Finding the Majority element in an array

In its simplest terms, the Majority element is the element that occurs the most within a given array of numbers. Here, we mean by “the most” that the number of occurrences of the element must be more than half the size of the array. In mathematical terms, given an unsorted array of integers from 1 to n, where n is the length of the array, an element is a Majority if it occurs at least (n/2) + 1 times.

The Majority element is an important problem in computer science and is used as a basis for more complex algorithms. We also encounter the majority element in our daily life. For example, the ownership of a large company is determined by shareholders having more than half the total share. Another example is the elections in your country where the winner is determined by obtaining more than half the voices.

In this article, we will look at a fast algorithm that runs in linear time, in exactly O(n) + O(n) to find the Majority element. The algorithm was discovered by Rengakrishnan Subramanian in December 2001. His official paper can be found here.

The algorithm approaches the problem in two steps, in the first step we will call the method FindElement that takes an array of integers as an input and returns an element that is “most likely” to be Majority. In the second step, the algorithm will call FindMajority which takes the same array as an input along with the most probable element we determined earlier and will return the element if it turns out to be Majority, or null if its not.

The FindElement algorithm works as follows:

1-declare two variables, we’ll call the first one count and the second one canbeMajority to store the candidate Majority.

2-assume that the first number in the array is Majority and assign its value to canbeMajority.

3-traverse the input array from start to end while applying the following conditions:

• If the next element in the array is equal to canbeMajority, then count is incremented by 1.
• If the next element in the array is not equal to canbeMajority, then count is decremented by 1.
• If at the next element, the count is equal to zero, then count is set to one and this element is considered as canbeMajority

Here is the implementation of FindElement and FindMajority in vb.net:

    Private Function FindElement(ByRef arr() As Integer) As Integer
        Dim count As Integer = 0
        Dim element As Integer = 0
        For i As Integer = 0 To arr.Length - 1
            If count = 0 Then
                count = 1
                element = arr(i)
            ElseIf arr(i) = element Then
                count += 1
            Else
                count -= 1
            End If
        Next
        Return element
    End Function

    Private Function FindMajority(ByRef arr() As Integer) As Integer
        Dim element As Integer = FindElement(arr)
        Dim count As Integer = 0
        For i As Integer = 0 To arr.Length - 1
            If arr(i) = element Then count += 1
        Next
        If count > arr.Length / 2 Then Return element Else Return Nothing
    End Function

I have created a simple application that reads or generates arbitrary array of numbers and find its Majority but I will leave the implementation of the application as an exercise to the reader . However you can compare your results with mine by looking at the picture below

MajorityElement

ChiMerge discretization algorithm

Chi merge is a simple algorithm that uses the chi-square statistic to discretize numeric attributes. It is a supervised, bottom-up data discretization method. It checks each pair of adjacent rows in order to determine if the class frequencies of the two intervals are significantly different. It tests the following hypothesis: that the two adjacent intervals are independent. If the hypothesis is confirmed the intervals are merged into a single interval, if not, they remain separated.

In this problem we select one of the following as an attribute: sepal length, sepal width, petal length, and petal width. Three Classes are available which are Iris-setosa, Iris-versicolor, Iris-virginica.

Chi merge algorithm is described below:

1. Set the values for minintervals and maxintervals in order not to result in a large number or small number of intervals. Choose minintervals and maxintervals such that:

2 ≤ minintervals < maxintervals

1. Look up the threshold value to use based on the significance level and the number of degrees of freedom. The threshold value can be found in statistical tables depending on two factors:

• The significance level, the higher the significance level, the higher the threshold value and the more likely the adjacent intervals will be merged.
• The number of degrees of freedom=(2-1)*([number of classes]-1) = [number of classes] -1

Here we have 3 classes thus the number of degrees of freedom is 2.

1. A) Sort the attributes into ascending numerical order.

B) Create a frequency table containing one row for each distinct attribute value and one column for each class. Enter the number of occurrences of each distinct value of the attribute for each possible class in the cells of the table. Then generate a set of distinct intervals. Set the interval lower bound equal to the attribute value (inclusive) that belongs to this interval, and set its upper bound to the attribute value (exclusive) belonging to the next interval.

