Implement both min and max heaps (max-heaps implemented as min-heaps wit...

examples/heap.jl

@@ -1,162 +1,312 @@
-# These functions define a heap
+# heap.jl
+#
+# This file defines Heap objects and a variety of utility functions.
+#
 # Example usage of the Heap object:
-# You can create an empty heap this way:
-#   h = Heap(Float64)
-# This defaults to using < for comparison, so is a min-heap.
+# You can create an empty min-heap this way:
+#   h = MinHeap(Float64)
 #
-# A more complex and complete example, yielding a max-heap:
-#   h = Heap(>,rand(3))  # initialize heap with 3 random points
+# A more complex and complete example with a max-heap:
+#   h = MaxHeap(rand(3))  # initialize heap with 3 random points
 #   for i = 1:10
 #     push!(h,rand())    # add more points
 #   end
 #   length(h)
 #   max_val = pop!(h)
-#  
-# Example using pure vectors (avoids making any copies):
+# 
+# You can also work with indexed heaps:
+#   z = rand(8)
+#   h = MinHeapIndirect(typeof(z))
+#   for i = 1:length(z)
+#     push!(h,z[i])
+#   end
+#   min_index, min_val = pop!(h)
+#
+# Finally, you can do min-heaps using pure vectors. This avoids the
+# need to make a copy of the data:
 #   z = rand(8)
 #   vector2heap!(z)
 #   isheap(z)
 #   min_val = heap_pop!(z)
 #   heap2rsorted!(z)
+# You can also do indirect min-heaps this way, by supplying an index
+# vector as the first argument.
+
+# Timothy E. Holy, 2012
 
 ## Function definitions for operating on vectors ##
 
+# A "direct heap" is represented as a vector, each entry containing
+# the "value" (key) of a particular item.  An "indirect heap" is
+# stored as a vector of "integer pointers" (denoted by the variable
+# iptr), and the key of node i is value[iptr[i]]. Representing the
+# heap indirectly (requiring lookup of the value) has some cost, but
+# is useful when the item index is the more fundamentally-interesting
+# quantity.
+
+# For an indirect heap, note that value stores the entire set of
+# values ever added to the heap, whereas iptr reflects the current
+# state of the heap.  Hence, value may be a longer vector than iptr,
+# if items have been popped off the heap.
+
+# These functions implement a min-heap. You can get a max-heap using
+# the Heap objects below.
+
 
 # This is a "percolate down" function, used by several other
 # functions. "percolate up" is implemented in heap_push.
-function _heapify!{T}(cmp::Function,x::Vector{T},i::Int,len::Int)
-    il = 2*i      # index of left child
+# direct version:
+function _heapify!{T}(value::Vector{T},i::Int,len::Int)
+    il = 2*i      # node index of left child
     while il <= len
-        # Among node i and its two children, find the extreme value
-        #(e.g., the smallest when using < for comparison)
-        iextreme = cmp(x[il],x[i]) ? il : i  # index of extreme value
+        # Among node i and its two children, find the smallest value
+        ismallest = isless(value[il],value[i]) ? il : i  # node index of smallest value
         if il < len
-            iextreme = cmp(x[il+1],x[iextreme]) ? il+1 : iextreme
+            ismallest = isless(value[il+1],value[ismallest]) ? il+1 : ismallest
         end
-        if iextreme == i
+        if ismallest == i
             return   # The heap below this node is fine
         end
-        # Put the extreme value at i via a swap
-        tmp = x[iextreme]
-        x[iextreme] = x[i]
-        x[i] = tmp
+        # Put the smallest value at i via a swap of their iptrs
+        value[ismallest], value[i] = value[i], value[ismallest]
         # Descend to the modified child
-        i = iextreme
+        i = ismallest
+        il = 2*i
+    end
+end
+# indirect version:
+function _heapify!{T}(iptr::Vector{Int},value::Vector{T},i::Int,len::Int)
+    il = 2*i
+    while il <= len
+        ismallest = isless(value[iptr[il]],value[iptr[i]]) ? il : i
+        if il < len
+            ismallest = isless(value[iptr[il+1]],value[iptr[ismallest]]) ? il+1 : ismallest
+        end
+        if ismallest == i
+            return
+        end
+        iptr[ismallest], iptr[i] = iptr[i], iptr[ismallest]
+        i = ismallest
         il = 2*i
     end
 end
 
 
 # Convert an arbitrary vector into heap storage format
-function vector2heap!{T}(cmp::Function, x::Vector{T})
-    for i = convert(Int,ifloor(length(x)/2)):-1:1
-        _heapify!(cmp,x,i,length(x))
+function vector2heap!{T}(value::Vector{T})
+    for i = convert(Int,ifloor(length(value)/2)):-1:1
+        _heapify!(value,i,length(value))
     end
 end
-function vector2heap!{T}(x::Vector{T})
-    vector2heap!(<,x)
+function vector2heap!{T}(iptr::Vector{Int},value::Vector{T})
+    for i = convert(Int,ifloor(length(iptr)/2)):-1:1
+        _heapify!(iptr,value,i,length(iptr))
+    end
 end
 
