[SPARK-8999] [MLLIB] PrefixSpan non-temporal sequences

mengxr Extends PrefixSpan to non-temporal itemsets. Continues work by zhangjiajin

 * Internal API uses List[Set[Int]] which is likely not efficient; will need to refactor during QA

Closes #7646

Author: zhangjiajin <zhangjiajin@huawei.com>
Author: Feynman Liang <fliang@databricks.com>
Author: zhang jiajin <zhangjiajin@huawei.com>

Closes #7818 from feynmanliang/SPARK-8999-nonTemporal and squashes the following commits:

4ded81d [Feynman Liang] Replace all filters to filter nonempty
350e67e [Feynman Liang] Code review feedback
03156ca [Feynman Liang] Fix tests, drop delimiters at boundaries of sequences
d1fe0ed [Feynman Liang] Remove comments
86ca4e5 [Feynman Liang] Fix style
7c7bf39 [Feynman Liang] Fixed itemSet sequences
6073b10 [Feynman Liang] Basic itemset functionality, failing test
1a7fb48 [Feynman Liang] Add delimiter to results
5db00aa [Feynman Liang] Working for items, not itemsets
6787716 [Feynman Liang] Working on temporal sequences
f1114b9 [Feynman Liang] Add -1 delimiter
00fe756 [Feynman Liang] Reset base files for rebase
f486dcd [zhangjiajin] change maxLocalProjDBSize and fix a bug (remove -3 from frequent items).
60a0b76 [zhangjiajin] fixed a scala style error.
740c203 [zhangjiajin] fixed a scala style error.
5785cb8 [zhangjiajin] support non-temporal sequence
a5d649d [zhangjiajin] restore original version
09dc409 [zhangjiajin] Merge branch 'master' of https://github.com/apache/spark into multiItems_2
ae8c02d [zhangjiajin] Fixed some Scala style errors.
216ab0c [zhangjiajin] Support non-temporal sequence in PrefixSpan
b572f54 [zhangjiajin] initialize file before rebase.
f06772f [zhangjiajin] fix a scala style error.
a7e50d4 [zhangjiajin] Add feature: Collect enough frequent prefixes before projection in PrefixSpan.
c1d13d0 [zhang jiajin] Delete PrefixspanSuite.scala
d9d8137 [zhang jiajin] Delete Prefixspan.scala
c6ceb63 [zhangjiajin] Add new algorithm PrefixSpan and test file.

3 files changed, 302 insertions, 92 deletions

diff --git a/mllib/src/main/scala/org/apache/spark/mllib/fpm/LocalPrefixSpan.scala b/mllib/src/main/scala/org/apache/spark/mllib/fpm/LocalPrefixSpan.scala
index 0ea7920810..ccebf951c8 100644
--- a/mllib/src/main/scala/org/apache/spark/mllib/fpm/LocalPrefixSpan.scala
+++ b/mllib/src/main/scala/org/apache/spark/mllib/fpm/LocalPrefixSpan.scala
@@ -25,7 +25,7 @@ import org.apache.spark.Logging
  * Calculate all patterns of a projected database in local.
  */
 private[fpm] object LocalPrefixSpan extends Logging with Serializable {
-
+  import PrefixSpan._
   /**
    * Calculate all patterns of a projected database.
    * @param minCount minimum count
@@ -39,12 +39,19 @@ private[fpm] object LocalPrefixSpan extends Logging with Serializable {
   def run(
       minCount: Long,
       maxPatternLength: Int,
-      prefixes: List[Int],
-      database: Iterable[Array[Int]]): Iterator[(List[Int], Long)] = {
-    if (prefixes.length == maxPatternLength || database.isEmpty) return Iterator.empty
-    val frequentItemAndCounts = getFreqItemAndCounts(minCount, database)
-    val filteredDatabase = database.map(x => x.filter(frequentItemAndCounts.contains))
-    frequentItemAndCounts.iterator.flatMap { case (item, count) =>
+      prefixes: List[Set[Int]],
+      database: Iterable[List[Set[Int]]]): Iterator[(List[Set[Int]], Long)] = {
+    if (prefixes.length == maxPatternLength || database.isEmpty) {
+      return Iterator.empty
+    }
+    val freqItemSetsAndCounts = getFreqItemAndCounts(minCount, database)
+    val freqItems = freqItemSetsAndCounts.keys.flatten.toSet
+    val filteredDatabase = database.map { suffix =>
+      suffix
+        .map(item => freqItems.intersect(item))
+        .filter(_.nonEmpty)
+    }
+    freqItemSetsAndCounts.iterator.flatMap { case (item, count) =>
       val newPrefixes = item :: prefixes
       val newProjected = project(filteredDatabase, item)
       Iterator.single((newPrefixes, count)) ++
@@ -54,20 +61,23 @@ private[fpm] object LocalPrefixSpan extends Logging with Serializable {
 