C) If the number of rows = minintervals then stop otherwise go to next step.

D) For each pair of adjacent rows in the frequency table calculate e the expected frequency value for that combination from the product of row and column sums divided by the total number of occurrences in the two rows combined. The expected value is the frequency value that would be expected to occur by chance given the assumption of independence. Calculate the value of (O – E) ²/E (if E<0.5, replace E by 0.5) where O is the observed frequency value for that combination, then add the values obtained to give chi ² for that pair of adjacent rows. No chi² is calculated for the final interval because there is not one below it.

E) Chi merge selects the smallest value of chi ².This value is compared     with a threshold, the two intervals are merged based on the following condition if chi2 is less than threshold or if the number of rows > maxintervals, if the condition is met then the two intervals are merged and the attribute for the merged row to that of the first of the two consistent rows and the upper interval bound to the upper bound interval of the next   attribute. Then go to step C, but if the condition is not met then the algorithm stops.

The chi² values are calculated for the revised frequency table, chi merge proceeds iteratively in this way merging two intervals at each stage until the chi² for the remaining pairs of intervals are greater than the threshold value and the number of intervals is less than the maximum number of intervals, hence no further merging of intervals is possible and the discretisation is complete.

Check these ChiMerge powerpoint slides that visualizes the above algorithm.

Now lets take the IRIS data set, obtained from http://www.ics.uci.edu/~mlearn/MLRepository.html (UC-Irvine Machine Learning Data Repository), as a data set to be discretized. We will perform data discretization for each of the four numerical attributes using the Chi-Merge method having the stopping criteria be: max-interval = 6.

5.1,3.5,1.4,0.2,Iris-setosa
4.9,3.0,1.4,0.2,Iris-setosa
4.7,3.2,1.3,0.2,Iris-setosa
4.6,3.1,1.5,0.2,Iris-setosa
5.0,3.6,1.4,0.2,Iris-setosa
5.4,3.9,1.7,0.4,Iris-setosa
4.6,3.4,1.4,0.3,Iris-setosa
5.0,3.4,1.5,0.2,Iris-setosa
4.4,2.9,1.4,0.2,Iris-setosa
4.9,3.1,1.5,0.1,Iris-setosa
5.4,3.7,1.5,0.2,Iris-setosa
4.8,3.4,1.6,0.2,Iris-setosa
4.8,3.0,1.4,0.1,Iris-setosa

These attributes refer to Sepal Length, Sepal Width, Petal Length,Petal Width & Class respectively. Lets create two classes that i’ll call “IRISDataItem” and “IRISWorkItem”. The former will be used to store the attribute values for each when the data file is read while the latter will be used for later processing and storing the ChiSquare for each distinct attribute.

Public Class IrisDataItem

    Public Enum IRISClass As Byte
        IrisSetosa = 1
        IrisVersicolour = 2
        IrisVirginica = 3
    End Enum

    Public Property SepalLength As Single
    Public Property SepalWidth As Single
    Public Property PetalLength As Single
    Public Property PetalWidth As Single
    Public Property [Class] As IRISClass

End Class

Public Class IrisWorkItem
    Public Property Attribute As Decimal
    Public Property Frequency As Integer
    Public Property SetosaCount As Integer
    Public Property VersicolorCount As Integer
    Public Property VirginicaCount As Integer
    Public Property Start As Decimal
    Public Property [End] As Decimal
    Public Property ChiSquare As Decimal

    'Operator overloading for merging of two adjacent items
    Public Shared Operator ^(ByVal Item1 As IrisWorkItem, ByVal Item2 As IrisWorkItem) As IrisWorkItem
        Item1.SetosaCount += Item2.SetosaCount
        Item1.VirginicaCount += Item2.VirginicaCount
        Item1.VersicolorCount += Item2.VersicolorCount
        Item1.End = Item2.End
        Item2.Frequency = -1
        Return Item1
    End Operator

    'Operator overloading for calculating the ChiSqaure for two items
    Public Shared Operator &(ByVal Item1 As IrisWorkItem, ByVal Item2 As IrisWorkItem) As Decimal
        Return ChiMerge.ChiSquare(Item1, Item2)
    End Operator
End Class

