Fault Tolerant Frequent Pattern Mining

Fault Tolerant Frequent Pattern Mining

Abstract

FP-Growth algorithm is a Frequent Pattern Mining (FPM) algorithm that has been extensively used to study correlations and patterns in large scale datasets. While several researchers have designed distributed memory FP-Growth algorithms, it is pivotal to consider fault tolerant FP-Growth, which can address the increasing fault rates in large scale systems. In this work, we propose a novel parallel, algorithm-level fault-tolerant FP-Growth algorithm. We leverage algorithmic properties and MPI advanced features to guarantee an space complexity, achieved by using the dataset memory space itself for checkpointing. We also propose a recovery algorithm that can use in-memory and disk-based checkpointing, though in many cases the recovery can be completed without any disk access, and incurring no memory overhead for checkpointing. We evaluate our FT algorithm on a large scale InfiniBand cluster with several large datasets using up to 2K cores. Our evaluation demonstrates excellent efficiency for checkpointing and recovery in comparison to the disk-based approach. We have also observed 20x average speed-up in comparison to Spark, establishing that a well designed algorithm can easily outperform a solution based on a general fault-tolerant programming model.

\IEEEpeerreviewmaketitle

1 Introduction

Machine Learning and Data Mining (MLDM) algorithms are becoming ubiquitous in analysing large volume of data produced in science areas (instrument and simulation data) as well as other areas such as social networks and financial transactions. Frequent Pattern Mining (FPM) is an important MLDM algorithm, which is used for finding attributes that frequently occur together. Due to its high applicability, several FPM algorithms have been proposed in the literature such as Apriori [10], Eclat [39], FP-Growth [17], and GenMax [15]. However, FP-Growth has become extremely popular due to its relatively small space and time complexity requirements.

To address increasing data volumes, several researchers have proposed large scale distributed memory FP-Growth algorithms [26, 13, 20, 11, 35]. One of the challenges that arise with execution on large-scale parallel systems is the increased likelihood (and frequency) of faults. Large scale systems frequently suffer from faults of several types in many components  [32, 33, 29, 34, 7, 30].

Driven by these trends, several recent programming models such as Hadoop, Spark [38], and MillWheel [3] have considered fault tolerance to be one of the most important design consideration. Hadoop achieves fault tolerance by using multiple replicas of the data structures in permanent storage — possibly resulting in a significant amount of I/O in the critical path. Spark addresses this limitation by using Resilient Distributed Datasets (RDDs), such that in-memory replication can be used for fault tolerance. However, for very large datasets, in-memory replication is infeasible. In several cases, Spark considers disk as the backend for checkpointing — which can again significantly slow-down the computation and increase data movement. Similarly, MillWheel is used for fault tolerant stream processing and uses the disk as the backend for checkpointing. Naturally, an advantage of using fault tolerant programming model is the fact that checkpointing and recovery is automated. However, the performance penalty of a fault tolerant programming model (due to disk-based checkpointing) or space overhead (due to in-memory checkpointing) is unattractive for scaling several MLDM algorithms at large volume and computing scale.

In the context of general-purpose programming systems, recently proposed methods such as Scalable Checkpoint Restart (SCR)  [25] are able to provide in-memory checkpointing for multi-level hierarchical file systems using non-blocking methods. SCR also allows using spare main memory for in-memory checkpointing. Similarly, other researchers have proposed programming model/runtime extensions to Charm++, and X10 for supporting fault tolerance. While these approaches provide non-blocking checkpointing, the overall memory requirements increase, since the implementations need to use spare memory for checkpointing. This can very well make the approach infeasible, especially with weak scaling executions, where spare memory is scarce.

Figure 1: Pattern of Memory Requirements of FP-Tree and Dataset during FP-Tree build phase. As more transactions are processed, lesser memory is required for dataset — which can be used for checkpointing

In this paper, we present an in-depth study of FP-Growth algorithm for fault tolerance. Considering its two-pass properties (impact shown in Figure 1), we propose a novel algorithm, which requires space complexity for saving critical data structures, i.e., FP-Tree, in memory of other computing nodes. The proposed algorithm incrementally leverages the memory allocated for the default algorithm for checkpointing FP-Trees – and possibly partial replica of transactions from other computing nodes – ensuring an space overhead of our proposed algorithms. To further minimize time overhead for checkpointing, our solution not only leverages non-blocking properties, but use MPI-Remote Memory Access ( MPI-RMA) in addition to minimize any involvement of remote process for checkpointing. By using MPI-RMA and contiguous data structures for implementing our proposed algorithms, we are able to leverage Remote Direct Memory Access (RDMA) effectively. We believe that our proposed extensions may be included with existing solutions such as SCR, where a class of algorithms may re-use already allocated memory for checkpointing and recovery.

1.1 Contributions

Specifically, we make the following contributions in the paper:

  • We propose an in-memory checkpointing based FP-Growth algorithm for large scale systems. The proposed algorithm leverages overlapping communication with FP-Tree build phase — such that the overhead of checkpointing is minimized.

  • We propose three different fault tolerance parallel FP-Growth mechanisms: a default Disk-based Fault tolerant FP-Growth (DFT), Synchronous Memory-based Fault tolerant FP-Growth (SMFT), and an Asynchronous Memory-based Fault tolerant FP-Growth (AMFT).