   /**
    * Calculate suffix sequence immediately after the first occurrence of an item.
-   * @param item item to get suffix after
+   * @param item itemset to get suffix after
    * @param sequence sequence to extract suffix from
    * @return suffix sequence
    */
-  def getSuffix(item: Int, sequence: Array[Int]): Array[Int] = {
-    val index = sequence.indexOf(item)
+  def getSuffix(item: Set[Int], sequence: List[Set[Int]]): List[Set[Int]] = {
+    val itemsetSeq = sequence
+    val index = itemsetSeq.indexWhere(item.subsetOf(_))
     if (index == -1) {
-      Array()
+      List()
     } else {
-      sequence.drop(index + 1)
+      itemsetSeq.drop(index + 1)
     }
   }
 
-  def project(database: Iterable[Array[Int]], prefix: Int): Iterable[Array[Int]] = {
+  def project(
+      database: Iterable[List[Set[Int]]],
+      prefix: Set[Int]): Iterable[List[Set[Int]]] = {
     database
       .map(getSuffix(prefix, _))
       .filter(_.nonEmpty)
@@ -81,14 +91,16 @@ private[fpm] object LocalPrefixSpan extends Logging with Serializable {
    */
   private def getFreqItemAndCounts(
       minCount: Long,
-      database: Iterable[Array[Int]]): mutable.Map[Int, Long] = {
+      database: Iterable[List[Set[Int]]]): Map[Set[Int], Long] = {
     // TODO: use PrimitiveKeyOpenHashMap
-    val counts = mutable.Map[Int, Long]().withDefaultValue(0L)
+    val counts = mutable.Map[Set[Int], Long]().withDefaultValue(0L)
     database.foreach { sequence =>
-      sequence.distinct.foreach { item =>
+      sequence.flatMap(nonemptySubsets(_)).distinct.foreach { item =>
         counts(item) += 1L
       }
     }
-    counts.filter(_._2 >= minCount)
+    counts
+      .filter { case (_, count) => count >= minCount }
+      .toMap
   }
 }
diff --git a/mllib/src/main/scala/org/apache/spark/mllib/fpm/PrefixSpan.scala b/mllib/src/main/scala/org/apache/spark/mllib/fpm/PrefixSpan.scala
index e6752332cd..22b4ddb8b3 100644
--- a/mllib/src/main/scala/org/apache/spark/mllib/fpm/PrefixSpan.scala
+++ b/mllib/src/main/scala/org/apache/spark/mllib/fpm/PrefixSpan.scala
@@ -17,7 +17,7 @@
 
 package org.apache.spark.mllib.fpm
 
-import scala.collection.mutable.ArrayBuffer
+import scala.collection.mutable.ArrayBuilder
 
 import org.apache.spark.Logging
 import org.apache.spark.annotation.Experimental
@@ -44,13 +44,14 @@ import org.apache.spark.storage.StorageLevel
 class PrefixSpan private (
     private var minSupport: Double,
     private var maxPatternLength: Int) extends Logging with Serializable {
+  import PrefixSpan._
 
   /**
    * The maximum number of items allowed in a projected database before local processing. If a
    * projected database exceeds this size, another iteration of distributed PrefixSpan is run.
    */
-  // TODO: make configurable with a better default value, 10000 may be too small
-  private val maxLocalProjDBSize: Long = 10000
+  // TODO: make configurable with a better default value
+  private val maxLocalProjDBSize: Long = 32000000L
 
