I have got a pretty hard task I find difficult to handle.
I'm trying to write a recursive function (no loops at all) that, given an array and its length, will print a pair of sub arrays, each of which's sum will be half of the sum of the entire array. In other words, the array is to be divided into two groups of integers so that their sum is equal.
For example, given the array {1,2,2,0,5}, the function should output {1,2,2} {0,5}
I have to do it recursively, with a function that gets only the array itself and its size. I am also only allowed to use one more additional recursive function to solve the problem.
Any thoughts or ideas will be most appreciated.
Hello again!
we got a code in class that goes like this
int SubsetSum(int arr[], int idx, int n, int S) {
if (S==0) return 1; //This is stopping condition #1.
if (S<0 || n==0) return 0; //This is stopping condition #2.
return SubsetSum(arr, idx+1, n-1, S-arr[idx]) || SubsetSum(arr, idx+1, n-1, S);
}
what does the " || " operator means in terms of recursion?
thanks ppl!
You first calculate the sum of the whole array (it'll better be even), and this gives you the half-sum, which you can reach using a binary knapsack routine.
There is also a recursive implementation on Stack Overflow in Java, which isn't too different from C and satisfies your requirements.
Divide the array and sub-arrays from the middle until there is only one element.
Compare the first element with second and find sum of these integers.
After that use these sums for binary elements comparing. When doing this compare find the sum. For example your sub-arrays are {1,2} and {0,3}. Look the first elements 0 and 1. When you see the little element take the other element in that sub-array ( {0,3} ). After that sum is now 3. The other part's sum is 1 and take the other element (2). Now the sum is 3 for both. And you can use this for all sub-arrays.
I'm not sure about the solution. But I think it seems like divide-and-conquer algorithm.
Related
**I have 3 arrays a[1...n] b[1...n] c[1....n] which contain integers.
It is not mentioned if the arrays are sorted or if each array has or has not duplicates.
The task is to check if there is any common number in the given arrays and return true or false.
For example : these arrays a=[3,1,5,10] b=[4,2,6,1] c=[5,3,1,7] have one common number : 1
I need to write an algorithm with time complexity O(n^2).
I let the current element traversed in a[] be x, in b[] be y and in c[] be z and have following cases inside the loop : If x, y and z are same, I can simply return true and stop the program,something like:
for(x=1;x<=n;x++)
for(y=1;y<=n;y++)
for(z=1;z<=n;z++)
if(a[x]==b[y]==c[z])
return true
But this algorithm has time complexity O(n^3) and I need O(n^2).Any suggestions?
There is a pretty simple and efficient solution for this.
Sort a and b. Complexity = O(NlogN)
For each element in c, use binary search to check if it exists in both a and b. Complexity = O(NlogN).
That'll give you a total complexity of O(NlogN), better than O(N^2).
Create a new array, and save common elements in a and b arrays. Then find common elements in this array with c array.
python solution
def find_d(a, b, c):
for i in a:
for j in b:
if i==j:
d.append(i)
def findAllCommon(c, d):
for i in c:
for j in d:
if i==j:
e.append(i)
break
a = [3,1,5,10]
b = [4,2,6,1]
c = [5,3,1,7]
d = []
e = []
find_d(a, b, c)
findAllCommon(c, d)
if len(e)>0:
print("True")
else:
print("False")
Since I haven't seen a solution based on sets, so I suggest looking for how sets are implemented in your language of choice and do the equivalent of this:
set(a).intersection(b).intersection(c) != set([])
This evaluates to True if there is a common element, False otherwise. It runs in O(n) time.
All solutions so far either require O(n) additional space (creating a new array/set) or change the order of the arrays (sorting).
If you want to solve the problem in O(1) additional space and without changing the original arrays, you indeed can't do better than O(n^2) time:
foreach(var x in a) { // n iterations
if (b.Contains(x) && c.Contains(x)) return true; // max. 2n
} // O(n^2)
return false;
A suggestion:
Combine into a single array(z) where z = sum of the entries in each array. Keep track of how many entries there were in Array 1, Array 2, Array 3.
For each entry Z traverse the array to see how many duplicates there are within the combined array and where they are. For those which have 2 or more (ie there are 3 or more of the same number), check that the location of those duplicates correspond to having been in different arrays to begin with (ruling our duplicates within the original arrays). If your number Z has 2 or more duplicates and they are all in different arrays (checked through their position in the array) then store that number Z in result array.
Report result array.