  • We study the limitations of existing programming models (Hadoop MapReduce, Spark and MillWheel) and implement our algorithms using Message Passing Interface (MPI) [16, 14]. Specifically, we use MPI-RMA mechanism to checkpoint critical data structures of FP-Growth asynchronously. With recent developments in MPI-RMA Fault tolerance [5], it is possible to use MPI for handling faults, while providing native performance.

  • We perform an in-depth evaluation of our proposed approaches using up to 200M transactions and 2048 cores. Using 100M transactions on 2048 cores, the checkpointing overhead is 5%, while the recovery cost for multiple failures is independent of the number of processes.

  • We also show the effectiveness of our fault-tolerant FP-Growth implementation – implementations outperforms Spark implementations of the same algorithm by providing 20x average speed-up.

2 Preliminaries

2.1 Frequent Pattern Mining

Frequent Pattern Mining (FPM) algorithms find items that frequently occur together within transactions of a database. An item or itemset is defined as frequent if its frequency is higher than a user-defined threshold. Several FPM algorithms have been proposed in the literature including Apriori, Eclat, GenMax and FP-Growth. The FP-Growth algorithm is very popular since it requires only two passes on the dataset, does not involve candidate generation (unlike Apriori) and provides a compressed representation of the frequent items using a Frequent Pattern (FP)-Tree. We specifically focus on designing parallel fault-tolerant versions of the FP-Growth algorithm, due to its attractive properties.

During the first pass, FP-Growth algorithm finds items that occur frequently. In the second pass, it creates an FP-Tree, which is a modified Trie. The first pass requires a simple scan through the given dataset to find all single frequent items. FP-Tree creation step (the second pass) is the most time consuming part of the overall calculation [35]. Hence, we focus on fault tolerant FP-Tree creation step of the algorithm, since longer execution time also implies higher fault probability.

2.2 Faults

Large scale systems suffer from several fault types — permanent, transient, and intermittent. A permanent fault typically requires a device (such as a compute node) to be replaced. We consider fault tolerance for permanent process faults in this paper.We assume a fail-stop fault model — once a process is perceived as dead/faulty, it is presumed unavailable for the rest of the computation.

Since permanent node faults are commonplace in large scale systems, several researchers have proposed techniques for addressing these faults. Typically, checkpoint-restart [28, 8] based methodologies are used. Application-independent methods checkpoint the entire application space on a permanent disk — however, they have been shown to scale only on small size systems [8]. Application-dependent methods — also known as Algorithm Based Fault Tolerance (ABFT) [12, 23, 31, 1] methods reduce this overhead by selectively checkpointing important data structures periodically. However, depending up on the application characteristics, checkpointing of critical data structures may still require disk access.

2.3 Fault Tolerant Programming Models

Recently, there has been a surge of large scale and fault tolerant functional programming models such as Hadoop, Spark, and MillWheel. Functional programming, in turn, uses the concept of single assignment, where every mutation of a variable is recorded, saved (on a permanent storage/memory of another node), and replayed when a fault occurs.

Now, let us examine the implication of such a framework for an algorithm like FP-Tree. Every change or mutation needs to be recorded locally, and such records can be eventually saved to permanent storage. In many cases, the step of saving a new version of the FP-Tree on the disk is carried-out at the end of the Reduce phase (of the MapReduce implementation). For a two-phase algorithm such as FP-Tree, where most of the time is spent on the second phase, no advantage is achieved. Another possible implementation may choose to divide the overall computation into multiple MapReduce steps. The checkpointing can be executed at the end of each Reduce phase. However, now the overall execution time will increase, since saving a new version will either involve writing to a disk (expensive) or neighbor’s memory. Since the reduce phase is a blocking phase, the application will observe a significant overhead of checkpointing, which will degrade the overall performance. Naturally, a scalable algorithm should harness best possible performance by using native execution, while minimizing the cost of checkpointing, by using non-blocking methods.

Now, in examining an alternate programming model, we consider the Message Passing Interface (MPI) [16, 14], which has been readily available and widely used on supercomputers and clusters, and beginning to find its place on cloud computing systems. While MPI has been frequently criticized for lack of fault tolerance support, recent literature and implementations indicate that fault tolerance is addressed well for permanent process faults [5]. More importantly, recently introduced MPI One-sided - MPI one-sided communication (also known as MPI-Remote Memory Access (MPI-RMA)) [16, 14]- primitives provide necessary tools for overlapping communication with computation. With this observation, we focus on using MPI for designing fault tolerant FP-Growth algorithm in this paper.

3 Parallel Baseline Algorithm

Algorithm 1 shows the key steps of the parallel FP-Growth algorithm, which we have used as the baseline for designing fault tolerant FP-Growth algorithms.

A brief explanation of the steps is presented here: The first step is to distribute the input database transactions among processes (Line 3) (Each process is a worker, which is involved in computing its local FP-Tree). Each process () scans the local transactions and records the frequency of each item (Line 4). To collect the global frequency, an all-to-all reduction (by MPI_Allreduce) is used (incurring time complexity) (Line 5). After all-to-all reduction, the items with frequency greater than support threshold are saved, and other items are discarded. Then, each generates a local FP-Tree () using its local transactions, which have at least one frequent item (Line 6). Later, each merges its local FP-Tree with the FP-Trees from other processes to produce a global FP-Tree () by using a ring communication algorithm [35] (Line 7). Finally, frequent itemsets () are produced using the output global FP-Tree (Line 8).

