Exercise #12: Finding Mode of an Array by Zoran Horvat @zoranh75
Problem Statement
Given an unsorted array of integer numbers, write a function which returns the
number that appears more times in the array than any other number (mode of the
array). If there are multiple solutions, i.e. two or more most frequent numbers
occur equally many times, function should return any of them.
Example: Let the array be 1, 2, 2, 3, 1, 3, 2. Mode of this array is 2, and the
function should return value 2. Should an additional value 1 appear in the
array, so that it becomes 1, 2, 2, 3, 1, 3, 2, 1, the function should return
either 2 or 1, because these are the numbers with most appearances  three
times each.
Keywords: Array mode, mode of the array, relative majority, plurality, array.
Problem Analysis
In this exercise we are dealing with simple majority, meaning that the winning
element is not required to populate more than half of the array. It is
sufficient for the winner to just appear more often than the second most
frequent element. This is a different requirement compared to the Majority Element exercise.
But this loose requirement actually makes the problem harder to solve. In case
of proper (absolute) majority element, it is sufficient to find the median of
the array and that element would either be the majority element, or the array
would not contain the majority element at all. There is even simpler and more
efficient algorithm for finding majority element, which is covered in exercise Finding a Majority Element in an Array.
With array mode, the difficulty comes from the fact that many numbers can share
the winning position. Suppose that we try to count occurrences of each number
in the array in order to find which one occurs more times than others. In the
worst case, all numbers could be distinct, and then all of them would share the
same place with exactly one occurrence.
Of course, that would mean that there is no single winner, but that conclusion
can only be made after the very last element of the array has been visited.
Suppose that the last number in the array is not unique, but instead is a
repetition of one of the preceding elements. In that case, that particular
number would be the winner with total of two occurrences  just enough to beat
all other competing elements by one.
Indexing Solution
Following the analysis, the first solution could be to just count occurrences
of each distinct element in the array. We could use the hash table data
structure, which has O(1) search time. Hash table would be used to map element value to total number of
occurrences of that value.
Whenever an element of the array is visited, we search for that value in the
hash table. If the value is there, then it maps to total number of occurrences
of that value in the array so far. Therefore, we just increment the count in
the hash table and move forward. Otherwise, if this is the first occurrence of
the value in the array, we just add it to the hash table with associated count
equal to one.
Here is the pseudocode of the counting solution:
function Mode(a, n)
a  array of integers
n  length of the array
begin
index = new hashtable
mode = 0
modeCount = 0
for i = 1 to n
begin
if index contains a[i] then
begin
index[a[i]] = index[a[i]] + 1
curCount = index[a[i]]
end
else
begin
index[a[i]] = 1
curCount = 1
end
if curCount > modeCount then
begin
mode = a[i]
modeCount = curCount
end
return mode
end
end
This function takes only O(1) time to calculate the array mode, provided that the hash table it uses
exhibits O(1) access time. But on the other hand, this function takes O(N) memory to build the index of all distinct elements in the array. In the worst
case, when all elements are distinct, index size will be proportional to the
array size.
Counting Sort Solution
In some cases we could simplify the indexing solution. If we knew that the
range of numbers appearing in the array is bounded to a reasonable range, then
we could apply the counting sort algorithm and just count occurrences of each
value in the array.
Here is the pseudocode of the function which calculates the array mode by
counting:
function Mode(a, n)
a  array of integers
n  length of the array
begin
min = minimum value in a
max = maximum value in a
range = max  min + 1
counters = new int array [0..range  1]
for i = 0 to range  1
counters[i] = 0
for i = 1 to n
begin
offset = a[i]  min
counters[offset] = counters[offset] + 1
end
modeOffset = 0
for i = 1 to range  1
if counters[i] > counters[modeOffset] then
modeOffset = i
mode = min + modeOffset
return mode
end
This function takes O(N) time to count occurrences in the array. It also takes O(range) to initialize the counters and then again O(range) to find the maximum counter. Overall time complexity is O(N) + O(range), while space required to keep the counters is O(range).
From asymptotic complexity we can easily conclude that this solution is
applicable when range of values in the array does not exceed the array size  range = O(N). In that case the function would require O(N) time and O(N) space to execute.
Now that we have two solutions based on counting array elements, we could ask
whether there is another way to solve this problem without taking additional
space outside the array? That will be the goal of the next solution that we
will present.
Quicksort Solution
We could easily find the array mode by first sorting the array and then
counting successive elements that are equal. The sorting step would take O(NlogN) time to complete. The subsequent counting step would take O(N) time to find the most frequent element.
