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|
// Copyright 2019+ Klaus Post. All rights reserved.
// License information can be found in the LICENSE file.
// Based on work by Yann Collet, released under BSD License.
package zstd
import (
"errors"
"fmt"
"math"
"math/bits"
"github.com/klauspost/compress/huff0"
)
type blockEnc struct {
size int
literals []byte
sequences []seq
coders seqCoders
litEnc *huff0.Scratch
wr bitWriter
extraLits int
last bool
output []byte
recentOffsets [3]uint32
prevRecentOffsets [3]uint32
}
// init should be used once the block has been created.
// If called more than once, the effect is the same as calling reset.
func (b *blockEnc) init() {
if cap(b.literals) < maxCompressedLiteralSize {
b.literals = make([]byte, 0, maxCompressedLiteralSize)
}
const defSeqs = 200
b.literals = b.literals[:0]
if cap(b.sequences) < defSeqs {
b.sequences = make([]seq, 0, defSeqs)
}
if cap(b.output) < maxCompressedBlockSize {
b.output = make([]byte, 0, maxCompressedBlockSize)
}
if b.coders.mlEnc == nil {
b.coders.mlEnc = &fseEncoder{}
b.coders.mlPrev = &fseEncoder{}
b.coders.ofEnc = &fseEncoder{}
b.coders.ofPrev = &fseEncoder{}
b.coders.llEnc = &fseEncoder{}
b.coders.llPrev = &fseEncoder{}
}
b.litEnc = &huff0.Scratch{WantLogLess: 4}
b.reset(nil)
}
// initNewEncode can be used to reset offsets and encoders to the initial state.
func (b *blockEnc) initNewEncode() {
b.recentOffsets = [3]uint32{1, 4, 8}
b.litEnc.Reuse = huff0.ReusePolicyNone
b.coders.setPrev(nil, nil, nil)
}
// reset will reset the block for a new encode, but in the same stream,
// meaning that state will be carried over, but the block content is reset.
// If a previous block is provided, the recent offsets are carried over.
func (b *blockEnc) reset(prev *blockEnc) {
b.extraLits = 0
b.literals = b.literals[:0]
b.size = 0
b.sequences = b.sequences[:0]
b.output = b.output[:0]
b.last = false
if prev != nil {
b.recentOffsets = prev.prevRecentOffsets
}
}
// reset will reset the block for a new encode, but in the same stream,
// meaning that state will be carried over, but the block content is reset.
// If a previous block is provided, the recent offsets are carried over.
func (b *blockEnc) swapEncoders(prev *blockEnc) {
b.coders.swap(&prev.coders)
b.litEnc, prev.litEnc = prev.litEnc, b.litEnc
}
// blockHeader contains the information for a block header.
type blockHeader uint32
// setLast sets the 'last' indicator on a block.
func (h *blockHeader) setLast(b bool) {
if b {
*h = *h | 1
} else {
const mask = (1 << 24) - 2
*h = *h & mask
}
}
// setSize will store the compressed size of a block.
func (h *blockHeader) setSize(v uint32) {
const mask = 7
*h = (*h)&mask | blockHeader(v<<3)
}
// setType sets the block type.
func (h *blockHeader) setType(t blockType) {
const mask = 1 | (((1 << 24) - 1) ^ 7)
*h = (*h & mask) | blockHeader(t<<1)
}
// appendTo will append the block header to a slice.
func (h blockHeader) appendTo(b []byte) []byte {
return append(b, uint8(h), uint8(h>>8), uint8(h>>16))
}
// String returns a string representation of the block.
func (h blockHeader) String() string {
return fmt.Sprintf("Type: %d, Size: %d, Last:%t", (h>>1)&3, h>>3, h&1 == 1)
}
// literalsHeader contains literals header information.
type literalsHeader uint64
// setType can be used to set the type of literal block.
func (h *literalsHeader) setType(t literalsBlockType) {
const mask = math.MaxUint64 - 3
*h = (*h & mask) | literalsHeader(t)
}
// setSize can be used to set a single size, for uncompressed and RLE content.
