Ranges

Cargo.toml

@@ -20,6 +20,7 @@ path = "./bin/run.rs"
 bit-set = "0.4"
 num-rational = { version = "0.1", default-features = false }
 num-traits = "0.1"
+ranges = "0.3.3"
 thiserror = "1.0.30"
 
 [badges]

src/constraint/alldifferent.rs

@@ -1,10 +1,11 @@
 //! All different implementation.
 
-use std::collections::HashMap;
+use std::collections::{hash_map::Entry, HashMap};
 use std::rc::Rc;
 
 use crate::{Constraint, Error, PsResult, PuzzleSearch, Val, VarToken};
 
+#[derive(Debug)]
 pub struct AllDifferent {
     vars: Vec<VarToken>,
 }
@@ -32,7 +33,7 @@ impl AllDifferent {
 }
 
 impl Constraint for AllDifferent {
-    fn vars<'a>(&'a self) -> Box<dyn Iterator<Item = &'a VarToken> + 'a> {
+    fn vars(&self) -> Box<dyn Iterator<Item = &'_ VarToken> + '_> {
         Box::new(self.vars.iter())
     }
 
@@ -54,10 +55,11 @@ impl Constraint for AllDifferent {
 
             for val in search.get_unassigned(var) {
                 match all_candidates.entry(val) {
-                    std::collections::hash_map::Entry::Occupied(mut e) => {
+                    Entry::Occupied(mut e) => {
                         e.insert(None);
                     }
-                    std::collections::hash_map::Entry::Vacant(e) => {
+
+                    Entry::Vacant(e) => {
                         e.insert(Some(var));
                     }
                 }

src/constraint/equality.rs

@@ -6,6 +6,7 @@ use std::rc::Rc;
 
 use crate::{Constraint, Error, LinExpr, PsResult, PuzzleSearch, Val, VarToken};
 
+#[derive(Debug)]
 pub struct Equality {
     // The equation: 0 = constant + coef1 * var1 + coef2 * var2 + ...
     eqn: LinExpr,
@@ -30,11 +31,11 @@ impl Equality {
 }
 
 impl Constraint for Equality {
-    fn vars<'a>(&'a self) -> Box<dyn Iterator<Item = &'a VarToken> + 'a> {
+    fn vars(&self) -> Box<dyn Iterator<Item = &'_ VarToken> + '_> {
         Box::new(self.eqn.coef.keys())
     }
 
-    fn on_assigned(&self, search: &mut PuzzleSearch, _: VarToken, _: Val) -> PsResult<()> {
+    fn on_assigned(&self, search: &mut PuzzleSearch, _var: VarToken, _val: Val) -> PsResult<()> {
         let mut sum = self.eqn.constant;
         let mut unassigned_var = None;
 
@@ -101,23 +102,25 @@ impl Constraint for Equality {
             }
 
             let (min_val, max_val) = search.get_min_max(var)?;
-            let (min_bnd, max_bnd);
+            let (min_bnd, max_bnd) = if coef > Ratio::zero() {
-
+                (
-            if coef > Ratio::zero() {
+                    ((coef * Ratio::from_integer(max_val) - sum_max) / coef)
-                min_bnd = ((coef * Ratio::from_integer(max_val) - sum_max) / coef)
+                        .ceil()
-                    .ceil()
+                        .to_integer(),
-                    .to_integer();
+                    ((coef * Ratio::from_integer(min_val) - sum_min) / coef)
-                max_bnd = ((coef * Ratio::from_integer(min_val) - sum_min) / coef)
+                        .floor()
-                    .floor()
+                        .to_integer(),
-                    .to_integer();
+                )
             } else {
-                min_bnd = ((coef * Ratio::from_integer(max_val) - sum_min) / coef)
+                (
-                    .ceil()
+                    ((coef * Ratio::from_integer(max_val) - sum_min) / coef)
-                    .to_integer();
+                        .ceil()
-                max_bnd = ((coef * Ratio::from_integer(min_val) - sum_max) / coef)
+                        .to_integer(),
-                    .floor()
+                    ((coef * Ratio::from_integer(min_val) - sum_max) / coef)
-                    .to_integer();
+                        .floor()
-            }
+                        .to_integer(),
+                )
+            };
 
             if min_val < min_bnd || max_bnd < max_val {
                 let (new_min, new_max) = search.bound_candidate_range(var, min_bnd, max_bnd)?;

src/constraint/mod.rs

@@ -5,14 +5,15 @@
 //! cannot store additional information about the state (e.g. caches)
 //! in the constraint to reuse later.
 