   /**
    * Constructs a default instance with default parameters
@@ -90,35 +91,41 @@ class PrefixSpan private (
 
   /**
    * Find the complete set of sequential patterns in the input sequences.
-   * @param sequences input data set, contains a set of sequences,
-   *                  a sequence is an ordered list of elements.
+   * @param data ordered sequences of itemsets. Items are represented by non-negative integers.
+   *                  Each itemset has one or more items and is delimited by [[DELIMITER]].
    * @return a set of sequential pattern pairs,
    *         the key of pair is pattern (a list of elements),
    *         the value of pair is the pattern's count.
    */
-  def run(sequences: RDD[Array[Int]]): RDD[(Array[Int], Long)] = {
-    val sc = sequences.sparkContext
+  // TODO: generalize to arbitrary item-types and use mapping to Ints for internal algorithm
+  def run(data: RDD[Array[Int]]): RDD[(Array[Int], Long)] = {
+    val sc = data.sparkContext
 
-    if (sequences.getStorageLevel == StorageLevel.NONE) {
+    if (data.getStorageLevel == StorageLevel.NONE) {
       logWarning("Input data is not cached.")
     }
 
+    // Use List[Set[Item]] for internal computation
+    val sequences = data.map { seq => splitSequence(seq.toList) }
+
     // Convert min support to a min number of transactions for this dataset
     val minCount = if (minSupport == 0) 0L else math.ceil(sequences.count() * minSupport).toLong
 
     // (Frequent items -> number of occurrences, all items here satisfy the `minSupport` threshold
     val freqItemCounts = sequences
-      .flatMap(seq => seq.distinct.map(item => (item, 1L)))
+      .flatMap(seq => seq.flatMap(nonemptySubsets(_)).distinct.map(item => (item, 1L)))
       .reduceByKey(_ + _)
-      .filter(_._2 >= minCount)
+      .filter { case (item, count) => (count >= minCount) }
       .collect()
+      .toMap
 
     // Pairs of (length 1 prefix, suffix consisting of frequent items)
     val itemSuffixPairs = {
-      val freqItems = freqItemCounts.map(_._1).toSet
+      val freqItemSets = freqItemCounts.keys.toSet
+      val freqItems = freqItemSets.flatten
       sequences.flatMap { seq =>
-        val filteredSeq = seq.filter(freqItems.contains(_))
-        freqItems.flatMap { item =>
+        val filteredSeq = seq.map(item => freqItems.intersect(item)).filter(_.nonEmpty)
+        freqItemSets.flatMap { item =>
           val candidateSuffix = LocalPrefixSpan.getSuffix(item, filteredSeq)
           candidateSuffix match {
             case suffix if !suffix.isEmpty => Some((List(item), suffix))
@@ -130,14 +137,15 @@ class PrefixSpan private (
 
     // Accumulator for the computed results to be returned, initialized to the frequent items (i.e.
     // frequent length-one prefixes)
-    var resultsAccumulator = freqItemCounts.map(x => (List(x._1), x._2))
+    var resultsAccumulator = freqItemCounts.map { case (item, count) => (List(item), count) }.toList
 
     // Remaining work to be locally and distributively processed respectfully
     var (pairsForLocal, pairsForDistributed) = partitionByProjDBSize(itemSuffixPairs)
 
     // Continue processing until no pairs for distributed processing remain (i.e. all prefixes have
-    // projected database sizes <= `maxLocalProjDBSize`)
-    while (pairsForDistributed.count() != 0) {
+    // projected database sizes <= `maxLocalProjDBSize`) or `maxPatternLength` is reached
+    var patternLength = 1
+    while (pairsForDistributed.count() != 0 && patternLength < maxPatternLength) {
       val (nextPatternAndCounts, nextPrefixSuffixPairs) =
         extendPrefixes(minCount, pairsForDistributed)
       pairsForDistributed.unpersist()
@@ -146,6 +154,7 @@ class PrefixSpan private (
       pairsForDistributed.persist(StorageLevel.MEMORY_AND_DISK)
       pairsForLocal ++= smallerPairsPart
       resultsAccumulator ++= nextPatternAndCounts.collect()
+      patternLength += 1 // pattern length grows one per iteration
     }
 