func (h *literalsHeader) setSize(regenLen int) {
inBits := bits.Len32(uint32(regenLen))
// Only retain 2 bits
const mask = 3
lh := uint64(*h & mask)
switch {
case inBits < 5:
lh |= (uint64(regenLen) << 3) | (1 << 60)
if debug {
got := int(lh>>3) & 0xff
if got != regenLen {
panic(fmt.Sprint("litRegenSize = ", regenLen, "(want) != ", got, "(got)"))
}
}
case inBits < 12:
lh |= (1 << 2) | (uint64(regenLen) << 4) | (2 << 60)
case inBits < 20:
lh |= (3 << 2) | (uint64(regenLen) << 4) | (3 << 60)
default:
panic(fmt.Errorf("internal error: block too big (%d)", regenLen))
}
*h = literalsHeader(lh)
}
// setSizes will set the size of a compressed literals section and the input length.
func (h *literalsHeader) setSizes(compLen, inLen int, single bool) {
compBits, inBits := bits.Len32(uint32(compLen)), bits.Len32(uint32(inLen))
// Only retain 2 bits
const mask = 3
lh := uint64(*h & mask)
switch {
case compBits <= 10 && inBits <= 10:
if !single {
lh |= 1 << 2
}
lh |= (uint64(inLen) << 4) | (uint64(compLen) << (10 + 4)) | (3 << 60)
if debug {
const mmask = (1 << 24) - 1
n := (lh >> 4) & mmask
if int(n&1023) != inLen {
panic(fmt.Sprint("regensize:", int(n&1023), "!=", inLen, inBits))
}
if int(n>>10) != compLen {
panic(fmt.Sprint("compsize:", int(n>>10), "!=", compLen, compBits))
}
}
case compBits <= 14 && inBits <= 14:
lh |= (2 << 2) | (uint64(inLen) << 4) | (uint64(compLen) << (14 + 4)) | (4 << 60)
if single {
panic("single stream used with more than 10 bits length.")
}
case compBits <= 18 && inBits <= 18:
lh |= (3 << 2) | (uint64(inLen) << 4) | (uint64(compLen) << (18 + 4)) | (5 << 60)
if single {
panic("single stream used with more than 10 bits length.")
}
default:
panic("internal error: block too big")
}
*h = literalsHeader(lh)
}
// appendTo will append the literals header to a byte slice.
func (h literalsHeader) appendTo(b []byte) []byte {
size := uint8(h >> 60)
switch size {
case 1:
b = append(b, uint8(h))
case 2:
b = append(b, uint8(h), uint8(h>>8))
case 3:
b = append(b, uint8(h), uint8(h>>8), uint8(h>>16))
case 4:
b = append(b, uint8(h), uint8(h>>8), uint8(h>>16), uint8(h>>24))
case 5:
b = append(b, uint8(h), uint8(h>>8), uint8(h>>16), uint8(h>>24), uint8(h>>32))
default:
panic(fmt.Errorf("internal error: literalsHeader has invalid size (%d)", size))
}
return b
}
// size returns the output size with currently set values.
func (h literalsHeader) size() int {
return int(h >> 60)
}
func (h literalsHeader) String() string {
return fmt.Sprintf("Type: %d, SizeFormat: %d, Size: 0x%d, Bytes:%d", literalsBlockType(h&3), (h>>2)&3, h&((1<<60)-1)>>4, h>>60)
}
// pushOffsets will push the recent offsets to the backup store.
func (b *blockEnc) pushOffsets() {
b.prevRecentOffsets = b.recentOffsets
}
// pushOffsets will push the recent offsets to the backup store.
func (b *blockEnc) popOffsets() {
b.recentOffsets = b.prevRecentOffsets
}
// matchOffset will adjust recent offsets and return the adjusted one,
// if it matches a previous offset.
func (b *blockEnc) matchOffset(offset, lits uint32) uint32 {
// Check if offset is one of the recent offsets.
// Adjusts the output offset accordingly.