\setstretch

1.0

Algorithm 1 : Parallel FP-Growth Algorithm
1:Input: Set of transactions , Support threshold
2:Output: Set of frequent itemsets
3:L.Trans getLocalTrans()
4:L.FreqList findLocalFreqItems(, )
5:G.FreqList Reduce Local Freq items through all processes
6:L.Tree generateLocalFPTree(, )
7:G.Tree generateGlobalFPTree()
8:FreqItemSet miningGFPTree()

Further, we summarize the symbols we have used to model the time and space complexity of the proposed fault tolerant algorithms in Table 1.

Name Symbol
Process Set
Transaction Database
Average Local Transaction Size
Minimum Support Threshold
Local FP-Tree Set
Average Local FP-Tree size
Average time to merge two local FP-Trees
Number of Checkpoints
Disk Access Bandwidth
Network Bandwidth
Table 1: Symbols used for Time-Space Complexity Modeling

4 Proposed FP-Growth Fault Tolerant Algorithms

In this section, we present several approaches for designing fault tolerant FP-Growth algorithm. Our baseline algorithm uses the disk as the safe storage for saving intermediate FP-Trees, whereas the optimized algorithms use the memory originally allocated to the database transactions for checkpointing intermediate FP-Trees and transactions of other processes (with a high overlap of communication with computation achieved using MPI-RMA methodology).

To design a fault tolerant FP-Growth algorithm, there are several design choices. Since we consider fail-stop model, it is important to understand the design choices between re-spawning a new set of processes on a spare node versus continued-execution with existing processes and nodes. We use continued-execution, primarily because for most systems, it is intricate to re-spawn, attach the processes/node to the existing set of processes, and continue recovery. Instead, continued-execution provides a simple mechanism to conduct recovery, without significant dependence on external software.

4.1 Disk-based Fault Tolerant (DFT) FP-Growth

The Disk-based Fault Tolerant (DFT) algorithm is the baseline for other approaches presented in this paper.

Checkpointing Algorithm and Complexity: In the FP-Growth algorithm, there are two critical data structures that are needed during the recovery process — database transactions themselves and intermediate FP-Trees generated by the processes. Under the DFT approach, the intermediate FP-Trees generated by each process are periodically saved on disk. For many supercomputers, the disks are located remotely, such as a remote storage. In other cases, locally available SSDs can be used as well. The database transactions are already resident on the disk. Hence, it is not necessary to checkpoint the database transactions.

Let us consider an equal distribution of database transactions to processes ( transactions are available on each process). Let be the number of checkpoints, which are executed by the application. The number of checkpoints are derived as a function of , and , such that the cost of checkpointing can be amortized over the FP-Tree creation phase. The DFT algorithm also needs to save metadata file associated with FP-Tree, which may be used during recovery. The space complexity of the metadata file is negligible, since only a few integers need to be saved.

Let represent the average size of an FP-Tree generated by each process (calculated as ). The time complexity for checkpointing intermediate FP-Trees is . However, the actual time to checkpoint can escalate due to the contention from multiple processes writing the checkpoint file simultaneously. The space complexity incurred by each process is , which can be reduced further by recycling existing checkpoints.

Recovery Algorithm and Complexity: In the DFT approach, the recovery is initiated by the master () (In our implementation we use the default process — process with the first rank in MPI as the master). reads the metadata file associated with the faulty process (), which provides the necessary information for conducting recovery. A recovery process () is selected, which reads checkpointed FP-Tree of from the disk and merges the checkpointed FP-Tree of with its FP-Tree, while reads dead process transactions from disk, and re-distributes them among remaining processes.

The time complexity of the recovery algorithm is a function of reading the partial dataset and executing the recovery algorithm. In the worst case, the entire transactions of the faulty process need to be re-executed. Hence, the worst case time complexity is (reading the dataset) (re-distributing among process) + (re-computation), where is the average cost of merging a transaction in an existing FP-Tree (In the worst case, the FP-Tree is null, since all transactions are re-executed).

Implementation Details: As mentioned earlier, each process saves a copy of local FP-Tree in a safe storage. Thus, our implementation depends on checkpointing local FP-Tree on disk —  file. This file associated with another file describes the checkpointed FP-Tree by storing a set of description values such as: checkpoint timestamp and last processed transaction. Each process asynchronously updates both files, during the execution. In the case of failure, the recovery operation is performed in two steps: The pre-determined recovery process process reads the last checkpointed FP-Tree of the faulty process from the disk and merges it with its local FP-Tree. At the same time, the master process reads the metadata file of to decide the set of transactions to be recovered from the disk. The master process recovers unprocessed transactions and redistributes them to the remaining processes.

Advantages and Limitations of DFT: The proposed DFT algorithm is largely equivalent to designing a fault tolerant FP-Growth algorithm using MapReduce programming models such as Hadoop/Spark. However, an advantage is that it can specifically take advantage of native communication by using MPI, especially when high performance interconnects are available. Disk-based approach makes DFT suffer from several limitations: These include prohibitive I/O cost for checkpointing/recovering local FP-Trees and recovering unprocessed transactions, and centralized bottleneck of the master process in the case of failure to re-read unprocessed transactions from the disk.

4.2 Synchronous Memory-based Fault Tolerant (SMFT) FP-Growth

As discussed above, the primary limitation of the DFT approach is that it uses disk-based checkpointing and recovery, which is prohibitive for scaling the FP-Growth algorithm. Hence, it is important to consider memory based fault tolerant FP-Growth algorithm.

