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
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
//! Partially ordered elements with a least upper bound.
//!
//! Lattices form the basis of differential dataflow's efficient execution in the presence of
//! iterative sub-computations. All logical times in differential dataflow must implement the
//! `Lattice` trait, and all reasoning in operators are done it terms of `Lattice` methods.

use timely::order::PartialOrder;
use timely::progress::{Antichain, frontier::AntichainRef};

/// A bounded partially ordered type supporting joins and meets.
pub trait Lattice : PartialOrder {

    /// The smallest element greater than or equal to both arguments.
    ///
    /// # Examples
    ///
    /// ```
    /// # use timely::PartialOrder;
    /// # use timely::order::Product;
    /// # use differential_dataflow::lattice::Lattice;
    /// # fn main() {
    ///
    /// let time1 = Product::new(3, 7);
    /// let time2 = Product::new(4, 6);
    /// let join = time1.join(&time2);
    ///
    /// assert_eq!(join, Product::new(4, 7));
    /// # }
    /// ```
    fn join(&self, other: &Self) -> Self;

    /// Updates `self` to the smallest element greater than or equal to both arguments.
    ///
    /// # Examples
    ///
    /// ```
    /// # use timely::PartialOrder;
    /// # use timely::order::Product;
    /// # use differential_dataflow::lattice::Lattice;
    /// # fn main() {
    ///
    /// let mut time1 = Product::new(3, 7);
    /// let time2 = Product::new(4, 6);
    /// time1.join_assign(&time2);
    ///
    /// assert_eq!(time1, Product::new(4, 7));
    /// # }
    /// ```
    fn join_assign(&mut self, other: &Self) where Self: Sized {
        *self = self.join(other);
    }

    /// The largest element less than or equal to both arguments.
    ///
    /// # Examples
    ///
    /// ```
    /// # use timely::PartialOrder;
    /// # use timely::order::Product;
    /// # use differential_dataflow::lattice::Lattice;
    /// # fn main() {
    ///
    /// let time1 = Product::new(3, 7);
    /// let time2 = Product::new(4, 6);
    /// let meet = time1.meet(&time2);
    ///
    /// assert_eq!(meet, Product::new(3, 6));
    /// # }
    /// ```
    fn meet(&self, other: &Self) -> Self;

    /// Updates `self` to the largest element less than or equal to both arguments.
    ///
    /// # Examples
    ///
    /// ```
    /// # use timely::PartialOrder;
    /// # use timely::order::Product;
    /// # use differential_dataflow::lattice::Lattice;
    /// # fn main() {
    ///
    /// let mut time1 = Product::new(3, 7);
    /// let time2 = Product::new(4, 6);
    /// time1.meet_assign(&time2);
    ///
    /// assert_eq!(time1, Product::new(3, 6));
    /// # }
    /// ```
    fn meet_assign(&mut self, other: &Self) where Self: Sized  {
        *self = self.meet(other);
    }

    /// Advances self to the largest time indistinguishable under `frontier`.
    ///
    /// This method produces the "largest" lattice element with the property that for every
    /// lattice element greater than some element of `frontier`, both the result and `self`
    /// compare identically to the lattice element. The result is the "largest" element in
    /// the sense that any other element with the same property (compares identically to times
    /// greater or equal to `frontier`) must be less or equal to the result.
    ///
    /// When provided an empty frontier `self` is not modified.
    ///
    /// # Examples
    ///
    /// ```
    /// # use timely::PartialOrder;
    /// # use timely::order::Product;
    /// # use differential_dataflow::lattice::Lattice;
    /// # fn main() {
    ///
    /// use timely::progress::frontier::{Antichain, AntichainRef};
    ///
    /// let time = Product::new(3, 7);
    /// let mut advanced = Product::new(3, 7);
    /// let frontier = Antichain::from(vec![Product::new(4, 8), Product::new(5, 3)]);
    /// advanced.advance_by(frontier.borrow());
    ///
    /// // `time` and `advanced` are indistinguishable to elements >= an element of `frontier`
    /// for i in 0 .. 10 {
    ///     for j in 0 .. 10 {
    ///         let test = Product::new(i, j);
    ///         // for `test` in the future of `frontier` ..
    ///         if frontier.less_equal(&test) {
    ///             assert_eq!(time.less_equal(&test), advanced.less_equal(&test));
    ///         }
    ///     }
    /// }
    ///
    /// assert_eq!(advanced, Product::new(4, 7));
    /// # }
    /// ```
    #[inline]
    fn advance_by(&mut self, frontier: AntichainRef<Self>) where Self: Sized {
        let mut iter = frontier.iter();
        if let Some(first) = iter.next() {
            let mut result = self.join(first);
            for f in iter {
                result.meet_assign(&self.join(f));
            }
            *self = result;
        }
    }
}

