1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186

use crate::nodes::{LazyNode, Node, PersistentNode};


/// Lazy persistent segment tree, it saves every version of itself, it has range queries and range updates.
/// It uses `O(n+q*log(n))` space, where `q` is the amount of updates, and assuming that each node uses `O(1)` space.
pub struct LazyPersistentSegmentTree<T: PersistentNode + LazyNode> {
    nodes: Vec<T>,
    roots: Vec<usize>,
    n: usize,
}

impl<T> LazyPersistentSegmentTree<T>
where
    T: PersistentNode + LazyNode + Clone, // + std::fmt::Debug,
{
    /// Builds a lazy persistent segment tree from slice, each element of the slice will correspond to a leaf of the segment tree.
    /// It has time complexity of `O(n*log(n))`, assuming that [combine](Node::combine) has constant time complexity.
    pub fn build(values: &[T]) -> Self {
        let n = values.len();
        let mut temp = Self {
            nodes: Vec::with_capacity(4*n),
            roots: Vec::with_capacity(1),
            n,
        };
        let root = temp.build_helper(values, 0, n - 1);
        temp.roots.push(root);
        temp
    }

    fn build_helper(&mut self, values: &[T], i: usize, j: usize) -> usize {
        let mid = (i + j) / 2;
        if i == j {
            let curr_node = self.nodes.len();
            self.nodes.push(values[i].clone());
            return curr_node;
        }
        let left_node = self.build_helper(values, i, mid);
        let right_node = self.build_helper(values, mid + 1, j);
        let curr_node = self.nodes.len();
        self.nodes
            .push(T::combine(&self.nodes[left_node], &self.nodes[right_node]));
        self.nodes[curr_node].set_children(left_node, right_node);
        curr_node
    }

    /// Returns the result from the range \[left,right\] from the version of the segment tree.
    /// It returns None if and only if range is empty.
    /// It will **panic** if left or right are not in [0,n), or if version is not in [0,[versions](LazyPersistentSegmentTree::versions)).
    /// It has time complexity of `O(log(n))`, assuming that [combine](Node::combine), [update_lazy_value](LazyNode::update_lazy_value) and [update_lazy_value](LazyNode::lazy_update) have constant time complexity.
    pub fn query(&mut self, version: usize, left: usize, right: usize) -> Option<T> {
        self.query_helper(self.roots[version], left, right, 0, self.n - 1)
    }

    fn push(&mut self, curr_node: usize, i: usize, j: usize) {
        if self.nodes[curr_node].lazy_value().is_some() && i != j {
            let left_node = self.nodes.len();
            let right_node = self.nodes.len() + 1;
            self.nodes
                .push(self.nodes[self.nodes[curr_node].left_child()].clone());
            self.nodes
                .push(self.nodes[self.nodes[curr_node].right_child()].clone());
            let (parent_slice, sons_slice) = self.nodes.split_at_mut(curr_node + 1);
            let value = parent_slice[curr_node].lazy_value().unwrap();
            sons_slice[left_node - curr_node - 1].update_lazy_value(value);
            sons_slice[right_node - curr_node - 1].update_lazy_value(value);
        }
        self.nodes[curr_node].lazy_update(i, j);
    }

    fn query_helper(
        &mut self,
        curr_node: usize,
        left: usize,
        right: usize,
        i: usize,
        j: usize,
    ) -> Option<T> {
        if j < left || right < i {
            return None;
        }
        if self.nodes[curr_node].lazy_value().is_some() {
            self.push(curr_node, i, j);
        }
        if left <= i && j <= right {
            return Some(self.nodes[curr_node].clone());
        }
        let mid = (i + j) / 2;
        let left_node = self.nodes[curr_node].left_child();
        let right_node = self.nodes[curr_node].right_child();
        match (
            self.query_helper(left_node, left, right, i, mid),
            self.query_helper(right_node, left, right, mid + 1, j),
        ) {
            (Some(ans_left), Some(ans_right)) => Some(T::combine(&ans_left, &ans_right)),
            (Some(ans_left), None) => Some(ans_left),
            (None, Some(ans_right)) => Some(ans_right),
            (None, None) => None,
        }
    }