     // Process the small projected databases locally
@@ -153,7 +162,7 @@ class PrefixSpan private (
       minCount, sc.parallelize(pairsForLocal, 1).groupByKey())
 
     (sc.parallelize(resultsAccumulator, 1) ++ remainingResults)
-      .map { case (pattern, count) => (pattern.toArray, count) }
+      .map { case (pattern, count) => (flattenSequence(pattern.reverse).toArray, count) }
   }
 
 
@@ -163,8 +172,8 @@ class PrefixSpan private (
    * @return prefix-suffix pairs partitioned by whether their projected database size is <= or
    *         greater than [[maxLocalProjDBSize]]
    */
-  private def partitionByProjDBSize(prefixSuffixPairs: RDD[(List[Int], Array[Int])])
-    : (Array[(List[Int], Array[Int])], RDD[(List[Int], Array[Int])]) = {
+  private def partitionByProjDBSize(prefixSuffixPairs: RDD[(List[Set[Int]], List[Set[Int]])])
+    : (List[(List[Set[Int]], List[Set[Int]])], RDD[(List[Set[Int]], List[Set[Int]])]) = {
     val prefixToSuffixSize = prefixSuffixPairs
       .aggregateByKey(0)(
         seqOp = { case (count, suffix) => count + suffix.length },
@@ -176,12 +185,12 @@ class PrefixSpan private (
       .toSet
     val small = prefixSuffixPairs.filter { case (prefix, _) => smallPrefixes.contains(prefix) }
     val large = prefixSuffixPairs.filter { case (prefix, _) => !smallPrefixes.contains(prefix) }
-    (small.collect(), large)
+    (small.collect().toList, large)
   }
 
   /**
-   * Extends all prefixes by one item from their suffix and computes the resulting frequent prefixes
-   * and remaining work.
+   * Extends all prefixes by one itemset from their suffix and computes the resulting frequent
+   * prefixes and remaining work.
    * @param minCount minimum count
    * @param prefixSuffixPairs prefix (length N) and suffix pairs,
    * @return (frequent length N+1 extended prefix, count) pairs and (frequent length N+1 extended
@@ -189,15 +198,16 @@ class PrefixSpan private (
    */
   private def extendPrefixes(
       minCount: Long,
-      prefixSuffixPairs: RDD[(List[Int], Array[Int])])
-    : (RDD[(List[Int], Long)], RDD[(List[Int], Array[Int])]) = {
+      prefixSuffixPairs: RDD[(List[Set[Int]], List[Set[Int]])])
+    : (RDD[(List[Set[Int]], Long)], RDD[(List[Set[Int]], List[Set[Int]])]) = {
 
-    // (length N prefix, item from suffix) pairs and their corresponding number of occurrences
+    // (length N prefix, itemset from suffix) pairs and their corresponding number of occurrences
     // Every (prefix :+ suffix) is guaranteed to have support exceeding `minSupport`
     val prefixItemPairAndCounts = prefixSuffixPairs
-      .flatMap { case (prefix, suffix) => suffix.distinct.map(y => ((prefix, y), 1L)) }
+      .flatMap { case (prefix, suffix) =>
+      suffix.flatMap(nonemptySubsets(_)).distinct.map(y => ((prefix, y), 1L)) }
       .reduceByKey(_ + _)
-      .filter(_._2 >= minCount)
+      .filter { case (item, count) => (count >= minCount) }
 
     // Map from prefix to set of possible next items from suffix
     val prefixToNextItems = prefixItemPairAndCounts
@@ -207,7 +217,6 @@ class PrefixSpan private (
       .collect()
       .toMap
 
