Core idea: O(n) scan with three local cases. Simulating the whole path is“O(n²)”; the key insight is that segment“i” can only ever intersect segment“i-3”,“i-4”, or“i-5” — everything older is walled off. So one pass with a sliding window of the last 6 lengths is enough.class Solution {public boolean isSelfCrossing(int[] x) {int n x.length;for (int i 3; i n; i) {// Case 1: segment i crosses segment i-3 (transverse crossing)if (x[i] x[i - 2] x[i - 3] x[i - 1]) return true;// Case 2: segment i overlaps/touches segment i-4 (they are parallel) if (i 4 x[i - 1] x[i - 3] x[i] x[i - 4] x[i - 2]) return true; // Case 3: segment i crosses segment i-5 (shrinking spiral closing in) if (i 5 x[i - 2] x[i - 4] // i-1 sits inside i-3 x[i - 1] x[i - 5] x[i - 3] x[i - 3] x[i - 1] x[i] x[i - 4] x[i - 2]) return true; } return false; }}What the three cases meanCase 1 — the spiral has started shrinking: step“i” is long enough to reach back past“i-2” (“x[i] x[i-2]”), and“i-1” is short enough that“i” can slide over it (“x[i-3] x[i-1]”). Example“[2,1,1,2]”: the 4th side cuts straight through the 1st.Case 2 —“i” and“i-4” run parallel on the same line (that’s what“x[i-1] x[i-3]” expresses), and their spans overlap. This catches touching, not just crossing, so“[1,1,1,1]” returns“true”.Case 3 — the innermost coil:“i” reaches back over“i-5” while“i-1” is nested inside“i-3”. All four inequalities together pin the overlap in both axes.Key pointsThe loop starts at“i 3” since fewer than 4 segments can never self-cross.Conditions are“”, not“” — a path that merely touches an earlier point counts as crossing.“x[i] x[i-4]” stays well within“int” for the given constraints (“x[i] ≤ 10^5”); switch to“long” if values can approach“10^9”.Complexity:“O(n)” time,“O(1)” space.Verified against a brute-force geometry check (build the polyline, test every non-adjacent segment pair, including parallel overlap and endpoint touching): exhaustive over all arrays of length ≤ 7 with values 1–4 (21,844 cases), plus 260,000 random cases up to length 32. All match. Official samples:“[2,1,1,2] → true”,“[1,2,3,4] → false”,“[1,1,1,1] → true”.“n 100,000” runs in under 1 ms.