Suppose we have an array A of n different numbers and another positive integer K, we have to find the longest sub-sequence in the array having Least Common Multiple (LCM) at most K. After than return the LCM and the length of the sub-sequence, following the indexes (starting from 0) of the elements of the obtained sub-sequence. Otherwise, return -1.
So, if the input is like A = [3, 4, 5, 6], K = 20, then the output will be LCM = 12, Length = 3, Indexes = [0,1,3]
To solve this, we will follow these steps −
n := size of A
my_dict := a map
for i in range 0 to n, do
my_dict[A[i]] := my_dict[A[i]] + 1
count := an array of size (k + 1) fill with 0
for each key in my_dict, do
if key <= k, then
i := 1
while key * i <= k, do
count[key * i] := count[key * i] + my_dict[key]
i := i + 1
otherwise,
come out from the loop
lcm := 0, size := 0
for i in range 1 to k + 1, do
if count[i] > size, then
size := count[i]
lcm := i
if lcm is same as 0, then
return -1
otherwise,
display lcm and size
for i in range 0 to n, do
if lcm mod A[i] is same as 0, then
display i
Example
Let us see the following implementation to get better understanding −
from collections import defaultdict def get_seq(A,k): n = len(A) my_dict = defaultdict(lambda:0) for i in range(0, n): my_dict[A[i]] += 1 count = [0] * (k + 1) for key in my_dict: if key <= k: i = 1 while key * i <= k: count[key * i] += my_dict[key] i += 1 else: break lcm = 0 size = 0 for i in range(1, k + 1): if count[i] > size: size = count[i] lcm = i if lcm == 0: print(-1) else: print("LCM = {0}, Length = {1}".format(lcm, size)) print("Index values: ", end = "") for i in range(0, n): if lcm % A[i] == 0: print(i, end = " ") k = 20 A = [3, 4, 5, 6] get_seq(A,k)
Input
[3, 4, 5, 6] , 20
Output
LCM = 12, Length = 3 Index values: 0 1 3