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The problem I've seen is as bellow, anyone has some idea on it?
http://judgecode.com/problems/1011
Given a permutation of integers from 0 to n - 1, sorting them is easy. But what if you can only swap a pair of integers every time?
Please calculate the minimal number of swaps
One classic algorithm seems to be permutation cycles (https://en.wikipedia.org/wiki/Cycle_notation#Cycle_notation). The number of swaps needed equals the total number of elements subtracted by the number of cycles.
For example:
1 2 3 4 5
2 5 4 3 1
Start with 1 and follow the cycle:
1 down to 2, 2 down to 5, 5 down to 1.
1 -> 2 -> 5 -> 1
3 -> 4 -> 3
We would need to swap index 1 with 5, then index 5 with 2; as well as index 3 with index 4. Altogether 3 swaps or n - 2. We subtract n by the number of cycles since cycle elements together total n and each cycle represents a swap less than the number of elements in it.
Here is a simple implementation in C for the above problem. The algorithm is similar to User גלעד ברקן:
Store the position of every element of a[] in b[]. So, b[a[i]] = i
Iterate over the initial array a[] from left to right.
At position i, check if a[i] is equal to i. If yes, then keep iterating.
If no, then it's time to swap. Look at the logic in the code minutely to see how the swapping takes place. This is the most important step as both array a[] and b[] needs to be modified. Increase the count of swaps.
Here is the implementation:
long long sortWithSwap(int n, int *a) {
int *b = (int*)malloc(sizeof(int)*n); //create a temporary array keeping track of the position of every element
int i,tmp,t,valai,posi;
for(i=0;i<n;i++){
b[a[i]] = i;
}
long long ans = 0;
for(i=0;i<n;i++){
if(a[i]!=i){
valai = a[i];
posi = b[i];
a[b[i]] = a[i];
a[i] = i;
b[i] = i;
b[valai] = posi;
ans++;
}
}
return ans;
}
The essence of solving this problem lies in the following observation
1. The elements in the array do not repeat
2. The range of elements is from 0 to n-1, where n is the size of the array.
The way to approach
After you have understood the way to approach the problem ou can solve it in linear time.
Imagine How would the array look like after sorting all the entries ?
It will look like arr[i] == i, for all entries . Is that convincing ?
First create a bool array named FIX, where FIX[i] == true if ith location is fixed, initialize this array with false initially
Start checking the original array for the match arr[i] == i, till the time this condition holds true, eveything is okay. While going ahead with traversal of array also update the FIX[i] = true. The moment you find that arr[i] != i you need to do something, arr[i] must have some value x such that x > i, how do we guarantee that ? The guarantee comes from the fact that the elements in the array do not repeat, therefore if the array is sorted till index i then it means that the element at position i in the array cannot come from left but from right.
Now the value x is essentially saying about some index , why so because the array only has elements till n-1 starting from 0, and in the sorted arry every element i of the array must be at location i.
what does arr[i] == x means is that , not only element i is not at it's correct position but also the element x is missing from it's place.
Now to fix ith location you need to look at xth location, because maybe xth location holds i and then you will swap the elements at indices i and x, and get the job done. But wait, it's not necessary that the index x will hold i (and you finish fixing these locations in just 1 swap). Rather it may be possible that index x holds value y, which again will be greater than i, because array is only sorted till location i.
Now before you can fix position i , you need to fix x, why ? we will see later.
So now again you try to fix position x, and then similarly you will try fixing till the time you don't see element i at some location in the fashion told .
The fashion is to follow the link from arr[i], untill you hit element i at some index.
It is guaranteed that you will definitely hit i at some location while following in this way . Why ? try proving it, make some examples, and you will feel it
Now you will start fixing all the index you saw in the path following from index i till this index (say it j). Now what you see is that the path which you have followed is a circular one and for every index i, the arr[i] is tored at it's previous index (index from where you reached here), and Once you see that you can fix the indices, and mark all of them in FIX array to be true. Now go ahead with next index of array and do the same thing untill whole array is fixed..
This was the complete idea, but to only conunt no. of swaps, you se that once you have found a cycle of n elements you need n swaps, and after doing that you fix the array , and again continue. So that's how you will count the no. of swaps.
Please let me know if you have some doubts in the approach .
