1- //! A generic implementation of Mo's algorithm, aka Query Sqrt Decomposition.
2- //! It answers q offline queries over intervals in 0..n by shifting the query
3- //! interval's endpoints by one position at a time.
4- //! Each endpoint incurs a total cost of at most n * sqrt(q * L_OP * R_OP).
5-
1+ /// A generic implementation of Mo's algorithm, aka Query Sqrt Decomposition.
2+ /// It answers q offline queries over intervals in 0..n by shifting the query
3+ /// interval's endpoints by one position at a time.
4+ /// Each endpoint incurs a total cost of at most n * sqrt(q * L_OP * R_OP).
65pub trait MoState {
76 type Q ;
87 type A ;
@@ -102,6 +101,53 @@ impl MoState for DistinctVals {
102101 }
103102}
104103
104+ /// Represents a minimum (lower envelope) of a collection of linear functions of a variable,
105+ /// evaluated using the convex hull trick with square root decomposition.
106+ pub struct PiecewiseLinearFn {
107+ sorted_lines : Vec < ( i64 , i64 ) > ,
108+ recent_lines : Vec < ( i64 , i64 ) > ,
109+ merge_threshold : usize ,
110+ }
111+ 112+ impl PiecewiseLinearFn {
113+ /// For N inserts interleaved with Q queries, a threshold of N/sqrt(Q) yields
114+ /// O(N sqrt Q + Q log N) time complexity. If all queries come after all inserts,
115+ /// a threshold of 0 yields O(N + Q log N) time complexity.
116+ pub fn with_merge_threshold ( merge_threshold : usize ) -> Self {
117+ Self {
118+ sorted_lines : vec ! [ ] ,
119+ recent_lines : vec ! [ ] ,
120+ merge_threshold,
121+ }
122+ }
123+ 124+ /// Replaces this function with the minimum of itself and a provided line
125+ pub fn min_with ( & mut self , slope : i64 , intercept : i64 ) {
126+ self . recent_lines . push ( ( slope, intercept) ) ;
127+ }
128+ 129+ fn update_envelope ( & mut self ) {
130+ self . recent_lines . extend ( self . sorted_lines . drain ( ..) ) ;
131+ self . recent_lines . sort_unstable ( ) ;
132+ for ( slope, intercept) in self . recent_lines . drain ( ..) {
133+ // TODO: do convex hull trick algorithm
134+ self . sorted_lines . push ( ( slope, intercept) ) ;
135+ }
136+ }
137+ 138+ fn eval_helper ( & self , x : i64 ) -> i64 {
139+ 0 // TODO: pick actual minimum, or infinity if empty
140+ }
141+ 142+ /// Evaluates the function at x
143+ pub fn evaluate ( & mut self , x : i64 ) -> i64 {
144+ if self . recent_lines . len ( ) > self . merge_threshold {
145+ self . update_envelope ( ) ;
146+ }
147+ self . eval_helper ( x)
148+ }
149+ }
150+ 105151#[ cfg( test) ]
106152mod test {
107153 use super :: * ;
@@ -115,4 +161,10 @@ mod test {
115161
116162 assert_eq ! ( answers, vec![ 2 , 1 , 5 , 5 ] ) ;
117163 }
164+ 165+ #[ test]
166+ fn test_convex_hull_trick ( ) {
167+ let mut func = PiecewiseLinearFn :: with_merge_threshold ( 3 ) ;
168+ // TODO: make test
169+ }
118170}
0 commit comments