// Gives a tiny bit of compression, typically around 1%.
if true {
if lits > 0 {
switch offset {
case b.recentOffsets[0]:
offset = 1
case b.recentOffsets[1]:
b.recentOffsets[1] = b.recentOffsets[0]
b.recentOffsets[0] = offset
offset = 2
case b.recentOffsets[2]:
b.recentOffsets[2] = b.recentOffsets[1]
b.recentOffsets[1] = b.recentOffsets[0]
b.recentOffsets[0] = offset
offset = 3
default:
b.recentOffsets[2] = b.recentOffsets[1]
b.recentOffsets[1] = b.recentOffsets[0]
b.recentOffsets[0] = offset
offset += 3
}
} else {
switch offset {
case b.recentOffsets[1]:
b.recentOffsets[1] = b.recentOffsets[0]
b.recentOffsets[0] = offset
offset = 1
case b.recentOffsets[2]:
b.recentOffsets[2] = b.recentOffsets[1]
b.recentOffsets[1] = b.recentOffsets[0]
b.recentOffsets[0] = offset
offset = 2
case b.recentOffsets[0] - 1:
b.recentOffsets[2] = b.recentOffsets[1]
b.recentOffsets[1] = b.recentOffsets[0]
b.recentOffsets[0] = offset
offset = 3
default:
b.recentOffsets[2] = b.recentOffsets[1]
b.recentOffsets[1] = b.recentOffsets[0]
b.recentOffsets[0] = offset
offset += 3
}
}
} else {
offset += 3
}
return offset
}
// encodeRaw can be used to set the output to a raw representation of supplied bytes.
func (b *blockEnc) encodeRaw(a []byte) {
var bh blockHeader
bh.setLast(b.last)
bh.setSize(uint32(len(a)))
bh.setType(blockTypeRaw)
b.output = bh.appendTo(b.output[:0])
b.output = append(b.output, a...)
if debug {
println("Adding RAW block, length", len(a))
}
}
// encodeLits can be used if the block is only litLen.
func (b *blockEnc) encodeLits(raw bool) error {
var bh blockHeader
bh.setLast(b.last)
bh.setSize(uint32(len(b.literals)))
// Don't compress extremely small blocks
if len(b.literals) < 32 || raw {
if debug {
println("Adding RAW block, length", len(b.literals))
}
bh.setType(blockTypeRaw)
b.output = bh.appendTo(b.output)
b.output = append(b.output, b.literals...)
return nil
}
var (
out []byte
reUsed, single bool
err error
)
if len(b.literals) >= 1024 {
// Use 4 Streams.
out, reUsed, err = huff0.Compress4X(b.literals, b.litEnc)
if len(out) > len(b.literals)-len(b.literals)>>4 {
// Bail out of compression is too little.
err = huff0.ErrIncompressible
}
} else if len(b.literals) > 32 {
// Use 1 stream
single = true
out, reUsed, err = huff0.Compress1X(b.literals, b.litEnc)
if len(out) > len(b.literals)-len(b.literals)>>4 {
// Bail out of compression is too little.
err = huff0.ErrIncompressible
}
} else {
err = huff0.ErrIncompressible
}
switch err {
case huff0.ErrIncompressible:
if debug {
println("Adding RAW block, length", len(b.literals))
}
bh.setType(blockTypeRaw)
b.output = bh.appendTo(b.output)
b.output = append(b.output, b.literals...)
return nil
case huff0.ErrUseRLE:
if debug {
println("Adding RLE block, length", len(b.literals))
}
bh.setType(blockTypeRLE)
b.output = bh.appendTo(b.output)
b.output = append(b.output, b.literals[0])
return nil
default:
return err
case nil:
}
// Compressed...
// Now, allow reuse
b.litEnc.Reuse = huff0.ReusePolicyAllow
bh.setType(blockTypeCompressed)
var lh literalsHeader
if reUsed {
if debug {
println("Reused tree, compressed to", len(out))
}
lh.setType(literalsBlockTreeless)
} else {
if debug {
println("New tree, compressed to", len(out), "tree size:", len(b.litEnc.OutTable))
}
lh.setType(literalsBlockCompressed)
}
// Set sizes
lh.setSizes(len(out), len(b.literals), single)
bh.setSize(uint32(len(out) + lh.size() + 1))
// Write block headers.
b.output = bh.appendTo(b.output)
b.output = lh.appendTo(b.output)
// Add compressed data.
b.output = append(b.output, out...)