You may also ask for C/C++ code help. Happy to help :-)
When drawing in random from a set of values in succession, where a drawn value is allowed to
be drawn again, a given value has (of course) a small chance of being drawn twice (or more) in immediate succession, but that causes an issue (for the purposes of a given application) and we would like to eliminate this chance. Any algorithmic ideas on how to do so (simple/efficient)?
Ideally we would like to set a threshold say as a percentage of the size of the data set:
Say the size of the set of values N=100, and the threshold T=10%, then if a given value is drawn in the current draw, it is guaranteed not to show up again in the next N*T=10 draws.
Obviously this restriction introduces bias in the random selection. We don't mind that a
proposed algorithm introduces further bias into the randomness of selection, what really
matters for this application is that the selection is just random enough to appear so
for a human observer.
As an implementation detail, the values are stored as database records, so database table flags/values can be used, or maybe external memory structures. Answers about the abstract case are welcome too.
Edit:
I just hit this other SO question here, which has good overlap with my own. Going through the good points there.
Here's an implementation that does the whole process in O(1) (for a single element) without any bias:
The idea is to treat the last K elements in the array A (which contains all the values) like a queue, we draw a value from the first N-k values in A, which is the random value, and swap it with an element in position N-Pointer, when Pointer represents the head of the queue, and it resets to 1 when it crosses K elements.
To eliminate any bias in the first K draws, the random value will be drawn between 1 and N-Pointer instead of N-k, so this virtual queue is growing in size at each draw until reaching the size of K (e.g. after 3 draws the number of possible values appear in A between indexes 1 and N-3, and the suspended values appear in indexes N-2 to N.
All operations are O(1) for drawing a single elemnent and there's no bias throughout the entire process.
void DrawNumbers(val[] A, int K)
{
N = A.size;
random Rnd = new random;
int Drawn_Index;
int Count_To_K = 1;
int Pointer = K;
while (stop_drawing_condition)
{
if (Count_To_K <= K)
{
Drawn_Index = Rnd.NextInteger(1, N-Pointer);
Count_To_K++;
}
else
{
Drawn_Index = Rnd.NextInteger(1, N-K)
}
Print("drawn value is: " + A[Drawn_Index])
Swap(A[Drawn_Index], A[N-Pointer])
Pointer--;
if (Pointer < 1) Pointer = K;
}
}
My previous suggestion, by using a list and an actual queue, is dependent on the remove method of the list, which I believe can be at best O(logN) by using an array to implement a self balancing binary tree, as the list has to have direct access to indexes.
void DrawNumbers(list N, int K)
{
queue Suspended_Values = new queue;
random Rnd = new random;
int Drawn_Index;
while (stop_drawing_condition)
{
if (Suspended_Values.count == K)
N.add(Suspended_Value.Dequeue());
Drawn_Index = Rnd.NextInteger(1, N.size) // random integer between 1 and the number of values in N
Print("drawn value is: " + N[Drawn_Index]);
Suspended_Values.Enqueue(N[Drawn_Index]);
N.Remove(Drawn_Index);
}
}
I assume you have an array, A, that contains the items you want to draw. At each time period you randomly select an item from A.
You want to prevent any given item, i, from being drawn again within some k iterations.
Let's say that your threshold is 10% of A.
So create a queue, call it drawn, that can hold threshold items. Also create a hash table that contains the drawn items. Call the hash table hash.
Then:
do
{
i = Get random item from A
if (i in hash)
{
// we have drawn this item recently. Don't draw it.
continue;
}
draw(i);
if (drawn.count == k)
{
// remove oldest item from queue
temp = drawn.dequeue();
// and from the hash table
hash.remove(temp);
}
// add new item to queue and hash table
drawn.enqueue(i);
hash.add(i);
} while (forever);
The hash table exists solely to increase lookup speed. You could do without the hash table if you're willing to do a sequential search of the queue to determine if an item has been drawn recently.
Say you have n items in your list, and you don't want any of the k last items to be selected.
Select at random from an array of size n-k, and use a queue of size k to stick the items you don't want to draw (adding to the front and removing from the back).
All operations are O(1).
---- clarification ----
Give n items, and a goal of not redrawing any of the last k draws, create an array and queue as follows.
Create an array A of size n-k, and put n-k of your items in the list (chosen at random, or seeded however you like).