+use std::fmt::Debug;
 use std::rc::Rc;
 
 use crate::{PsResult, PuzzleSearch, Val, VarToken};
 
 /// Constraint trait.
-pub trait Constraint {
+pub trait Constraint: Debug {
     /// An iterator over the variables that are involved in the constraint.
-    fn vars<'a>(&'a self) -> Box<dyn Iterator<Item = &'a VarToken> + 'a>;
+    fn vars(&self) -> Box<dyn Iterator<Item = &'_ VarToken> + '_>;
 
     /// Applied after a variable has been assigned.
     fn on_assigned(&self, _search: &mut PuzzleSearch, _var: VarToken, _val: Val) -> PsResult<()> {

src/constraint/unify.rs

@@ -5,6 +5,7 @@ use std::rc::Rc;
 
 use crate::{Constraint, PsResult, PuzzleSearch, VarToken};
 
+#[derive(Debug)]
 pub struct Unify {
     var1: VarToken,
     var2: VarToken,
@@ -25,15 +26,12 @@ impl Unify {
     /// puzzle_solver::constraint::Unify::new(m, carry[3]);
     /// ```
     pub fn new(var1: VarToken, var2: VarToken) -> Self {
-        Unify {
+        Unify { var1, var2 }
-            var1,
-            var2,
-        }
     }
 }
 
 impl Constraint for Unify {
-    fn vars<'a>(&'a self) -> Box<dyn Iterator<Item = &'a VarToken> + 'a> {
+    fn vars(&self) -> Box<dyn Iterator<Item = &'_ VarToken> + '_> {
         if self.var1 != self.var2 {
             Box::new(iter::once(&self.var1).chain(iter::once(&self.var2)))
         } else {
@@ -52,10 +50,7 @@ impl Constraint for Unify {
     fn substitute(&self, from: VarToken, to: VarToken) -> PsResult<Rc<dyn Constraint>> {
         let var1 = if self.var1 == from { to } else { self.var1 };
         let var2 = if self.var2 == from { to } else { self.var2 };
-        Ok(Rc::new(Unify {
+        Ok(Rc::new(Unify { var1, var2 }))
-            var1,
-            var2,
-        }))
     }
 }
 

src/lib.rs

@@ -7,7 +7,9 @@ mod error;
 mod linexpr;
 mod puzzle;
 
+use core::fmt;
 use num_rational::Rational32;
+use num_traits::Signed;
 use std::collections::HashMap;
 use std::ops;
 
@@ -41,6 +43,34 @@ pub struct LinExpr {
     coef: HashMap<VarToken, Coef>,
 }
 
+impl fmt::Display for LinExpr {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        write!(f, "{}", self.constant)?;
+
+        for (tok, coef) in self.coef.iter() {
+            if coef.is_negative() {
+                if coef.abs() == Rational32::from_integer(1) {
+                    write!(f, " - x{}", tok.0)?;
+                } else {
+                    write!(f, " - {} * x{}", coef.abs(), tok.0)?;
+                }
+            } else if coef.abs() == Rational32::from_integer(1) {
+                write!(f, " + x{}", tok.0)?;
+            } else {
+                write!(f, " + {} * x{}", coef, tok.0)?;
+            }
+        }
+
+        Ok(())
+    }
+}
+
+impl fmt::Debug for LinExpr {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        write!(f, "LinExpr {{ {} }}", self)
+    }
+}
+
 /// A result during a puzzle solution search (Err = contradiction).
 pub type PsResult<T> = Result<T, Error>;
 

src/puzzle.rs

@@ -1,24 +1,30 @@
 //! The puzzle's state and rules.
 
 use bit_set::BitSet;
+use ranges::GenericRange;
 use std::cell::Cell;
 use std::collections::BTreeSet;
 use std::fmt;
 use std::iter;
 use std::mem;
 use std::ops;
+use std::ops::Bound;
+use std::ops::RangeBounds;
 use std::rc::Rc;
 
 use crate::constraint;
 use crate::Error;
 use crate::{Constraint, LinExpr, PsResult, Solution, Val, VarToken};
 
