# Find elements occurring even number of times in an integer array

I came across an interview question, which is like this:

You have an array of integers, such that each integer is present an odd number of time, except 3 of them. Find the three numbers.

I tried implementing it using a HashMap<Integer, Integer>, by mapping each integer element to a number that would identify if that number occurs even number of times. I used the XOR logic for that. 5 ^ 5 = 0, and 5 ^ anyNumberOtherThan5 != 0.

Here's my code:

public static void findEvenOccurringNum(int[] arr) {
Map<Integer, Integer> map = new HashMap<Integer, Integer>();

for (int num: arr) {
if (map.containsKey(num)) {
map.put(num, map.get(num) ^ 1);
} else {
map.put(num, 1);
}
}

for (Entry<Integer, Integer> entry: map.entrySet()) {
if (entry.getValue() == 0) {
System.out.println(entry.getKey());
}
}
}


What I don't like about this solution is the 2nd iteration. That would iterate over all the unique numbers in the array. Can there be any possible improvement here? May be a solution that avoids using a map? Or is it the best I can do?

Can there be any possible improvement here?

There is always a way to improve things. But, sometimes improvements are opinion based... ;-)

In general I dislike solutions that use generous amounts of AutoBoxing (int to Integer, boolean to Boolean, etc.). In your case, the actual work that Java does (and memory it uses) to manage the objects required to solve the solution is far more than the actual work of the problem itself.

I believe the following solution is faster, smaller, and all those things, but it may need a little explaining. I also like that it returns the values in sorted order, and as an int[] array which is the same format as the input.

It's only heap-space data usage is for the copy of the input array and the small result array... (unlike all the Integers and Booleans and Map.Objects of the Collection-based solutions.

public static final int[] findDuplicates(final int[] data) {
// take a copy of the data.
int[] sorted = Arrays.copyOf(data, data.length);
// sort it. This is O(n log(n)) which will be the bulk of our time.
Arrays.sort(sorted);

// now scan the sorted data for even-numbered sequences of values.
int len = 0;
boolean even = false;
// sorted[0] is our first sequence, and currently it is not even.
// note that the scan starts at position 1.
for (int i = 1; i < sorted.length; i++) {
if (sorted[i] == sorted[len]) {
// this element is the same as the sequence, so 'count' the even-ness.
even = !even;
} else {
// this element is different to our sequence, it's a new sequence.
if (even) {
// the old sequence had an even count, so we 'preserve' it.
len++;
}
// move the value of the new sequence to the 'beginning'.
sorted[len] = sorted[i];
// and the new sequence is odd.
even = false;
}
}
if (even) {
// the last sequence was even, we preserve it.
len++;
}
// return the first part of the array, which contains the even sequence values.
return Arrays.copyOf(sorted, len);
}

• Is it really faster, especially for long inputs? Any general-purpose sort will be at least O(n log n). Both the original and my Set solution are O(n). – 200_success Dec 7 '13 at 17:04
• @200_success - I just plotted the performance metrics for our systems... FYI.... and, no, I don't know why (I can guess it's the HashSets...but) your solution is not O(n) for sparse data – rolfl Dec 7 '13 at 19:38

## Wiki Answer - discussion of scalability.

After some investigation I thought it would be interesting to publish some of the performance results I have seen.... there are some interesting observations.

I have taken two algorithms, both on this page, the 200_Success one, and the rolfl one. I have run both algorithms through two types of tests, and for each type, I ran it through data of different sizes. I measured the actual performance.

The two types of test are 'data complexity' related. The first test I call 'sparse', and there are very few duplicates of data. The second set of data is 'freq' which has many, many duplicates. The two data sets are constructed as (1 << 18 is 256K elements and runs in an 'about-right' time on my computer):

    int[] sparse = new int[1<<18];
Random rand = new Random(1); // repeatable 'random' numbers
for (int i = 0; i < sparse.length; i++) {
sparse[i] = rand.nextInt(sparse.length);
}

int[] freq = new int[1<<18];
for (int i = 0; i < freq.length; i++) {
freq[i] = rand.nextInt(10);
}


So, two different datasets.... and then, I use the data in 5 different sizes.... each size is half the size of the next size.... so, I have data of 16K, 32K, 64K, 128K, and 256K.