-
     // Frequent patterns with length N+1 and their corresponding counts
     val extendedPrefixAndCounts = prefixItemPairAndCounts
       .map { case ((prefix, item), count) => (item :: prefix, count) }
@@ -216,9 +225,12 @@ class PrefixSpan private (
     val extendedPrefixAndSuffix = prefixSuffixPairs
       .filter(x => prefixToNextItems.contains(x._1))
       .flatMap { case (prefix, suffix) =>
-        val frequentNextItems = prefixToNextItems(prefix)
-        val filteredSuffix = suffix.filter(frequentNextItems.contains(_))
-        frequentNextItems.flatMap { item =>
+        val frequentNextItemSets = prefixToNextItems(prefix)
+        val frequentNextItems = frequentNextItemSets.flatten
+        val filteredSuffix = suffix
+          .map(item => frequentNextItems.intersect(item))
+          .filter(_.nonEmpty)
+        frequentNextItemSets.flatMap { item =>
           LocalPrefixSpan.getSuffix(item, filteredSuffix) match {
             case suffix if !suffix.isEmpty => Some(item :: prefix, suffix)
             case _ => None
@@ -237,13 +249,38 @@ class PrefixSpan private (
    */
   private def getPatternsInLocal(
       minCount: Long,
-      data: RDD[(List[Int], Iterable[Array[Int]])]): RDD[(List[Int], Long)] = {
+      data: RDD[(List[Set[Int]], Iterable[List[Set[Int]]])]): RDD[(List[Set[Int]], Long)] = {
     data.flatMap {
-      case (prefix, projDB) =>
-        LocalPrefixSpan.run(minCount, maxPatternLength, prefix.toList.reverse, projDB)
-          .map { case (pattern: List[Int], count: Long) =>
-          (pattern.reverse, count)
-        }
+      case (prefix, projDB) => LocalPrefixSpan.run(minCount, maxPatternLength, prefix, projDB)
+    }
+  }
+
+}
+
+private[fpm] object PrefixSpan {
+  private[fpm] val DELIMITER = -1
+
+  /** Splits a sequence of itemsets delimited by [[DELIMITER]]. */
+  private[fpm] def splitSequence(sequence: List[Int]): List[Set[Int]] = {
+    sequence.span(_ != DELIMITER) match {
+      case (x, xs) if xs.length > 1 => x.toSet :: splitSequence(xs.tail)
+      case (x, xs) => List(x.toSet)
+    }
+  }
+
+  /** Flattens a sequence of itemsets into an Array, inserting[[DELIMITER]] between itemsets. */
+  private[fpm] def flattenSequence(sequence: List[Set[Int]]): List[Int] = {
+    val builder = ArrayBuilder.make[Int]()
+    for (itemSet <- sequence) {
+      builder += DELIMITER
+      builder ++= itemSet.toSeq.sorted
     }
+    builder.result().toList.drop(1) // drop trailing delimiter
+  }
+
+  /** Returns an iterator over all non-empty subsets of `itemSet` */
+  private[fpm] def nonemptySubsets(itemSet: Set[Int]): Iterator[Set[Int]] = {
+    // TODO: improve complexity by using partial prefixes, considering one item at a time
+    itemSet.subsets.filter(_ != Set.empty[Int])
   }
 }
diff --git a/mllib/src/test/scala/org/apache/spark/mllib/fpm/PrefixSpanSuite.scala b/mllib/src/test/scala/org/apache/spark/mllib/fpm/PrefixSpanSuite.scala
index 6dd2dc926a..457f32670f 100644
--- a/mllib/src/test/scala/org/apache/spark/mllib/fpm/PrefixSpanSuite.scala
+++ b/mllib/src/test/scala/org/apache/spark/mllib/fpm/PrefixSpanSuite.scala
@@ -21,7 +21,7 @@ import org.apache.spark.mllib.util.MLlibTestSparkContext
 
 class PrefixSpanSuite extends SparkFunSuite with MLlibTestSparkContext {
 
-  test("PrefixSpan using Integer type") {
+  test("PrefixSpan using Integer type, singleton itemsets") {
 
