Minimum numbers needed to express every integer below N as a sum
We have an integer N. We need to express N as a sum of K integers such that by adding some(or all) of these integers we can get all the numbers in the range[1, N]. What is the minimum value of K?
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Input : N = 7 Output : 3 Explanation : Three integers are 1, 2, 4. By adding some(or all) of these groups we can get all number in the range 1 to N. 1; 2; 1+2=3; 4; 1+4=5; 2+4=6; 1+2+4=7 Input : N = 32 Output : 6 Explanation : Six integers are 1, 2, 4, 8, 16, 1.
1st we solve the problem for small numbers by hand.
n=1 : 1
n=2 : 1, 1
n=3 : 1, 2
n=4 : 1, 2, 1
n=5 : 1, 2, 2
n=6 : 1, 2, 3
n=7 : 1, 2, 4
n=8 : 1, 2, 4, 1
If we inspect this closely we can see that if then the integers are . Which is just another way of saying .So now we know for minimum value of K is m.
Now we inspect what happens for .For we just add a new integer 1 to our list of integers. Realize that for every number from we can increase the newly added integer by 1 and that will be the optimal list of integers. To verify look at N=4 to N=7, minimum K does not change; only the last integer is increased in each step.
Of course we can implement this in iterative manner in O(log N) time (by inserting successive powers of 2 in the list and the last element will be of the form N-(2^n-1)). But this is exactly same as finding the length of binary expression of N which also can be done in O(log N) time.
Minimum value of K is = 6
Please see count set bits for more efficient methods to count set bits in an integer.