Since available memory size is relatively small in comparison to the disk size, it is also unattractive to incur additional space complexity for in-memory checkpointing of FP-Trees and database transactions from other processes. SMFT involves checkpointing method where the overall space complexity of the algorithm remains constant. Additionally, we overlap the checkpointing of FP-Trees and database transactions by using non-blocking primitives provided by the MPI one-sided model. We present the checkpointing, and recovery methods with their time-space complexity analysis in the ensuing sections.

Figure 2: SMFT FP-Tree Checkpointing Operation Overview

Checkpointing Algorithm: The premise of constant space complexity is based on the two-pass properties of the FP-Growth algorithm. During the FP-Tree creation phase, once a database transaction is processed, the memory occupied by the transaction can be used for checkpointing. We leverage this property of the algorithm to checkpoint the FP-Trees and database transactions. Specifically, once a transaction is processed, we reclaim the memory consumed by the transaction and allocate a separate window of memory, which can be used by other processes for checkpointing their FP-Trees and database transactions. With this technique, the overall space complexity of the algorithm is .

Besides optimal space complexity, the objective of SMFT algorithm is to minimize the time complexity of checkpointing both the FP-Trees and database transactions. Considering as the number of checkpoints, under a naive algorithm, each process can checkpoint its existing FP-Tree to another process at every steps. Since the time overhead of checkpointing is non-negligible, as this step blocks for the communication to complete before continuing to process remaining transactions, at every checkpointing step — with increasing FP-Tree size — the overhead of blocking increases. Hence, it is important to consider non-blocking methods of checkpointing, such that communication cost of checkpointing can be overlapped with computation.

SMFT algorithm uses MPI one-sided non-blocking methods for checkpointing. Specifically, as the database transactions are processed, a similar amount of memory is added to a checkpoint window. The algorithm uses dynamic allocation feature in MPI-RMA , MPI_Win_create_dynamic, that allows incremental increase in the size of the checkpointing memory space during the execution. However, this dynamic allocation technique requires synchronization between both cooperated processes to perform each single checkpoint which adds more overhead to the checkpoint process. SMFT checkpoint overhead comes from different sources: waiting time till synchronization, communication —which is negligible based on well known communication model LogGP [4]—, and memory allocation and de-allocation cost.

Figure 2 shows an overview over the FP-Tree checkpointing operation in SMFT approach. Assuming process needs to checkpoint on process memory, each time period, i.e., , , …, , process re-initiates a checkpoint space that can handle process checkpointed local FP-Tree. In this case, process can remotely checkpoint its local FP-Tree to the new assigned location without communicating with checkpoint process .

Recovery Algorithm: Assuming a process fails while executing the FP-Tree phase. On fault recovery, the recovery process (in the simplistic case, a neighbor such as ) merges checkpointed FP-Tree of stored on its memory to its local FP-Tree. If has also stored part of the database transactions from , it re-distributes these transactions to other processes, which are still active in the computation. The recovered transactions can be gathered from the memory of , if they were checkpointed by before failure. In the case of disk recovery, lost transactions can be read from the disk using two different ways. First, dataset transactions may be read from the disk by using the master process and re-distributed evenly among the remaining processes. However, in this case, disk access will be the most expensive part of the overall recovery algorithm. So, we suggest using all available processes to read samples of failed process () from the disk in parallel. With this, each process will only access the disk to read transactions. Further, since failed process held the data checkpointed by process , process performs a critical checkpoint on process  — in the simplest case, the processes can be assumed to be connected in a virtual ring topology. Using this methodology, there is always at least one replica of the FP-Tree of each process.

Advantages and Limitations of SMFT: The primary advantage of SMFT is that it avoids reading/writing from the disk. Naturally, SMFT achieves native performance using MPI and is expected to incur low overhead for checkpointing with non-blocking MPI one-sided communication. The recovery algorithm uses memory to recover the database transactions, if possible. By distributing the transactions of a failed process to other active processes, the algorithm is able to minimize the recovery overhead. In the case of disk-based transactions recovery, SMFT uses all processes to read recovered transactions from the disk in parallel to avoid master process bottleneck.

SMFT approach has two main limitations. First, each two processes and need to synchronize in all checkpoints to share the address of checkpoint vector and the size of checkpointed FP-Tree or checkpointed transactions. Second, SMFT algorithm requires de-allocating existing space and allocating new space for checkpointing window. The overhead of synchronization, de-allocation and allocation are observed during FP-Tree creation phase. We address these two limitations in the AMFT approach, presented later.

Implementation Details: In SMFT, each process allocates three memory vectors. These vectors are used to handle checkpoints from process namely: vector to handle local FP-Tree of proceeding process , vector to handle transactions checkpoint of , and vector that includes a set of parameters to describe both checkpoint vectors. These vectors are allocated and exposed for read/update by each process using MPI-RMA primitives.

For in-memory checkpointing, SMFT requires that each process selects another process for checkpointing. While SMFT supports any arbitrary topology, in the simplest case, the processes can be assumed to be connected in a virtual ring topology. Each process uses the memory of adjacent processor for checkpointing its local FP-Tree and transactions. Therefore, each process should prepare its checkpoint buffers ( FPT checkpoints and transaction checkpoints vectors) to handle the data checkpointed by process , when needed during recovery.

