path: root/drivers/webp/enc/backward_references.c

// Copyright 2012 Google Inc. All Rights Reserved.
//
// Use of this source code is governed by a BSD-style license
// that can be found in the COPYING file in the root of the source
// tree. An additional intellectual property rights grant can be found
// in the file PATENTS. All contributing project authors may
// be found in the AUTHORS file in the root of the source tree.
// -----------------------------------------------------------------------------
//
// Author: Jyrki Alakuijala (jyrki@google.com)
//

#include <assert.h>
#include <math.h>

#include "./backward_references.h"
#include "./histogram.h"
#include "../dsp/lossless.h"
#include "../dsp/dsp.h"
#include "../utils/color_cache.h"
#include "../utils/utils.h"

#define VALUES_IN_BYTE 256

#define MIN_BLOCK_SIZE 256  // minimum block size for backward references

#define MAX_ENTROPY    (1e30f)

// 1M window (4M bytes) minus 120 special codes for short distances.
#define WINDOW_SIZE_BITS 20
#define WINDOW_SIZE ((1 << WINDOW_SIZE_BITS) - 120)

// Bounds for the match length.
#define MIN_LENGTH 2
// If you change this, you need MAX_LENGTH_BITS + WINDOW_SIZE_BITS <= 32 as it
// is used in VP8LHashChain.
#define MAX_LENGTH_BITS 12
// We want the max value to be attainable and stored in MAX_LENGTH_BITS bits.
#define MAX_LENGTH ((1 << MAX_LENGTH_BITS) - 1)
#if MAX_LENGTH_BITS + WINDOW_SIZE_BITS > 32
#error "MAX_LENGTH_BITS + WINDOW_SIZE_BITS > 32"
#endif

// -----------------------------------------------------------------------------

static const uint8_t plane_to_code_lut[128] = {
 96,   73,  55,  39,  23,  13,   5,  1,  255, 255, 255, 255, 255, 255, 255, 255,
 101,  78,  58,  42,  26,  16,   8,  2,    0,   3,  9,   17,  27,  43,  59,  79,
 102,  86,  62,  46,  32,  20,  10,  6,    4,   7,  11,  21,  33,  47,  63,  87,
 105,  90,  70,  52,  37,  28,  18,  14,  12,  15,  19,  29,  38,  53,  71,  91,
 110,  99,  82,  66,  48,  35,  30,  24,  22,  25,  31,  36,  49,  67,  83, 100,
 115, 108,  94,  76,  64,  50,  44,  40,  34,  41,  45,  51,  65,  77,  95, 109,
 118, 113, 103,  92,  80,  68,  60,  56,  54,  57,  61,  69,  81,  93, 104, 114,
 119, 116, 111, 106,  97,  88,  84,  74,  72,  75,  85,  89,  98, 107, 112, 117
};

static int DistanceToPlaneCode(int xsize, int dist) {
  const int yoffset = dist / xsize;
  const int xoffset = dist - yoffset * xsize;
  if (xoffset <= 8 && yoffset < 8) {
    return plane_to_code_lut[yoffset * 16 + 8 - xoffset] + 1;
  } else if (xoffset > xsize - 8 && yoffset < 7) {
    return plane_to_code_lut[(yoffset + 1) * 16 + 8 + (xsize - xoffset)] + 1;
  }
  return dist + 120;
}

// Returns the exact index where array1 and array2 are different. For an index
// inferior or equal to best_len_match, the return value just has to be strictly
// inferior to best_len_match. The current behavior is to return 0 if this index
// is best_len_match, and the index itself otherwise.
// If no two elements are the same, it returns max_limit.
static WEBP_INLINE int FindMatchLength(const uint32_t* const array1,
                                       const uint32_t* const array2,
                                       int best_len_match, int max_limit) {
  // Before 'expensive' linear match, check if the two arrays match at the
  // current best length index.
  if (array1[best_len_match] != array2[best_len_match]) return 0;