    // Creates a new segment tree version from version were the p-th element of the segment tree to value T and update the segment tree correspondingly.
    /// It will panic if p is not in \[0,n), or if version is not in [0,[versions](LazyPersistentSegmentTree::versions)).
    /// It has time complexity of `O(log(n))`, assuming that [combine](Node::combine), [update_lazy_value](LazyNode::update_lazy_value) and [update_lazy_value](LazyNode::lazy_update) have constant time complexity.
    pub fn update(&mut self, version: usize, left: usize, right: usize, value: <T as Node>::Value) {
        let new_root = self.update_helper(self.roots[version], left, right, &value, 0, self.n - 1);
        self.roots.push(new_root);
    }

    fn update_helper(
        &mut self,
        curr_node: usize,
        left: usize,
        right: usize,
        value: &<T as Node>::Value,
        i: usize,
        j: usize,
    ) -> usize {
        if j < left || right < i {
            return curr_node;
        }
        let x = self.nodes.len();
        self.nodes.push(self.nodes[curr_node].clone());
        if left <= i && j <= right {
            self.nodes[x].update_lazy_value(value);
            self.push(x, i, j);
            return x;
        }
        let mid = (i + j) / 2;
        let left_node = self.update_helper(self.nodes[x].left_child(), left, right, value, i, mid);
        let right_node =
            self.update_helper(self.nodes[x].right_child(), left, right, value, mid + 1, j);
        self.nodes[x] = Node::combine(&self.nodes[left_node], &self.nodes[right_node]);
        self.nodes[x].set_children(left_node, right_node);
        x
    }

    /// Return the amount of different versions the current segment tree has.
    pub fn versions(&self) -> usize {
        self.roots.len()
    }
}
#[cfg(test)]
mod tests {
    use crate::{
        default::Sum, nodes::Node, segment_tree::persistent_lazy::LazyPersistentSegmentTree,
    };

    #[test]
    fn non_empty_query_returns_some() {
        let nodes: Vec<Sum<usize>> = (0..=10).map(|x| Sum::initialize(&x)).collect();
        let mut segment_tree = LazyPersistentSegmentTree::build(&nodes);
        assert!(segment_tree.query(0, 0, 10).is_some());
    }
    #[test]
    fn empty_query_returns_none() {
        let nodes: Vec<Sum<usize>> = (0..=10).map(|x| Sum::initialize(&x)).collect();
        let mut segment_tree = LazyPersistentSegmentTree::build(&nodes);
        assert!(segment_tree.query(0, 10, 0).is_none());
    }
    #[test]
    fn normal_update_works() {
        let nodes: Vec<Sum<usize>> = (0..=10).map(|x| Sum::initialize(&x)).collect();
        let mut segment_tree = LazyPersistentSegmentTree::build(&nodes);
        let value = 20;
        segment_tree.update(0, 0, 0, value);
        assert_eq!(segment_tree.query(1, 0, 0).unwrap().value(), &value);
    }

    #[test]
    fn branched_update_works() {
        let nodes: Vec<Sum<usize>> = (0..=10).map(|x| Sum::initialize(&x)).collect();
        let mut segment_tree = LazyPersistentSegmentTree::build(&nodes);
        let value = 20;
        segment_tree.update(0, 0, 10, value);
        segment_tree.update(0, 1, 1, value);
        assert_eq!(segment_tree.query(2, 0, 0).unwrap().value(), &0);
        assert_eq!(segment_tree.query(2, 1, 1).unwrap().value(), &(value + 1));
    }

    #[test]
    fn query_works() {
        let nodes: Vec<Sum<usize>> = (0..=10).map(|x| Sum::initialize(&x)).collect();
        let mut segment_tree = LazyPersistentSegmentTree::build(&nodes);
        assert_eq!(segment_tree.query(0, 0, 10).unwrap().value(), &55);
    }
}