// No sequences.
b.output = append(b.output, 0)
return nil
}
// fuzzFseEncoder can be used to fuzz the FSE encoder.
func fuzzFseEncoder(data []byte) int {
if len(data) > maxSequences || len(data) < 2 {
return 0
}
enc := fseEncoder{}
hist := enc.Histogram()[:256]
maxSym := uint8(0)
for i, v := range data {
v = v & 63
data[i] = v
hist[v]++
if v > maxSym {
maxSym = v
}
}
if maxSym == 0 {
// All 0
return 0
}
maxCount := func(a []uint32) int {
var max uint32
for _, v := range a {
if v > max {
max = v
}
}
return int(max)
}
cnt := maxCount(hist[:maxSym])
if cnt == len(data) {
// RLE
return 0
}
enc.HistogramFinished(maxSym, cnt)
err := enc.normalizeCount(len(data))
if err != nil {
return 0
}
_, err = enc.writeCount(nil)
if err != nil {
panic(err)
}
return 1
}
// encode will encode the block and put the output in b.output.
func (b *blockEnc) encode(raw bool) error {
if len(b.sequences) == 0 {
return b.encodeLits(raw)
}
// We want some difference
if len(b.literals) > (b.size - (b.size >> 5)) {
return errIncompressible
}
var bh blockHeader
var lh literalsHeader
bh.setLast(b.last)
bh.setType(blockTypeCompressed)
b.output = bh.appendTo(b.output)
var (
out []byte
reUsed, single bool
err error
)
if len(b.literals) >= 1024 && !raw {
// Use 4 Streams.
out, reUsed, err = huff0.Compress4X(b.literals, b.litEnc)
} else if len(b.literals) > 32 && !raw {
// Use 1 stream
single = true
out, reUsed, err = huff0.Compress1X(b.literals, b.litEnc)
} else {
err = huff0.ErrIncompressible
}
switch err {
case huff0.ErrIncompressible:
lh.setType(literalsBlockRaw)
lh.setSize(len(b.literals))
b.output = lh.appendTo(b.output)
b.output = append(b.output, b.literals...)
if debug {
println("Adding literals RAW, length", len(b.literals))
}
case huff0.ErrUseRLE:
lh.setType(literalsBlockRLE)
lh.setSize(len(b.literals))
b.output = lh.appendTo(b.output)
b.output = append(b.output, b.literals[0])
if debug {
println("Adding literals RLE")
}
default:
if debug {
println("Adding literals ERROR:", err)
}
return err
case nil:
// Compressed litLen...
if reUsed {
if debug {
println("reused tree")
}
lh.setType(literalsBlockTreeless)
} else {
if debug {
println("new tree, size:", len(b.litEnc.OutTable))
}
lh.setType(literalsBlockCompressed)
if debug {
_, _, err := huff0.ReadTable(out, nil)
if err != nil {
panic(err)
}
}
}
lh.setSizes(len(out), len(b.literals), single)
if debug {
printf("Compressed %d literals to %d bytes", len(b.literals), len(out))
println("Adding literal header:", lh)
}
b.output = lh.appendTo(b.output)
b.output = append(b.output, out...)
b.litEnc.Reuse = huff0.ReusePolicyAllow
if debug {
println("Adding literals compressed")
}
}
// Sequence compression
// Write the number of sequences
switch {
case len(b.sequences) < 128:
b.output = append(b.output, uint8(len(b.sequences)))
case len(b.sequences) < 0x7f00: // TODO: this could be wrong
n := len(b.sequences)
b.output = append(b.output, 128+uint8(n>>8), uint8(n))
default:
n := len(b.sequences) - 0x7f00
b.output = append(b.output, 255, uint8(n), uint8(n>>8))
}
if debug {
println("Encoding", len(b.sequences), "sequences")
}
b.genCodes()
llEnc := b.coders.llEnc
ofEnc := b.coders.ofEnc
mlEnc := b.coders.mlEnc
err = llEnc.normalizeCount(len(b.sequences))
if err != nil {
return err
}
err = ofEnc.normalizeCount(len(b.sequences))
if err != nil {
return err
}
err = mlEnc.normalizeCount(len(b.sequences))
if err != nil {
return err
}
// Choose the best compression mode for each type.
// Will evaluate the new vs predefined and previous.
chooseComp := func(cur, prev, preDef *fseEncoder) (*fseEncoder, seqCompMode) {
// See if predefined/previous is better
hist := cur.count[:cur.symbolLen]
nSize := cur.approxSize(hist) + cur.maxHeaderSize()
predefSize := preDef.approxSize(hist)
prevSize := prev.approxSize(hist)
// Add a small penalty for new encoders.
