

periodic generators from a dataset, called PerMiner. Wealso present a tool called Competitors Finder that helps toidentify pairs of “competitors”, pairs of periodic patternswhere a pattern breaks the periodicity of another pattern.This tool helps to identify possible conflicts between differententities in the system.

Algorithm 1 Periodic Pattern Miner

1: procedure PerMiner(I,D,min sup)Input: Itemset I, dataset D, minimum support min supOutput: All frequent periodic patterns that occur in D2: TS := TripleMiner(I,D,min sup)3: TC := Data-Peeler (TS,min sup)4: MPG := MPGMiner(TC)5: FPP := PostProcess(MPG)6: Print(FPP )7: end procedure

The PerMiner algorithm in Algorithm 1, is divided intothree steps:

1. The set of triples TS is generated. All possible periodswith their transactions are calculated for individualitems and then outputted in the form of triples.

2. The set of triadic concepts TC is generated from theset of triples TS. For this step an existing algorithmcalled Data-Peeler [3] has been used.

3. The set of minimal periodic generators MPG is gen-erated from the set of triadic concepts TC.

The set of minimal periodic generators MPG set is thentransformed into frequent periodic patterns before being out-putted.

The procedure TripleMiner in Algorithm 2 generates alltriples (i, p, t), i.e. for each item and for each possible periodin the range [1..|D|/min sup] it searches all cycles of the se-lected period with transactions containing the selected item.The objective of this function is to transform our datasetinto a triadic context exploitable by Data-Peeler.

The function build triples in Algorithm 3 is used by Tri-pleMiner to generate all triples of a given item and a givenperiod. For this, it scans the support set of the item i, given

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Algorithm 2 Triple Miner

1: procedure TripleMiner(I,D,min sup)Input: Itemset I, dataset D, minimum support min supOutput: All triples that occur in D2: TS ⇐ ∅3: for all i ∈ I do4: for all p ∈ [1, D.size/min sup] do5: TS := TS ∪ build triples(i, p,D)6: end for7: end for8: return TS9: end procedure

by D[{i}], which is the set of transactions of D in which ioccurs. For each transaction in D[{i}], the function tries tobuild a cycle of the given period starting in that transaction.If a cycle is found, its length being at least two transactions,the cycle is transformed in triples and added to the outputlist. Visited transactions are marked to avoid generatingredundant cycles.

Algorithm 3 Build triples

1: function build triples(i, p,D)Input: Item i, period p, dataset DOutput: Set of triples of item i and period p2: LT ← ∅3: for all t ∈ D[{i}] do4: if t not visited yet then5: t← visited; len← 0; next← tid(t)6: while tnext ∈ D[{i}] do7: tnext ← visited; len := len + 1; next := next + p8: end while9: if len > 2 then

10: while len > 0 do11: LT := LT ∪ (i, p, tid(t) + ((len− 1) ∗ p))12: end while13: end if14: end if15: end for16: return LT

17: end function

Data-Peeler generates all triadic concepts, containingat least min sup transactions, contained in the set of triplesgenerated by TripleMiner. Then, the procedure MPG-Miner in Algorithm 4 is in charge of generating the set ofminimal periodic generators from the set of triadic conceptsgenerated in the previous step. Following Definition 8, thefunction compares the set of triadic concepts two by two,and if it finds a triadic concept that is “included” in anothertriadic concept in the set, the former is deleted from the list.As explained in section 4.2, by“included”we mean that theyhave the same itemset, the set of periods of the latter is in-cluded in the set of periods of the former, and the set oftransactions of the former is included in the set of periodsof the latter.

The procedure PostProcessMPG in Algorithm 5 gener-ates a set of frequent periodic patterns from the set of mini-mal periodic generators MPG. For each period p in the setof periods P of each minimal periodic generator (I, P, T ),build cycles generates all possible cycles of period p that

Algorithm 4 Minimal Periodic Generators Miner

1: procedure MPGMiner(CTS)Input: Set of triadic concepts TCOutput: Set of minimal periodic generators MPG2: MPG⇐ TC3: for all (I, P, T ) ∈MPG do4: for all (I ′, P ′, T ′) ∈MPG

with (I, P, T ) 6= (I ′, P ′, T ′) do5: if I == I ′ AND P ⊂ P ′ AND T ⊃ T ′ then6: MPG := MPG \ (I ′, P ′, T ′)7: end if8: end for9: end for

10: return MPG11: end function

can be formed using the set of transactions T of the mini-mal periodic generator.