You will traverse the entire combined array once and then almost (no need to check if Z is a duplicate of itself) traverse it again for each Z, so n^2 complexity.
Also worth noting that the time complexity will now be a function of total number of entries and not of number of arrays (your nested loops would become n^4 with 4 arrays - this will stay as n^2)
You could make it more efficient by always checking the already found duplicates before checking for a new Z - if the new Z is already found as a duplicate to an earlier Z you need not traverse to check for that number again. This will make it more efficient the more duplicates there are - with few duplicates the reduction in number of traverses is probably not worth the extra complexity.
Of course you could also do this without actually combining the values into a single array - you would just need to make sure that your traversing routine looks through the arrays and keeps track of what it finds the in the right order.
Edit
Thinking about it, the above is doing way more than you want. It would allow you to report on doubles, quads etc. as well.
If you just want triples, then it is much easier/quicker. Since a triple needs to be in all 3 arrays, you can start by finding those numbers which are in any of the 2 arrays (if they are different lengths, compare the 2 shortest arrays first) and then to check any doublets found against the third array. Not sure what that brings the complexity down to but it will be less than n^2...
there are many ways to solve this here few selected ones sorted by complexity (descending) assuming n is average size of your individual arrays:
Brute force O(n^3)
its basicaly the same as you do so test any triplet combination by 3 nested for loops
for(x=1;x<=n;x++)
for(y=1;y<=n;y++)
for(z=1;z<=n;z++)
if(a[x]==b[y]==c[z])
return true;
return false;
slightly optimized brute force O(n^2)
simply check if each element from a is in b and if yes then check if it is also in c which is O(n*(n+n)) = O(n^2) as the b and c loops are not nested anymore:
for(x=1;x<=n;x++)
{
for(ret=false,y=1;y<=n;y++)
if(a[x]==b[y])
{ ret=true; break; }
if (!ret) continue;
for(ret=false,z=1;z<=n;z++)
if(a[x]==c[z])
{ ret=true; break; }
if (ret) return true;
}
return false;
exploit sorting O(n.log(n))
simply sort all arrays O(n.log(n)) and then just traverse all 3 arrays together to test if each element is present in all arrays (single for loop, incrementing the smallest element array). This can be done also with binary search like one of the other answers suggest but that is slower still not exceeding n.log(n). Here the O(n) traversal:
for(x=1,y=1,z=1;(x<=n)&&(y<=n)&&(z<=n);)
{
if(a[x]==b[y]==c[z]) return true;
if ((a[x]<b[y])&&(a[x]<c[z])) x++;
else if ((b[y]<a[x])&&(b[y]<c[z])) y++;
else z++;
}
return false;
however this needs to change the contents of arrays or need additional arrays for index sorting instead (so O(n) space).
histogram based O(n+m)
this can be used only if the range of elements in your array is not too big. Let say the arrays can hold numbers 1 .. m then you add (modified) histogram holding set bit for each array where value is presen and simply check if value is present in all 3:
int h[m]; // histogram
for(i=1;i<=m;i++) h[i]=0; // clear histogram
for(x=1;x<=n;x++) h[a[x]]|=1;
for(y=1;y<=n;y++) h[b[y]]|=2;
for(z=1;z<=n;z++) h[c[z]]|=4;
for(i=1;i<=m;i++) if (h[i]==7) return true;
return false;
This needs O(m) space ...
So you clearly want option #2
Beware all the code is just copy pasted yours and modified directly in answer editor so there might be typos or syntax error I do not see right now...
Goal
I would like to write an algorithm (in C) which returns TRUE or FALSE (1 or 0) depending whether the array A given in input can “sum and/or sub” to x (see below for clarification). Note that all values of A are integers bounded between [1,x-1] that were randomly (uniformly) sampled.
Clarification and examples
By “sum and/or sub”, I mean placing "+" and "-" in front of each element of array and summing over. Let's call this function SumSub.
int SumSub (int* A,int x)
{
...
}
SumSub({2,7,5},10)
should return TRUE as 7-2+5=10. You will note that the first element of A can also be taken as negative so that the order of elements in A does not matter.
SumSub({2,7,5,2},10)
should return FALSE as there is no way to “sum and/or sub” the elements of array to reach the value of x. Please note, this means that all elements of A must be used.