+use ranges::Ranges;
+
 /// A collection of candidates.
 #[derive(Clone, Debug, Eq, PartialEq)]
 enum Candidates {
     None,                   // A variable with no candidates.
     Value(Val),             // A variable set to its initial value.
     Set(Rc<BTreeSet<Val>>), // A variable with a list of candidates.
+    Range(Ranges<Val>),     // A variable with candidate ranges.
 }
 
 /// The state of a variable during the solution search.
@@ -75,15 +81,38 @@ impl Candidates {
             Candidates::None => 0,
             Candidates::Value(_) => 1,
             Candidates::Set(ref rc) => rc.len(),
+            Candidates::Range(r) => {
+                let mut total = 0;
+
+                for i in r.as_slice() {
+                    let min = match i.start_bound() {
+                        Bound::Included(val) => *val,
+                        Bound::Excluded(val) => val + 1,
+                        Bound::Unbounded => unreachable!(),
+                    };
+
+                    let max = match i.end_bound() {
+                        Bound::Included(val) => *val,
+                        Bound::Excluded(val) => val - 1,
+                        Bound::Unbounded => unreachable!(),
+                    };
+
+                    total += max - min + 1;
+                }
+                total as _
+            }
         }
     }
 
     /// Get an iterator over all of the candidates of a variable.
-    fn iter<'a>(&'a self) -> Box<dyn Iterator<Item = Val> + 'a> {
+    fn iter(&self) -> Box<dyn Iterator<Item = Val> + '_> {
         match self {
             Candidates::None => Box::new(iter::empty()),
             Candidates::Value(val) => Box::new(iter::once(*val)),
             Candidates::Set(ref rc) => Box::new(rc.iter().cloned()),
+            Candidates::Range(range) => {
+                Box::new(range.as_slice().iter().map(|x| x.into_iter()).flatten())
+            }
         }
     }
 }
@@ -138,6 +167,21 @@ impl Puzzle {
         var
     }
 
+    /// Allocate a new puzzle variable, initialising it with potential
+    /// candidates.
+    ///
+    /// # Examples
+    ///
+    /// ```
+    /// let mut send_more_money = puzzle_solver::Puzzle::new();
+    /// send_more_money.new_var_with_candidates(&[0,1,2,3,4,5,6,7,8,9]);
+    /// ```
+    pub fn new_var_with_range<R: RangeBounds<Val>>(&mut self, range: R) -> VarToken {
+        let var = self.new_var();
+        self.insert_range(var, range);
+        var
+    }
+
     /// Allocate a 1d vector of puzzle variables, each initialised to
     /// have the same set of potential candidates.
     ///
@@ -155,6 +199,27 @@ impl Puzzle {
         vars
     }
 
+    /// Allocate a 1d vector of puzzle variables, each initialised to
+    /// have the same set of potential candidates.
+    ///
+    /// # Examples
+    ///
+    /// ```
+    /// let mut send_more_money = puzzle_solver::Puzzle::new();
+    /// send_more_money.new_vars_with_candidates_1d(8, &[0,1,2,3,4,5,6,7,8,9]);
+    /// ```
+    pub fn new_vars_with_range_1d<R: RangeBounds<Val> + Clone>(
+        &mut self,
+        n: usize,
+        range: R,
+    ) -> Vec<VarToken> {
+        let mut vars = Vec::with_capacity(n);
+        for _ in 0..n {
+            vars.push(self.new_var_with_range(range.clone()));
+        }
+        vars
+    }
+
     /// Allocate a 2d array of puzzle variables, each initialised to
     /// have the same set of potential candidates.
     ///
@@ -177,6 +242,28 @@ impl Puzzle {
         vars
     }
 
