Li chao

src/li_chao.rs

@@ -0,0 +1,110 @@
+/// A structure for answering maximum queries on a set of linear functions. Supports two
+/// operations: inserting a linear function and querying for maximum at a given point. 
+/// The queries can be done in any order, and we can do all the calculations using integers.
+/// https://cp-algorithms.com/geometry/convex_hull_trick.html#li-chao-tree
+/// Compared to the code in the above link, this implementation further improves the algorithm by
+/// reducing the number of nodes to (right - left). This is done by removing the midpoint of a
+/// segment from both children. Even better, this allows the index of a node to just be the
+/// midpoint of the interval!
+
+/// Just like normal segment trees, this could be modified to a dynamic tree when the range is
+/// huge, or if the queries are known in advance the x-coordinates can be compressed.
+/// (it can also be made persistent!).
+
+pub struct LiChaoTree {
+    left: i64,
+    right: i64,
+    lines: Vec<(i64, i64)>,
+}
+
+impl LiChaoTree {
+    /// Creates a new tree, built to handle queries on the interval [left, right).
+    pub fn new(left: i64, right: i64) -> Self {
+        Self {
+            left,
+            right,
+            lines: vec![(0, std::i64::MIN); (right - left) as usize],
+        }
+    }
+
+    /// Every node in the tree has the property that the line that maximizes its midpoint is found
+    /// either in the node or one of its ancestors.  When we visit a node, we compute the winner at
+    /// the midpoint of the node. The winner is stored in the node. The loser can still possibly
+    /// beat the winner on some segment, either to the left or to the right of the current
+    /// midpoint, so we propagate it to that segment. This sequence ensures that the invariant is
+    /// kept.
+    fn max_with_impl(&mut self, mut m: i64, mut b: i64, l: i64, r: i64) {
+        if r <= l {
+            return;
+        }
+        let ix = ((r - self.left + l - self.left) / 2) as usize;
+        let mid = self.left + (ix as i64);
+        let (ref mut m_ix, ref mut b_ix) = self.lines[ix];
+        if m * mid + b > *m_ix * mid + *b_ix {
+            std::mem::swap(&mut m, m_ix);
+            std::mem::swap(&mut b, b_ix);
+        }
+        if m < *m_ix {
+            self.max_with_impl(m, b, l, mid);
+        } else if m > *m_ix {
+            self.max_with_impl(m, b, mid + 1, r);
+        }
+    }
+
+    /// Adds the line with slope m and intercept b. O(log N) complexity.
+    pub fn max_with(&mut self, m: i64, b: i64) {
+        self.max_with_impl(m, b, self.left, self.right);
+    }
+
+    /// Because of the invariant established by add_line, we know that the best line for a given
+    /// point is stored in one of the ancestors of its node. So we accumulate the maximum answer as
+    /// we go back up the tree.
+    fn evaluate_impl(&self, x: i64, l: i64, r: i64) -> i64 {
+        if r == l {
+            return i64::MIN;
+        }
+        let ix = ((r - self.left + l - self.left) / 2) as usize;
+        let mid = ix as i64 + self.left;
+        let y = self.lines[ix].0 * x + self.lines[ix].1;
+        if x == mid {
+            y
+        } else if x < mid {
+            self.evaluate_impl(x, l, mid).max(y)
+        } else {
+            self.evaluate_impl(x, mid + 1, r).max(y)
+        }
+    }
+
+    /// Finds the maximum mx+b among all lines in the structure. O(log N) complexity.
+    pub fn evaluate(&self, x: i64) -> i64 {
+        self.evaluate_impl(x, self.left, self.right)
+    }
+}
+
+#[cfg(test)]
+mod test {
+    use super::*;
+
+    #[test]
+    fn test_li_chao_tree() {
+        let lines = [(0, -3), (-1, 0), (1, -8), (-2, 1), (1, -4)];
+        let xs = [0, 1, 2, 3, 4, 5];
+        // results[i] consists of the expected y-coordinates after processing
+        // the first i+1 lines.
+        let results = [
+            [-3, -3, -3, -3, -3, -3],
+            [0, -1, -2, -3, -3, -3],
+            [0, -1, -2, -3, -3, -3],
+            [1, -1, -2, -3, -3, -3],
+            [1, -1, -2, -1, 0, 1],
+        ];
+        let mut li_chao = LiChaoTree::new(0, 6);
+
+        assert_eq!(li_chao.evaluate(0), std::i64::MIN);
+        for (&(slope, intercept), expected) in lines.iter().zip(results.iter()) {
+            li_chao.max_with(slope, intercept);
+            let ys: Vec<i64> = xs.iter().map(|&x| li_chao.evaluate(x)).collect();
+            assert_eq!(expected, &ys[..]);
+        }
+    }
+}

src/lib.rs

@@ -2,6 +2,7 @@
 
 pub mod caching;
 pub mod graph;
+pub mod li_chao;
 pub mod math;
 pub mod order;
 pub mod range_query;