  return VP8LVectorMismatch(array1, array2, max_limit);
}

// -----------------------------------------------------------------------------
//  VP8LBackwardRefs

struct PixOrCopyBlock {
  PixOrCopyBlock* next_;   // next block (or NULL)
  PixOrCopy* start_;       // data start
  int size_;               // currently used size
};

static void ClearBackwardRefs(VP8LBackwardRefs* const refs) {
  assert(refs != NULL);
  if (refs->tail_ != NULL) {
    *refs->tail_ = refs->free_blocks_;  // recycle all blocks at once
  }
  refs->free_blocks_ = refs->refs_;
  refs->tail_ = &refs->refs_;
  refs->last_block_ = NULL;
  refs->refs_ = NULL;
}

void VP8LBackwardRefsClear(VP8LBackwardRefs* const refs) {
  assert(refs != NULL);
  ClearBackwardRefs(refs);
  while (refs->free_blocks_ != NULL) {
    PixOrCopyBlock* const next = refs->free_blocks_->next_;
    WebPSafeFree(refs->free_blocks_);
    refs->free_blocks_ = next;
  }
}

void VP8LBackwardRefsInit(VP8LBackwardRefs* const refs, int block_size) {
  assert(refs != NULL);
  memset(refs, 0, sizeof(*refs));
  refs->tail_ = &refs->refs_;
  refs->block_size_ =
      (block_size < MIN_BLOCK_SIZE) ? MIN_BLOCK_SIZE : block_size;
}

VP8LRefsCursor VP8LRefsCursorInit(const VP8LBackwardRefs* const refs) {
  VP8LRefsCursor c;
  c.cur_block_ = refs->refs_;
  if (refs->refs_ != NULL) {
    c.cur_pos = c.cur_block_->start_;
    c.last_pos_ = c.cur_pos + c.cur_block_->size_;
  } else {
    c.cur_pos = NULL;
    c.last_pos_ = NULL;
  }
  return c;
}

void VP8LRefsCursorNextBlock(VP8LRefsCursor* const c) {
  PixOrCopyBlock* const b = c->cur_block_->next_;
  c->cur_pos = (b == NULL) ? NULL : b->start_;
  c->last_pos_ = (b == NULL) ? NULL : b->start_ + b->size_;
  c->cur_block_ = b;
}

// Create a new block, either from the free list or allocated
static PixOrCopyBlock* BackwardRefsNewBlock(VP8LBackwardRefs* const refs) {
  PixOrCopyBlock* b = refs->free_blocks_;
  if (b == NULL) {   // allocate new memory chunk
    const size_t total_size =
        sizeof(*b) + refs->block_size_ * sizeof(*b->start_);
    b = (PixOrCopyBlock*)WebPSafeMalloc(1ULL, total_size);
    if (b == NULL) {
      refs->error_ |= 1;
      return NULL;
    }
    b->start_ = (PixOrCopy*)((uint8_t*)b + sizeof(*b));  // not always aligned
  } else {  // recycle from free-list
    refs->free_blocks_ = b->next_;
  }
  *refs->tail_ = b;
  refs->tail_ = &b->next_;
  refs->last_block_ = b;
  b->next_ = NULL;
  b->size_ = 0;
  return b;
}

static WEBP_INLINE void BackwardRefsCursorAdd(VP8LBackwardRefs* const refs,
                                              const PixOrCopy v) {
  PixOrCopyBlock* b = refs->last_block_;
  if (b == NULL || b->size_ == refs->block_size_) {
    b = BackwardRefsNewBlock(refs);
    if (b == NULL) return;   // refs->error_ is set
  }
  b->start_[b->size_++] = v;
}

int VP8LBackwardRefsCopy(const VP8LBackwardRefs* const src,
                         VP8LBackwardRefs* const dst) {
  const PixOrCopyBlock* b = src->refs_;
  ClearBackwardRefs(dst);
  assert(src->block_size_ == dst->block_size_);
  while (b != NULL) {
    PixOrCopyBlock* const new_b = BackwardRefsNewBlock(dst);
    if (new_b == NULL) return 0;   // dst->error_ is set
    memcpy(new_b->start_, b->start_, b->size_ * sizeof(*b->start_));
    new_b->size_ = b->size_;
    b = b->next_;
  }
  return 1;
}