Algorithm 5 Post processing of the set of Minimal PeriodicGenerators1: procedure PostProcessMPG(MPG)

Input: Set of minimal periodic generators MPGOutput: Set of frequent periodic patterns FPP2: FPP ⇒ ∅3: for all (I, P, T ) ∈MPG do4: for all p ∈ P do5: FPP.add(I, p, build cycles(p, T ))6: end for7: end for8: return FPP9: end function

The function add includes the itemset I, with the periodp and the associated cycles to the list of frequent periodicpatterns in the form of frequent periodic pattern. If thereexists a frequent periodic pattern in the set with the sameitemset and the same period, the sets of cycles are joined,checking if there exist overlap between any two cycles of theset. If an overlap between two cycles is found, the frequentperiodic pattern is split in as many patterns as needed inorder not to have overlaps between the cycles of the sameset. Functions build cycles and add are not detailed here dueto lack of space, but their implementation is straightforward.

5.1 Competitors FinderHere we introduce a new analysis tool called Competitors

Finder that helps to identify pairs of competitor periodicpatterns from the set of frequent periodic patterns deducedfrom the set of minimal periodic generators. This tool al-lows the developer to easily identify possible conflicts be-tween different parts of the system, i.e. between the appli-cation and the operating system, between different modulesor drivers of the operating system, and so on, saving a sig-nificant amount of time to the developer. In this sense, weconsider that a conflict or competition between two patternsis simply the inverse of the overlap between the two patterns.This is, if one pattern P1 disrupts another pattern P2, thenduring the disruption P1 will be active but not P2. So, ourobjective is to automatically identify these situations.

The procedure CompetitorsF inder in Algorithm 6 re-ceives the set of frequent periodic patterns deduced from

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Algorithm 6 Competitors Finder

1: procedure CompetitorsF inder(MPG′,min ratio)Input: Set of frequent periodic patterns MPG′, minimum

rate of competition min ratioOutput: Set of pairs of competitors MaxComp2: MaxComp⇐ ∅3: for all p ∈MPG′ do4: max ratio := 05: for all p′ ∈MPG′ with p 6= p′ do6: comp ratio := competition ratio(p, p′)7: if comp ratio > max ratio then8: max ratio⇐ comp ratio; max pat⇐ p′

9: end if10: end for11: if (max pat, p, ∗) /∈ MaxComp AND max ratio ≥

min ratio then12: MaxComp.add(p,max pat,max ratio)13: end if14: end for15: return MaxComp16: end procedure

the set of minimal periodic generators, and for each fre-quent periodic pattern it searches the pattern that is themost in competition with it. For each pattern, this proce-dure compares it with all other patterns and calculates thecompetition ratio by invoking competition ratio function.In max ratio and max pat, the procedure stores the maxi-mum ratio found and the pattern linked to that ratio.

When the pattern has been compared to all other pat-terns, the procedure checks if the competition ratio is big-ger than the competition ratio threshold given as an input,and as well, whether the two patterns were already stated ascompetitors, in which case they are not outputted to avoidrepetitions. Finally the list of competitors with their com-petition ratio is returned by this procedure.

Algorithm 7 Competition Ratio Calculator

1: function competition ratio(P, P ′)Input: Two periodic patternsOutput: The ratio of competition between P and P ′

2: for all ci, ci+1 ∈ P.cycles do3: for all c′ ∈ P ′.cycles do4: match += calculate coexecution(c, c′)5: match += calculate cogap(ci, ci+1, c

′)6: end for7: end for8: return (1− ((num trans−match)/num trans))∗100;9: end function

The function competition ratio in Algorithm 7 calculatesthe competition ratio between two frequent periodic pat-terns. By comparing the cycles between them, it uses thefunction calculate coexecution to calculate the area of co-execution of two cycles, and the function calculate cogap tocalculate the area between two cycles of the first periodicpatterns (gap) not occupied by a cycle of the second peri-odic pattern. The sum of all the output values of these twofunctions is called matching ratio, and the competition ra-tio is the result of calculating just the opposite, i.e. 100% -matching ratio.

6. EXPERIMENTAL RESULTSIn this section, we present several experiments carried out

on embedded multimedia application traces. The experi-ments show that periodic pattern mining can help to debugapplications by automatically extracting representative in-formation from their execution traces. These experimentscould have been carried out by analyzing exclusively the setof frequent periodic patterns but it would have taken a lotlonger since the difference in size between the set of frequentperiodic patterns and the set of minimal periodic generatorsis considerable as stated below.

