4. (20 Points) Provide best-case and worst-case running time and space complexit

Question

4. (20 Points) Provide best-case and worst-case running time and space complexity analysis in Big-Oh notation for the following pow 2 method. For each case, provide an example input pair and brief explanation, Big-O Notation Example Input Explanation Best-Case Running Time Worst-Case Running Time Best-Case Space Complexity Worst-Case Space Complexity public static long pow_2(long x, int n) { if (n-0) if (n1) if(n % 2-0) { return 1; return x; return pow 2( x, n / 2) pow2x, n/ 2 ); return pow 2(x * x, n / 2) * x; else

Explanation / Answer

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Big -O Notation Example Input Explanation Best Case Running Time O(1) n=0 or n=1 and any value of x As per the code,for n=0 and n=1, we can return a constant value which is 1(n=0) and x(n=1). Constant value can be directly picked and returned in constant time,for constant time O(1) is the best explanation. Worst Case Running Time O(n) x=2000,n=7000 For any large value,we try to find out the power of x raised to n by breaking n into smaller ranks(n/2), after that we call the function recursively. So for any large number n ,there can be as many calls to the recursive function,until the value of n/2 in first call,n/2^2 in second call.... becomes minimal. So the worst case time complexity is respresentable as O(n/2**) as 2** is constant it can be written as O(n). Best Case Space Complexity. O(1) for any value of x and n As the code is the recursive call,stack is used to store the intermediate values,which is emptied at the time of obtaining results. So the total space required will be the space required to store results only,which is constant,hence O(1) space complexity is required either in best case or worst case scenario. Worst Case Space Complexity. O(1) for any value of x and n As the code is the recursive call,stack is used to store the intermediate values,which is emptied at the time of obtaining results. So the total space required will be the space required to store results only,which is constant,hence O(1) space complexity is required either in best case or worst case scenario.

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