Create a queue (linked list) Q and populate it with the remaining k items, again in random order or whatever order you like.
Now, each time you want to select an item at random:
Choose a random index from your array, call this i.
Give A[i] to whomever is asking for it, and add it to the front of Q.
Remove the element from the back of Q, and store it in A[i].
Everything is O(1) after the array and linked list are created, which is a one-time O(n) operation.
Now, you might wonder, what do we do if we want to change n (i.e. add or remove an element).
Each time we add an element, we either want to grow the size of A or of Q, depending on our logic for deciding what k is (i.e. fixed value, fixed fraction of n, whatever...).
If Q increases then the result is trivial, we just append the new element to Q. In this case I'd probably append it to the end of Q so that it gets in play ASAP. You could also put it in A, kicking some element out of A and appending it to the end of Q.
If A increases, you can use a standard technique for increasing arrays in amortized constant time. E.g., each time A fills up, we double it in size, and keep track of the number of cells of A that are live. (look up 'Dynamic Arrays' in Wikipedia if this is unfamiliar).
Set-based approach:
If the threshold is low (say below 40%), the suggested approach is:
Have a set and a queue of the last N*T generated values.
When generating a value, keep regenerating it until it's not contained in the set.
When pushing to the queue, pop the oldest value and remove it from the set.
Pseudo-code:
generateNextValue:
// once we're generated more than N*T elements,
// we need to start removing old elements
if queue.size >= N*T
element = queue.pop
set.remove(element)
// keep trying to generate random values until it's not contained in the set
do
value = getRandomValue()
while set.contains(value)
set.add(value)
queue.push(value)
return value
If the threshold is high, you can just turn the above on its head:
Have the set represent all values not in the last N*T generated values.
Invert all set operations (replace all set adds with removes and vice versa and replace the contains with !contains).
Pseudo-code:
generateNextValue:
if queue.size >= N*T
element = queue.pop
set.add(element)
// we can now just get a random value from the set, as it contains all candidates,
// rather than generating random values until we find one that works
value = getRandomValueFromSet()
//do
// value = getRandomValue()
//while !set.contains(value)
set.remove(value)
queue.push(value)
return value
Shuffled-based approach: (somewhat more complicated that the above)
If the threshold is a high, the above may take long, as it could keep generating values that already exists.
In this case, some shuffle-based approach may be a better idea.
Shuffle the data.
Repeatedly process the first element.
When doing so, remove it and insert it back at a random position in the range [N*T, N].
Example:
Let's say N*T = 5 and all possible values are [1,2,3,4,5,6,7,8,9,10].
Then we first shuffle, giving us, let's say, [4,3,8,9,2,6,7,1,10,5].
Then we remove 4 and insert it back in some index in the range [5,10] (say at index 5).
Then we have [3,8,9,2,4,6,7,1,10,5].
And continue removing the next element and insert it back, as required.
Implementation:
An array is fine if we don't care about efficient a whole lot - to get one element will cost O(n) time.
To make this efficient we need to use an ordered data structure that supports efficient random position inserts and first position removals. The first thing that comes to mind is a (self-balancing) binary search tree, ordered by index.
We won't be storing the actual index, the index will be implicitly defined by the structure of the tree.
At each node we will have a count of children (+ 1 for itself) (which needs to be updated on insert / remove).
An insert can be done as follows: (ignoring the self-balancing part for the moment)
// calling function
insert(node, value)
insert(node, N*T, value)
insert(node, offset, value)
// node.left / node.right can be defined as 0 if the child doesn't exist
leftCount = node.left.count - offset
rightCount = node.right.count
// Since we're here, it means we're inserting in this subtree,
// thus update the count
node.count++
// Nodes to the left are within N*T, so simply go right
// leftCount is the difference between N*T and the number of nodes on the left,
// so this needs to be the new offset (and +1 for the current node)
if leftCount < 0
insert(node.right, -leftCount+1, value)
else
// generate a random number,
// on [0, leftCount), insert to the left
// on [leftCount, leftCount], insert at the current node
// on (leftCount, leftCount + rightCount], insert to the right
sum = leftCount + rightCount + 1
random = getRandomNumberInRange(0, sum)
if random < leftCount
insert(node.left, offset, value)
else if random == leftCount
// we don't actually want to update the count here
node.count--
newNode = new Node(value)
newNode.count = node.count + 1
// TODO: swap node and newNode's data so that node's parent will now point to newNode
newNode.right = node
newNode.left = null
else
insert(node.right, -leftCount+1, value)
To visualize inserting at the current node:
If we have something like:
4
/
1
/ \
2 3
And we want to insert 5 where 1 is now, it will do this:
4
/
5
\
1
/ \
2 3
Note that when a red-black tree, for example, performs operations to keep itself balanced, none of these involve comparisons, so it doesn't need to know the order (i.e. index) of any already-inserted elements. But it will have to update the counts appropriately.
