update

java/src/PointInPolygon.java

@@ -102,9 +102,11 @@ static int[][] getRandomPolygon(int n, int maxWidth, int maxHeight) {
 			}
 			boolean ok = true;
 			for (int i = 0; i < n; i++) {
-				int i1 = (i - 1 + n) % n;
-				int i2 = (i + 1) % n;
-				ok &= x[i1] * y[i2] - x[i2] * y[i1] != 0 || x[i1] * x[i2] + y[i1] * y[i2] <= 0;
+				long x1 = x[(i - 1 + n) % n] - x[i];
+				long y1 = y[(i - 1 + n) % n] - y[i];
+				long x2 = x[(i + 1) % n] - x[i];
+				long y2 = y[(i + 1) % n] - y[i];
+				ok &= x1 * y2 - x2 * y1 != 0 || x1 * x2 + y1 * y2 <= 0;
 			}
 			for (int i2 = 0, i1 = p.length - 1; i2 < p.length; i1 = i2++)
 				for (int j2 = 0, j1 = p.length - 1; j2 < p.length; j1 = j2++)

java/src/PointLocation.java

@@ -0,0 +1,373 @@
+import java.util.*;
+
+public class PointLocation {
+
+	public static int[] locatePoints(int[][] polygonX, int[][] polygonY, int[] qx, int[] qy) {
+		Map<Long, Integer> endPoints = new HashMap<>();
+		Map<Integer, NavigableSet<Segment>> verticalSegments = new HashMap<>();
+		List<Event> events = new ArrayList<>();
+		for (int k = 0; k < polygonX.length; ++k) {
+			int n = polygonX[k].length;
+			for (int i = 0, j = n - 1; i < n; j = i++) {
+				endPoints.put(((long) polygonX[k][i] << 32) + polygonY[k][i], k);
+				Segment s = new Segment(polygonX[k][i], polygonY[k][i], polygonX[k][j], polygonY[k][j], k);
+				if (s.x1 != s.x2) {
+					events.add(new Event(s.x1, s.y1, 1, s));
+					events.add(new Event(s.x2, s.y2, -1, s));
+				} else {
+					if (!verticalSegments.containsKey(s.x1)) {
+						verticalSegments.put(s.x1, new TreeSet<>(segmentComparator));
+					}
+					verticalSegments.get(s.x1).add(s);
+				}
+			}
+		}
+		Collections.sort(events, eventComparator);
+		Treap[] treaps = new Treap[events.size()];
+		int[] eventsX = new int[events.size()];
+		int cnt = 0;
+		Treap treap = null;
+		for (Event event : events) {
+			if (cnt == 0 || eventsX[cnt - 1] < event.x)
+				++cnt;
+			if (event.type == 1) {
+				treap = insert(treap, event.segment);
+			} else if (event.type == -1) {
+				treap = remove(treap, event.segment);
+			}
+			eventsX[cnt - 1] = event.x;
+			treaps[cnt - 1] = treap;
+		}
+		int[] res = new int[qx.length];
+		for (int i = 0; i < qx.length; i++) {
+			Integer polygonId = endPoints.get(((long) qx[i] << 32) + qy[i]);
+			if (polygonId != null) {
+				res[i] = polygonId;
+				continue;
+			}
+			NavigableSet<Segment> vSegments = verticalSegments.get(qx[i]);
+			if (vSegments != null) {
+				SortedSet<Segment> head = vSegments.headSet(new Segment(qx[i], qy[i], qx[i], qy[i], 0), true);
+				if (!head.isEmpty() && head.last().contains(qx[i], qy[i])) {
+					res[i] = head.last().polygonId;
+					continue;
+				}
+			}
+
+			int lo = Arrays.binarySearch(eventsX, 0, cnt, qx[i]);
+			if (lo < 0) lo = -lo - 2;
+			if (lo == -1 || lo == cnt - 1) {
+				res[i] = -1;
+				continue;
+			}
+
+			Segment minRight = new Segment(qx[i], qy[i], qx[i], qy[i], 0);