To perform a single checkpoint, each pair of processes need to perform three operations. First, increases the size of the and data structure, such that the new checkpoint from can be handled. The operation of determining the size of the checkpointed local FP-Tree requires synchronization between and . Specifically, sends a checkpointing request to including the volume of data to be checkpointed. uses MPI_Win_create_dynamic mechanism to increase the size of the checkpointed space. The new virtual address is communicated to , which is used by for checkpointing the actual data using MPI_Put operation.

A process may also checkpoint its remaining local transactions on memory to avoid reading it from disk in the case of failure. If the fault occurs before checkpointing the transactions, remaining transactions are recovered from the disk. However, if fails after dataset transactions have been checkpointed, they can be redistributed directly by to other available processes. Transactions checkpointing can be performed similar to FP-Tree checkpointing on vector of the target process.

\setstretch

1.1

Algorithm 2 : SMFT FP-Growth Algorithm
1:Procedure: initialization( = SMFT)
2:
3:Create , and vectors on (initially-empty).
4:Expose , and addresses for read/update //using MPI-RMA.
Procedure: performLFPChk ()
5:
1:Synchronize with to resize the vector.
2:Add (, )  (MPI_Put)
3:Update vector  (MPI_Put)
Procedure: performTransChk ()
4:
1:Synchronize with to resize the vector.
2:Add (, )  (MPI_Put)
3:Update vector  (MPI_Put)
Procedure: performRecovery (, , )
4:
1: process: merge (, , )
2:if  is NULL then
3:      diskTransRec()
4:else
5:      memTransRec(, )
6:end if

Algorithm 2 shows the checkpointing and recovery algorithms for SMFT. In procedure, each process create three vectors , and vectors to handle proceeding process checkpoints (Line 1). These vectors are allocated and exposed using MPI-RMA technology for facilitating remote read/update (Line 2). Both procedure and procedures, illustrate checkpoint operation in SMFT for both local FP-Tree and transactions, respectively. Process synchronizes with its source process by receiving its checkpoint size and resizing its checkpoint buffer to handle data. Process finalizes the synchronization operation by sending the new checkpoint vector address to the source process (Line 1). Next, process uses MPI_Put function to checkpoint its data and updates the metadata vector on target process memory (Lines 2-3).

The procedure shows the recovery algorithm in SMFT. The predetermined recovery process is used to recover failed process by merging checkpointed local FP-Tree of it has on its memory to local FP-Tree (Line 1). Further, failed process transactions can be recovered with the aid of metadata vector directly from recovery process memory if available or from the disk if not (Lines 2-6). Disk-based recovery should be performed in parallel to speed-up the total recovery time.

4.3 Asynchronous Memory-based Fault Tolerant (AMFT) FP-Growth

In the SMFT approach, we observed the advantages of using in-memory checkpointing of FP-Tree and database transactions. However, there are a few limitations of SMFT. Specifically, a pair of processes need to synchronize for memory allocation and address exchange — which reduces the overall effectiveness of the MPI One-sided model.

We address the limitations of SMFT by proposing a truly one-sided mechanism for checkpointing, i.e., Asynchronous Memory-based Fault Tolerant (AMFT). Under AMFT, we use the memory of already processed transactions for checkpointing instead of allocating new space. Similar to SMFT, under the AMFT approach, it is possible to checkpoint the FP-Trees and a portion of the database transactions. We describe the checkpointing, recovery and implementation details of the AMFT approach as follows.

Checkpointing Algorithm: Consider a subset of two processes  —  and . The checkpoint from is stored on . To enable truly one-sided mechanism for checkpointing, must ensure that its checkpoint size is less than the size of the already processed transactions in . In AMFT, we achieve this objective by using atomic operations on variables allocated using MPI-RMA and exposing it to read/update by other processes. The original parallel FP-Growth algorithm is slightly modified to atomically update the size of available checkpointing space — this step does not require communication with any other process. When decides to checkpoint its FP-Tree, it atomically reads the value of available checkpointing space on . By carefully designing the checkpointing interval, it is highly likely that the size of the available checkpointing space on is greater than the size required by . In the pathological case, periodically reads the available checkpointing space, till the condition is satisfied — in practice, this situation is not observed. In the common case, simply initiates the checkpoint using MPI_Put. Besides local FP-Tree, remaining (unprocessed) transactions of process can also be checkpointed to memory if there is enough space. Checkpointing transactions is one-time operation that improves the recovery process by reading failed process’s transactions directly from checkpoint memory space instead of disk.

Figure 3 illustrates AMFT checkpointing operation by showing two different cases. In Figure 3 only local FP-Tree of process is checkpointed on available transactions space. However, in Figure 3 both remaining transactions and local FP-Tree of process are checkpointed to memory (i.e., memory space availability is required).

\subcaptionbox

Local FP-Tree Checkpointing     \subcaptionboxUnprocessed Transactions and Local FP-Tree Checkpointing

Figure 3: AMFT Checkpointing Operation Overview

The effectiveness of AMFT checkpointing algorithm is in its simplicity. Unlike SMFT, there is no synchronization required between any pair of processes, and memory allocation is not required as well. By using MPI-RMA on high performance interconnects such as InfiniBand, we expect AMFT to be a near-optimal checkpointing algorithm for designing large scale FP-Growth algorithm. As expected, since each process simply initiates the communication for the checkpoint, the expected time complexity of the checkpointing is , using the LogGP model [4].