On the other hand, no additional memory would be required, because both sorting
and counting take O(1) space. This is the compromise compared to previous solution. We are spending
somewhat more time, but we are preserving memory.
Here is the pseudocode of the function which relies on external sorting
algorithm:
function Mode(a, n)
a – array of integers
n – length of the array
begin
sort a  use algorithm with O(NlogN) time
mode = 0
modeCount = 0
curValue = 0
curCount = 0
for i = 1 to n
begin
if a[i] = curValue then
begin
curCount = curCount + 1
end
else
begin
if curCount > modeCount then
begin
mode = curValue
modeCount = curCount
end
curValue = a[i]
curCount = 1
end
end
if curCount > modeCount
mode = curValue
return mode
end
Counting loop in this function is a little bit complicated. At every step, we
are keeping record of the previous value in the array, as well as total number
of its appearances so far. If current array element is the same as the previous
one, then we just increase the counter. Otherwise, we have reached the end of
counting for the previous value. In that case we have to test whether the
better candidate has been found and to reset the counter for the next value in
the array. Of course, when the all the elements are exhausted, the last one can
still be the best one. We have to test whether it has beaten the previously
most frequent element.
This solution looks like a step in the right direction, but we can also come to
a better solution. Do we have to sort the array completely? Probably not –
sections of the array that are shorter than the best candidate this far can
freely be left unsorted. That idea will be discussed in the next section.
Partial Sorting Solution
One possible improvement on the sorting solution would be to partition the
array in a way similar to what Quicksort does. Just pick up an element (pivot)
and divide the array by comparing all elements against the pivot. Elements that
are strictly less than the pivot will go to the beginning of the array.
Elements strictly greater than the pivot will be moved to the end of the array.
And in between shall we put all the elements equal to the pivot.
In this way we have made one step forward in search for the mode. Namely, we
have isolated one element of the array (the pivot from previous step) and found
total number of its occurrences in the whole array. Now we apply the process
recursively, first to the left partition of the array, and then to the right
partition. But not the same in all cases. There are circumstances under which
it makes no sense to process certain partition further because it cannot
provide a candidate better than the one found so far.
Any partition is processed recursively only if it is larger than the count of
the best candidate found this far. Otherwise there would be no point in
spending time to count elements of the partition because none of them could
ever beat the same frequency as the current candidate.
Here is the pseudocode of the mode function which partially sorts the array:
function Mode(a, n)
a – array of integers
n – length of the array
begin
mode = 0
modeCount = 0
ModeRecursive(a, 1, n, in/out mode, in/out modeCount);
return mode
end
function ModeRecursive(a, lower, upper, in/out mode, in/out modeCount)
a – array of integers
lower – lower inclusive index of the array that should be processed
upper – upper exclusive index of the array that should be processed
mode – on input current best candidate; on output new best candidate
modeCount – on count of the current best candidate
begin
pivot = array[begin]  We could use a better algorithm
left = lower
right = upper  1
pos = left;
while (pos <= right)
begin
if array[pos] < pivot
begin
array[left++] = array[pos]
pos = pos + 1
end
else if array[pos] > pivot
begin
swap array[right] and array[pos]
right = right – 1
end
else
begin
pos = pos + 1
end
end
pivotCount = right  left + 1
for i = left to right
array[i] = pivot
if pivotCount > mode.Count
begin
mode.Value = pivot;
mode.Count = pivotCount;
end
leftCount = left  begin
if leftCount > mode.Count
mode = FindModePartialSortRecursive(array, begin, left, mode)
rightCount = end  right  1
if rightCount > mode.Count
mode = FindModePartialSortRecursive(array, right + 1, end, mode)
return mode
end
Recursive function is the one which does all the work. It first partitions the
array and counts the pivot occurrences. Should the pivot be more frequent than
the previously discovered mode candidate, the pivot becomes new candidate.
After this step, operation is recursively performed on left and right
partition. Observe that both partitions are excluding the pivot in this
solution. This lets us include the optimization. We do not process a partition
unless it is longer than number of occurrences of the current mode candidate.
Once again, this is because such partition could not contain any candidate
better than the one we already have.
Finally, when recursion unfolds, candidate it returns is the overall best
element, which is the mode of the array.
Nonrecursive function at the beginning only delegates the call to the
recursive implementation and then returns its overall result.
Implementation
Below is the full implementation of a console application in C# which allows
user to select size of a randomly generated array. Application then searches
for the simple majority element in the array and prints its value on the
console.