+			TreapPair treapPair = split(treaps[lo], minRight, true);
+			Treap max = max(treapPair.left);
+			Treap min = min(treapPair.right);
+			if (min != null && min.key.contains(qx[i], qy[i])) {
+				res[i] = min.key.polygonId;
+			} else if (max != null && max.key.contains(qx[i], qy[i])) {
+				res[i] = max.key.polygonId;
+			} else if (getSize(treapPair.left) % 2 == 1) {
+				res[i] = max.key.polygonId;
+			} else {
+				res[i] = -1;
+			}
+		}
+		return res;
+	}
+
+	static final Comparator<Segment> segmentComparator = (a, b) -> {
+		if (a.x1 == b.x1 && a.y1 == b.y1) {
+			long v = cross(a.x1, a.y1, a.x2, a.y2, b.x2, b.y2);
+			if (v != 0)
+				return v > 0 ? -1 : 1;
+		} else if (a.x1 < b.x1) {
+			long v = cross(a.x1, a.y1, a.x2, a.y2, b.x1, b.y1);
+			if (v != 0)
+				return v > 0 ? -1 : 1;
+		} else {
+			long v = cross(b.x1, b.y1, b.x2, b.y2, a.x1, a.y1);
+			if (v != 0)
+				return v < 0 ? -1 : 1;
+		}
+		return Integer.compare(a.y1, b.y1);
+	};
+
+	static class Segment {
+		final int x1, y1, x2, y2, polygonId;
+
+		public Segment(int x1, int y1, int x2, int y2, int polygonId) {
+			if (x1 < x2 || x1 == x2 && y1 < y2) {
+				this.x1 = x1;
+				this.y1 = y1;
+				this.x2 = x2;
+				this.y2 = y2;
+			} else {
+				this.x1 = x2;
+				this.y1 = y2;
+				this.x2 = x1;
+				this.y2 = y1;
+			}
+			this.polygonId = polygonId;
+		}
+
+		public boolean contains(int x, int y) {
+			long a = y2 - y1;
+			long b = x1 - x2;
+			long c = -a * x1 - b * y1;
+			return x >= x1 && x <= x2 && (y >= y1 || y >= y2) && (y <= y1 || y <= y2) && a * x + b * y + c == 0;
+		}
+	}
+
+	static final Comparator<Event> eventComparator = (a, b) -> {
+		if (a.x != b.x)
+			return a.x < b.x ? -1 : 1;
+		if (a.type != b.type)
+			return a.type < b.type ? -1 : 1;
+		return Integer.compare(a.y, b.y);
+	};
+
+	static class Event {
+		final int x, y;
+		final int type;
+		final Segment segment;
+
+		public Event(int x, int y, int type, Segment segment) {
+			this.x = x;
+			this.y = y;
+			this.type = type;
+			this.segment = segment;
+		}
+	}
+
+	static long cross(long ax, long ay, long bx, long by, long cx, long cy) {
+		return (bx - ax) * (cy - ay) - (by - ay) * (cx - ax);
+	}
+
+	static Random random = new Random();
+
+	// persistent treap
+	static class Treap {
+		final Segment key;
+		final Treap left;
+		final Treap right;
+		final int size;
+
+		Treap(Segment key) {
+			this.key = key;
+			left = null;
+			right = null;
+			size = 1;
+		}
+
+		public Treap(Segment key, Treap left, Treap right) {
+			this.key = key;
+			this.left = left;
+			this.right = right;
+			this.size = getSize(left) + getSize(right) + 1;
+		}
+	}
+
+	static int getSize(Treap root) {
+		return root == null ? 0 : root.size;
+	}
+
+	static class TreapPair {
+		Treap left;
+		Treap right;
+
+		TreapPair(Treap left, Treap right) {
+			this.left = left;
+			this.right = right;
+		}
+	}
+
+	static TreapPair split(Treap root, Segment minRight, boolean inclusive) {