use timely::order::Product;

impl<T1: Lattice, T2: Lattice> Lattice for Product<T1, T2> {
    #[inline]
    fn join(&self, other: &Product<T1, T2>) -> Product<T1, T2> {
        Product {
            outer: self.outer.join(&other.outer),
            inner: self.inner.join(&other.inner),
        }
    }
    #[inline]
    fn meet(&self, other: &Product<T1, T2>) -> Product<T1, T2> {
        Product {
            outer: self.outer.meet(&other.outer),
            inner: self.inner.meet(&other.inner),
        }
    }
}

/// A type that has a unique maximum element.
pub trait Maximum {
    /// The unique maximal element of the set.
    fn maximum() -> Self;
}

/// Implements `Maximum` for elements with a `MAX` associated constant.
macro_rules! implement_maximum {
    ($($index_type:ty,)*) => (
        $(
            impl Maximum for $index_type {
                fn maximum() -> Self { Self::MAX }
            }
        )*
    )
}

implement_maximum!(usize, u128, u64, u32, u16, u8, isize, i128, i64, i32, i16, i8, Duration,);
impl Maximum for () { fn maximum() -> () { () }}

use timely::progress::Timestamp;

// Tuples have the annoyance that they are only a lattice for `T2` with maximal elements,
// as the `meet` operator on `(x, _)` and `(y, _)` would be `(x meet y, maximum())`.
impl<T1: Lattice+Clone, T2: Lattice+Clone+Maximum+Timestamp> Lattice for (T1, T2) {
    #[inline]
    fn join(&self, other: &(T1, T2)) -> (T1, T2) {
        if self.0.eq(&other.0) {
            (self.0.clone(), self.1.join(&other.1))
        } else if self.0.less_than(&other.0) {
            other.clone()
        } else if other.0.less_than(&self.0) {
            self.clone()
        } else {
            (self.0.join(&other.0), T2::minimum())
        }
    }
    #[inline]
    fn meet(&self, other: &(T1, T2)) -> (T1, T2) {
        if self.0.eq(&other.0) {
            (self.0.clone(), self.1.meet(&other.1))
        } else if self.0.less_than(&other.0) {
            self.clone()
        } else if other.0.less_than(&self.0) {
            other.clone()
        } else {
            (self.0.meet(&other.0), T2::maximum())
        }
    }
}

macro_rules! implement_lattice {
    ($index_type:ty, $minimum:expr) => (
        impl Lattice for $index_type {
            #[inline] fn join(&self, other: &Self) -> Self { ::std::cmp::max(*self, *other) }
            #[inline] fn meet(&self, other: &Self) -> Self { ::std::cmp::min(*self, *other) }
        }
    )
}

use std::time::Duration;

implement_lattice!(Duration, Duration::new(0, 0));
implement_lattice!(usize, 0);
implement_lattice!(u128, 0);
implement_lattice!(u64, 0);
implement_lattice!(u32, 0);
implement_lattice!(u16, 0);
implement_lattice!(u8, 0);
implement_lattice!(isize, 0);
implement_lattice!(i128, 0);
implement_lattice!(i64, 0);
implement_lattice!(i32, 0);
implement_lattice!(i16, 0);
implement_lattice!(i8, 0);
implement_lattice!((), ());