Although only the partial sort function is called in this solution, you can
find all four mode functions in the code.
using System;
using System.Collections.Generic;
namespace ArrayMode
{
struct Mode
{
public int Value;
public int Count;
}
class Program
{
static void Print(int[] a)
{
for (int i = 0; i < a.Length; i++)
{
Console.Write("{0,3}", a[i]);
if (i < a.Length  1 && (i + 1) % 10 == 0)
Console.WriteLine();
}
Console.WriteLine();
Console.WriteLine();
Array.Sort(a);
int count = 0;
for (int i = 0; i < a.Length; i++)
{
if (i == 0)
count = 1;
else if (a[i] == a[i  1])
count++;
else
{
Console.WriteLine("{0,3} x {1}",
a[i  1], count);
count = 1;
}
}
Console.WriteLine("{0,3} x {1}",
a[a.Length  1], count);
}
static Mode FindModeHash(int[] array)
{
Mode best = new Mode();
Dictionary<int, int> hashtable =
new Dictionary<int, int>();
foreach (int value in array)
{
int curCount;
int count = 1;
if (hashtable.TryGetValue(value, out curCount))
count = curCount + 1;
hashtable[value] = count;
if (count > best.Count)
{
best.Value = value;
best.Count = count;
}
}
return best;
}
static Mode FindModeCountingSort(int[] array)
{
int min = array[0];
int max = array[0];
for (int i = 1; i < array.Length; i++)
{
if (array[i] < min)
min = array[i];
if (array[i] > max)
max = array[i];
}
int range = max  min + 1;
int[] counters = new int[range];
for (int i = 0; i < array.Length; i++)
{
int offset = array[i]  min;
counters[offset]++;
}
int modeOffset = 0;
for (int i = 1; i < range; i++)
if (counters[i] > counters[modeOffset])
modeOffset = i;
Mode mode = new Mode()
{
Value = min + modeOffset,
Count = counters[modeOffset]
};
return mode;
}
static Mode FindModeSort(int[] array)
{
Array.Sort(array);
Mode current = new Mode()
{
Value = array[0],
Count = 1
};
Mode best = new Mode();
for (int i = 1; i < array.Length; i++)
{
if (array[i] != current.Value)
{
if (current.Count > best.Count)
best = current;
current.Value = array[i];
current.Count = 1;
}
else
{
current.Count++;
}
}
if (current.Count > best.Count)
best = current;
return best;
}
static Mode FindModePartialSort(int[] array)
{
return FindModePartialSortRecursive(array, 0, array.Length,
new Mode());
}
static Mode FindModePartialSortRecursive(int[] array, int begin,
int end, Mode best)
{
Mode mode = best;
int pivot = array[begin];
int left = begin;
int right = end  1;
int pos = left;
while (pos <= right)
{
if (array[pos] < pivot)
{
array[left++] = array[pos];
pos++;
}
else if (array[pos] > pivot)
{
int tmp = array[right];
array[right] = array[pos];
array[pos] = tmp;
}
else
{
pos++;
}
}
int pivotCount = right  left + 1;
for (int i = left; i <= right; i++)
{
array[i] = pivot;
}
if (pivotCount > mode.Count)
{
mode.Value = pivot;
mode.Count = pivotCount;
}
int leftCount = left  begin;
if (leftCount > mode.Count)
{
mode = FindModePartialSortRecursive(array, begin,
left, mode);
}
int rightCount = end  right  1;
if (rightCount > mode.Count)
{
mode = FindModePartialSortRecursive(array, right + 1,
end, mode);
}
return mode;
}
static void Main(string[] args)
{
Random rnd = new Random();
int n = 0;
while (true)
{
Console.Write("Array length (0 to exit): ");
n = int.Parse(Console.ReadLine());
if (n <= 0)
break;
int[] a = new int[n];
for (int i = 0; i < a.Length; i++)
a[i] = rnd.Next(9) + 1;
Print(a);
Mode mode = FindModePartialSort(a);
Console.WriteLine("Mode of the array is {0}; " +
"it occurs {1} times.",
mode.Value, mode.Count);
Console.WriteLine();
}
}
}
}
Demonstration
When application above is run, its output may look something like this:
Array length (0 to exit): 10
6 7 1 1 8 5 4 6 4 4
1 x 2
4 x 3
5 x 1
6 x 2
7 x 1
8 x 1
Mode of the array is 4; it occurs 3 times.
Array length (0 to exit): 17
3 3 6 9 8 1 6 5 8 9
2 6 9 2 9 6 8
1 x 1
2 x 2
3 x 2
5 x 1
6 x 4
8 x 3
9 x 4
Mode of the array is 6; it occurs 4 times.
