This is the algorithm: I think its time complexity is O(nlogn) but i am not sure
k=1;
while (k<=n) do
j=1;
while (j<=k) do
sum=sum+1;
j=j+1;
k=k*2;
The inner loop at the first time performs 1 iteration, at the second 2 iterations. The sequence goes like 1, 2, 4, 8, 16, 32, ... as long as it is smaller than or equal to n
. The sequence may have Θ(log(n))
elements but its sum is Θ(n)
. This is because
1 + 2 + 4 + ... + 2^k = 2 * 2^k - 1
and we know n/2 < 2^k <= n
. So the inner loop is performed Θ(n)
times and each inner loop execution requires constant number of instructions.
The rest of the code is just log(n)
assignments to j = 1
and log(n)
doubles of k
.
So the time complexity of the algorithm is Θ(n)
.
// code | max times executed
k=1; | 1
while (k<=n) do | log n
j=1; | log n
while (j<=k) do | log n * n
sum=sum+1; | log n * n
j=j+1; | log n * n
k=k*2; | log n
So the O complexity would seem to be n log n.
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