Complexity
Let n be the length of A. Complexity of the problem is of order O(2^n) if one has to explore all possible combinations of pluses and minus. However, some combinations are more likely than others and therefore are worth being explored first (hoping the output will be TRUE). Typically, the combination which requires substracting all elements from the largest number is impossible (as all elements of A are lower than x). Also, if n>x, it makes no sense to try adding all the elements of A.
Question
How should I go about writing this function?
Unfortunately your problem can be reduced to subset-sum problem which is NP-Complete. Thus the exponential solution can't be avoided.
The original problem's solution is indeed exponential as you said. BUT with the given range[1,x-1] for numbers in A[] you can make the solution polynomial. There is a very simple dynamic programming solution.
With the order:
Time Complexity: O(n^2*x)
Memory Complexity: O(n^2*x)
where, n=num of elements in A[]
You need to use dynamic programming approach for this
You know the min,max range that can be made in in the range [-nx,nx]. Create a 2d array of size (n)X(2*n*x+1). Lets call this dp[][]
dp[i][j] = taking all elements of A[] from [0..i-1] whether its possible to make the value j
so
dp[10][3] = 1 means taking first 10 elements of A[] we CAN create the value 3
dp[10][3] = 0 means taking first 10 elements of A[] we can NOT create the value 3
Here is a kind of pseudo code for this:
int SumSub (int* A,int x)
{
bool dp[][];//set all values of this array 0
dp[0][0] = true;
for(i=1;i<=n;i++) {
int val = A[i-1];
for(j=-n*x;j<=n*x;j++) {
dp[i][j]=dp[ i-1 ][ j + val ] | dp[ i-1 ][ j - val ];
}
}
return dp[n][x];
}
Unfortunately this is NP-complete even when x is restricted to the value 0, so don't expect a polynomial-time algorithm. To show this I'll give a simple reduction from the NP-hard Partition Problem, which asks whether a given multiset of positive integers can be partitioned into two parts having equal sums:
Suppose we have an instance of the Partition Problem consisting of n positive integers B_1, ..., B_n. Create from this an instance of your problem in which A_i = B_i for each 1 <= i <= n, and set x = 0.
Clearly if there is a partition of B into two parts C and D having equal sums, then there is also a solution to the instance of your problem: Put a + in front of every number in C, and a - in front of every number in D (or the other way round). Since C and D have equal sums, this expression must equal 0.
OTOH, if the solution to the instance of your problem that we just created is YES (TRUE), then we can easily create a partition of B into two parts having equal sums: just put all the positive terms in one part (say, C), and all the negative terms (without the preceding - of course) in the other (say, D). Since we know that the total value of the expression is 0, it must be that the sum of the (positive) numbers in C is equal to the (negated) sum of the numbers in D.
Thus a YES to either problem instance implies a YES to the other problem instance, which in turn implies that a NO to either problem instance implies a NO to the other problem instance -- that is, the two problem instances have equal solutions. Thus if it were possible to solve your problem in polynomial time, it would be possible to solve the NP-hard Partition Problem in polynomial time too, by constructing the above instance of your problem, solving it with your poly-time algorithm, and reporting the result it gives.
where all other integers in this array appeared n times. we have n>m.
all elements in this array are integers. Can you design an algorithm that works in O(N) where N is the number of elements in the array, while minimizing the space complexity? In the best case space complexity can be limited to O(log(m)).
a special case is n=2 and m=1 (which is easy). Is there a generalized algorithm that can handle arbitrary m and n?
thanks
You can use a hashtable that maps numbers in the array to the number of occurrences. You can iterate through the array, incrementing the number of occurrences for each number. Then, you can iterate through the hashtable, searching for a key with n occurrences.
If the array has length > m, then pivot on a random element in the array. Find the half of the array that has length m (mod n), and repeat on that half.
This has expected run-time O(N), and requires O(1) additional storage.
The below algo might be what you're looking for. Basically, you create a map with key as the integer value and the value as the number of occurrence.you loop through your array just once and anytime a numbers occurs more than once, you increase it's count
public static void findCount (int[] array,int m, int n){
if(m>n){
thrown new IllegalArgumentException("m is greater than n");
}
Map<Integer,Integer> intCount = new HashMap<Integer,Integer>();
for(int i = 0; i<array.length; i++){
if (!intCount.containsKey(array[i])) intCount.put(array[i], 0);
intCount.put(array[i], intCount.get(array[i]) + 1);
}
for (Map.Entry<String,Integer> entry : words.entrySet()) {
Integer key = entry.getKey();
Integer value = entry.getValue();
if(value==m){
System.out.println("Value "+key+" Occurs "+value+" times");
}
}
}
The case n=2, m=1 can be done by xoring all the numbers together in the array A.