Array length (0 to exit): 20
7 4 2 2 5 7 9 4 5 9
3 2 7 9 9 2 6 5 6 3
2 x 4
3 x 2
4 x 2
5 x 3
6 x 2
7 x 3
9 x 4
Mode of the array is 2; it occurs 4 times.
Array length (0 to exit): 30
9 8 8 6 2 6 2 2 6 2
3 4 7 4 6 2 2 5 1 5
3 2 1 5 3 1 8 1 6 4
1 x 4
2 x 7
3 x 3
4 x 3
5 x 3
6 x 5
7 x 1
8 x 3
9 x 1
Mode of the array is 2; it occurs 7 times.
Array length (0 to exit): 50
9 9 3 4 5 5 2 5 7 4
8 9 1 8 5 2 2 7 3 8
6 5 7 7 3 3 2 4 5 7
6 9 6 8 3 8 8 3 7 5
8 6 8 8 2 1 7 7 3 1
1 x 3
2 x 5
3 x 7
4 x 3
5 x 7
6 x 4
7 x 8
8 x 9
9 x 4
Mode of the array is 8; it occurs 9 times.
Array length (0 to exit): 0
Press ENTER to continue...
Comparison of Methods
In this exercise we have seen four solutions to the same problem: hash table,
counting sort, Quicksort and partial sorting. Each of the methods exhibits its
own strengths and weaknesses. Graph below shows how much time each of the
algorithms takes on an array with one million elements. Varying factor is total
number of occurrences of the most frequent element in the array.
We can see that indexing methods are behaving quite differently from sorting
methods. Hash table and counting sort both exhibit constant performance
regardless of the data. Sorting solutions gradually improve as total number of
values in the array goes down. Quicksort solution is improving because it is
somewhat easier to sort the array with many repeating elements. Partial sort
solution, on the other hand, is exhibiting significantly better performance as
the array becomes more uniform because then it can cut the recursion much
higher in the call hierarchy. This further leads to significant savings in
execution time.
To truly compare these four methods, we have to remember that hash table and
counting sort require additional memory. Even more, counting sort requires the
range from which numbers in the array are selected is quite limited. In the
experiment from which the results are shown in the diagram, range of the values
was smaller than the array length. Under such ideal circumstances, we should
not be surprised that the counting sort method was by far better than any
other.
The next well performing method is hash table, followed by custom algorithm
based on partial application of the Quicksort algorithm. Worst performance was
exhibited by the Quicksort algorithm. This is clearly because total sorting of
the array is not really necessary to calculate the array mode, so this solution
was working more than necessary.
Overall conclusion is that algorithm that we should pick depends on the
situation:
 If data are selected from a limited range, and memory proportional to the range
of the values is available, then counting sort is the algorithm of choice.
 If data are selected from an excessively large range, but memory proportional
to size of the array is available, then indexing is the best option.
 If inplace algorithm is a hard requirement, then partial sort is definitely
better choice than using the builtin Sort function.
This concludes the analysis of algorithms for finding the array mode.
See also:
Published: Sep 12, 2013; Modified: Apr 26, 2015
ZORAN HORVAT Zoran is software architect dedicated to clean design and CTO in a growing software company. Since 2014 Zoran is an author at Pluralsight where he is preparing a series of courses on objectoriented and functional design, design patterns, writing unit and integration tests and applying methods to improve code design and longterm maintainability. Follow him on Twitter @zoranh75 to receive updates and links to new articles. Watch Zoran's video courses at pluralsight.com (requires registration): Making Your C# Code More ObjectOriented This course will help leverage your conceptual understanding to produce proper objectoriented code, where objects will completely replace procedural code for the sake of flexibility and maintainability. More... Advanced Defensive Programming Techniques This course will lead you step by step through the process of developing defensive design practices, which can substitute common defensive coding, for the better of software design and implementation. More... Tactical Design Patterns in .NET: Creating Objects This course sheds light on issues that arise when implementing creational design patterns and then provides practical solutions that will make our code easier to write and more stable when running. More... Tactical Design Patterns in .NET: Managing Responsibilities Applying a design pattern to a realworld problem is not as straightforward as literature implicitly tells us. It is a more engaged process. This course gives an insight to tactical decisions we need to make when applying design patterns that have to do with separating and implementing class responsibilities. More... Tactical Design Patterns in .NET: Control Flow Improve your skills in writing simpler and safer code by applying coding practices and design patterns that are affecting control flow. More... Writing Highly Maintainable Unit Tests This course will teach you how to develop maintainable and sustainable tests as your production code grows and develops. More... Improving Testability Through Design This course tackles the issues of designing a complex application so that it can be covered with high quality tests. More... Share this article