 # Test whether a vector is a valid heap
-function isheap{T}(cmp::Function, x::Vector{T})
-    for i = 1:convert(Int,ifloor(length(x)/2))
+function isheap{T}(value::Vector{T})
+    for i = 1:convert(Int,ifloor(length(iptr)/2))
         i2 = 2*i
-        if !cmp(x[i],x[i2])
+        if isless(value[i2],value[i])
             return false
         end
-        if i2 < length(x) && !cmp(x[i],x[i2+1])
+        if i2 < length(x) && isless(value[i2+1],value[i])
             return false
         end
     end
     return true
 end
-function isheap{T}(x::Vector{T})
-    isheap(<,x)
+function isheap{T}(iptr::Vector{Int},value::Vector{T})
+    for i = 1:convert(Int,ifloor(length(iptr)/2))
+        i2 = 2*i
+        if isless(value[iptr[i2]],value[iptr[i]])
+            return false
+        end
+        if i2 < length(x) && isless(value[iptr[i2+1]],value[iptr[i]])
+            return false
+        end
+    end
+    return true
 end
 
 # Add a new item to a heap
-function heap_push!{T}(cmp::Function, x::Vector{T},item::T)
-    # Append new element at the bottom
-    push(x,item)
+function heap_push!{T}(value::Vector{T},newvalue::T)
+    # Append the new value at the bottom
+    push(value,newvalue)
     # Let the new item percolate up until stopped by a more-extreme parent
-    i = length(x)
+    i = length(value)
     ip = convert(Int,ifloor(i/2))  # index of parent
-    while i > 1 && cmp(x[i],x[ip])
+    while i > 1 && isless(value[i],value[ip])
         # Swap i and its parent
-        tmp = x[ip]
-        x[ip] = x[i]
-        x[i] = tmp
+        value[i], value[ip] = value[ip], value[i]
         # Traverse up the tree
         i = ip
         ip = convert(Int,ifloor(i/2))
     end
 end
-function heap_push!{T}(x::Vector{T},item::T)
-    heap_push!(<,x,item)
+function heap_push!{T}(iptr::Vector{Int},value::Vector{T},newvalue::T)
+    push(value,newvalue)
+    push(iptr,length(value))
+    i = length(iptr)
+    ip = convert(Int,ifloor(i/2))
+    while i > 1 && isless(value[iptr[i]],value[iptr[ip]])
+        iptr[i], iptr[ip] = iptr[ip], iptr[i]
+        i = ip
+        ip = convert(Int,ifloor(i/2))
+    end
 end
 
 # Remove the root node from the heap, leaving the remaining values in
 # a valid heap
-function heap_pop!{T}(cmp::Function, x::Vector{T})
-    # Save the value we want to return
-    extreme_item = x[1]
+function heap_pop!{T}(value::Vector{T})
+    # Save the item we want to return
+    extreme_item = value[1]
     # We need to shorten the list, so replace the former top with the
     # last item, then let it percolate down
-    x[1] = x[end]
-    pop(x)
-    _heapify!(cmp,x,1,length(x))
+    value[1] = value[end]
+    pop(value)
+    _heapify!(value,1,length(value))
     return extreme_item
 end
-function heap_pop!{T}(x::Vector{T})
-    extreme_item = heap_pop!(<,x)
+function heap_pop!{T}(iptr::Vector{Int},value::Vector{T})
+    extreme_item = iptr[1]
+    iptr[1] = iptr[end]
+    pop(iptr)
+    _heapify!(iptr,value,1,length(iptr))
     return extreme_item
 end
 
 # From a heap, return a sorted vector. This is implemented efficiently
 # if the sorting is in the reverse order of the comparison function.
-function heap2rsorted!{T}(cmp::Function, x::Vector{T})
-    for i = length(x):-1:2
+function heap2rsorted!{T}(value::Vector{T})
+    for i = length(value):-1:2
         # Swap the root with i, the last unsorted position
-        tmp = x[1]
-        x[1] = x[i]
-        x[i] = tmp
+        value[1], value[i] = value[i], value[1]
         # The heap portion now has length i-1, but needs fixing up
         # starting with the root
-        _heapify!(cmp,x,1,i-1)
+        _heapify!(value,1,i-1)
     end
 end
-function heap2rsorted!{T}(x::Vector{T})
-    heap2rsorted!(<,x)
+function heap2rsorted!{T}(iptr::Vector{Int},value::Vector{T})
+    for i = length(iptr):-1:2
+        iptr[1], iptr[i] = iptr[i], iptr[1]
+        _heapify!(iptr,value,1,i-1)
+    end
 end
 
