src/BitTree.bend
mylsm-lsm-store@0.6.0.0 · 261 lines · 0 definitions
1import Base
2import 0x9ee2e9a299991dcc089fe22c7f3ceb5f/src/containers/bitset.bend as BitWords
3import 0x9ee2e9a299991dcc089fe22c7f3ceb5f/src/math/pow2.bend as Pow2
4
5# Pure indexed bit tree.
6#
7# A non-empty BitTree is a perfect binary tree whose leaves are U32 words.
8# `words` is the logical word count; `capacity` is the power-of-two leaf
9# capacity, and `height` is log2(capacity). Leaves at indexes >= words are
10# padding. A word index follows its `height`-bit binary representation from
11# most-significant bit at the root to least-significant bit at the leaf.
12#
13# The empty value is represented by Empty{} with all metadata zero. Bit
14# indexes outside [0, words * 32) are total: test returns False{} and set is
15# the identity.
16
17# Represent WordTree data used by the packed bit-tree operations.
18type WordTree is Data:
19 Empty{}
20 Leaf{word: U32}
21 Fork{left: WordTree, right: WordTree}
22
23# Represent BitTree data used by the packed bit-tree operations.
24type BitTree is Data:
25 BT{root: WordTree, words: Nat, capacity: Nat, height: Nat}
26
27# Represent Plan data used by the packed bit-tree operations.
28type Plan is Data:
29 Plan{capacity: Nat, height: Nat}
30
31# Find the least power of two >= need. `fuel = need` is a conservative
32# termination bound; each non-final step doubles capacity.
33def plan_go(fuel: Nat, +need: Nat, +capacity: Nat, +height: Nat, done: Bool) -> Plan:
34 match fuel:
35 case 0n:
36 Plan{capacity, height}
37 case 1n+f:
38 match done:
39 case True{}:
40 Plan{capacity, height}
41 case False{}:
42 +next = Nat.double(capacity)
43 plan_go(f, need, next, Nat.add(height, 1n), Nat.is_le(need, next))
44
45# Handle plan in the packed bit-tree operations.
46def plan(+need: Nat) -> Plan:
47 match need:
48 case 0n:
49 Plan{0n, 0n}
50 case 1n+n:
51 plan_go(need, need, 1n, 0n, Nat.is_le(need, 1n))
52
53# Independent subtrees are built in parallel.
54def zero_tree(+height: Nat) -> WordTree:
55 match height:
56 case 0n:
57 Leaf{0}
58 case 1n+h:
59 left right = zero_tree(h) zero_tree(h)
60 Fork{left, right}
61
62# Create from plan for the packed bit-tree operations.
63def new_from_plan(words: Nat, plan: Plan) -> BitTree:
64 match plan:
65 case Plan{capacity, +height}:
66 BT{zero_tree(height), words, capacity, height}
67
68# Handle new in the packed bit-tree operations.
69def new(+words: Nat) -> BitTree:
70 match words:
71 case 0n:
72 BT{Empty{}, 0n, 0n, 0n}
73 case 1n+n:
74 new_from_plan(words, plan(words))
75
76# Metadata helpers intended to keep representation laws independent of field
77# layout.
78def word_count(tree: BitTree) -> Nat:
79 match tree:
80 case BT{root, words, capacity, height}:
81 words
82
83# Handle bit count in the packed bit-tree operations.
84def bit_count(tree: BitTree) -> Nat:
85 Nat.mul(word_count(tree), 32n)
86
87# Check whether bit holds for the packed bit-tree operations.
88def valid_bit(tree: BitTree, bit: Nat) -> Bool:
89 Nat.is_lt(bit, bit_count(tree))
90
91# Build the root-to-leaf path by consuming low bits while prepending them.
92def path_bit(rem: Nat) -> Bool:
93 match rem:
94 case 0n:
95 False{}
96 case 1n+r:
97 True{}
98
99# Handle index path in the packed bit-tree operations.
100def index_path(height: Nat, +index: Nat, acc: List<&2, Bool>) -> List<&2, Bool>:
101 match height:
102 case 0n:
103 acc
104 case 1n+h:
105 index_path(h, Nat.div(index, 2n), Con{path_bit(Nat.mod(index, 2n)), acc})
106
107# Handle tree word in the packed bit-tree operations.
108def tree_word(path: List<&2, Bool>, tree: WordTree) -> U32:
109 match path tree:
110 case Nil{} Empty{}:
111 0
112 case Nil{} Leaf{word}:
113 word
114 case Nil{} Fork{left, right}:
115 0
116 case Con{side, rest} Empty{}:
117 0
118 case Con{side, rest} Leaf{word}:
119 0
120 case Con{False{}, rest} Fork{left, right}:
121 tree_word(rest, left)
122 case Con{True{}, rest} Fork{left, right}:
123 tree_word(rest, right)
124
125# Handle tree set word in the packed bit-tree operations.
126def tree_set_word(path: List<&2, Bool>, value: U32, tree: WordTree) -> WordTree:
127 match path tree:
128 case Nil{} Empty{}:
129 Empty{}
130 case Nil{} Leaf{word}:
131 Leaf{value}
132 case Nil{} Fork{left, right}:
133 Fork{left, right}
134 case Con{side, rest} Empty{}:
135 Empty{}
136 case Con{side, rest} Leaf{word}:
137 Leaf{word}
138 case Con{False{}, rest} Fork{left, right}:
139 Fork{tree_set_word(rest, value, left), right}
140 case Con{True{}, rest} Fork{left, right}:
141 Fork{left, tree_set_word(rest, value, right)}
142
143# Handle word at if in the packed bit-tree operations.