// -----------------------------------------------------------------------------
// Hash chains

int VP8LHashChainInit(VP8LHashChain* const p, int size) {
  assert(p->size_ == 0);
  assert(p->offset_length_ == NULL);
  assert(size > 0);
  p->offset_length_ =
      (uint32_t*)WebPSafeMalloc(size, sizeof(*p->offset_length_));
  if (p->offset_length_ == NULL) return 0;
  p->size_ = size;

  return 1;
}

void VP8LHashChainClear(VP8LHashChain* const p) {
  assert(p != NULL);
  WebPSafeFree(p->offset_length_);

  p->size_ = 0;
  p->offset_length_ = NULL;
}

// -----------------------------------------------------------------------------

#define HASH_MULTIPLIER_HI (0xc6a4a793U)
#define HASH_MULTIPLIER_LO (0x5bd1e996U)

static WEBP_INLINE uint32_t GetPixPairHash64(const uint32_t* const argb) {
  uint32_t key;
  key  = argb[1] * HASH_MULTIPLIER_HI;
  key += argb[0] * HASH_MULTIPLIER_LO;
  key = key >> (32 - HASH_BITS);
  return key;
}

// Returns the maximum number of hash chain lookups to do for a
// given compression quality. Return value in range [8, 86].
static int GetMaxItersForQuality(int quality) {
  return 8 + (quality * quality) / 128;
}

static int GetWindowSizeForHashChain(int quality, int xsize) {
  const int max_window_size = (quality > 75) ? WINDOW_SIZE
                            : (quality > 50) ? (xsize << 8)
                            : (quality > 25) ? (xsize << 6)
                            : (xsize << 4);
  assert(xsize > 0);
  return (max_window_size > WINDOW_SIZE) ? WINDOW_SIZE : max_window_size;
}

static WEBP_INLINE int MaxFindCopyLength(int len) {
  return (len < MAX_LENGTH) ? len : MAX_LENGTH;
}

int VP8LHashChainFill(VP8LHashChain* const p, int quality,
                      const uint32_t* const argb, int xsize, int ysize) {
  const int size = xsize * ysize;
  const int iter_max = GetMaxItersForQuality(quality);
  const int iter_min = iter_max - quality / 10;
  const uint32_t window_size = GetWindowSizeForHashChain(quality, xsize);
  int pos;
  uint32_t base_position;
  int32_t* hash_to_first_index;
  // Temporarily use the p->offset_length_ as a hash chain.
  int32_t* chain = (int32_t*)p->offset_length_;
  assert(p->size_ != 0);
  assert(p->offset_length_ != NULL);

  hash_to_first_index =
      (int32_t*)WebPSafeMalloc(HASH_SIZE, sizeof(*hash_to_first_index));
  if (hash_to_first_index == NULL) return 0;

  // Set the int32_t array to -1.
  memset(hash_to_first_index, 0xff, HASH_SIZE * sizeof(*hash_to_first_index));
  // Fill the chain linking pixels with the same hash.
  for (pos = 0; pos < size - 1; ++pos) {
    const uint32_t hash_code = GetPixPairHash64(argb + pos);
    chain[pos] = hash_to_first_index[hash_code];
    hash_to_first_index[hash_code] = pos;
  }
  WebPSafeFree(hash_to_first_index);

