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Practice
Contest

Author:Praveen Dhinwa
Tester:Shiplu Hawlader
Editorialist:Lalit Kundu

DIFFICULTY:

EASY

PRE-REQUISITES:

GCD, Precomputation

PROBLEM:

You are given an array A of integers of size N. You will be given Q queries where each query is represented by two integers L, R. You have to find the gcd(Greatest Common Divisor) of the array after excluding the part from range L to R inclusive (1 Based indexing). You are guaranteed that after excluding the part of the array remaining array is non empty.
1 ≤ N, Q≤ 100000

EXPLANATION:

If you do it naively(ie. calculating GCD of remaining array for each query), the worstcase complexity will be O(N * Q).
Let's denote by G(L, R), the GCD of A_L, A_L+1 ... A_R. We can observe that for query [L, R], we need GCD of G(1, L-1) and G(R+1, N).
So, we precalculate prefix and suffix gcd arrays.
If we have:
Prefix_i = GCD of A₁, A₂ ... A_i
Suffix_i = GCD of A_N, A_N-1 ... A_i
answer to query [L, R], would be GCD of Prefix_L-1 and Suffix_R+1.

We can calculate prefix and suffix arrays in O(N) if we notice that:
Prefix_i = GCD(Prefix_i-1, A_i)
Suffix_i = GCD(Suffix_i+1, A_i)

Pseudo Code for building prefix and suffix arrays:

n,a=input
pre[n],suf[n]

//base case
pre[1]=a[1]
suf[n]=a[n]

for i=2 to n:
    pre[i] = gcd(pre[i-1], a[i])

for i=n-1 to 1:
    suf[i] = gcd(suf[i+1], a[i])

So, overall complexity would be O((N + Q) * K), where K is a constant factor for gcd calculation.

ALTERNATIVE SOLUTION:

Use segment trees for range gcd queries. But note that a factor of log N will be increased in complexity.

SOLUTIONS:

Setter's solution
Tester's solution