+		if (root == null)
+			return new TreapPair(null, null);
+		int cmp = segmentComparator.compare(root.key, minRight);
+		if (cmp > 0 || cmp == 0 && inclusive) {
+			TreapPair leftSplit = split(root.left, minRight, inclusive);
+			return new TreapPair(leftSplit.left, new Treap(root.key, leftSplit.right, root.right));
+		} else {
+			TreapPair rightSplit = split(root.right, minRight, inclusive);
+			return new TreapPair(new Treap(root.key, root.left, rightSplit.left), rightSplit.right);
+		}
+	}
+
+	static Treap merge(Treap left, Treap right) {
+		if (left == null)
+			return right;
+		if (right == null)
+			return left;
+		if (random.nextInt(left.size + right.size) < left.size) {
+			return new Treap(left.key, left.left, merge(left.right, right));
+		} else {
+			return new Treap(right.key, merge(left, right.left), right.right);
+		}
+	}
+
+	static Treap insert(Treap root, Segment x) {
+		TreapPair t = split(root, x, true);
+		return merge(merge(t.left, new Treap(x)), t.right);
+	}
+
+	static Treap remove(Treap root, Segment x) {
+		TreapPair t = split(root, x, true);
+		return merge(t.left, split(t.right, x, false).right);
+	}
+
+	static Treap min(Treap root) {
+		while (root != null && root.left != null)
+			root = root.left;
+		return root;
+	}
+
+	static Treap max(Treap root) {
+		while (root != null && root.right != null)
+			root = root.right;
+		return root;
+	}
+
+	// random test
+	public static void main(String[] args) {
+		for (int step = 0; step < 10_000; step++) {
+			int range = 20;
+			int R = 3;
+			int C = 3;
+			int[][] polygonX = new int[R * C][];
+			int[][] polygonY = new int[R * C][];
+			for (int r = 0; r < R; r++) {
+				for (int c = 0; c < C; c++) {
+					int n = random.nextInt(10) + 3;
+					int[][] xy = getRandomPolygon(n, range, range);
+					polygonX[r * C + c] = xy[0];
+					polygonY[r * C + c] = xy[1];
+					for (int i = 0; i < n; i++) {
+						polygonX[r * C + c][i] += c * (range + 1);
+						polygonY[r * C + c][i] += r * (range + 1);
+					}
+				}
+			}
+			int queries = random.nextInt(10) + 1;
+			int[] qx = new int[queries];
+			int[] qy = new int[queries];
+			int[] res1 = new int[queries];
+			Arrays.fill(res1, -1);
+			for (int i = 0; i < queries; i++) {
+				qx[i] = random.nextInt((C + 2) * (range + 1)) - range;
+				qy[i] = random.nextInt((R + 2) * (range + 1)) - range;
+				for (int j = 0; j < polygonX.length; j++) {
+					int v = pointInPolygon(qx[i], qy[i], polygonX[j], polygonY[j]);
+					if (v >= 0) {
+						res1[i] = j;
+						break;
+					}
+				}
+			}
+			int[] res2 = locatePoints(polygonX, polygonY, qx, qy);
+			if (!Arrays.equals(res1, res2))
+				throw new RuntimeException();
+		}
+	}
+
+	static int pointInPolygon(int qx, int qy, int[] x, int[] y) {
+		int n = x.length;
+		int cnt = 0;
+		for (int i = 0, j = n - 1; i < n; j = i++) {
+			if (y[i] == qy && (x[i] == qx || y[j] == qy && (x[i] <= qx || x[j] <= qx) && (x[i] >= qx || x[j] >= qx)))