The overall efficiency will be O(log n) to get one element.
I'd put all "values" into a "list" of size N, then shuffle the list and retrieve values from the top of the list. Then you "insert" the retrieved value at a random position with any index >= N*T.
Unfortunately I'm not truly a math-guy :( So I simply tried it (in VB, so please take it as pseudocode ;) )
Public Class BiasedRandom
Private prng As New Random
Private offset As Integer
Private l As New List(Of Integer)
Public Sub New(ByVal size As Integer, ByVal threshold As Double)
If threshold <= 0 OrElse threshold >= 1 OrElse size < 1 Then Throw New System.ArgumentException("Check your params!")
offset = size * threshold
' initial fill
For i = 0 To size - 1
l.Add(i)
Next
' shuffle "Algorithm p"
For i = size - 1 To 1 Step -1
Dim j = prng.Next(0, i + 1)
Dim tmp = l(i)
l(i) = l(j)
l(j) = tmp
Next
End Sub
Public Function NextValue() As Integer
Dim tmp = l(0)
l.RemoveAt(0)
l.Insert(prng.Next(offset, l.Count + 1), tmp)
Return tmp
End Function
End Class
Then a simple check:
Public Class Form1
Dim z As Integer = 10
Dim k As BiasedRandom
Private Sub Form1_Load(sender As Object, e As EventArgs) Handles MyBase.Load
k = New BiasedRandom(z, 0.5)
End Sub
Private Sub Button1_Click(sender As Object, e As EventArgs) Handles Button1.Click
Dim j(z - 1)
For i = 1 To 10 * 1000 * 1000
j(k.NextValue) += 1
Next
Stop
End Sub
End Class
And when I check out the distribution it looks okay enough for an unarmed eye ;)
EDIT:
After thinking about RonTeller's argumentation, I have to admit that he is right. I don't think that there is a performance friendly way to achieve the wanted and to pertain a good (not more biased than required) random order.
I came to the follwoing idea:
Given a list (array whatever) like this:
0123456789 ' not shuffled to make clear what I mean
We return the first element which is 0. This one must not come up again for 4 (as an example) more draws but we also want to avoid a strong bias. Why not simply put it to the end of the list and then shuffle the "tail" of the list, i.e. the last 6 elements?
1234695807
We now return the 1 and repeat the above steps.
2340519786
And so on and so on. Since removing and inserting is kind of unnecessary work, one could use a simple array and a "pointer" to the actual element. I have changed the code from above to give a sample. It's slower than the first one, but should avoid the mentioned bias.
Public Function NextValue() As Integer
Static current As Integer = 0
' only shuffling a part of the list
For i = current + l.Count - 1 To current + 1 + offset Step -1
Dim j = prng.Next(current + offset, i + 1)
Dim tmp = l(i Mod l.Count)
l(i Mod l.Count) = l(j Mod l.Count)
l(j Mod l.Count) = tmp
Next
current += 1
Return l((current - 1) Mod l.Count)
End Function
EDIT 2:
Finally (hopefully), I think the solution is quite simple. The below code assumes that there is an array of N elements called TheArray which contains the elements in random order (could be rewritten to work with sorted array). The value DelaySize determines how long a value should be suspended after it has been drawn.
Public Function NextValue() As Integer
Static current As Integer = 0
Dim SelectIndex As Integer = prng.Next(0, TheArray.Count - DelaySize)
Dim ReturnValue = TheArray(SelectIndex)
TheArray(SelectIndex) = TheArray(TheArray.Count - 1 - current Mod DelaySize)
TheArray(TheArray.Count - 1 - current Mod DelaySize) = ReturnValue
current += 1
Return ReturnValue
End Function
You are given an array of n+2 elements. All elements of the array are in range 1 to n. And all elements occur once except two numbers which occur twice. Find the two repeating numbers.