Recovery Algorithm: The recovery algorithm for AMFT is similar to SMFT. Assuming is the recovery process . When a fault occurs (on ), recovery process merges the checkpointed FP-Tree of with its FP-Tree and re-distributes the dead process transactions among a subset of available processes (such as ), if an in-memory checkpoint is available locally. Otherwise, all available processes recovered unprocessed transactions of the failed process from the disk in parallel.

The worst case time complexity of AMFT approach is similar to SMFT. In the worst case, the entire transactions are read from disk in parallel as mentioned in SMFT approach with () time complexity, and recomputed by processes in (). However, in many cases — especially when the fault occurs during later stages of FP-Tree build phase — disk will be completely avoided, resulting in much faster recovery in comparison to the worst case scenario.

Implementation Details:

\setstretch

1.1

Algorithm 3 : AMFT FP-Growth Algorithm
1:Procedure: initialization( = AMFT)
2:
3:Assume vector is the memory space contains on
4:Create vector on to describe checkpoint (initially-empty).
5:Expose and vectors addresses for read/update  //using MPI-RMA
Procedure: performChk (, , )
6:
1:if  has enough space for of  then
2:      add(, ) (  (MPI_Put)
3:end if
4:if  has enough space for remaining of (Only one time) then
5:      add(, )  (MPI_Put)
6:end if
7:Update vector  (MPI_Put)
Procedure: performRecovery (, , )
8:
1: process: merge (, , )
2:if  is NULL then
3:      diskTransRecv()
4:else
5:      memTransRecv(, )
6:end if

Algorithm 3 illustrates the checkpointing and recovery procedures for AMFT algorithm. During the initialization procedure, each process has its own vector that contains local set of transactions (Line 1). In line 2, each process creates a single vector, i.e., , that represents a set of parameters to describe the status of vector and checkpointed data of source process stored on memory. In line 3, MPI-RMA technology is used to shared both vectors, i.e., and , to other processes.

Both and remaining transactions can be checkpointed using procedure. Each process should read on target process to check for space availability before checkpointing (Lines 1-6). Remaining transaction checkpointing is only performed one time once a space is available.

The procedure shows the recovery algorithm in AMFT approach. Like the SMFT recovery algorithm, the recovery process process is used to recover by merging latest checkpointed FP-Tree it has with its local FP-Tree. unprocessed transactions can be recover from recovery process memory if it was checkpointed before failure or directly from disk (Lines 2-6).

\subcaptionbox

100M Trans. =\subcaptionbox100M Trans. =\subcaptionbox200M Trans. =\subcaptionbox200M Trans. =

Figure 4: Proposed FT mechanisms checkpointing overhead with different number of transactions, support threshold, and cores

5 Performance Evaluation

In this section, we present a detailed performance evaluation of the proposed fault tolerant FP-Growth algorithms, i.e., DFT, SMFT, and AMFT that were presented in section 4. For each fault tolerant algorithm, we present a detailed performance analysis of the checkpointing and recovery overhead. We use up to 200 million transactions and a large scale evaluation using up to 2048 cores. At the end of this section, a comparison against a fault-tolerant version executed on Spark is presented.

5.1 Setup

Experimental Testbed

We use Stampede supercomputer at the Texas Advanced Computing Center (TACC) for performance evaluation. The Stampede supercomputer is Dell PowerEdge C8220 cluster with 6,400 Dell PowerEdge server nodes, each with 32GB memory, (2) Intel Xeon E5 (8-core Sandy Bridge) processors. We use MVAPICH2-2.1, a high performance MPI library available on Remote Direct Memory Access (RDMA) interconnects such as InfiniBand. We use aggressive compiler optimizations with Intel compiler v15.0.1 for performance evaluation.

Datasets

To evaluate different proposed fault tolerant FP-Growth algorithms, we use IBM Quest dataset generator [2] for generating large scale synthetic datasets. IBM Quest dataset generator has been widely used in several studies, and accurately reflects the pattern of transactions in real-world datasets [37, 9, 36, 22]. For experimental evaluation, we use two synthetic datasets with 100 and 200 million transactions. The number of items per transaction is 15-20. A total of 1000 item-ids are used.

5.2 Overhead of Supporting FP-Growth Fault Tolerance

Checkpointing Overhead Evaluation

While the recovery algorithm is executed only during faults, the cost of checkpointing is incurred even in the absence of faults. Naturally, it is critical to minimize the checkpointing time — especially, when the fault rates are low.

# Cores Sup. DFT (%) SMFT (%) AMFT (%)
100M 200M 100M 200M 100M 200M
256 0.03 19.76 35.59 10.85 12.23 6.08 10.4
0.05 67.31 69.01 31.02 40.8 21.62 29.09
512 0.03 15.28 25.87 9.76 10.01 5.66 9.32
0.05 54.11 58.07 29.77 38.88 18.50 25.5
1024 0.03 13.77 22.22 6.67 11.52 4.47 9.14
0.05 41.17 52.93 27.87 41.34 12.21 18.8
2048 0.03 10.13 17.56 6.05 8.69 4.21 7.9
0.05 27.39 35.61 15.83 30.15 5.11 9.92
Table 2: DFT, AMFT, and SMFT systems slowdowns related to w/o FT FP-Growth algorithm

