Difference Array

Assume we have an array A of size N, and given a list of updates of length K, each of which is in the form of [from, to, diff] which means adding diff to all elements with indexes in range [from, to].

For example:

A: [0, 0, 0, 0]
updates: [[0,1,2], [1,2,1],[3,3,6]]

// after updates
A: [2, 3, 1, 6]

The brute force way is to visit all elements in range [from, to] and add diff to them for each update. It will take O(NK) time.

We can improve this by using difference array, i.e. only storing the delta value at break points. Let delta[i] be U[i] - U[i-1] where U[i] is the sum of all diffs to A[i].

A: [0, 0, 0, 0]
updates: [[0,1,2], [1,2,1],[3,3,6]]

U: [2, 3, 1, 6]
delta: [2, 1, -2, 5]
// after update
A: [2, 2+1, 2+1-2, 2+1-2+5] = [2, 3, 1, 6]

For update[i] = [from, to, diff], we do:

• delta[from] += diff

• delta[to+1] -= diff

updates: [[0,1,2], [1,2,1],[3,3,6]]
// apply first update
delta: [2, 0, -2, 0]
// apply second update
delta: [2, 1, -2, -1]
// apply third update
delta: [2, 1, -2, 5]

So in the end we just need to traverse delta array to recover the U array.

Problems

• 1674. Minimum Moves to Make Array Complementary (Medium)

• 1854. Maximum Population Year (Easy)