     /*
       library("arulesSequences")
@@ -35,12 +35,12 @@ class PrefixSpanSuite extends SparkFunSuite with MLlibTestSparkContext {
     */
 
     val sequences = Array(
-      Array(1, 3, 4, 5),
-      Array(2, 3, 1),
-      Array(2, 4, 1),
-      Array(3, 1, 3, 4, 5),
-      Array(3, 4, 4, 3),
-      Array(6, 5, 3))
+      Array(1, -1, 3, -1, 4, -1, 5),
+      Array(2, -1, 3, -1, 1),
+      Array(2, -1, 4, -1, 1),
+      Array(3, -1, 1, -1, 3, -1, 4, -1, 5),
+      Array(3, -1, 4, -1, 4, -1, 3),
+      Array(6, -1, 5, -1, 3))
 
     val rdd = sc.parallelize(sequences, 2).cache()
 
@@ -50,64 +50,225 @@ class PrefixSpanSuite extends SparkFunSuite with MLlibTestSparkContext {
     val result1 = prefixspan.run(rdd)
     val expectedValue1 = Array(
       (Array(1), 4L),
-      (Array(1, 3), 2L),
-      (Array(1, 3, 4), 2L),
-      (Array(1, 3, 4, 5), 2L),
-      (Array(1, 3, 5), 2L),
-      (Array(1, 4), 2L),
-      (Array(1, 4, 5), 2L),
-      (Array(1, 5), 2L),
+      (Array(1, -1, 3), 2L),
+      (Array(1, -1, 3, -1, 4), 2L),
+      (Array(1, -1, 3, -1, 4, -1, 5), 2L),
+      (Array(1, -1, 3, -1, 5), 2L),
+      (Array(1, -1, 4), 2L),
+      (Array(1, -1, 4, -1, 5), 2L),
+      (Array(1, -1, 5), 2L),
       (Array(2), 2L),
-      (Array(2, 1), 2L),
+      (Array(2, -1, 1), 2L),
       (Array(3), 5L),
-      (Array(3, 1), 2L),
-      (Array(3, 3), 2L),
-      (Array(3, 4), 3L),
-      (Array(3, 4, 5), 2L),
-      (Array(3, 5), 2L),
+      (Array(3, -1, 1), 2L),
+      (Array(3, -1, 3), 2L),
+      (Array(3, -1, 4), 3L),
+      (Array(3, -1, 4, -1, 5), 2L),
+      (Array(3, -1, 5), 2L),
       (Array(4), 4L),
-      (Array(4, 5), 2L),
+      (Array(4, -1, 5), 2L),
       (Array(5), 3L)
     )
-    assert(compareResults(expectedValue1, result1.collect()))
+    compareResults(expectedValue1, result1.collect())
 
     prefixspan.setMinSupport(0.5).setMaxPatternLength(50)
     val result2 = prefixspan.run(rdd)
     val expectedValue2 = Array(
       (Array(1), 4L),
       (Array(3), 5L),
-      (Array(3, 4), 3L),
+      (Array(3, -1, 4), 3L),
       (Array(4), 4L),
       (Array(5), 3L)
     )
-    assert(compareResults(expectedValue2, result2.collect()))
+    compareResults(expectedValue2, result2.collect())
 