  // Find the best match interval at each pixel, defined by an offset to the
  // pixel and a length. The right-most pixel cannot match anything to the right
  // (hence a best length of 0) and the left-most pixel nothing to the left
  // (hence an offset of 0).
  p->offset_length_[0] = p->offset_length_[size - 1] = 0;
  for (base_position = size - 2 < 0 ? 0 : size - 2; base_position > 0;) {
    const int max_len = MaxFindCopyLength(size - 1 - base_position);
    const uint32_t* const argb_start = argb + base_position;
    int iter = iter_max;
    int best_length = 0;
    uint32_t best_distance = 0;
    const int min_pos =
        (base_position > window_size) ? base_position - window_size : 0;
    const int length_max = (max_len < 256) ? max_len : 256;
    uint32_t max_base_position;

    for (pos = chain[base_position]; pos >= min_pos; pos = chain[pos]) {
      int curr_length;
      if (--iter < 0) {
        break;
      }
      assert(base_position > (uint32_t)pos);

      curr_length =
          FindMatchLength(argb + pos, argb_start, best_length, max_len);
      if (best_length < curr_length) {
        best_length = curr_length;
        best_distance = base_position - pos;
        // Stop if we have reached the maximum length. Otherwise, make sure
        // we have executed a minimum number of iterations depending on the
        // quality.
        if ((best_length == MAX_LENGTH) ||
            (curr_length >= length_max && iter < iter_min)) {
          break;
        }
      }
    }
    // We have the best match but in case the two intervals continue matching
    // to the left, we have the best matches for the left-extended pixels.
    max_base_position = base_position;
    while (1) {
      assert(best_length <= MAX_LENGTH);
      assert(best_distance <= WINDOW_SIZE);
      p->offset_length_[base_position] =
          (best_distance << MAX_LENGTH_BITS) | (uint32_t)best_length;
      --base_position;
      // Stop if we don't have a match or if we are out of bounds.
      if (best_distance == 0 || base_position == 0) break;
      // Stop if we cannot extend the matching intervals to the left.
      if (base_position < best_distance ||
          argb[base_position - best_distance] != argb[base_position]) {
        break;
      }
      // Stop if we are matching at its limit because there could be a closer
      // matching interval with the same maximum length. Then again, if the
      // matching interval is as close as possible (best_distance == 1), we will
      // never find anything better so let's continue.
      if (best_length == MAX_LENGTH && best_distance != 1 &&
          base_position + MAX_LENGTH < max_base_position) {
        break;
      }
      if (best_length < MAX_LENGTH) {
        ++best_length;
        max_base_position = base_position;
      }
    }
  }
  return 1;
}

static WEBP_INLINE int HashChainFindOffset(const VP8LHashChain* const p,
                                           const int base_position) {
  return p->offset_length_[base_position] >> MAX_LENGTH_BITS;
}

static WEBP_INLINE int HashChainFindLength(const VP8LHashChain* const p,
                                           const int base_position) {
  return p->offset_length_[base_position] & ((1U << MAX_LENGTH_BITS) - 1);
}

static WEBP_INLINE void HashChainFindCopy(const VP8LHashChain* const p,
                                          int base_position,
                                          int* const offset_ptr,
                                          int* const length_ptr) {
  *offset_ptr = HashChainFindOffset(p, base_position);
  *length_ptr = HashChainFindLength(p, base_position);
}

static WEBP_INLINE void AddSingleLiteral(uint32_t pixel, int use_color_cache,
                                         VP8LColorCache* const hashers,
                                         VP8LBackwardRefs* const refs) {
  PixOrCopy v;
  if (use_color_cache) {
    const uint32_t key = VP8LColorCacheGetIndex(hashers, pixel);
    if (VP8LColorCacheLookup(hashers, key) == pixel) {
      v = PixOrCopyCreateCacheIdx(key);
    } else {
      v = PixOrCopyCreateLiteral(pixel);
      VP8LColorCacheSet(hashers, key, pixel);
    }
  } else {
    v = PixOrCopyCreateLiteral(pixel);
  }
  BackwardRefsCursorAdd(refs, v);
}

static int BackwardReferencesRle(int xsize, int ysize,
                                 const uint32_t* const argb,
                                 int cache_bits, VP8LBackwardRefs* const refs) {
  const int pix_count = xsize * ysize;
  int i, k;
  const int use_color_cache = (cache_bits > 0);
  VP8LColorCache hashers;