Our next task is to actually read the data file and fill create a List(Of IRISDataItem) object. This can be easily done by reading the file from disc, or you can embed the file in the application resources and read from there. Here is a couple of functions that handle the two ways of reading the file:

    Private Function ReadDataFile(ByVal Path As String) As StringCollection
        If Not String.IsNullOrWhiteSpace(Path) Then
            Dim reader As New IO.StreamReader(Path)
            Dim data As New StringCollection
            While Not reader.EndOfStream
                data.Add(reader.ReadLine)
            End While
            reader.Close()
            Return data
        Else
            Throw New Exception("Invalid IRIS File Path!")
        End If
    End Function

    Private Function ReadDataFileFromResource() As StringCollection
        Dim strFile As String = System.Text.Encoding.ASCII.GetString(My.Resources.iris)
        Dim reader As New IO.StringReader(strFile)
        Dim data As New StringCollection
        While reader.Peek > 0
            data.Add(reader.ReadLine)
        End While
        reader.Close()
        Return data
    End Function

    Public Shared Function CreateIrisDataItems(ByRef StringItems As StringCollection) As List(Of IrisDataItem)
        If StringItems IsNot Nothing AndAlso StringItems.Count > 0 Then
            Dim items As New List(Of IrisDataItem)
            For Each i In StringItems
                If Not String.IsNullOrWhiteSpace(i) Then
                    Dim item As New IrisDataItem
                    Dim strItem() As String = i.Split(",")
                    item.SepalLength = strItem(0)
                    item.SepalWidth = strItem(1)
                    item.PetalLength = strItem(2)
                    item.PetalWidth = strItem(3)
                    Select Case strItem(4)
                        Case "Iris-setosa"
                            item.Class = IrisDataItem.IRISClass.IrisSetosa
                        Case "Iris-versicolor"
                            item.Class = IrisDataItem.IRISClass.IrisVersicolour
                        Case "Iris-virginica"
                            item.Class = IrisDataItem.IRISClass.IrisVirginica
                    End Select
                    items.Add(item)
                End If
            Next
            Return items
        End If
        Return Nothing
    End Function

    Public Shared Function CreateIrisWorkItems(ByVal AttributeIndex As Integer, ByVal DataItems As List(Of IrisDataItem)) As List(Of IrisWorkItem)
        Dim workItems As New List(Of IrisWorkItem)
        DataItems.ForEach(Sub(dataItem)
                              Dim workItem As IrisWorkItem = Nothing
                              Dim value As Decimal
                              Select Case AttributeIndex
                                  Case 0
                                      workItem = workItems.Find(Function(w) w.Attribute = dataItem.SepalLength)
                                      value = dataItem.SepalLength
                                  Case 1
                                      workItem = workItems.Find(Function(w) w.Attribute = dataItem.SepalWidth)
                                      value = dataItem.SepalWidth
                                  Case 2
                                      workItem = workItems.Find(Function(w) w.Attribute = dataItem.PetalLength)
                                      value = dataItem.PetalLength
                                  Case 3
                                      workItem = workItems.Find(Function(w) w.Attribute = dataItem.PetalWidth)
                                      value = dataItem.PetalWidth
                              End Select
                              If workItem IsNot Nothing Then
                                  workItem.Frequency += 1
                              Else
                                  workItem = New IrisWorkItem
                                  workItem.Frequency = 1
                                  workItems.Add(workItem)
                              End If
                              workItem.Attribute = value
                              Select Case dataItem.Class
                                  Case IrisDataItem.IRISClass.IrisSetosa
                                      workItem.SetosaCount += 1
                                  Case IrisDataItem.IRISClass.IrisVersicolour
                                      workItem.VersicolorCount += 1
                                  Case IrisDataItem.IRISClass.IrisVirginica
                                      workItem.VirginicaCount += 1
                              End Select
                          End Sub)
        Return workItems.OrderBy(Function(w) w.Attribute).ToList
    End Function