     prefixspan.setMinSupport(0.33).setMaxPatternLength(2)
     val result3 = prefixspan.run(rdd)
     val expectedValue3 = Array(
       (Array(1), 4L),
-      (Array(1, 3), 2L),
-      (Array(1, 4), 2L),
-      (Array(1, 5), 2L),
-      (Array(2, 1), 2L),
+      (Array(1, -1, 3), 2L),
+      (Array(1, -1, 4), 2L),
+      (Array(1, -1, 5), 2L),
+      (Array(2, -1, 1), 2L),
       (Array(2), 2L),
       (Array(3), 5L),
-      (Array(3, 1), 2L),
-      (Array(3, 3), 2L),
-      (Array(3, 4), 3L),
-      (Array(3, 5), 2L),
+      (Array(3, -1, 1), 2L),
+      (Array(3, -1, 3), 2L),
+      (Array(3, -1, 4), 3L),
+      (Array(3, -1, 5), 2L),
       (Array(4), 4L),
-      (Array(4, 5), 2L),
+      (Array(4, -1, 5), 2L),
       (Array(5), 3L)
     )
-    assert(compareResults(expectedValue3, result3.collect()))
+    compareResults(expectedValue3, result3.collect())
+  }
+
+  test("PrefixSpan using Integer type, variable-size itemsets") {
+    val sequences = Array(
+      Array(1, -1, 1, 2, 3, -1, 1, 3, -1, 4, -1, 3, 6),
+      Array(1, 4, -1, 3, -1, 2, 3, -1, 1, 5),
+      Array(5, 6, -1, 1, 2, -1, 4, 6, -1, 3, -1, 2),
+      Array(5, -1, 7, -1, 1, 6, -1, 3, -1, 2, -1, 3))
+    val rdd = sc.parallelize(sequences, 2).cache()
+    val prefixspan = new PrefixSpan().setMinSupport(0.5).setMaxPatternLength(5)
+    val result = prefixspan.run(rdd)
+
+    /*
+      To verify results, create file "prefixSpanSeqs" with content
+      (format = (transactionID, idxInTransaction, numItemsinItemset, itemset)):
+        1 1 1 1
+        1 2 3 1 2 3
+        1 3 2 1 3
+        1 4 1 4
+        1 5 2 3 6
+        2 1 2 1 4
+        2 2 1 3
+        2 3 2 2 3
+        2 4 2 1 5
+        3 1 2 5 6
+        3 2 2 1 2
+        3 3 2 4 6
+        3 4 1 3
+        3 5 1 2
+        4 1 1 5
+        4 2 1 7
+        4 3 2 1 6
+        4 4 1 3
+        4 5 1 2
+        4 6 1 3
+      In R, run:
+        library("arulesSequences")
+        prefixSpanSeqs = read_baskets("prefixSpanSeqs", info = c("sequenceID","eventID","SIZE"))
+        freqItemSeq = cspade(prefixSpanSeqs,
+                             parameter = list(support = 0.5, maxlen = 5 ))
+        resSeq = as(freqItemSeq, "data.frame")
+        resSeq
+
+                    sequence support
+        1              <{1}>    1.00
+        2              <{2}>    1.00
+        3              <{3}>    1.00
+        4              <{4}>    0.75
+        5              <{5}>    0.75
+        6              <{6}>    0.75
+        7          <{1},{6}>    0.50
+        8          <{2},{6}>    0.50
+        9          <{5},{6}>    0.50
+        10       <{1,2},{6}>    0.50
+        11         <{1},{4}>    0.50
+        12         <{2},{4}>    0.50
+        13       <{1,2},{4}>    0.50
+        14         <{1},{3}>    1.00
+        15         <{2},{3}>    0.75
+        16           <{2,3}>    0.50
+        17         <{3},{3}>    0.75
+        18         <{4},{3}>    0.75
+        19         <{5},{3}>    0.50
+        20         <{6},{3}>    0.50
+        21     <{5},{6},{3}>    0.50
+        22     <{6},{2},{3}>    0.50
+        23     <{5},{2},{3}>    0.50
+        24     <{5},{1},{3}>    0.50
+        25     <{2},{4},{3}>    0.50
+        26     <{1},{4},{3}>    0.50
+        27   <{1,2},{4},{3}>    0.50
+        28     <{1},{3},{3}>    0.75
+        29       <{1,2},{3}>    0.50
+        30     <{1},{2},{3}>    0.50
