specs.rs

//! A collection of example ArqSpec implementations

pub trait ArqSpec {
    /// Type of underlying array elements.
    type S: Clone;
    /// Type of data representing an endomorphism.
    // Note that while a Fn(S) -> S may seem like a more natural representation
    // for an endomorphism, compositions would then have to delegate to each of
    // their parts. This representation is more efficient.
    type F: Clone;

    /// Must satisfy the Associative Law:
    /// For all a,b,c, op(a, op(b, c)) = op(op(a, b), c)
    fn op(a: &Self::S, b: &Self::S) -> Self::S;
    /// Must satisfy the Identity Law:
    /// For all a, op(a, identity()) = op(identity(), a) = a
    fn identity() -> Self::S;
    /// Must satisfy the Composition Law:
    /// For all f,g,a, apply(compose(f, g), a) = apply(f, apply(g, a))
    fn compose(f: &Self::F, g: &Self::F) -> Self::F;
    /// Must satisfy the Distributive Law:
    /// For all f,a,b, apply(f, op(a, b), s+t) = op(apply(f, a, s), apply(f, b, t))
    /// The `size` parameter makes this law easier to satisfy in certain cases.
    fn apply(f: &Self::F, a: &Self::S, size: i64) -> Self::S;

    // The following relaxations to the laws may apply.
    // If only point updates are made, the Composition and Distributive Laws
    // no longer apply.
    // - compose() is never called, so it can be left unimplemented!().
    // - apply() is only ever called on leaves, i.e., with size == 1.
    // If only point queries are made, the Associative and Distributive Laws
    // no longer apply.
    // - op()'s result only matters when identity() is an argument.
    // - apply()'s result only matters on leaves, i.e., with size == 1.
}

/// Range Minimum Query (RMQ), a classic application of ARQ.
/// update(l, r, &f) sets all entries a[l..=r] to f.
/// query(l, r) finds the minimum value in a[l..=r].
//
// Exercises: try augmenting this struct to find the index of a minimum element
// in a range query, as well as the number of elements equal to the minimum.
// Then instead of overwriting values with a constant assignment a[i] = f,
// try supporting addition: a[i] += f.
pub enum AssignMin {}
impl ArqSpec for AssignMin {
    type S = i64;
    type F = i64;
    fn op(&a: &Self::S, &b: &Self::S) -> Self::S {
        a.min(b)
    }
    fn identity() -> Self::S {
        i64::MAX
    }
    fn compose(&f: &Self::F, _: &Self::F) -> Self::F {
        f
    }
    fn apply(&f: &Self::F, _: &Self::S, _: i64) -> Self::S {
        f
    }
}

/// Range Sum Query, a slightly trickier classic application of ARQ.
/// update(l, r, &f) sets all entries a[l..=r] to f.
/// query(l, r) sums all the entries a[l..=r].
///
/// # Panics
///
/// Associated functions will panic on overflow.
//
// Note that while the `size` parameter seems necessary to satisfy the
// Distributive Law, it is merely a convenience: in essence what we've done
// is move to the product monoid of tuples (value, size_of_subtree).
//
// In mathematical jargon, we say that constant assignment f(a) = f is not an
// endomorphism on (i64, +) because f(a+b) = f != 2*f = f(a) + f(b).
// On the other hand, f((a, s)) = (f*s, s) is indeed an endomorphism on pairs
// with vector addition: f((a, s) + (b, t)) = f((a+b, s+t)) = (f*(s+t), s+t)
//                       = (f*s, s) + (f*t, t) = f((a,s)) + f((b,t)).
pub enum AssignSum {}
impl ArqSpec for AssignSum {
    type S = i64;
    type F = i64;
    fn op(&a: &Self::S, &b: &Self::S) -> Self::S {
        a + b
    }
    fn identity() -> Self::S {
        0
    }
    fn compose(&f: &Self::F, _: &Self::F) -> Self::F {
        f
    }
    fn apply(&f: &Self::F, _: &Self::S, size: i64) -> Self::S {
        f * size
    }
}

/// Supply & Demand, based on https://codeforces.com/gym/102218/problem/F
/// update(i, i, &(p, o)) increases supply by p and demand by o at time i.
/// query(l, r) computes total supply and demand at times l to r, as well as
//              how much of the supply is subsequently met by the demand.
//
// Note that the apply() operation is only correct when applied to leaf nodes.
// Therefore, update() must only be used in "eager" mode, i.e., with l == r.
// compose() should be unimplemented!() to prevent accidental "lazy" updates.
pub enum SupplyDemand {}
impl ArqSpec for SupplyDemand {
    type S = (i64, i64, i64); // production, orders, sales
    type F = (i64, i64);
    fn op((p1, o1, s1): &Self::S, (p2, o2, s2): &Self::S) -> Self::S {
        let extra = (p1 - s1).min(o2 - s2);
        (p1 + p2, o1 + o2, s1 + s2 + extra)
    }
    fn identity() -> Self::S {
        (0, 0, 0)
    }
    fn compose(_: &Self::F, _: &Self::F) -> Self::F {
        unimplemented!()
    }
    fn apply(&(p_add, o_add): &Self::F, &(p, o, _): &Self::S, s: i64) -> Self::S {
        assert_eq!(s, 1);
        let p = p + p_add;
        let o = o + o_add;
        (p, o, p.min(o))
    }
}