Figure 4 shows the checkpointing overhead of DFT, SMFT and AMFT algorithms using 100M, 200M transactions and support threshold () values of 0.03 and 0.05. Table 2 presents the data in a tabular form, by showing the percentage of slowdown in comparison to the default parallel algorithm that is not fault-tolerant. In Figure 4(a), if we focus on strong scaling evaluation (keeping the overall work constant and increasing the number of processes), the algorithm scales very well (scaling from 256 -512 processes, we observe super-linear speed-up due to better cache utilization). Similar speed-ups are observed for DFT, SMFT, and AMFT algorithms, respectively. Since the support threshold is high (0.05), the number of frequent item-ids is relatively small. Hence, the overall computation time is less than 50s. Naturally, the slow down observed by DFT and SMFT is high — 67% and 31%, respectively. AMFT only experiences a slowdown of 21%. We expected negligible overhead for AMFT. However, we experienced slowdown, because for small scales such as 256 processes, the size of individual FP-Tree is larger (in comparison to larger process counts). Unfortunately, current MPI-RMA implementations are not always optimized for bulk data transfers. To validate this argument, we observe the column for AMFT with 100M transactions. On 2048 cores — with strong scaling — the overhead of checkpointing reduced to 5%. For lower support threshold, as shown in Figure 4(b), the overall slowdown for AMFT is 4-6%, while DFT overhead is 10-20%, for different process counts.

Figure 4(c) shows the performance comparison of DFT, SMFT, and AMFT algorithms using 200M transactions and 0.05 support threshold. We observe similar pattern as Figure 4(a). While we expect relatively high overheads for DFT and SMFT approaches, we observe higher relative overhead for AMFT approach as well. We argue that for larger transactions per process, the size of the FP-Tree is larger. Since MPI-RMA runtimes are less optimized for bulk transfer, the slowdown is smaller, but non-negligible.

\subcaptionbox

100M Trans. =\subcaptionbox100M Trans. =\subcaptionbox200M Trans. =\subcaptionbox200M Trans. =

Figure 5: SMFT and AMFT Recovery Speed up Compared to DFT Approach with Different Number of transactions, Support Threshold , and Cores

Figure 4(d) illustrates the performance of the proposed approaches with 200M transactions and 0.03 support threshold. The DFT approach observes a slowdown of 17-35% in comparison to the basic parallel algorithm, while AMFT only observes up to 10% overhead.

Clearly AMFT outperforms other approaches, especially the disk-based approach easily without incurring any additional space complexity. We also observe that with strong scaling, which is usually a problem for distributed memory algorithms, the relative overhead of AMFT decreases. We argue that it is due to the unoptimized MPI-RMA protocols for bulk data transfer. With further optimizations, as expected in near future, these overheads are expected to reduce further. With space complexity and still acceptable checkpointing overhead such as 10% for AMFT, we expect the proposed algorithm to be used as the basis for future research and practical deployments.

Recovery Overhead Evaluation

The effectiveness of any fault tolerance mechanism is related to failure recovery overhead besides the checkpointing overhead. In this subsection, we evaluate the recovery overhead in the case of failure by injecting faults into FP-Growth parallel execution. To simulate faults, we select a process to fail and the point of failure. When reaching failure point, that process is eliminated from the execution. We assume failure point after processing 80% of dataset transactions to fairly comparing recovering algorithm for DFT, SMFT , and AMFT approaches.

In the case of failure, DFT recovery algorithm needs to recover FP-Tree of failed process from the disk comparing to both SMFT and AMFT approaches where FP-Tree is recovered from memory. In the first set of experiments, we calculate the speed-up using both SMFT and AMFT approaches compared to DFT approach to recovery one failure process as shown in Figure 5. In Figures 5 and  5, with 0.05 support threshold, the average speed-up by SMFT algorithm is 1.36x while average gained speed-up by AMFT algorithm is 1.41x using 100M dataset in the recovery process. In the case of 200M synthetic dataset, both SMFT and AMFT recovery algorithms speed-up the total execution time with recovery by 1.55x and 1.59x, respectively, compared to DFT algorithm. In Figure 5 and 5, with 0.03 support threshold, the recovered FP-Tree becomes larger which negatively impacts the performance of DFT approach compared to the other two approaches (i.e., SMFT and AMFT). Thus, with 100M dataset, compared to DFT approach, SMFT speeds-up the recovery process by 1.39x while AMFT speeds-up the recovery process with 1.46x. Using 200M dataset, SMFT speeds-up the algorithm execution with recovery by 1.51x while AMFT speeds-up the algorithm with 1.68x.

# Cores Sup. DFT Time (Sec) SMFT Time (Sec) AMFT Time (Sec)
100M 200M 100M 200M 100M 200M
256 0.03 2312.65 8860.26 2049.68 6945.23 1972.01 6822.59
0.05 67.12 182.685 56.57 132.52 54.23 119.16
512 0.03 948.125 3227.25 722.19 2268.12 701.12 2226.65
0.05 34.59 92.36 26.95 64.12 24.83 59.63
1024 0.03 609.52 1762.34 415.12 1038.23 399.52 1022.52
0.05 15.88 45.48 11.06 31.68 9.95 27.23
2048 0.03 438.85 1151.12 280.23 629.62 272.85 609.62
0.05 10.55 27.04 6.97 15.12 6.40 13.78
Table 3: DFT, SMFT and AMFT Total Execution Time Including The Recovery Time