+				return 0; // boundary
+			if ((y[i] > qy) != (y[j] > qy)) {
+				long det = (long) (x[i] - qx) * (y[j] - qy) - (long) (x[j] - qx) * (y[i] - qy);
+				if (det == 0)
+					return 0; // boundary
+				if ((det > 0) != (y[j] - y[i] > 0))
+					++cnt;
+			}
+		}
+		return cnt % 2 == 0 ? -1 /* exterior */ : 1 /* interior */;
+	}
+
+	static int[][] getRandomPolygon(int n, int maxWidth, int maxHeight) {
+		int[] x = new int[n];
+		int[] y = new int[n];
+		int[] p = new int[n];
+		while (true) {
+			for (int i = 0; i < n; i++) {
+				x[i] = random.nextInt(maxWidth);
+				y[i] = random.nextInt(maxHeight);
+				p[i] = i;
+			}
+			for (boolean improved = true; improved; ) {
+				improved = false;
+				for (int i = 0; i < n; i++) {
+					for (int j = 0; j < n; j++) {
+						int[] p1 = p.clone();
+						reverse(p1, i, j);
+						if (len(x, y, p) > len(x, y, p1)) {
+							p = p1;
+							improved = true;
+						}
+					}
+				}
+			}
+			int[] tx = x.clone();
+			int[] ty = y.clone();
+			for (int i = 0; i < n; i++) {
+				x[i] = tx[p[i]];
+				y[i] = ty[p[i]];
+			}
+			boolean ok = true;
+			for (int i = 0; i < n; i++) {
+				long x1 = x[(i - 1 + n) % n] - x[i];
+				long y1 = y[(i - 1 + n) % n] - y[i];
+				long x2 = x[(i + 1) % n] - x[i];
+				long y2 = y[(i + 1) % n] - y[i];
+				ok &= x1 * y2 - x2 * y1 != 0 || x1 * x2 + y1 * y2 <= 0;
+			}
+			for (int i2 = 0, i1 = p.length - 1; i2 < p.length; i1 = i2++)
+				for (int j2 = 0, j1 = p.length - 1; j2 < p.length; j1 = j2++)
+					ok &= i1 == j1 || i1 == j2 || i2 == j1
+							|| !isCrossOrTouchIntersect(x[i1], y[i1], x[i2], y[i2], x[j1], y[j1], x[j2], y[j2]);
+			if (ok)
+				return new int[][]{x, y};
+		}
+	}
+
+	// http://en.wikipedia.org/wiki/2-opt
+	static void reverse(int[] p, int i, int j) {
+		int n = p.length;
+		// reverse order from i to j
+		while (i != j) {
+			int t = p[j];
+			p[j] = p[i];
+			p[i] = t;
+			i = (i + 1) % n;
+			if (i == j) break;
+			j = (j - 1 + n) % n;
+		}
+	}
+
+	static double len(int[] x, int[] y, int[] p) {
+		double res = 0;
+		for (int i = 0, j = p.length - 1; i < p.length; j = i++) {
+			double dx = x[p[i]] - x[p[j]];
+			double dy = y[p[i]] - y[p[j]];
+			res += Math.sqrt(dx * dx + dy * dy);
+		}
+		return res;
+	}
+
+	static boolean isCrossOrTouchIntersect(long x1, long y1, long x2, long y2, long x3, long y3, long x4, long y4) {
+		if (Math.max(x1, x2) < Math.min(x3, x4) || Math.max(x3, x4) < Math.min(x1, x2)
+				|| Math.max(y1, y2) < Math.min(y3, y4) || Math.max(y3, y4) < Math.min(y1, y2))
+			return false;
+		long z1 = (x2 - x1) * (y3 - y1) - (y2 - y1) * (x3 - x1);
+		long z2 = (x2 - x1) * (y4 - y1) - (y2 - y1) * (x4 - x1);
+		long z3 = (x4 - x3) * (y1 - y3) - (y4 - y3) * (x1 - x3);
+		long z4 = (x4 - x3) * (y2 - y3) - (y4 - y3) * (x2 - x3);
+		return (z1 <= 0 || z2 <= 0) && (z1 >= 0 || z2 >= 0) && (z3 <= 0 || z4 <= 0) && (z3 >= 0 || z4 >= 0);
+	}
+}