power_set [directed_power_set ( []return power_set the number of recursive callsc(nsatisfies the recurrence ( ( )which solves to (no( since we spend (ntime within callthe time complexity is ( the space complexity is ( )since there are subsetsand the average subset size is / if we just want to print the subsetsrather than returning all of themwe simply perform print instead of adding the subset to the resultwhich reduces the space complexity to ( )--the time complexity remains the same for given ordering of the elements of sthere exists one-to-one correspondence between the bit arrays of length and the set of all subsets of --the in the -length bit array indicate the elements of in the subset corresponding to for exampleif {abcd}the bit array denotes the subset {acdthis observation can be used to derive nonrecursive algorithm for enumerating subsets in particularwhen is less than or equal to the width of an integer on the architecture (or languagewe are working onwe can enumerate bit arrays by enumerating integers in [ and examining the indices of bits set in these integers these indices are determined by first isolating the lowest set bit by computing &~( )and then getting the index by computing log def generate_power_set ( )power_set [for int_for_subset in range ( <len( ))bit_array int_for_subset subset [while bit_array subset append (int(math log bit_array ~bit_array )))bit_array &bit_array power_set append subset return power_set since each set takes (ntime to computethe time complexity is ( in practicethis approach is very fast furthermoreits space complexity is (nwhen we want to just enumerate subsetse to print themrather that to return all the subsets variantsolve this problem when the input array may have duplicatesi denotes multiset you should not repeat any multiset for exampleif id then you should return hih ih ih , ih ih ih , ih ih ih