// Don't bother with extremely small (<2 byte gains).
nSize = nSize + (nSize+2*8*16)>>4
switch {
case predefSize <= prevSize && predefSize <= nSize || forcePreDef:
if debug {
println("Using predefined", predefSize>>3, "<=", nSize>>3)
}
return preDef, compModePredefined
case prevSize <= nSize:
if debug {
println("Using previous", prevSize>>3, "<=", nSize>>3)
}
return prev, compModeRepeat
default:
if debug {
println("Using new, predef", predefSize>>3, ". previous:", prevSize>>3, ">", nSize>>3, "header max:", cur.maxHeaderSize()>>3, "bytes")
println("tl:", cur.actualTableLog, "symbolLen:", cur.symbolLen, "norm:", cur.norm[:cur.symbolLen], "hist", cur.count[:cur.symbolLen])
}
return cur, compModeFSE
}
}
// Write compression mode
var mode uint8
if llEnc.useRLE {
mode |= uint8(compModeRLE) << 6
llEnc.setRLE(b.sequences[0].llCode)
if debug {
println("llEnc.useRLE")
}
} else {
var m seqCompMode
llEnc, m = chooseComp(llEnc, b.coders.llPrev, &fsePredefEnc[tableLiteralLengths])
mode |= uint8(m) << 6
}
if ofEnc.useRLE {
mode |= uint8(compModeRLE) << 4
ofEnc.setRLE(b.sequences[0].ofCode)
if debug {
println("ofEnc.useRLE")
}
} else {
var m seqCompMode
ofEnc, m = chooseComp(ofEnc, b.coders.ofPrev, &fsePredefEnc[tableOffsets])
mode |= uint8(m) << 4
}
if mlEnc.useRLE {
mode |= uint8(compModeRLE) << 2
mlEnc.setRLE(b.sequences[0].mlCode)
if debug {
println("mlEnc.useRLE, code: ", b.sequences[0].mlCode, "value", b.sequences[0].matchLen)
}
} else {
var m seqCompMode
mlEnc, m = chooseComp(mlEnc, b.coders.mlPrev, &fsePredefEnc[tableMatchLengths])
mode |= uint8(m) << 2
}
b.output = append(b.output, mode)
if debug {
printf("Compression modes: 0b%b", mode)
}
b.output, err = llEnc.writeCount(b.output)
if err != nil {
return err
}
start := len(b.output)
b.output, err = ofEnc.writeCount(b.output)
if err != nil {
return err
}
if false {
println("block:", b.output[start:], "tablelog", ofEnc.actualTableLog, "maxcount:", ofEnc.maxCount)
fmt.Printf("selected TableLog: %d, Symbol length: %d\n", ofEnc.actualTableLog, ofEnc.symbolLen)
for i, v := range ofEnc.norm[:ofEnc.symbolLen] {
fmt.Printf("%3d: %5d -> %4d \n", i, ofEnc.count[i], v)
}
}
b.output, err = mlEnc.writeCount(b.output)
if err != nil {
return err
}
// Maybe in block?
wr := &b.wr
wr.reset(b.output)
var ll, of, ml cState
// Current sequence
seq := len(b.sequences) - 1
s := b.sequences[seq]
llEnc.setBits(llBitsTable[:])
mlEnc.setBits(mlBitsTable[:])
ofEnc.setBits(nil)
llTT, ofTT, mlTT := llEnc.ct.symbolTT[:256], ofEnc.ct.symbolTT[:256], mlEnc.ct.symbolTT[:256]
// We have 3 bounds checks here (and in the loop).
// Since we are iterating backwards it is kinda hard to avoid.
llB, ofB, mlB := llTT[s.llCode], ofTT[s.ofCode], mlTT[s.mlCode]
ll.init(wr, &llEnc.ct, llB)
of.init(wr, &ofEnc.ct, ofB)
wr.flush32()
ml.init(wr, &mlEnc.ct, mlB)
// Each of these lookups also generates a bounds check.
wr.addBits32NC(s.litLen, llB.outBits)
wr.addBits32NC(s.matchLen, mlB.outBits)
wr.flush32()
wr.addBits32NC(s.offset, ofB.outBits)
if debugSequences {
println("Encoded seq", seq, s, "codes:", s.llCode, s.mlCode, s.ofCode, "states:", ll.state, ml.state, of.state, "bits:", llB, mlB, ofB)
}
seq--
if llEnc.maxBits+mlEnc.maxBits+ofEnc.maxBits <= 32 {
// No need to flush (common)
for seq >= 0 {
s = b.sequences[seq]
wr.flush32()
llB, ofB, mlB := llTT[s.llCode], ofTT[s.ofCode], mlTT[s.mlCode]
// tabelog max is 8 for all.
