How to adapt Fenwick tree to answer range minimum queries

Question

Fenwick tree is a data-structure that gives an efficient way to answer to main queries:

• add an element to a particular index of an array update(index, value)

Answer 1

The Fenwick tree structure works for addition because addition is invertible. It doesn't work for minimum, because as soon as you have a cell that's supposed to be the minimum of two or more inputs, you've lost information potentially.

If you're willing to double your storage requirements, you can support RMQ with a segment tree that is constructed implicitly, like a binary heap. For an RMQ with n values, store the n values at locations [n, 2n) of an array. Locations [1, n) are aggregates, with the formula A(k) = min(A(2k), A(2k+1)). Location 2n is an infinite sentinel. The update routine should look something like this.

def update(n, a, i, x):  # value[i] = x
    i += n
    a[i] = x
    # update the aggregates
    while i > 1:
        i //= 2
        a[i] = min(a[2*i], a[2*i+1])

The multiplies and divides here can be replaced by shifts for efficiency.

The RMQ pseudocode is more delicate. Here's another untested and unoptimized routine.

def rmq(n, a, i, j):  # min(value[i:j])
    i += n
    j += n
    x = inf
    while i < j:
        if i%2 == 0:
            i //= 2
        else:
            x = min(x, a[i])
            i = i//2 + 1
        if j%2 == 0:
            j //= 2
        else:
            x = min(x, a[j-1])
            j //= 2
    return x