This idea can be generalised by counting (modulo n) the number of elements in A with the i'th bit set. That count is non-zero if and only if the i'th bit of the answer is set.
This gives you an O(N.log(max(A))) way to compute the solution using O(log(n)) additional storage.
This doesn't achieve the O(log(N)) run-time and O(log(m)) storage complexities given in the question, but it seems an interesting approach.
I have an array
A[4]={4,5,9,1}
I need it would give the first 3 top elements like 9,5,4
I know how to find the max element but how to find the 2nd and 3rd max?
i.e if
max=A[0]
for(i=1;i<4;i++)
{
if (A[i]>max)
{
max=A[i];
location=i+1;
}
}
actually sorting will not be suitable for my application because,
the position number is also important for me i.e. I have to know in which positions the first 3 maximum is occurring, here it is in 0th,1th and 2nd position...so I am thinking of a logic
that after getting the max value if I could put 0 at that location and could apply the same steps for that new array i.e.{4,5,0,1}
But I am bit confused how to put my logic in code
Consider using the technique employed in the Python standard library. It uses an underlying heap data structure:
def nlargest(n, iterable):
"""Find the n largest elements in a dataset.
Equivalent to: sorted(iterable, reverse=True)[:n]
"""
if n < 0:
return []
it = iter(iterable)
result = list(islice(it, n))
if not result:
return result
heapify(result)
for elem in it:
heappushpop(result, elem)
result.sort(reverse=True)
return result
The steps are:
Make an n length fixed array to hold the results.
Populate the array with the first n elements of the input.
Transform the array into a minheap.
Loop over remaining inputs, replacing the top element of the heap if new data element is larger.
If needed, sort the final n elements.
The heap approach is memory efficient (not requiring more memory than the target output) and typically has a very low number of comparisons (see this comparative analysis).
You can use the selection algorithm
Also to mention that the complexity will be O(n) ie, O(n) for selection and O(n) for iterating, so the total is also O(n)
What your essentially asking is equivalent to sorting your array in descending order. The fastest way to do this is using heapsort or quicksort depending on the size of your array.
Once your array is sorted your largest number will be at index 0, your second largest will be at index 1, ...., in general your nth largest will be at index n-1
you can follw this procedure,
1. Add the n elements to another array B[n];
2. Sort the array B[n]
3. Then for each element in A[n...m] check,
A[k]>B[0]
if so then number A[k] is among n large elements so,
search for proper position for A[k] in B[n] and replace and move the numbers on left in B[n] so that B[n] contains n large elements.
4. Repeat this for all elements in A[m].
At the end B[n] will have the n largest elements.
As mentioned in the title, I want to find the pairs of elements whose difference is K
example k=4 and a[]={7 ,6 23,19,10,11,9,3,15}
output should be :
7,11
7,3
6,10
19,23
15,19
15,11
I have read the previous posts in SO " find pair of numbers in array that add to given sum"
In order to find an efficient solution, how much time does it take? Is the time complexity O(nlogn) or O(n)?
I tried to do this by a divide and conquer technique, but i'm not getting any clue of exit condition...
If an efficient solution includes sorting the input array and manipulating elements using two pointers, then I think I should take minimum of O(nlogn)...
Is there any math related technique which brings solution in O(n). Any help is appreciated..
You can do it in O(n) with a hash table. Put all numbers in the hash for O(n), then go through them all again looking for number[i]+k. Hash table returns "Yes" or "No" in O(1), and you need to go through all numbers, so the total is O(n). Any set structure with O(1) setting and O(1) checking time will work instead of a hash table.
A simple solution in O(n*Log(n)) is to sort your array and then go through your array with this function:
void find_pairs(int n, int array[], int k)
{
int first = 0;
int second = 0;
while (second < n)
{
while (array[second] < array[first]+k)
second++;
if (array[second] == array[first]+k)
printf("%d, %d\n", array[first], array[second]);
first++;
}
}
This solution does not use extra space unlike the solution with a hashtable.
One thing may be done using indexing in O(n)
Take a boolean array arr indexed by the input list.
For each integer i is in the input list then set arr[i] = true
Traverse the entire arr from the lowest integer to the highest integer as follows:
whenever you find a true at ith index, note down this index.
see if there arr[i+k] is true. If yes then i and i+k numbers are the required pair
else continue with the next integer i+1