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src/datastructure/slope_trick.cr
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- Last update: 2021-10-17 21:54:43+09:00
Depends on
src/datastructure/binary_heap.cr
Code
require "./binary_heap"
# reference: https://maspypy.com/slope-trick-1-解説編
class SlopeTrick(T)
getter min : T, shift : T, left, right
# Lets `f(x) = 0`
def initialize
@min = T.zero
@shift = T.zero
@left = BinaryHeap(T).new { |a, b| b <=> a }
@right = BinaryHeap(T).new
end
# Calculates `f(x)`
def f(x : T) : T
min +
@left.sum { |l| Math.max(l - x, T.zero) } +
@right.sum { |r| Math.max(x - r, T.zero) }
end
def l0 : T?
@left.top?.try &.+(shift)
end
def r0 : T?
@right.top?.try &.+(shift)
end
# Adds constant function `a`
def add(a : T) : self
@min += a
self
end
# Adds `max(x - a, 0)`
def add_r(a : T) : self
if @left.empty?
@right << a - shift
else
@min += Math.max(l0.not_nil! - a, T.zero)
@left << a - shift
@right << @left.pop.not_nil!
end
self
end
# Adds `max(a - x, 0)`
def add_l(a : T) : self
if @right.empty?
@left << a - shift
else
@min += Math.max(a - r0.not_nil!, T.zero)
@right << a - shift
@left << @right.pop.not_nil!
end
self
end
# Adds `|x - a|`
def add_abs(a : T) : self
add_r(a)
add_l(a)
end
# `g(x) := min_{y <= x} f(y)`
def accumulate_min : self
@right.clear
self
end
# `g(x) := min_{y >= x} f(y)`
def accumulate_min_right : self
@left.clear
self
end
# `g(x) := f(x - a)`
def slide(a : T) : self
@shift += a
self
end
end# require "./binary_heap"
class BinaryHeap(T)
# Creates a new empty heap.
def initialize
@heap = Array(T).new
@compare_proc = nil
end
# Creates a new empty heap backed by a buffer that is initially *initial_capacity* big (default: `0`).
#
# ```
# a = BinaryHeap.new(3)
# a << 3 << 1 << 2
# a.pop # => 1
# a.pop # => 2
# a.pop # => 3
# ```
def initialize(initial_capacity : Int = 0)
@heap = Array(T).new(initial_capacity)
@compare_proc = nil
end
# Creates a new heap from the elements in *enumerable*.
#
# ```
# a = BinaryHeap.new [3, 1, 2]
# a.pop # => 1
# a.pop # => 2
# a.pop # => 3
# ```
def initialize(enumerable : Enumerable(T))
initialize
enumerable.each { |x| add(x) }
end
# Creates a new empty heap with the custom comperator.
#
# The block must implement a comparison between two elements *a* and *b*, where `a < b` returns `-1`,
# `a == b` returns `0`, and `a > b` returns `1`. The comparison operator `#<=>` can be used for this.
#
# ```
# a = BinaryHeap.new [3, 1, 2]
# a.pop # => 1
# b = BinaryHeap.new [3, 1, 2] { |a, b| b <=> a }
# b.pop # => 3
# ```
def initialize(initial_capacity : Int = 0, &block : T, T -> Int32?)
@heap = Array(T).new(initial_capacity)
@compare_proc = block
end
# :ditto:
def initialize(enumerable : Enumerable(T), &block : T, T -> Int32?)
initialize &block
enumerable.each { |x| add(x) }
end
include Enumerable(T)
include Iterable(T)
def_clone
# Returns true if both heap have the same elements.
def ==(other : BinaryHeap(T)) : Bool
return false if size != other.size
@heap.sort == other.@heap.sort
end
# Returns the number of elements in the heap.
def size : Int32
@heap.size
end
# Returns `true` if `self` is empty, `false` otherwise.
def empty? : Bool
@heap.empty?
end
# Removes all elements from the heap and returns `self`.
def clear : self
@heap.clear
self
end
# Returns the lowest value in the `self`.
# If the `self` is empty, calls the block and returns its value.
def top(&block)
@heap.first { yield }
end
# Returns the lowest value in the `self`.
# If the `self` is empty, returns `nil`.
def top? : T?
top { nil }
end
# Returns the lowest value in the `self`.
# If the `self` is empty, raises `IndexError`.
def top : T
top { raise IndexError.new }
end
# Requires `0 <= i < size`, `0 <= j < size`.
private def compare(i : Int32, j : Int32)
x, y = @heap.unsafe_fetch(i), @heap.unsafe_fetch(j)
if @compare_proc
v = @compare_proc.not_nil!.call(x, y)
raise ArgumentError.new("Comparison of #{x} and #{y} failed") if v.nil?