Figures 5 and 6 correspond to two visualization tools de-veloped to facilitate the analysis of the periodic patterns.In Figure 6 the list of periodic patterns can be explored bynavigating the list of itemsets. When the user selects anitemset from the list, the periodic patterns associated withthat itemset, one for each period, are shown on the bottompart of the tool. Each line corresponds to a periodic patternwith the period value on the left part of the figure and theoccurrences on the right. The occurrences are visualized bya sequence of vertical lines representing all possible transac-tions on the dataset (from left to right). A line is coloredwhen the corresponding transaction forms part of the se-lected periodic pattern. Similarly, Figure 5 shows the list ofpairs of competitors. When a pair of competitors is selected,the occurrences of both patterns are shown, pattern 1 on topand pattern 2 at the bottom.

6.1 HNDTest ApplicationIn this example, we retrieved a trace from an execution of

a video and audio decoding test application called HNDTest,test application for STMicroelectronics development boards,on an ST40 processor, introduced in Section 2.1. Then, wepreprocessed it, as explained in Section 2.2, in order to splitthe trace into frames. The execution trace occupied 7.2 MBof memory. After preprocessing, the dataset contained 240frames, and with a support threshold of 10% our algorithmmined 859 minimal periodic generators, faster to analyzethan 7.2 MB of execution trace. This trace would produce109,668 triadic concepts and 38,459 frequent periodic pat-terns. Therefore, we can observe that mining minimal peri-odic generators considerably reduces the number of patternsto analyze.

The bigger number of triadic concepts with respect to thenumber of frequent periodic patterns can be explained by thefact that the number of periods in frequent periodic patternsis limited to one per pattern while in the triadic conceptsis not limited. Consequently, all possible combinations ofall lengths of the set of possible periods are generated astriadic concepts. All these extra patterns are then removedby the minimal periodic generator miner. Also, the transac-tions of the dataset used in this example have an average of8 items per transaction which produces patterns with rela-tively short itemsets that consequently do not produce manycombinations in terms of frequent periodic patterns.

As part of the analysis of the set of frequent periodicpatterns mined, we used the tool Competitors Finder (seeSection 5.1) to identify possible conflicts between differententities of the system. We highlight here a pair of competi-tors found by the tool involving on one hand, Interrupt 16,which is the clock of the processor, and Interrupt 168, whichis a USB port interrupt, and on the other hand, a context

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Figure 5: Conflict between application and operating system

switch and a system call (try_to_wake_up) involving threadHNDTest.

In Figure 5 we can observe that these two patterns arein competition: the top pattern (interrupts) stops executingwhen the bottom pattern (HNDTest) increases its frequency.In this context, the processor periodically polls the deviceconnected to the USB port to check whether it has datato transfer. This is achieved using interrupt 168. WhenHNDTest increases its activity, it masks the interrupts andtherefore prevents the processor from transferring any in-coming data from the USB port, which causes a delay intransmission. This might have further repercussions if theUSB reception buffer becomes full while waiting to transferthe data, causing the data to be overwritten.

6.2 Gstreamer ApplicationGStreamer is a pipeline-based multimedia framework that

has been adopted by many different corporations such asNokia, Texas Instruments and so on. In this experiment,we used a trace that registered an audio decoding applica-tions, using GStreamer, while playing back an audio file.The application was executed on a platform that containedan ARM processor over an Orly SoC [20]. Our algorithm,with a support threshold of 10%, mined 1,467 minimal pe-riodic generators. This trace would have produced 21,588triadic concepts and 3,086,321 frequent periodic patterns.

In this example, the number of frequent periodic patternsis much bigger than the number of triadic concepts. This isbecause the transactions in the dataset of this example arevery long, 35 items per transaction in average, and thereforegenerate patterns with a long itemset. For this reason, thenumber of frequent periodic patterns per itemset and perperiod is large since there is a frequent periodic pattern foreach combination (of any length) of the items in the itemset.

Figure 6: Trace visualizationIn this context, the mixer maps audio samples every 32ms,

so we decided to split the trace using a time interval of 32msto see whether the expected period was preserved or not.In the minimal periodic generators mined, we expected tofind some patterns with a period of 1 (32ms) over all thedata. Surprisingly, the patterns found exhibit gaps in theperiodicity (vertical white lines in period visualization inFigure 6) showing that the application is unable to keep upthe expected mixing rate.

The application developers investigated these gaps andfound that there was a bug in the calculation of an interruptperiod. This interrupt was in charge of flushing a bufferof samples read from memory but it was being generatedtoo late causing buffer overflows and higher level drivers tounderflow when operating double buffered.

