F. Good Pairs
binary search; data structures; dp
You are given an array \(a\) consisting of \(n\) integers and an integer \(k\).A pair \((l,r)\) is good if there exists a sequence of indices \(i_1, i_2, \dots, i_m\) such that \(i_1=l\) and \(i_m=r\); \(i_j < i_{j+1}\) for all \(1 \leq j < m\); and \(|a_{i_j}-a_{i_{j+1}}| \leq k\) for all \(1 \leq j < m\). Find the number of pairs \((l,r)\) (\(1 \leq l \leq r \leq n\)) that are good.
Each test contains multiple test cases. The first line contains a single integer \(t\) (\(1 \leq t \leq 10^5\)) — the number of test cases. The description of the test cases follows.The first line of each test case contains two space-separated integers \(n\) and \(k\) (\(1 \leq n \leq 5 \cdot 10^5\); \(0 \leq k \leq 10^5\)) — the length of the array \(a\) and the integer \(k\).The second line of each test case contains \(n\) space-separated integers \(a_1,a_2,\ldots,a_n\) (\(1 \leq a_i \leq 10^5\)) — representing the array \(a\).It is guaranteed that the sum of \(n\) over all test cases does not exceed \(5 \cdot 10^5\).
For each test case, print the number of good pairs.
In the first test case, good pairs are \((1,1)\), \((1,2)\), \((1,3)\), \((2,2)\), \((2,3)\), and \((3,3)\).In the second test case, good pairs are \((1,1)\), \((1,3)\), \((1,4)\), \((2,2)\), \((2,3)\), \((2,4)\), \((3,3)\), \((3,4)\) and \((4,4)\). Pair \((1,4)\) is good because there exists a sequence of indices \(1, 3, 4\) which satisfy the given conditions.