v > 0
else
x > y
end
end
# Adds *object* to the heap and returns `self`.
def add(object : T) : self
@heap << object
i = size - 1
parent = i.pred // 2
while i > 0 && compare(parent, i)
@heap.swap(parent, i)
i, parent = parent, parent.pred // 2
end
self
end
# :ditto:
def <<(object : T) : self
add(object)
end
# Removes the lowest value from `self` and returns the removed value.
# If the array is empty, the given block is called.
def pop(&block)
case size
when 0
yield
when 1
@heap.pop
else
value = @heap.unsafe_fetch(0)
@heap[0] = @heap.pop
i = 0
loop do
left, right = i * 2 + 1, i * 2 + 2
j = if right < size && compare(i, right)
compare(left, right) ? right : left
elsif left < size && compare(i, left)
left
else
break
end
@heap.swap(i, j)
i = j
end
value
end
end
# Like `#pop`, but returns `nil` if `self` is empty.
def pop? : T?
pop { nil }
end
# Removes the lowest value from `self` and returns the removed value.
# Raises `IndexError` if heap is of 0 size.
def pop : T
pop { raise IndexError.new }
end
# Removes the last *n* values from `self` ahd returns the removed values.
def pop(n : Int) : Array(T)
raise ArgumentError.new unless n >= 0
n = Math.min(n, size)
Array.new(n) { pop }
end
# Yields each element of the heap, and returns `nil`.
def each(&) : Nil
@heap.each { |elem| yield elem }
end
# Returns an iterator for each element of the heap.
def each
@heap.each
end
# Returns a new array with all elements sorted.
#
# ```
# a = BinaryHeap.new [3, 1, 2]
# a.sort # => [1, 2, 3]
# b = BinaryHeap.new [3, 1, 2] { |a, b| b <=> a }
# b.sort # => [3, 2, 1]
# ```
def sort : Array(T)
if @compare_proc
@heap.sort { |a, b| @compare_proc.not_nil!.call(a, b) }
else
@heap.sort
end
end
# Returns the elements as an Array.
#
# ```
# BinaryHeap{3, 1, 2}.to_a # => [1, 3, 2]
# ```
def to_a : Array(T)
@heap.dup
end
# Writes a string representation of the heap to `io`.
#
# ```
# BinaryHeap{1, 2}.to_s # => "BinaryHeap{1, 2}"
# ```
def to_s(io : IO) : Nil
io << "BinaryHeap{"
# TODO: use join
each_with_index do |x, i|
io << ", " if i > 0
io << x
end
io << '}'
end
# Writes a string representation of the heap to `io`.
#
# ```
# BinaryHeap{1, 2}.inspect # => "BinaryHeap{1, 2}"
# ```
def inspect(io : IO) : Nil
to_s(io)
end
end
# reference: https://maspypy.com/slope-trick-1-解説編
class SlopeTrick(T)
getter min : T, shift : T, left, right
# Lets `f(x) = 0`
def initialize
@min = T.zero
@shift = T.zero
@left = BinaryHeap(T).new { |a, b| b <=> a }
@right = BinaryHeap(T).new
end
# Calculates `f(x)`
def f(x : T) : T
min +
@left.sum { |l| Math.max(l - x, T.zero) } +
@right.sum { |r| Math.max(x - r, T.zero) }
end
def l0 : T?
@left.top?.try &.+(shift)
end
def r0 : T?
@right.top?.try &.+(shift)
end
# Adds constant function `a`
def add(a : T) : self
@min += a
self
end
# Adds `max(x - a, 0)`
def add_r(a : T) : self
if @left.empty?
@right << a - shift
else
@min += Math.max(l0.not_nil! - a, T.zero)
@left << a - shift
@right << @left.pop.not_nil!
end
self
end
# Adds `max(a - x, 0)`
def add_l(a : T) : self
if @right.empty?
@left << a - shift
else
@min += Math.max(a - r0.not_nil!, T.zero)
@right << a - shift
@left << @right.pop.not_nil!
end
self
end
# Adds `|x - a|`
def add_abs(a : T) : self
add_r(a)
add_l(a)
end
# `g(x) := min_{y <= x} f(y)`
def accumulate_min : self
@right.clear
self
end
# `g(x) := min_{y >= x} f(y)`
def accumulate_min_right : self
@left.clear
self
end
# `g(x) := f(x - a)`
def slide(a : T) : self
@shift += a
self
end
end