For example, array = {4, 2, 4, 5, 2, 3, 1} and n = 5
Guys I know 4 probable solutions to this problem but recently i encountered a solution which i am not able to interpret .Below is an algorithm for the solution
traverse the list for i= 1st to n+2 elements
{
check for sign of A[abs(A[i])] ;
if positive then
make it negative by A[abs(A[i])]=-A[abs(A[i])];
else // i.e., A[abs(A[i])] is negative
this element (ith element of list) is a repetition
}
Example: A[] = {1,1,2,3,2}
i=1 ; A[abs(A[1])] i.e,A[1] is positive ; so make it negative ;
so list now is {-1,1,2,3,2}
i=2 ; A[abs(A[2])] i.e., A[1] is negative ; so A[i] i.e., A[2] is repetition,
now list is {-1,1,2,3,2}
i=3 ; list now becomes {-1,-1,2,3,2} and A[3] is not repeated
now list becomes {-1,-1,-2,3,2}
i=4 ;
and A[4]=3 is not repeated
i=5 ; we find A[abs(A[i])] = A[2] is negative so A[i]= 2 is a repetition,
This method modifies the original array.
How this algorithm is producing proper results i.e. how it is working.Guys don't take this as an Homework Question as this question has been recently asked in Microsoft's interview.
You are given an array of n+2
elements. All elements of the array
are in range 1 to n. And all elements
occur once except two numbers which
occur twice
Lets modify this slightly, and go with just n, not n+2, and the first part of the problem statement, it becomes
You are given an array of n
elements. All elements of the array
are in range 1 to n
So now you know you have an array, the numbers in the array start at 1 and go up by one for every item in the array. So if you have 10 items, the array will contain the numbers 1 to 10. 5 items, you have 1 to 5 and so forth.
It follows that the numbers stored in the array can be used to index the array. i.e. you can always say A[A[i]] where i <= size of A. e.g. A={5,3,4,1,2}; print A[A[2]]
Now, lets add in one duplicate number.
The algorithm takes the value of each number in the array, and visits that index. We know if we visit the same index twice, we know we have found a duplicate.
How do we know if we visit the same index twice?
Yup, we change the sign of the number in each index we visit, if the sign has already changed, we know we've already been here, ergo, this index (not the value stored at the index) is a duplicate number.
You could achieve the same result by keeping a second array of booleans, initialised to false. That algroithm becomes
A={123412}
B={false, false, false, false}
for(i = 1; i <= 4; i++)
{
if(B[A[i]])
// Duplicate
else
B[A[i]] = true;
}
However in the MS question you're changing the sign of the element in A instead of setting a boolean value in B.
Hope this helps,
What you are doing is using the array values in two ways: they have a number AND they have a sign. You 'store' the fact that you've seen a number n on the n-th spot in your array, without loosing the origional value in that spot: you're just changing the sign.
You start out with all positives, and if you find that your index you want to 'save' the fact you've seen your current value to is allready negative, then this value has allready be seen.
example:
So if you see 4 for the first time, you change the sign on the fourth spot to negative. That doesn't change the 4th spot, because you are using [abs] on that when you would go there, so no worries there.
If you see another 4, you check the 4th spot again, see that it is negative: presto: a double.
When you find some element in position i, let's say n, then you make A[abs(A(i))]=A[abs(n)] negative. So if you find another position j containing n, you will also check A[abs(A(j))]=A[abs(n)]. Since you find it negative, then n is repeated :)
the best approach for finding two repeated elements would be using XOR method.
This solution works only if array has positive integers and all the elements in the array are in range from 1 to n.
As we know A XOR A = 0. We have n + 2 elements in array with 2 repeated elements (say repeated elements are X and Y) and we know the range of elements are from 1 to n.
XOR all the numbers in array numbers from 1 to n. Result be X XOR Y.
1 XOR 1 = 0 and 1 XOR 0 = 1 with this logic in the result of X XOR Y if any kth bit is set to 1 implies either kth bit is 1 either in X or in Y not in both.
Use the above step to divide all the elements in array and from 1 to n into 2 groups, one group which has the elements for which the kth bit is set to 1 and second group which has the elements for which the kth bit is 0.