+    /// Allocate a 2d array of puzzle variables, each initialised to
+    /// have the same set of potential candidates.
+    ///
+    /// # Examples
+    ///
+    /// ```
+    /// let mut magic_square = puzzle_solver::Puzzle::new();
+    /// magic_square.new_vars_with_candidates_2d(3, 3, &[1,2,3,4,5,6,7,8,9]);
+    /// ```
+    pub fn new_vars_with_range_2d<R: RangeBounds<Val> + Clone>(
+        self: &mut Puzzle,
+        width: usize,
+        height: usize,
+        range: R,
+    ) -> Vec<Vec<VarToken>> {
+        let mut vars = Vec::with_capacity(height);
+        for _ in 0..height {
+            vars.push(self.new_vars_with_range_1d(width, range.clone()));
+        }
+        vars
+    }
+
     /// Set a variable to a known value.
     ///
     /// This is useful when the variable is given as part of the
@@ -220,6 +307,7 @@ impl Puzzle {
 
         match self.candidates[idx] {
             Candidates::Value(_) => panic!("attempt to set fixed variable"),
+            Candidates::Range(_) => panic!("attempt to insert candidates into range"),
 
             Candidates::None => {
                 self.candidates[idx] = Candidates::Set(Rc::new(BTreeSet::new()));
@@ -252,10 +340,9 @@ impl Puzzle {
         let VarToken(idx) = var;
 
         match self.candidates[idx] {
-            Candidates::Value(_) => panic!("attempt to set fixed variable"),
-
             Candidates::None => (),
-
+            Candidates::Value(_) => panic!("attempt to set fixed variable"),
+            Candidates::Range(_) => panic!("attempt to remove candidates from range"),
             Candidates::Set(ref mut rc) => {
                 let cs = Rc::get_mut(rc).expect("unique");
                 for c in candidates.iter() {
@@ -282,10 +369,9 @@ impl Puzzle {
         let VarToken(idx) = var;
 
         match self.candidates[idx] {
-            Candidates::Value(_) => panic!("attempt to set fixed variable"),
-
             Candidates::None => (),
-
+            Candidates::Value(_) => panic!("attempt to set fixed variable"),
+            Candidates::Range(_) => panic!("attempt to intersect candidates on the range"),
             Candidates::Set(ref mut rc) => {
                 let cs = Rc::get_mut(rc).expect("unique");
                 let mut set = BTreeSet::new();
@@ -295,6 +381,31 @@ impl Puzzle {
         }
     }
 
+    ///
+    /// Insert range
+    ///
+    ///
+    pub fn insert_range<R: RangeBounds<Val>>(&mut self, var: VarToken, range: R) {
+        let VarToken(idx) = var;
+
+        match self.candidates[idx] {
+            ref mut x @ Candidates::None => {
+                *x = Candidates::Range(Ranges::from((
+                    range.start_bound().cloned(),
+                    range.end_bound().cloned(),
+                )));
+            }
+            Candidates::Value(_) => panic!("attempt to set fixed variable"),
+            Candidates::Range(ref mut r) => {
+                r.insert(GenericRange::new_with_bounds(
+                    range.start_bound().cloned(),
+                    range.end_bound().cloned(),
+                ));
+            }
+            Candidates::Set(_) => panic!("attempt to insert range on set"),
+        }
+    }
+
     /// Add a constraint to the puzzle solution.
     pub fn add_constraint<T>(&mut self, constraint: T)
     where
@@ -570,6 +681,28 @@ impl<'a> PuzzleSearch<'a> {
             VarState::Unassigned(ref cs) => match cs {
                 Candidates::None => Err(Error::Default),
                 Candidates::Value(val) => Ok((*val, *val)),
+                Candidates::Range(r) => {
+                    let slice = r.as_slice();
+                    let first = slice.get(0).ok_or(Error::Default)?.start_bound();
+                    let last = slice
+                        .get(slice.len() - 1)
+                        .ok_or(Error::Default)?
+                        .end_bound();
+
+                    let min = match first {
+                        Bound::Included(val) => *val,
+                        Bound::Excluded(val) => val + 1,
+                        Bound::Unbounded => unreachable!(),
+                    };
+
+                    let max = match last {
+                        Bound::Included(val) => *val,
+                        Bound::Excluded(val) => val - 1,
+                        Bound::Unbounded => unreachable!(),
+                    };
+
+                    Ok((min, max))
+                }
                 Candidates::Set(ref rc) => rc
                     .iter()
                     .cloned()
@@ -587,30 +720,34 @@ impl<'a> PuzzleSearch<'a> {
     pub fn set_candidate(&mut self, var: VarToken, val: Val) -> PsResult<()> {
         let VarToken(idx) = var;
 