I then run each of the algorithms a number of times for each data size. The number of times depends on the size of the data.... The bigger the data is, the fewer times I run the loop, and, as the data size doubles, I halve the number of iterations.

iterations   256    128    64     32      16
data size  16384  32768 65536 131072  262144


In theory, in a perfectly scalable system, the data-size and iteration combinations should result in constant-time execution....

Then I timed the actual execution times, and here are the results (on a warmed up system).

RLsparse,130.632,164.163,225.850,271.158,289.436,322.981,343.293,360.966,390.827,412.959
RLfreq,60.455,66.364,53.215,76.341,82.661,85.065,88.740,87.201,89.798,87.361
2Ssparse,157.926,184.521,214.407,242.943,256.200,286.686,352.467,438.048,420.164,666.043
2Sfreq,152.263,155.817,158.332,160.107,158.856,157.523,159.182,158.238,158.978,157.142
datasize,512,1024,2048,4096,8192,16384,32768,65536,131072,262144
iterations,8192,4096,2048,1024,512,256,128,64,32,16


Plotting this as a graph, using the datasize as the X axis, it looks like:

The 'dip' in the performance of the 2Ssparse 'curve' is consistent... there must be some anomoly that happens in the data of that size that 'wins' for whatever reason.

The 'obvious' observations are:

1. both algorithms like lots of duplicate values.
2. for high-duplicate values the 200_Success algorithm is practically O(n) and it appears the HashMap is O(1) as stated.
3. for sparse-duplicate values the 200_Success scales worse than the rolfl version, and, although it started faster, for larger data sets it quickly degrades to be slower.
4. for sparse-duplicate values the rolfl version is faster (runs in 2/3 of the time), but is slightly degrading in performance.... I expect that for all reasonable data sizes it will be faster.....

## EDIT:

After a request for 'medium' amounts of duplication (the random numbers are taken from a pool of 1024 instead of just 10):

RLsparse,124.211,147.412,216.475,273.934,293.235,330.671,346.458,372.003,399.730,407.985
RLfreq,42.496,41.446,53.732,58.305,66.281,65.889,64.953,66.078,73.901,71.291
RLmedium,122.813,179.644,218.106,253.908,272.583,259.016,232.671,226.121,220.121,207.180
2Ssparse,155.091,180.669,205.877,236.890,254.193,288.193,351.495,450.381,426.626,665.759
2Sfreq,172.426,173.460,173.988,174.462,175.049,174.477,175.384,172.826,174.230,175.976
2Smedium,162.969,210.496,232.372,235.070,229.555,222.412,219.141,219.358,218.502,221.412


And the corresponding plot is:

## EDIT2:

The HashSet based implementation has been revised to include hints on th size of the HashSets. The revised performance results are:

RLsparse,113.848,156.549,195.834,264.010,280.561,314.784,332.941,360.607,387.728,401.440
RLfreq,50.688,45.955,61.013,59.733,74.000,68.274,69.208,70.487,76.905,74.185
RLmedium,116.118,177.327,223.468,258.029,275.395,263.365,234.839,225.775,221.798,210.071
2Ssparse,127.956,160.641,159.736,171.221,180.202,195.650,265.159,328.633,270.251,369.847
2Sfreq,154.762,154.865,157.780,157.265,158.705,159.562,160.342,161.276,160.924,163.296
2Smedium,128.366,152.183,187.613,198.995,270.550,198.450,200.526,202.244,205.382,207.368
datasize,512,1024,2048,4096,8192,16384,32768,65536,131072,262144
iterations,8192,4096,2048,1024,512,256,128,64,32,16


And the corresponding plot is (Note the scale is the same as other plots, max 700ms):

Here is the code I used to generate the above graph (the actual algorithms not included since they are above...).

This code is not neat, and please do not review it .... ;-) But feel free to make any changes you like to improve it... I know it is ugly... but it works.

package dupcnt;

import java.util.Arrays;
import java.util.HashSet;
import java.util.Random;
import java.util.Set;

public class CompareDups {

public static void main(String[] args) {

System.out.println("Building Data");

int[] sparse = new int[1<<18];
int[] freq = new int[sparse.length];
int[] medium = new int[sparse.length];
Random rand = new Random(1); // repeatable 'random' numbers
for (int i = 0; i < sparse.length; i++) {
sparse[i] = rand.nextInt(sparse.length);
}
for (int i = 0; i < freq.length; i++) {
freq[i] = rand.nextInt(10);
}
for (int i = 0; i < medium.length; i++) {
medium[i] = rand.nextInt(1024);
}