Table  3 summarizes the total execution time including the recovery time of DFT, SMFT, and AMFT algorithms to handle one failure using 256, 512, 1024, and 2048 cores with 0.03 and 0.05 support threshold, respectively. Several observations can be drawn from Figure 5 and Table 3. Both SMFT and AMFT algorithms speed-up the FP-Growth algorithm recovery process compared to DFT algorithm. With smaller support threshold (=0.05), the size of checkpointed local FP-Trees and dead process recovered FP-Tree is small. Thus, in SMFT the synchronization overhead can be clearly shown compared to AMFT algorithm. In this case, AMFT outperforms SMFT algorithm as shown. However, in the case of ( =0.03), the size of FP-Tree is larger and the synchronization overheads are small compared to checkpointing and recovery time. Thus, the speed-up difference between SMFT and AMFT decreases. Another observation that could be obvious is that the average speed-up for both SMFT and AMFT algorithms increases with larger dataset (i.e., 200M). The main reason of this that FP-Trees become larger and DFT algorithm needs more time to checkpoint or recover it from disk. Finally, with ( =0.3), we observe a super-linear speed-up from 256 to 512 cores due to better cash utilization.

5.3 Comparison Against Spark

We compare our proposed AMFT FP-Growth algorithm with Spark FP-Growth algorithm to show the effectiveness of our proposed system. Although, it is common for MPI-based implementations to outperform MapReduce-based implementations [18], we are particularly interested in absolute and relative overheads for handling failures. Spark has a built-in Machine Learning library (MLlib) that includes an FP-Growth algorithm, which we use in our comparison. A set of experiments has been conducted with different number of nodes and using 500K synthetic dataset to show the performance of both MPI-based and spark-based FP-Growth algorithms.

\subcaptionbox

500K Trans. = \subcaptionbox500K Trans. =

Figure 6: Spark and MPI-based (AMFT))with Different Support Threshold and using 500K Synthetic Dataset

Figure 6 shows the performance of AMFT algorithm compared to Spark. With the absence of a failure, AMFT algorithm outperforms spark FP-Growth version with an average speed-up of 20x with and an average speed-up of 8.6x with . The average speed-up in the case of smaller threshold () is larger because the size of checkpointed FP-Trees is larger. Moreover, when checkpointing, the scalability of AMFT algorithm is better than the Spark-based algorithm because AMFT only depends on checkpointing FP-Trees and a set of transactions periodically, which are both small with larger number of cores. However, Spark depends on the RDD mechanism by having in-memory replication of both FP-Trees and transactions, overhead of which increases with a larger number of cores.

In the case of a failure, the average gained speed-up from using AMFT compared to Spark is 15.3x with and 8.34x with . Performance of both AMFT and Spark-based algorithms becomes better with larger number of cores and/or smaller support threshold (i.e., because recovered FP-Tree is smaller in both cases.

6 Related Work

Several researchers have proposed FP-Growth algorithms for both single node and distributed memory systems [21, 24, 6, 40, 10, 27]. These algorithms have addressed several issues for scalable FP-Growth such as memory utilization, communication cost, and load-balancing. However, fault tolerance has not been considered in these efforts.

Several programming models proposed recently provide automatic fault tolerance using functional paradigms. These include MapReduce implementations like Hadoop and Spark, as well as MillWheel. There have been studies for using MapReduce to parallelize frequent pattern mining algorithms, including FP-Growth [21, 40, 19] and apriori [24, 6]. In these work, MapReduce achieves fault-tolerance by re-executing all the tasks of the failed node(s). As far as we are aware, recovery algorithm has to completely re-execute the FP-Tree generation from scratch in these implementations, which severely and negatively impacts the recovery performance.

Scalable Checkpoint/Restart library (SCR) is another way to support fault tolerant MPI-based applications through a multi-level checkpointing technique [25]. SCR handles hardware failures in MPI application by performing less frequent and inexpensive checkpoints on available compute nodes memory. Our work has somewhat similar ideas, but further specializes them by considering algorithm-specific properties.

7 Acknowledgement

The research described in this paper is part of the Analysis in Motion (AIM) Laboratory Directed Research and Development (LDRD) Initiative at Pacific Northwest National Laboratory.

8 Conclusion

This paper focuses on building a fault tolerance framework to support FP-Growth algorithm in parallel systems. Three fault tolerance algorithms have been proposed: Disk-based Fault Tolerance (DFT), Synchronous Memory-based Fault Tolerance (SMFT), Asynchronous Memory-based Fault Tolerance (AMFT). DFT algorithm represents the brute-force approach to build a fault tolerance system using periodically checkpoints on disk. However, the other two algorithms, i.e., SMFT and AMFT, perform periodically checkpoints on memory instead of disk to avoid I/O latency.

In SMFT algorithm, we shrink the processed transactions space and allocate a new space that can remotely be accessed by other processes to perform FP-Tree and transactions checkpoint. This algorithm requires synchronization between processes before any single checkpoint which adds more overhead to checkpointing operation. However, in AMFT algorithm, we use the transactions vector itself as checkpoint space to avoid any communication between processes during the checkpointing operation.

An extensive evaluation over 256, 512, 1024, and 2048 cores has been performed on large datasets, i.e., 100 and 200 million transactions datasets. Our evaluation demonstrates excellent efficiency for checkpointing and recovery in comparison to the disk-based algorithm. Our detailed experimental evaluation also shows low overheads and how we can outperform Spark by an average of 20x with and 8.6x with .

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