Having the data ready in our hands, we can now proceed to implement the ChiSquare function which is basically an implementation of the formula:

Where :

k = number of classes

A_ij = number of samples in ith interval, jth class

E_ij = expected frequency of A_ij = (R_i * C_j) / N

R_i = number of samples in ith interval

C_j = number of samples in jth class

N = total number of samples on the two intervals

If Eij=<0.5 then set Eij to 0.5

    Public Shared Function ChiSquare(ByVal Item1 As IrisWorkItem, ByVal Item2 As IrisWorkItem) As Decimal
        ChiSquare = 0
        Dim table(,) As Integer = {{Item1.SetosaCount, Item1.VersicolorCount, Item1.VirginicaCount}, {Item2.SetosaCount, Item2.VersicolorCount, Item2.VirginicaCount}}
        Dim rowSums() As Decimal = {Item1.SetosaCount + Item1.VersicolorCount + Item1.VirginicaCount, Item2.SetosaCount + Item2.VersicolorCount + Item2.VirginicaCount}
        Dim columnSums() As Decimal = {Item1.SetosaCount + Item2.SetosaCount, Item1.VersicolorCount + Item2.VersicolorCount, Item1.VirginicaCount + Item2.VirginicaCount}
        Dim totalSum As Decimal = rowSums.Sum
        Dim E As Decimal = 0
        For i As Integer = 0 To table.GetLength(0) - 1
            For j As Integer = 0 To table.GetLength(1) - 1
                E = rowSums(i) * columnSums(j) / totalSum
                If E > 0 Then ChiSquare += ((table(i, j) - E) ^ 2) / IIf(E < 0.5, 0.5, E)
            Next
        Next
    End Function

One last thing to do before we implement Chi-Merge algorithm is to setup the initial interval bounds and prepare them. This is easily achieved by taking the attribute value itself to be the lower bound of the interval and the next attribute value to be the upper bound of the interval. Below is an easy way of doing this:

    Public Shared Sub CreateIntervalBounds(ByVal Items As List(Of IrisWorkItem))
        For i As Integer = 0 To Items.Count - 2
            Items(i).Start = Items(i).Attribute
            Items(i).End = Items(i + 1).Attribute
        Next
        Items(Items.Count - 1).Start = Items(Items.Count - 1).Attribute
        Items(Items.Count - 1).End = Items(Items.Count - 1).Attribute + 1
    End Sub

Finally, we are ready to implement the Chi-Merge algorithm. One thing to note about the below implementation is the use of LINQ to Objects and Lambda Expressions to filter out tuples that need to be dropped and merged. So you could probably that the code below will compile only using Visual Studio 2010 and .Net Framework 4.0. I haven’t tested it on Visual Studio 2008 Sp1 but if you do let me know if it compiles well.

    Public Sub Compute(ByRef Items As List(Of IrisWorkItem))
        For i As Integer = 0 To Items.Count - 2
            Items(i).ChiSquare = Items(i) & Items(i + 1) 'calculate chiSquare
        Next
        Dim minChiSquare As Decimal = Items.Take(Items.Count - 1).Min(Function(x) x.ChiSquare)
        If minChiSquare < THRESHOLD Then
            Dim i As Integer = 0
            While i < Items.Count
                If Items(i).ChiSquare = minChiSquare Then
                    Items(i) ^= Items(i + 1) 'merge
                    Dim j As Integer = i
                    i += 1
                    While i < Items.Count - 1 AndAlso Items(i).ChiSquare = minChiSquare
                        Items(j) ^= Items(i + 1) 'merge
                        i += 1
                    End While
                End If
                i += 1
            End While
            Items.RemoveAll(Function(x) x.Frequency = -1)
            If Items.Count >= MIN_INTERVAL Then Compute(Items)
        Else
            If Items.Count > MAX_INTERVAL Then
                THRESHOLD += 0.5
                Compute(Items)
            End If
        End If
    End Sub

Once you run the algorithm, you can compare the obtained intervals and split points with the ones found on the internet. Below are my final results compared to the results on http://sci2s.ugr.es/keel/pdf/algorithm/congreso/1992-Kerber-ChimErge-AAAI92.pdf