Let’s have that kth bit as right most set bit (Read how to find right most set bit)
Now we can claim that these two groups are responsible to produce X and Y.
Group -1: XOR all the elements whose kth bit is 1 will produce either X or Y.
Group -2: XOR all the elements whose kth bit is 0 will produce either X or Y.
See the diagram below for more understanding.(Click on the diagram to see it larger).
public class TwoRepeatingXOR {
public static void twoRepeating(int [] A, int n){
int XOR = A[0];
int right_most_bit, X=0, Y=0, size = A.length;
for (int i = 1; i <=n ; i++)
XOR ^= i;
for (int i = 0; i <size ; i++)
XOR ^= A[i];
//Now XOR contains the X XOR Y
//get the right most bit number
right_most_bit = XOR & ~(XOR-1);
//divide the elements into 2 groups based on the right most set bit
for (int i = 0; i <size ; i++) {
if((A[i] & right_most_bit)!=0)
X = X^A[i];
else
Y = Y^A[i];
}
for (int i = 1; i <=n ; i++) {
if((i&right_most_bit)!=0)
X = X^i;
else
Y = Y^i;
}
System.out.println("Two Repeated elements are: " + X + " and " + Y);
}
public static void main(String[] args) {
int [] A = {1,4,5,6,3,2,5,2};
int n = 6;
twoRepeating(A, n);
}
}
credits go to https://algorithms.tutorialhorizon.com/find-the-two-repeating-elements-in-a-given-array-6-approaches/
Simple, use a Hashtable.
For each item, check if it already exists O(1) , and if not, add it to the hashtable O(1).
When you find an item that already exists... that's it.
I know this isn't really an answer to your question, but if I actually had to write this code on a real project, I would start with a sort algo like quicksort, and in my comparison function something like,
int Compare(int l, int r)
{
if(l == r)
{
// duplicate; insert into duplicateNumbers array if it doesn't exist already.
// if we found 2 dupes, quit the sort altogether
}
return r - l;
}
I would file this into the "good balance between performance and maintainability" bucket of possible solutions.
I have the following elements in a list/array
a1,a2,a3
and these elements are used to build another list in a predictable pattern
example
a1,a1,a2,a2,a3,a3,a1,a1,a2,a2,a3,a3...
The pattern may change but I will always know how many times each element repeats and all elements repeats the same number of times. And the elements always show up in the same order.
so another pattern might be
a1,a1,a1,a2,a2,a2,a3,a3,a3,a1,a1,a1,a2,a2,a2,a3,a3,a3...
or
a1,a2,a3,a1,a2,a3
it will never be
a2,a2,a1,a1,a3,a3... or a1,a2,a3,a2,a3,a1 etc
How I determine what element is at any index in the list?
I can't run through the generated list because it is what might be. It doesn't actually exist. And I need to get tbe answer for any index from 0 to infinity (actually integer.maxvalue)
Lets make some denotations:
n - number of elements in orginal array
k - how many times element is repeated
x- index
Array[x] == Array[(x mod (kn)) div k] - that is what You were searching for.
In other words element at index x is equal to element at index (x mod (kn)) div k
Run through the list until you find an element which is not the same as the first element.
After that you know how long each group is and you can determine the element at any index with a simple modulo statement.
Pseudo stuff:
determine(index){
firstelement = list[0]
i=0;
for i=0; element.count; i++
if element != firstelement
break;
m = index modulo (i*3)
switch(m)
case 0: return 'a1'
case 1: return 'a2'
case 2: return 'a3'
}
Original Problem:
I have 3 boxes each containing 200 coins, given that there is only one person who has made calls from all of the three boxes and thus there is one coin in each box which has same fingerprints and rest of all coins have different fingerprints. You have to find the coin which contains same fingerprint from all of the 3 boxes. So that we can find the fingerprint of the person who has made call from all of the 3 boxes.
Converted problem:
You have 3 arrays containing 200 integers each. Given that there is one and only one common element in these 3 arrays. Find the common element.
Please consider solving this for other than trivial O(1) space and O(n^3) time.
Some improvement in Pelkonen's answer:
From converted problem in OP:
"Given that there is one and only one common element in these 3 arrays."
We need to sort only 2 arrays and find common element.