Discussion. In these experiments, we have shown thatour approach allows developers to quickly discover certainproblems in the execution of their applications by automati-cally analyzing their execution traces, that otherwise wouldhave taken a long time to be discovered by manually ana-lyzing execution traces.

7. RELATED WORKPattern mining is starting to play an important role in sys-

tem analysis, even more in embedded systems, where tracingis widely used in order to analyze the system while avoidinghigh intrusiveness.

First uses of pattern mining in system analysis focused ondetecting bugs, e.g. introduced by copying-and-pasting ker-nel source code [12] or caused by the violation of programingrules [13], or more generally detecting systemic problems [9].Lo et al. [14] studied how to classify software behaviors, ob-tained by pattern mining of known normal and failing exe-cution traces, in order to detect failures in future executionsof the system.

Recently, Chang et al. [4] worked on system verificationusing pattern mining in order to extract assertions from sim-ulated traces of the system being validated. In terms of per-formance analysis, Zou et al. [23] used pattern mining toreduce vast amounts of hardware sample data into a set ofeasier-to-analyze frequent instruction sequences, in order tohelp to analyze the performance of the system.

Nevertheless, none of the previous studies have appliedperiodic pattern mining to system analysis.

The first studies carried out on periodicity focused on as-sociation rules. As an example Ozden et al. [17] looked forcyclic association rules in transactional databases. Their ob-jective was to find association rules that hold in all segmentsof the database over a given period. This kind of periodicity,called perfect periodicity, is very useful for certain contextsbut is too restrictive in our case.

Han et al. [6, 5] introduced a certain confidence in theperiodicity which means that the pattern does not need tobe found in all instances, but allows certain misses. Never-theless, when there is a gap in the periodicity of the pattern,this gap is always regular, i.e. multiple of the period.

The first study to include irregularity in the periodicitywas carried out by Ma et al. [15]. In their study, the authorsextended the concept of partial periodicity introduced byHan et al. considering that periodic behavior might be foundin part of the dataset and it is not necessarily expected tobe persistent. The authors allowed gaps in the periodicitybut they restricted their sizes to a certain time window.

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None of the previous studies regarding periodic patternmining have allowed for irregular gaps without any restric-tion on the length of the gap. It is important for us toallow irregularities in the gaps in order to discover differenttypes of periodic patterns, e.g. patterns that are periodicduring certain segments of the trace with big gaps betweenthose segments, or patterns with a regular periodicity butthat every time there is a gap the pattern gets shifted by npositions.

Triadic Concept Analysis was first introduced by RudolfWille in 1995 [22, 10]. Since then, several algorithms thatmine triadic concepts have been proposed, among them wecan find CubeMiner proposed by Ji et al. [12], Trias pro-posed by Jaschke et al. [7] and Data-Peeler proposed byCerf et al. [3]. The latter was chosen in this paper sincethe authors showed through an experimental comparisonthat their algorithm was more efficient than CubeMiner orTrias. Moreover, the authors generalized the computationof triples previously studied by [12] and [7] to a constraint-based approach for mining closed sets from n-ary relations.But none of the previous studies have applied ternary rela-tion theory to mining periodic patterns.

8. CONCLUSIONS AND FUTURE WORKIn the context of analyzing traces of multimedia embedded

applications, we have presented a new definition of periodicpatterns that allows unrestricted-sized gaps in the period-icity. Moreover, we have presented a lossless representationof the set of frequent periodic patterns, called minimal peri-odic generators, that simplifies the analysis. Also, we haveintroduced an algorithm for mining such minimal periodicgenerators and a tool that identifies pairs of competitorsfrom the set of minimal periodic generators.

Through experiments on execution traces, we have demon-strated how pattern mining can help to discover problemsmore quickly than manually analyzing raw execution traces.We have applied our mining method and our CompetitorsFinder tool to an execution trace and identified a conflictbetween the application and the operating system, demon-strating the interest of our approach.

We have also shown how discovering the periodic behaviorof multimedia applications can help to pinpoint when thisperiodicity is lost and therefore what affects the expectedexecution of the application.

The scope of this paper is to present our approach, itsapplicability and its interest. We intent to study the use ofother patterns such as sequences (ordered itemsets) whichwould help to discover other behaviors of the application,and therefore complete the analysis. Moreover, we are plan-ning to design an algorithm to mine minimal periodic gener-ators directly, without having to generate the set of triadicconcepts, in order to increase efficiency.

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