 
 
-## Heap object ##
+## Heap objects ##
 
-type Heap{T}
-    cmp::Function
-    data::Array{T,1}
+abstract Heap
+abstract HeapDirect <: Heap
+abstract HeapIndirect <: Heap
 
-    function Heap(cmp::Function,x::Array{T,1})
-        data = copy(x)
-        vector2heap!(cmp,data)
-        new(cmp,data)
+# MinHeap
+type MinHeap{T} <: HeapDirect
+    value::Vector{T}
+
+    function MinHeap(v::Vector{T})
+        value = copy(v)
+        vector2heap!(value)
+        new(value)
     end
 end
-Heap{T}(cmp::Function,x::Vector{T}) = Heap{T}(cmp,x)
-Heap{T}(x::Vector{T}) = Heap{T}(<,x)
-Heap{T}(cmp::Function,::Type{T}) = Heap{T}(cmp,zeros(T,0))
-Heap{T}(::Type{T}) = Heap{T}(<,zeros(T,0))
+MinHeap{T}(v::Vector{T}) = MinHeap{T}(v)
+MinHeap{T}(::Type{T}) = MinHeap{T}(zeros(T,0))
 
-function push!{T}(h::Heap{T},item::T)
-    heap_push!(h.cmp,h.data,item)
+function push!{T}(h::MinHeap{T},item::T)
+    heap_push!(h.data,item)
 end
 
-function pop!{T}(h::Heap{T})
-    extreme_item = heap_pop!(h.cmp,h.data)
-    return extreme_item
+function pop!{T}(h::MinHeap{T})
+    min_item = heap_pop!(h.data)
+    return min_item
+end
+
+function length{T}(h::HeapDirect)
+    return length(h.value)
+end
+
+function isempty{T}(h::HeapDirect)
+    return isempty(h.value)
+end
+
+# MaxHeap
+type MaxHeap{T} <: HeapDirect
+    value::Vector{T}
+
+    function MaxHeap(v::Vector{T})
+        value = copy(v)
+        vector2heap!(value)
+        new(value)
+    end
+end
+MaxHeap{T}(v::Vector{T}) = MaxHeap{T}(v)
+MaxHeap{T}(::Type{T}) = MaxHeap{T}(zeros(T,0))
+
+function push!{T}(h::MaxHeap{T},item::T)
+    heap_push!(h.data,-item)
+end
+
+function pop!{T}(h::MaxHeap{T})
+    max_item = -heap_pop!(h.data)
+    return max_item
+end
+
+# MinHeapIndirect (an indexed heap)
+type MinHeapIndirect{T} <: HeapIndirect
+    index::Vector{Int}
+    value::Vector{T}
+
+    function MinHeapIndirect(v::Vector{T})
+        value = copy(v)
+        index = linspace(1,length(v),length(v))
+        vector2heap!(index,value)
+        new(index,value)
+    end
+end
+#MinHeapIndirect{T}(i::Vector{Int},v::Vector{T}) = MinHeapIndirect{T}(i,v)
+MinHeapIndirect{T}(v::Vector{T}) = MinHeapIndirect{T}(v)
+MinHeapIndirect{T}(::Type{T}) = MinHeapIndirect{T}(zeros(T,0))
+
+function push!{T}(h::MinHeapIndirect{T},newvalue::T)
+    heap_push!(h.index,h.value,newvalue)
+end
+
+function pop!{T}(h::MinHeapIndirect{T})
+    min_index = heap_pop!(h.index,h.value)
+    return (min_index, h.value[min_index])
+end
+
+function length{T}(h::HeapIndirect)
+    return length(h.index)
+end
+function isempty{T}(h::HeapIndirect)
+    return isempty(h.index)
+end
+
+# MaxHeapIndirect (an indexed heap)
+type MaxHeapIndirect{T} <: HeapIndirect
+    index::Vector{Int}
+    value::Vector{T}
+
+    function HeapIndirect(v::Vector{T})
+        value = copy(v)
+        index = linspace(1,length(v),length(v))
+        vector2heap!(index,value)
+        new(index,value)
+    end
+end
+MaxHeapIndirect{T}(v::Vector{T}) = MaxHeapIndirect{T}(v)
+MaxHeapIndirect{T}(::Type{T}) = MaxHeapIndirect{T}(zeros(T,0))
+
+function push!{T}(h::MaxHeapIndirect{T},newvalue::T)
+    heap_push!(h.index,h.value,-newvalue)
 end
 
-function length{T}(h::Heap{T})
-    return length(h.data)
+function pop!{T}(h::MaxHeapIndirect{T})
+    min_index = heap_pop!(h.index,h.value)
+    return min_index, -h.value[index]
 end