of.encode(ofB)
ml.encode(mlB)
ll.encode(llB)
wr.flush32()
// We checked that all can stay within 32 bits
wr.addBits32NC(s.litLen, llB.outBits)
wr.addBits32NC(s.matchLen, mlB.outBits)
wr.addBits32NC(s.offset, ofB.outBits)
if debugSequences {
println("Encoded seq", seq, s)
}
seq--
}
} else {
for seq >= 0 {
s = b.sequences[seq]
wr.flush32()
llB, ofB, mlB := llTT[s.llCode], ofTT[s.ofCode], mlTT[s.mlCode]
// tabelog max is below 8 for each.
of.encode(ofB)
ml.encode(mlB)
ll.encode(llB)
wr.flush32()
// ml+ll = max 32 bits total
wr.addBits32NC(s.litLen, llB.outBits)
wr.addBits32NC(s.matchLen, mlB.outBits)
wr.flush32()
wr.addBits32NC(s.offset, ofB.outBits)
if debugSequences {
println("Encoded seq", seq, s)
}
seq--
}
}
ml.flush(mlEnc.actualTableLog)
of.flush(ofEnc.actualTableLog)
ll.flush(llEnc.actualTableLog)
err = wr.close()
if err != nil {
return err
}
b.output = wr.out
if len(b.output)-3 >= b.size {
// Maybe even add a bigger margin.
b.litEnc.Reuse = huff0.ReusePolicyNone
return errIncompressible
}
// Size is output minus block header.
bh.setSize(uint32(len(b.output)) - 3)
if debug {
println("Rewriting block header", bh)
}
_ = bh.appendTo(b.output[:0])
b.coders.setPrev(llEnc, mlEnc, ofEnc)
return nil
}
var errIncompressible = errors.New("incompressible")
func (b *blockEnc) genCodes() {
if len(b.sequences) == 0 {
// nothing to do
return
}
if len(b.sequences) > math.MaxUint16 {
panic("can only encode up to 64K sequences")
}
// No bounds checks after here:
llH := b.coders.llEnc.Histogram()[:256]
ofH := b.coders.ofEnc.Histogram()[:256]
mlH := b.coders.mlEnc.Histogram()[:256]
for i := range llH {
llH[i] = 0
}
for i := range ofH {
ofH[i] = 0
}
for i := range mlH {
mlH[i] = 0
}
var llMax, ofMax, mlMax uint8
for i, seq := range b.sequences {
v := llCode(seq.litLen)
seq.llCode = v
llH[v]++
if v > llMax {
llMax = v
}
v = ofCode(seq.offset)
seq.ofCode = v
ofH[v]++
if v > ofMax {
ofMax = v
}
v = mlCode(seq.matchLen)
seq.mlCode = v
mlH[v]++
if v > mlMax {
mlMax = v
if debug && mlMax > maxMatchLengthSymbol {
panic(fmt.Errorf("mlMax > maxMatchLengthSymbol (%d), matchlen: %d", mlMax, seq.matchLen))
}
}
b.sequences[i] = seq
}
maxCount := func(a []uint32) int {
var max uint32
for _, v := range a {
if v > max {
max = v
}
}
return int(max)
}
if mlMax > maxMatchLengthSymbol {
panic(fmt.Errorf("mlMax > maxMatchLengthSymbol (%d)", mlMax))
}
if ofMax > maxOffsetBits {
panic(fmt.Errorf("ofMax > maxOffsetBits (%d)", ofMax))
}
if llMax > maxLiteralLengthSymbol {
panic(fmt.Errorf("llMax > maxLiteralLengthSymbol (%d)", llMax))
}
b.coders.mlEnc.HistogramFinished(mlMax, maxCount(mlH[:mlMax+1]))
b.coders.ofEnc.HistogramFinished(ofMax, maxCount(ofH[:ofMax+1]))
b.coders.llEnc.HistogramFinished(llMax, maxCount(llH[:llMax+1]))
}
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