144def word_at_if(inside: Bool, tree: BitTree, index: Nat) -> U32:
145 match inside:
146 case False{}:
147 0
148 case True{}:
149 match tree:
150 case BT{root, words, capacity, height}:
151 tree_word(index_path(height, index, Nil{}), root)
152
153# Safe logical-word lookup; padding leaves are not externally addressable.
154def word_at(+tree: BitTree, +index: Nat) -> U32:
155 word_at_if(Nat.is_lt(index, word_count(tree)), tree, index)
156
157# Replace word if for the packed bit-tree operations.
158def replace_word_if(inside: Bool, tree: BitTree, index: Nat, value: U32) -> BitTree:
159 match inside:
160 case False{}:
161 tree
162 case True{}:
163 match tree:
164 case BT{root, words, capacity, +height}:
165 BT{tree_set_word(index_path(height, index, Nil{}), value, root), words, capacity, height}
166
167# Safe logical-word update. This helper makes metadata preservation and
168# unaffected-word laws direct to state.
169def replace_word(+tree: BitTree, +index: Nat, value: U32) -> BitTree:
170 replace_word_if(Nat.is_lt(index, word_count(tree)), tree, index, value)
171
172# Update if for the packed bit-tree operations.
173def set_if(inside: Bool, +tree: BitTree, +bit: Nat) -> BitTree:
174 match inside:
175 case False{}:
176 tree
177 case True{}:
178 +index = Nat.div(bit, 32n)
179 +old = word_at(tree, index)
180 replace_word(tree, index, BitWords.word_put(True{}, old, Nat.mod(bit, 32n)))
181
182# Set one logical bit. Out-of-range indexes leave the tree unchanged.
183def set(+tree: BitTree, +bit: Nat) -> BitTree:
184 set_if(valid_bit(tree, bit), tree, bit)
185
186# Handle test if in the packed bit-tree operations.
187def test_if(inside: Bool, +tree: BitTree, +bit: Nat) -> Bool:
188 match inside:
189 case False{}:
190 False{}
191 case True{}:
192 BitWords.word_get(word_at(tree, Nat.div(bit, 32n)), Nat.mod(bit, 32n))
193
194# Test one logical bit. Out-of-range indexes, including every index in the
195# zero-word tree, return False{}.
196def test(+tree: BitTree, +bit: Nat) -> Bool:
197 test_if(valid_bit(tree, bit), tree, bit)
198
199# Word-wise union for chunked Bloom builds. Both sides always share the exact
200# shape (same height from the same bit count); mismatched arms are unreachable
201# by construction and keep the Fork side.
202def union_bits(ta: WordTree, tb: WordTree) -> WordTree:
203 match ta tb:
204 case Empty{} Empty{}:
205 Empty{}
206 case Empty{} Leaf{wb}:
207 Leaf{wb}
208 case Empty{} Fork{lb, rb}:
209 Fork{lb, rb}
210 case Leaf{wa} Empty{}:
211 Leaf{wa}
212 case Leaf{wa} Leaf{wb}:
213 Leaf{U32.or(wa, wb)}
214 case Leaf{wa} Fork{lb, rb}:
215 Fork{lb, rb}
216 case Fork{la, ra} Empty{}:
217 Fork{la, ra}
218 case Fork{la, ra} Leaf{wb}:
219 Fork{la, ra}
220 case Fork{la, ra} Fork{lb, rb}:
221 Fork{union_bits(la, lb), union_bits(ra, rb)}
222
223# Structural checker for the exact perfect-tree shape described by metadata.
224def shape_ok(+height: Nat, tree: WordTree) -> Bool:
225 match height tree:
226 case 0n Empty{}:
227 False{}
228 case 0n Leaf{word}:
229 True{}
230 case 0n Fork{left, right}:
231 False{}
232 case 1n+h Empty{}:
233 False{}
234 case 1n+h Leaf{word}:
235 False{}
236 case 1n+h Fork{left, right}:
237 a b = shape_ok(h, left) shape_ok(h, right)
238 Bool.and(a, b)
239
240# Check whether whether metadata ok holds for the packed bit-tree operations.
241def empty_metadata_ok(words: Nat, capacity: Nat, height: Nat) -> Bool:
242 Bool.and(Nat.is_eq(words, 0n), Bool.and(Nat.is_eq(capacity, 0n), Nat.is_eq(height, 0n)))
243
244# Handle nonempty metadata ok in the packed bit-tree operations.
245def nonempty_metadata_ok(root: WordTree, +words: Nat, +capacity: Nat, +height: Nat) -> Bool:
246 Bool.and(
247 Nat.is_gt(words, 0n),
248 Bool.and(
249 Nat.is_le(words, capacity),
250 Bool.and(
251 Nat.is_eq(capacity, Pow2.pow2t(height)),
252 shape_ok(height, root))))
253
254# Checks zero canonicality, logical bounds, power-of-two capacity, height, and
255# exact tree shape. It does not require padding words to remain zero.
256def well_formed(tree: BitTree) -> Bool:
257 match tree:
258 case BT{Empty{}, words, capacity, height}:
259 empty_metadata_ok(words, capacity, height)
260 case BT{root, words, capacity, height}:
261 nonempty_metadata_ok(root, words, capacity, height)