src/nmutil/prefix_sum.py

   1 # SPDX-License-Identifier: LGPL-3-or-later
   2 # Copyright 2022 Jacob Lifshay programmerjake@gmail.com
   3
   4 # Funded by NLnet Assure Programme 2021-02-052, https://nlnet.nl/assure part
   5 # of Horizon 2020 EU Programme 957073.
   6
   7 from collections import defaultdict
   8 from dataclasses import dataclass
   9 import operator
  10
  11
  12 @dataclass(order=True, unsafe_hash=True, frozen=True)
  13 class Op:
  14     """An associative operation in a prefix-sum.
  15     The operation is `items[self.out] = fn(items[self.lhs], items[self.rhs])`.
  16     The operation is not assumed to be commutative.
  17     """
  18     out: int
  19     """index of the item to output to"""
  20     lhs: int
  21     """index of the item the left-hand-side input comes from"""
  22     rhs: int
  23     """index of the item the right-hand-side input comes from"""
  24     row: int
  25     """row in the prefix-sum diagram"""
  26
  27
  28 def prefix_sum_ops(item_count, *, work_efficient=False):
  29     """ Get the associative operations needed to compute a parallel prefix-sum
  30     of `item_count` items.
  31
  32     The operations aren't assumed to be commutative.
  33
  34     This has a depth of `O(log(N))` and an operation count of `O(N)` if
  35     `work_efficient` is true, otherwise `O(N*log(N))`.
  36
  37     The algorithms used are derived from:
  38     https://en.wikipedia.org/wiki/Prefix_sum#Algorithm_1:_Shorter_span,_more_parallel
  39     https://en.wikipedia.org/wiki/Prefix_sum#Algorithm_2:_Work-efficient
  40
  41     Parameters:
  42     item_count: int
  43         number of input items.
  44     work_efficient: bool
  45         True if the algorithm used should be work-efficient -- has a larger
  46         depth (about twice as large) but does only `O(N)` operations total
  47         instead of `O(N*log(N))`.
  48     Returns: Iterable[Op]
  49         output associative operations.
  50     """
  51     assert isinstance(item_count, int)
  52     # compute the partial sums using a set of binary trees
  53     # this is the first half of the work-efficient algorithm and the whole of
  54     # the non-work-efficient algorithm.
  55     dist = 1
  56     row = 0
  57     while dist < item_count:
  58         start = dist * 2 - 1 if work_efficient else dist
  59         step = dist * 2 if work_efficient else 1
  60         for i in reversed(range(start, item_count, step)):
  61             yield Op(out=i, lhs=i - dist, rhs=i, row=row)
  62         dist <<= 1
  63         row += 1
  64     if work_efficient:
  65         # express all output items in terms of the computed partial sums.
  66         dist >>= 1
  67         while dist >= 1:
  68             for i in reversed(range(dist * 3 - 1, item_count, dist * 2)):
  69                 yield Op(out=i, lhs=i - dist, rhs=i, row=row)
  70             row += 1
  71             dist >>= 1
  72
  73
  74 def prefix_sum(items, fn=operator.add, *, work_efficient=False):
  75     """ Compute the parallel prefix-sum of `items`, using associative operator
  76     `fn` instead of addition.
  77
  78     This has a depth of `O(log(N))` and an operation count of `O(N)` if
  79     `work_efficient` is true, otherwise `O(N*log(N))`.
  80
  81     The algorithms used are derived from:
  82     https://en.wikipedia.org/wiki/Prefix_sum#Algorithm_1:_Shorter_span,_more_parallel
  83     https://en.wikipedia.org/wiki/Prefix_sum#Algorithm_2:_Work-efficient
  84
  85     Parameters:
  86     items: Iterable[_T]
  87         input items.
  88     fn: Callable[[_T, _T], _T]
  89         Operation to use for the prefix-sum algorithm instead of addition.
  90         Assumed to be associative not necessarily commutative.
  91     work_efficient: bool
  92         True if the algorithm used should be work-efficient -- has a larger
  93         depth (about twice as large) but does only `O(N)` operations total
  94         instead of `O(N*log(N))`.
  95     Returns: list[_T]
  96         output items.
  97     """
  98     items = list(items)
  99     for op in prefix_sum_ops(len(items), work_efficient=work_efficient):
 100         items[op.out] = fn(items[op.lhs], items[op.rhs])
 101     return items
 102
 103
 104 @dataclass
 105 class _Cell:
 106     slant: bool
 107     plus: bool
 108     tee: bool
 109
 110
 111 def render_prefix_sum_diagram(item_count, *, work_efficient=False,
 112                               sp=" ", vbar="|", plus="⊕",
 113                               slant="\\", connect="●", no_connect="X",
 114                               padding=1,
 115                               ):
 116     """renders a prefix-sum diagram, matches `prefix_sum_ops`.