-        match &self.vars[idx] {
+        match self.vars[idx] {
-            VarState::Assigned(v) => return bool_to_result(*v == val),
+            VarState::Assigned(v) => bool_to_result(v == val),
-            VarState::Unassigned(ref cs) => match cs {
+            VarState::Unassigned(ref mut cs) => match cs {
-                Candidates::None => return Err(Error::Default),
+                Candidates::None => Err(Error::Default),
-                Candidates::Value(v) => return bool_to_result(*v == val),
+                Candidates::Value(v) => bool_to_result(*v == val),
-                Candidates::Set(_) => (),
+                Candidates::Range(ref mut r) => {
-            },
+                    if r.contains(&val) {
-            VarState::Unified(_) => (),
+                        *r = Ranges::from(val..=val);
-        }
+                        self.wake.union_with(&self.constraints.wake[idx]);
 
-        if let VarState::Unified(other) = self.vars[idx] {
+                        Ok(())
-            self.set_candidate(other, val)
+                    } else {
-        } else if let VarState::Unassigned(Candidates::Set(ref mut rc)) = self.vars[idx] {
+                        Err(Error::Default)
-            if rc.contains(&val) {
+                    }
-                let set = Rc::make_mut(rc);
+                }
-                set.clear();
+                Candidates::Set(rc) => {
-                set.insert(val);
+                    if rc.contains(&val) {
-                self.wake.union_with(&self.constraints.wake[idx]);
+                        let set = Rc::make_mut(rc);
-                Ok(())
+                        set.clear();
-            } else {
+                        set.insert(val);
-                Err(Error::Default)
+                        self.wake.union_with(&self.constraints.wake[idx]);
-            }
+                        Ok(())
-        } else {
+                    } else {
-            unreachable!();
+                        Err(Error::Default)
+                    }
+                }
+            },
+            VarState::Unified(other) => self.set_candidate(other, val),
         }
     }
 
@@ -618,27 +755,29 @@ impl<'a> PuzzleSearch<'a> {
     pub fn remove_candidate(&mut self, var: VarToken, val: Val) -> PsResult<()> {
         let VarToken(idx) = var;
 
-        match &self.vars[idx] {
+        match self.vars[idx] {
-            VarState::Assigned(v) => return bool_to_result(*v != val),
+            VarState::Assigned(v) => bool_to_result(v != val),
-            VarState::Unassigned(ref cs) => match cs {
+            VarState::Unassigned(ref mut cs) => match cs {
-                Candidates::None => return Err(Error::Default),
+                Candidates::None => Err(Error::Default),
-                Candidates::Value(v) => return bool_to_result(*v != val),
+                Candidates::Value(v) => bool_to_result(*v != val),
-                Candidates::Set(_) => (),
+                Candidates::Range(r) => {
-            },
+                    if r.contains(&val) {
-            VarState::Unified(_) => (),
+                        r.remove(val..=val);
-        }
+                        self.wake.union_with(&self.constraints.wake[idx]);
+                    }
 
-        if let VarState::Unified(other) = self.vars[idx] {
+                    bool_to_result(!r.is_empty())
-            self.remove_candidate(other, val)
+                }
-        } else if let VarState::Unassigned(Candidates::Set(ref mut rc)) = self.vars[idx] {
+                Candidates::Set(rc) => {
-            if rc.contains(&val) {
+                    if rc.contains(&val) {
-                let set = Rc::make_mut(rc);
+                        let set = Rc::make_mut(rc);
-                set.remove(&val);
+                        set.remove(&val);
-                self.wake.union_with(&self.constraints.wake[idx]);
+                        self.wake.union_with(&self.constraints.wake[idx]);
-            }
+                    }
-            bool_to_result(!rc.is_empty())
+                    bool_to_result(!rc.is_empty())
-        } else {
+                }
-            unreachable!();
+            },
+            VarState::Unified(other) => self.remove_candidate(other, val),
         }
     }
 