If you sort all the arrays first O(n log n) then it will be pretty easy to find the common element in less than O(n^3) time. You can for example use binary search after sorting them.
Let N = 200, k = 3,
Create a hash table H with capacity ≥ Nk.
For each element X in array 1, set H[X] to 1.
For each element Y in array 2, if Y is in H and H[Y] == 1, set H[Y] = 2.
For each element Z in array 3, if Z is in H and H[Z] == 2, return Z.
throw new InvalidDataGivenByInterviewerException();
O(Nk) time, O(Nk) space complexity.
Use a hash table for each integer and encode the entries such that you know which array it's coming from - then check for the slot which has entries from all 3 arrays. O(n)
Use a hashtable mapping objects to frequency counts. Iterate through all three lists, incrementing occurrence counts in the hashtable, until you encounter one with an occurrence count of 3. This is O(n), since no sorting is required. Example in Python:
def find_duplicates(*lists):
num_lists = len(lists)
counts = {}
for l in lists:
for i in l:
counts[i] = counts.get(i, 0) + 1
if counts[i] == num_lists:
return i
Or an equivalent, using sets:
def find_duplicates(*lists):
intersection = set(lists[0])
for l in lists[1:]:
intersection = intersection.intersect(set(l))
return intersection.pop()
O(N) solution: use a hash table. H[i] = list of all integers in the three arrays that map to i.
For all H[i] > 1 check if three of its values are the same. If yes, you have your solution. You can do this check with the naive solution even, it should still be very fast, or you can sort those H[i] and then it becomes trivial.
If your numbers are relatively small, you can use H[i] = k if i appears k times in the three arrays, then the solution is the i for which H[i] = 3. If your numbers are huge, use a hash table though.
You can extend this to work even if you can have elements that can be common to only two arrays and also if you can have elements repeating elements in one of the arrays. It just becomes a bit more complicated, but you should be able to figure it out on your own.
If you want the fastest* answer:
Sort one array--time is N log N.
For each element in the second array, search the first. If you find it, add 1 to a companion array; otherwise add 0--time is N log N, using N space.
For each non-zero count, copy the corresponding entry into the temporary array, compacting it so it's still sorted--time is N.
For each element in the third array, search the temporary array; when you find a hit, stop. Time is less than N log N.
Here's code in Scala that illustrates this:
import java.util.Arrays
val a = Array(1,5,2,3,14,1,7)
val b = Array(3,9,14,4,2,2,4)
val c = Array(1,9,11,6,8,3,1)
Arrays.sort(a)
val count = new Array[Int](a.length)
for (i <- 0 until b.length) {
val j =Arrays.binarySearch(a,b(i))
if (j >= 0) count(j) += 1
}
var n = 0
for (i <- 0 until count.length) if (count(i)>0) { count(n) = a(i); n+= 1 }
for (i <- 0 until c.length) {
if (Arrays.binarySearch(count,0,n,c(i))>=0) println(c(i))
}
With slightly more complexity, you can either use no extra space at the cost of being even more destructive of your original arrays, or you can avoid touching your original arrays at all at the cost of another N space.
Edit: * as the comments have pointed out, hash tables are faster for non-perverse inputs. This is "fastest worst case". The worst case may not be so unlikely unless you use a really good hashing algorithm, which may well eat up more time than your sort. For example, if you multiply all your values by 2^16, the trivial hashing (i.e. just use the bitmasked integer as an index) will collide every time on lists shorter than 64k....
//Begineers Code using Binary Search that's pretty Easy
// bool BS(int arr[],int low,int high,int target)
// {
// if(low>high)
// return false;
// int mid=low+(high-low)/2;
// if(target==arr[mid])
// return 1;
// else if(target<arr[mid])
// BS(arr,low,mid-1,target);
// else
// BS(arr,mid+1,high,target);
// }
// vector <int> commonElements (int A[], int B[], int C[], int n1, int n2, int n3)
// {
// vector<int> ans;
// for(int i=0;i<n2;i++)
// {
// if(i>0)
// {
// if(B[i-1]==B[i])
// continue;
// }
// //The above if block is to remove duplicates
// //In the below code we are searching an element form array B in both the arrays A and B;
// if(BS(A,0,n1-1,B[i]) && BS(C,0,n3-1,B[i]))
// {
// ans.push_back(B[i]);
// }
// }
// return ans;
// }