# HackerRank Minimum Multiple

## Problem statment

The origin statment of the problem can be found here.

In short, this problem ask you to do two operations on an array, one is to query the least common multipler(lcm) of an range [l, r], and another is to update one value of the array.

Since the maximum length of the array is $5 \times 10^4$, an naive $O(n^2)$ algorithm will definitively not acceptable. So we need to find more efficient way to do the query and update operations.

## Segment Tree

The solution is to use a structure called segment tree to make the query and update operations run in $O(log{n})$ time complexity.

Segment tree is often used in scenario of efficient range query and update. The following is an example of segment tree. In segment tree, all the leaf nodes store the information of the segment value, and the internal node store the result of a range [l, r].

The pseudo codes of query and update are:

function query(f, l, r, root):
if l <= root.left and root.right <= r:
return root.val
else if r <= root.leftChild.right:
return query(f, l, r, root.leftChild)
else if l >= root.rightChild.left:
return query(f, l, r, root.rightChild)
else:
return f(query(f, l, r, root.leftChild),
query(f, l, r, root.rightChild))

function update(f, idx, newVal, root):
if idx <= root.left and root.right <= idx:
root.val = newVal
else if idx <= root.leftChild.right:
root.val = f(newVal, root.val)
update(f, idx, newVal, root.leftChild)
else:
root.val = f(newVal, root.val)
update(f, idx, newVal, root.rightChild)


At first glance, the running time of query is not explicit $O(log{n})$. But you can find the proof from the stackoverflow answer, which shows that the running time of the query function is actually $O(log{n})$.

## Solution

Then the solution is trival. We use an segment tree to store the lcm information. The node cover range [l, r] is the lcm of the elements in array[l..r]. So both the update and query operator can be done in $O(log{n})$.

The following is a least common multiple segment tree example. The circle nodes are leaf nodes and store the numbers in the array, and the rectangle nodes store range lcm information as described in the brackets in the node.

Here is the haskell code of the segment tree structure

data SegTree =
Node {
val                   :: Integer
, left, right           :: Int
, leftChild, rightChild :: SegTree
} |
Leaf {
val         :: Integer
, left, right :: Int
}

initSegTree :: Int -> SegTree
initSegTree n = aux 0 (n - 1)
where aux l r
| l == r = Leaf {val = -1, left = l, right = r}
| otherwise =
let mid = (l + r) div 2
in Node { val = -1, left = l, right = r
, leftChild = aux l mid
, rightChild = aux (succ mid) r
}

query :: (Int, Int) -> SegTree -> Integer
query range@(l, r) root
| r < left root = 1
| l > right root = 1
| l <= left root && right root <= r = val root
| otherwise =
lcm (query range (leftChild root)) (query range (rightChild root))

update :: Int -> Integer -> SegTree -> SegTree
update idx newVal root
| left root <= idx && idx <= right root =
case root of
Leaf {} -> root {val = newVal }
_ -> root {val = lcm newVal (val root),
leftChild = lChild, rightChild = rChild }
| otherwise = root
where
lChild = update idx newVal $leftChild root rChild = update idx newVal$ rightChild root


In fact, since haskell is an pure functional languages, it should be a little trival to do programming related with states in it. But since the character of programming contests problems. We do not need to use state manipulation in haskell to get the problem accept, we can just use purely functional style to do the query and update operators using fold in haskell.

The following code is related with solving the problem.

processQueries
:: [(Char, Int, Int)] -> SegTree -> [Integer] -> [Integer]
processQueries [] _ acc = reverse acc
processQueries (('Q', l, r):q) root acc =
let ans = (query (l, r) root) mod modulo
in processQueries q root (ans : acc)
processQueries ((_, idx, value):q) root acc =
let oldVal = query (idx, idx) root
newVal = oldVal * (fromIntegral value)
nroot = update idx newVal root
in processQueries q nroot acc

solve :: Int -> [Integer] -> [(Char, Int, Int)] -> [Integer]
solve n arr queries = processQueries queries tree []
where
tree = foldl' (\ root (idx, v) -> update idx v root)
(initSegTree n)
(zip [0..] arr)


The only thing one should notice is that the depth limit of stack on HackerRank is quite small. So we need to avoid too deep recursive call, and use the strick version of foldl' call in the solve function, and also avoid explicit recursive call in processQueries, but to optimize the function as an tail recursive call.

The full code can be found here.

11 April 2015