/// Returns the "smallest" minimal antichain "greater or equal" to both inputs.
///
/// This method is primarily meant for cases where one cannot use the methods
/// of `Antichain`'s `PartialOrder` implementation, such as when one has only
/// references rather than owned antichains.
///
/// # Examples
///
/// ```
/// # use timely::PartialOrder;
/// # use timely::order::Product;
/// # use differential_dataflow::lattice::Lattice;
/// # use differential_dataflow::lattice::antichain_join;
/// # fn main() {
///
/// let f1 = &[Product::new(3, 7), Product::new(5, 6)];
/// let f2 = &[Product::new(4, 6)];
/// let join = antichain_join(f1, f2);
/// assert_eq!(&*join.elements(), &[Product::new(4, 7), Product::new(5, 6)]);
/// # }
/// ```
pub fn antichain_join<T: Lattice>(one: &[T], other: &[T]) -> Antichain<T> {
    let mut upper = Antichain::new();
    antichain_join_into(one, other, &mut upper);
    upper
}

/// Returns the "smallest" minimal antichain "greater or equal" to both inputs.
///
/// This method is primarily meant for cases where one cannot use the methods
/// of `Antichain`'s `PartialOrder` implementation, such as when one has only
/// references rather than owned antichains.
///
/// This function is similar to [antichain_join] but reuses an existing allocation.
/// The provided antichain is cleared before inserting elements.
///
/// # Examples
///
/// ```
/// # use timely::PartialOrder;
/// # use timely::order::Product;
/// # use timely::progress::Antichain;
/// # use differential_dataflow::lattice::Lattice;
/// # use differential_dataflow::lattice::antichain_join_into;
/// # fn main() {
///
/// let mut join = Antichain::new();
/// let f1 = &[Product::new(3, 7), Product::new(5, 6)];
/// let f2 = &[Product::new(4, 6)];
/// antichain_join_into(f1, f2, &mut join);
/// assert_eq!(&*join.elements(), &[Product::new(4, 7), Product::new(5, 6)]);
/// # }
/// ```
pub fn antichain_join_into<T: Lattice>(one: &[T], other: &[T], upper: &mut Antichain<T>) {
    upper.clear();
    for time1 in one {
        for time2 in other {
            upper.insert(time1.join(time2));
        }
    }
}

/// Returns the "greatest" minimal antichain "less or equal" to both inputs.
///
/// This method is primarily meant for cases where one cannot use the methods
/// of `Antichain`'s `PartialOrder` implementation, such as when one has only
/// references rather than owned antichains.
///
/// # Examples
///
/// ```
/// # use timely::PartialOrder;
/// # use timely::order::Product;
/// # use differential_dataflow::lattice::Lattice;
/// # use differential_dataflow::lattice::antichain_meet;
/// # fn main() {
///
/// let f1 = &[Product::new(3, 7), Product::new(5, 6)];
/// let f2 = &[Product::new(4, 6)];
/// let meet = antichain_meet(f1, f2);
/// assert_eq!(&*meet.elements(), &[Product::new(3, 7), Product::new(4, 6)]);
/// # }
/// ```
pub fn antichain_meet<T: Lattice+Clone>(one: &[T], other: &[T]) -> Antichain<T> {
    let mut upper = Antichain::new();
    for time1 in one {
        upper.insert(time1.clone());
    }
    for time2 in other {
        upper.insert(time2.clone());
    }
    upper
}

impl<T: Lattice+Clone> Lattice for Antichain<T> {
    fn join(&self, other: &Self) -> Self {
        let mut upper = Antichain::new();
        for time1 in self.elements().iter() {
            for time2 in other.elements().iter() {
                upper.insert(time1.join(time2));
            }
        }
        upper
    }
    fn meet(&self, other: &Self) -> Self {
        let mut upper = Antichain::new();
        for time1 in self.elements().iter() {
            upper.insert(time1.clone());
        }
        for time2 in other.elements().iter() {
            upper.insert(time2.clone());
        }
        upper
    }
}