+        31       <{1},{2,3}>    0.50
+        32         <{1},{2}>    1.00
+        33           <{1,2}>    0.50
+        34         <{3},{2}>    0.75
+        35         <{4},{2}>    0.50
+        36         <{5},{2}>    0.50
+        37         <{6},{2}>    0.50
+        38     <{5},{6},{2}>    0.50
+        39     <{6},{3},{2}>    0.50
+        40     <{5},{3},{2}>    0.50
+        41     <{5},{1},{2}>    0.50
+        42     <{4},{3},{2}>    0.50
+        43     <{1},{3},{2}>    0.75
+        44 <{5},{6},{3},{2}>    0.50
+        45 <{5},{1},{3},{2}>    0.50
+        46         <{1},{1}>    0.50
+        47         <{2},{1}>    0.50
+        48         <{3},{1}>    0.50
+        49         <{5},{1}>    0.50
+        50       <{2,3},{1}>    0.50
+        51     <{1},{3},{1}>    0.50
+        52   <{1},{2,3},{1}>    0.50
+        53     <{1},{2},{1}>    0.50
+     */
+    val expectedValue = Array(
+      (Array(1), 4L),
+      (Array(2), 4L),
+      (Array(3), 4L),
+      (Array(4), 3L),
+      (Array(5), 3L),
+      (Array(6), 3L),
+      (Array(1, -1, 6), 2L),
+      (Array(2, -1, 6), 2L),
+      (Array(5, -1, 6), 2L),
+      (Array(1, 2, -1, 6), 2L),
+      (Array(1, -1, 4), 2L),
+      (Array(2, -1, 4), 2L),
+      (Array(1, 2, -1, 4), 2L),
+      (Array(1, -1, 3), 4L),
+      (Array(2, -1, 3), 3L),
+      (Array(2, 3), 2L),
+      (Array(3, -1, 3), 3L),
+      (Array(4, -1, 3), 3L),
+      (Array(5, -1, 3), 2L),
+      (Array(6, -1, 3), 2L),
+      (Array(5, -1, 6, -1, 3), 2L),
+      (Array(6, -1, 2, -1, 3), 2L),
+      (Array(5, -1, 2, -1, 3), 2L),
+      (Array(5, -1, 1, -1, 3), 2L),
+      (Array(2, -1, 4, -1, 3), 2L),
+      (Array(1, -1, 4, -1, 3), 2L),
+      (Array(1, 2, -1, 4, -1, 3), 2L),
+      (Array(1, -1, 3, -1, 3), 3L),
+      (Array(1, 2, -1, 3), 2L),
+      (Array(1, -1, 2, -1, 3), 2L),
+      (Array(1, -1, 2, 3), 2L),
+      (Array(1, -1, 2), 4L),
+      (Array(1, 2), 2L),
+      (Array(3, -1, 2), 3L),
+      (Array(4, -1, 2), 2L),
+      (Array(5, -1, 2), 2L),
+      (Array(6, -1, 2), 2L),
+      (Array(5, -1, 6, -1, 2), 2L),
+      (Array(6, -1, 3, -1, 2), 2L),
+      (Array(5, -1, 3, -1, 2), 2L),
+      (Array(5, -1, 1, -1, 2), 2L),
+      (Array(4, -1, 3, -1, 2), 2L),
+      (Array(1, -1, 3, -1, 2), 3L),
+      (Array(5, -1, 6, -1, 3, -1, 2), 2L),
+      (Array(5, -1, 1, -1, 3, -1, 2), 2L),
+      (Array(1, -1, 1), 2L),
+      (Array(2, -1, 1), 2L),
+      (Array(3, -1, 1), 2L),
+      (Array(5, -1, 1), 2L),
+      (Array(2, 3, -1, 1), 2L),
+      (Array(1, -1, 3, -1, 1), 2L),
+      (Array(1, -1, 2, 3, -1, 1), 2L),
+      (Array(1, -1, 2, -1, 1), 2L))
+
+    compareResults(expectedValue, result.collect())
   }
 
   private def compareResults(
-    expectedValue: Array[(Array[Int], Long)],
-    actualValue: Array[(Array[Int], Long)]): Boolean = {
-    expectedValue.map(x => (x._1.toSeq, x._2)).toSet ==
-      actualValue.map(x => (x._1.toSeq, x._2)).toSet
+      expectedValue: Array[(Array[Int], Long)],
+      actualValue: Array[(Array[Int], Long)]): Unit = {
+    val expectedSet = expectedValue.map(x => (x._1.toSeq, x._2)).toSet
+    val actualSet = actualValue.map(x => (x._1.toSeq, x._2)).toSet
+    assert(expectedSet === actualSet)
+  }
+
+  private def insertDelimiter(sequence: Array[Int]): Array[Int] = {
+    sequence.zip(Seq.fill(sequence.length)(PrefixSpan.DELIMITER)).map { case (a, b) =>
+      List(a, b)
+    }.flatten
   }
 
 }