 117
 118     Parameters:
 119     item_count: int
 120         number of input items.
 121     work_efficient: bool
 122         True if the algorithm used should be work-efficient -- has a larger
 123         depth (about twice as large) but does only `O(N)` operations total
 124         instead of `O(N*log(N))`.
 125     sp: str
 126         character used for blank space
 127     vbar: str
 128         character used for a vertical bar
 129     plus: str
 130         character used for the addition operation
 131     slant: str
 132         character used to draw a line from the top left to the bottom right
 133     connect: str
 134         character used to draw a connection between a vertical line and a line
 135         going from the center of this character to the bottom right
 136     no_connect: str
 137         character used to draw two lines crossing but not connecting, the lines
 138         are vertical and diagonal from top left to the bottom right
 139     padding: int
 140         amount of padding characters in the output cells.
 141     Returns: str
 142         rendered diagram
 143     """
 144     assert isinstance(item_count, int)
 145     assert isinstance(padding, int)
 146     ops_by_row = defaultdict(set)
 147     for op in prefix_sum_ops(item_count, work_efficient=work_efficient):
 148         assert op.out == op.rhs, f"can't draw op: {op}"
 149         assert op not in ops_by_row[op.row], f"duplicate op: {op}"
 150         ops_by_row[op.row].add(op)
 151
 152     def blank_row():
 153         return [_Cell(slant=False, plus=False, tee=False)
 154                 for _ in range(item_count)]
 155
 156     cells = [blank_row()]
 157
 158     for row in sorted(ops_by_row.keys()):
 159         ops = ops_by_row[row]
 160         max_distance = max(op.rhs - op.lhs for op in ops)
 161         cells.extend(blank_row() for _ in range(max_distance))
 162         for op in ops:
 163             assert op.lhs < op.rhs and op.out == op.rhs, f"can't draw op: {op}"
 164             y = len(cells) - 1
 165             x = op.out
 166             cells[y][x].plus = True
 167             x -= 1
 168             y -= 1
 169             while op.lhs < x:
 170                 cells[y][x].slant = True
 171                 x -= 1
 172                 y -= 1
 173             cells[y][x].tee = True
 174
 175     lines = []
 176     for cells_row in cells:
 177         row_text = [[] for y in range(2 * padding + 1)]
 178         for cell in cells_row:
 179             # top padding
 180             for y in range(padding):
 181                 # top left padding
 182                 for x in range(padding):
 183                     is_slant = x == y and (cell.plus or cell.slant)
 184                     row_text[y].append(slant if is_slant else sp)
 185                 # top vertical bar
 186                 row_text[y].append(vbar)
 187                 # top right padding
 188                 for x in range(padding):
 189                     row_text[y].append(sp)
 190             # center left padding
 191             for x in range(padding):
 192                 row_text[padding].append(sp)
 193             # center
 194             center = vbar
 195             if cell.plus:
 196                 center = plus
 197             elif cell.tee:
 198                 center = connect
 199             elif cell.slant:
 200                 center = no_connect
 201             row_text[padding].append(center)
 202             # center right padding
 203             for x in range(padding):
 204                 row_text[padding].append(sp)
 205             # bottom padding
 206             for y in range(padding + 1, 2 * padding + 1):
 207                 # bottom left padding
 208                 for x in range(padding):
 209                     row_text[y].append(sp)
 210                 # bottom vertical bar
 211                 row_text[y].append(vbar)
 212                 # bottom right padding
 213                 for x in range(padding + 1, 2 * padding + 1):
 214                     is_slant = x == y and (cell.tee or cell.slant)
 215                     row_text[y].append(slant if is_slant else sp)
 216         for line in row_text:
 217             lines.append("".join(line))
 218
 219     return "\n".join(map(str.rstrip, lines))
 220
 221
 222 if __name__ == "__main__":
 223     print("the non-work-efficient algorithm, matches the diagram in wikipedia:"
 224           "\n"
 225           "https://commons.wikimedia.org/wiki/File:Hillis-Steele_Prefix_Sum.svg"
 226           "\n\n")
 227     print(render_prefix_sum_diagram(16, work_efficient=False))
 228     print()
 229     print()
 230     print("the work-efficient algorithm, matches the diagram in wikipedia:")
 231     print("https://en.wikipedia.org/wiki/File:Prefix_sum_16.svg")
 232     print()
 233     print(render_prefix_sum_diagram(16, work_efficient=True))
 234     print()
 235     print()