@@ -654,51 +793,77 @@ impl<'a> PuzzleSearch<'a> {
         match self.vars[idx] {
             VarState::Assigned(v) => {
                 if min <= v && v <= max {
-                    return Ok((v, v));
+                    Ok((v, v))
                 } else {
-                    return Err(Error::Default);
+                    Err(Error::Default)
                 }
             }
-            VarState::Unassigned(ref cs) => match cs {
+
-                Candidates::None => return Err(Error::Default),
+            VarState::Unassigned(ref mut cs) => match cs {
+                Candidates::None => Err(Error::Default),
                 Candidates::Value(v) => {
                     if min <= *v && *v <= max {
-                        return Ok((*v, *v));
+                        Ok((*v, *v))
                     } else {
-                        return Err(Error::Default);
+                        Err(Error::Default)
+                    }
+                }
+                Candidates::Range(ref mut r) => {
+                    if max < min {
+                        return Err(Error::Default);
+                    }
+
+                    *r = r.clone().intersect(min..=max);
+
+                    self.wake.union_with(&self.constraints.wake[idx]);
+
+                    let slice = r.as_slice();
+                    let first = slice.get(0).ok_or(Error::Default)?.start_bound();
+                    let last = slice
+                        .get(slice.len() - 1)
+                        .ok_or(Error::Default)?
+                        .end_bound();
+
+                    let min = match first {
+                        Bound::Included(val) => *val,
+                        Bound::Excluded(val) => val + 1,
+                        Bound::Unbounded => unreachable!(),
+                    };
+
+                    let max = match last {
+                        Bound::Included(val) => *val,
+                        Bound::Excluded(val) => val - 1,
+                        Bound::Unbounded => unreachable!(),
+                    };
+
+                    Ok((min, max))
+                }
+                Candidates::Set(rc) => {
+                    let &curr_min = rc.iter().min().expect("candidates");
+                    let &curr_max = rc.iter().max().expect("candidates");
+
+                    if curr_min < min || max < curr_max {
+                        let set = Rc::make_mut(rc);
+                        *set = set
+                            .iter()
+                            .filter(|&val| min <= *val && *val <= max)
+                            .cloned()
+                            .collect();
+                        self.wake.union_with(&self.constraints.wake[idx]);
+                        rc.iter()
+                            .cloned()
+                            .min()
+                            .into_iter()
+                            .zip(rc.iter().cloned().max())
+                            .next()
+                            .ok_or(Error::Default)
+                    } else {
+                        Ok((curr_min, curr_max))
                     }
                 }
-                Candidates::Set(_) => (),
             },
-            VarState::Unified(_) => (),
-        }
 
-        if let VarState::Unified(other) = self.vars[idx] {
+            VarState::Unified(other) => self.bound_candidate_range(other, min, max),
-            self.bound_candidate_range(other, min, max)
-        } else if let VarState::Unassigned(Candidates::Set(ref mut rc)) = self.vars[idx] {
-            let &curr_min = rc.iter().min().expect("candidates");
-            let &curr_max = rc.iter().max().expect("candidates");
-
-            if curr_min < min || max < curr_max {
-                let set = Rc::make_mut(rc);
-                *set = set
-                    .iter()
-                    .filter(|&val| min <= *val && *val <= max)
-                    .cloned()
-                    .collect();
-                self.wake.union_with(&self.constraints.wake[idx]);
-                rc.iter()
-                    .cloned()
-                    .min()
-                    .into_iter()
-                    .zip(rc.iter().cloned().max())
-                    .next()
-                    .ok_or(Error::Default)
-            } else {
-                Ok((curr_min, curr_max))
-            }
-        } else {
-            unreachable!();
         }
     }
 

tests/sudoku.rs

@@ -12,7 +12,7 @@ type Board = [[Val; SIZE]; SIZE];
 
 fn make_sudoku(board: &Board) -> (Puzzle, Vec<Vec<VarToken>>) {
     let mut sys = Puzzle::new();
-    let vars = sys.new_vars_with_candidates_2d(SIZE, SIZE, &[1, 2, 3, 4, 5, 6, 7, 8, 9]);
+    let vars = sys.new_vars_with_range_2d(SIZE, SIZE, 1..=9);
 
     for y in 0..SIZE {
         sys.all_different(&vars[y]);

tests/takuzu.rs

@@ -60,8 +60,8 @@ fn make_sums(size: usize) -> Vec<Val> {
         let mut v = val as usize;
 
         while v > 0 {
-            count = count + (v & 1);
+            count += (v & 1);
-            v = v >> 1;
+            v >>= 1;
         }
 
         if count == size / 2 {