LeetCode 446. Arithmetic Slices II - Subsequence

Problem Statement

(Source) A sequence of numbers is called arithmetic if it consists of at least three elements and if the difference between any two consecutive elements is the same.

For example, these are arithmetic sequences:

1, 3, 5, 7, 9
7, 7, 7, 7
3, -1, -5, -9

The following sequence is not arithmetic.

1, 1, 2, 5, 7

A zero-indexed array A consisting of N numbers is given. A subsequence slice of that array is any sequence of integers (P0, P1, …, Pk) such that 0 ≤ P0 < P1 < … < Pk < N.

A subsequence slice (P0, P1, …, Pk) of array A is called arithmetic if the sequence A[P0], A[P1], …, A[Pk-1], A[Pk] is arithmetic. In particular, this means that k ≥ 2.

The function should return the number of arithmetic subsequence slices in the array A.

The input contains N integers. Every integer is in the range of $-2^{31}$and $2^{31}-1$ and 0 ≤ N ≤ 1000. The output is guaranteed to be less than $2^{31}-1$.

Example:

Input: [2, 4, 6, 8, 10]

Output: 7

Explanation:
All arithmetic subsequence slices are:
[2,4,6]
[4,6,8]
[6,8,10]
[2,4,6,8]
[4,6,8,10]
[2,4,6,8,10]
[2,6,10]

Solution

Tags: Dynamic Programming.

For all possible combinations of positions j and i with j < i, choose A[i] - A[j] as the difference of the Arithmetic Slice with the last two elements being A[j], A[i], add the number of Arithmetic Slice that can be produced by this condition, and update dp[i][A[i] - A[j]], where dp[i][x] stores the number of points before position i that have distance x from A[i] (Condition 1), or recursively have distances x from those points meets Condition 1.

class Solution(object):
    def numberOfArithmeticSlices(self, A):
        """
        :type A: List[int]
        :rtype: int
        """
        res, n = 0, len(A)
        dp = [{} for i in xrange(n)]
        for i in xrange(1, n):
            for j in xrange(0, i):
                dist = A[i] - A[j]
                s = dp[j].get(dist, 0) + 1
                dp[i][dist] = dp[i].get(dist, 0) + s
                res += (s-1)
        return res

Complexity Analysis:

• Time Complexity: $O(n^2)$.
• Space Complexity: $O(n^2)$.

References:
(1) LeetCode Discussion.