System.out.println("Built data " + sparse.length);

final int dpoints = 10;
int[] iterations = new int[dpoints];
for (int i = 0; i < iterations.length; i++) {
iterations[i] = 1 << dpoints + 3 - i;
}

for (int i = 0; i < 10; i++) {
long[][] data = new long[8][dpoints];
for (int dp = 0; dp < dpoints; dp++) {
int[] input = null;
input = Arrays.copyOf(sparse, sparse.length >>> dpoints - 1 - dp);
data[0][dp] = processRL(input, iterations[dp]);
data[3][dp] = process2S(input, iterations[dp]);
input = Arrays.copyOf(freq, freq.length >>> dpoints - 1 - dp);
data[1][dp] = processRL(input, iterations[dp]);
data[4][dp] = process2S(input, iterations[dp]);
input = Arrays.copyOf(medium, medium.length >>> dpoints - 1 - dp);
data[2][dp] = processRL(input, iterations[dp]);
data[5][dp] = process2S(input, iterations[dp]);
data[6][dp] = input.length;
data[7][dp] = iterations[dp];
}
String[] series = {"RLsparse", "RLfreq", "RLmedium", "2Ssparse", "2Sfreq", "2Smedium", "datasize", "iterations"};
for (int s = 0; s < series.length; s++) {
StringBuilder sb = new StringBuilder();
sb.append(series[s]);
for (long l : data[s]) {
if (s < 6) {
sb.append(",").append(String.format("%.3f", l/1000000.0));
} else {
sb.append(",").append(String.format("%d", l));
}
}
System.out.println(sb.toString());
}
}
}

private static long processRL(final int[] input, final int iterations) {
System.gc();
int cnt = 0;
long nanos = System.nanoTime();
for (int i = 0; i < iterations; i++) {
cnt += findDuplicates(input).length;
}
if (cnt < 0) {
throw new IllegalStateException("This is just to keep things from being optimized out.");
}
return System.nanoTime() - nanos;
}

private static long process2S(final int[] input, final int iterations) {
System.gc();
int cnt = 0;
long nanos = System.nanoTime();
for (int i = 0; i < iterations; i++) {
cnt += findEvenOccurrences(input).size();
}
if (cnt < 0) {
throw new IllegalStateException("This is just to keep things from being optimized out.");
}
return System.nanoTime() - nanos;
}

}

• What about upping the number of entries for the 'duplicates' run? So, what happens if you cut the number of duplicates in half, but up the number of entries? Because the sparse one will potentially have only one of each number... – Clockwork-Muse Dec 8 '13 at 3:54
• @Clockwork-Muse Like adding a series for say 'medium' where the random numbers come from a range of about 256 instead of 10 or 256K? – rolfl Dec 8 '13 at 4:10
• Yeah, something like that. – Clockwork-Muse Dec 8 '13 at 4:12
• @Clockwork-Muse - are you asking out of curiosity or do you suspect something (because something odd has happened....). – rolfl Dec 8 '13 at 4:20
• Mostly, I'm curious as to the 'break even' point for @200's version; I'm a little worried about it thrashing during item shifts in the backing arrays. – Clockwork-Muse Dec 8 '13 at 4:36

Since your map values toggle between 0 and 1, you could just as well use a Map<Integer, Boolean> instead.

Usually, it is good practice not to mingle your output routines into your logic. In the absence of any specific instructions from your interviewer, it would have been better to return the found values rather than print them. A reasonable return type might be Set<Integer>, since the elements to be returned will be distinct, and the order is unspecified. Another possible return type would be int[], since it is similar to the array that was passed to you.

Since we want to return a set, it would be nice if we could build the set as we go along. Building the set of numbers with odd occurrences is easy: try to add a number to a set; if it was already there (Set.add() returns false), remove it instead. The trick is that removing an entry from the odd-occurrences set means you should transfer it to the even-occurrences set.

public static Set<Integer> findEvenOccurrences(int[] arr) {
// The HashSet capacity should scale with the size of the input
Set<Integer> oddOccurrences = new HashSet<Integer>(arr.length),
evenOccurrences = new HashSet<Integer>(arr.length / 2);
for (int num : arr) {

• #winces# - except, you're going to be swapping elements between them many times (assuming the odd entries are present more than one time); if the set condenses/shifts entries each add/remove, this is going to be less efficient than it could. Personally, I agree with the Map<Integer, Boolean> idea. – Clockwork-Muse Dec 8 '13 at 3:43