  if (use_color_cache && !VP8LColorCacheInit(&hashers, cache_bits)) {
    return 0;
  }
  ClearBackwardRefs(refs);
  // Add first pixel as literal.
  AddSingleLiteral(argb[0], use_color_cache, &hashers, refs);
  i = 1;
  while (i < pix_count) {
    const int max_len = MaxFindCopyLength(pix_count - i);
    const int kMinLength = 4;
    const int rle_len = FindMatchLength(argb + i, argb + i - 1, 0, max_len);
    const int prev_row_len = (i < xsize) ? 0 :
        FindMatchLength(argb + i, argb + i - xsize, 0, max_len);
    if (rle_len >= prev_row_len && rle_len >= kMinLength) {
      BackwardRefsCursorAdd(refs, PixOrCopyCreateCopy(1, rle_len));
      // We don't need to update the color cache here since it is always the
      // same pixel being copied, and that does not change the color cache
      // state.
      i += rle_len;
    } else if (prev_row_len >= kMinLength) {
      BackwardRefsCursorAdd(refs, PixOrCopyCreateCopy(xsize, prev_row_len));
      if (use_color_cache) {
        for (k = 0; k < prev_row_len; ++k) {
          VP8LColorCacheInsert(&hashers, argb[i + k]);
        }
      }
      i += prev_row_len;
    } else {
      AddSingleLiteral(argb[i], use_color_cache, &hashers, refs);
      i++;
    }
  }
  if (use_color_cache) VP8LColorCacheClear(&hashers);
  return !refs->error_;
}

static int BackwardReferencesLz77(int xsize, int ysize,
                                  const uint32_t* const argb, int cache_bits,
                                  const VP8LHashChain* const hash_chain,
                                  VP8LBackwardRefs* const refs) {
  int i;
  int i_last_check = -1;
  int ok = 0;
  int cc_init = 0;
  const int use_color_cache = (cache_bits > 0);
  const int pix_count = xsize * ysize;
  VP8LColorCache hashers;

  if (use_color_cache) {
    cc_init = VP8LColorCacheInit(&hashers, cache_bits);
    if (!cc_init) goto Error;
  }
  ClearBackwardRefs(refs);
  for (i = 0; i < pix_count;) {
    // Alternative#1: Code the pixels starting at 'i' using backward reference.
    int offset = 0;
    int len = 0;
    int j;
    HashChainFindCopy(hash_chain, i, &offset, &len);
    if (len > MIN_LENGTH + 1) {
      const int len_ini = len;
      int max_reach = 0;
      assert(i + len < pix_count);
      // Only start from what we have not checked already.
      i_last_check = (i > i_last_check) ? i : i_last_check;
      // We know the best match for the current pixel but we try to find the
      // best matches for the current pixel AND the next one combined.
      // The naive method would use the intervals:
      // [i,i+len) + [i+len, length of best match at i+len)
      // while we check if we can use:
      // [i,j) (where j<=i+len) + [j, length of best match at j)
      for (j = i_last_check + 1; j <= i + len_ini; ++j) {
        const int len_j = HashChainFindLength(hash_chain, j);
        const int reach =
            j + (len_j > MIN_LENGTH + 1 ? len_j : 1);  // 1 for single literal.
        if (reach > max_reach) {
          len = j - i;
          max_reach = reach;
        }
      }
    } else {
      len = 1;
    }
    // Go with literal or backward reference.
    assert(len > 0);
    if (len == 1) {
      AddSingleLiteral(argb[i], use_color_cache, &hashers, refs);
    } else {
      BackwardRefsCursorAdd(refs, PixOrCopyCreateCopy(offset, len));
      if (use_color_cache) {
        for (j = i; j < i + len; ++j) VP8LColorCacheInsert(&hashers, argb[j]);
      }
    }
    i += len;
  }

  ok = !refs->error_;
 Error:
  if (cc_init) VP8LColorCacheClear(&hashers);
  return ok;
}

// -----------------------------------------------------------------------------

typedef struct {
  double alpha_[VALUES_IN_BYTE];
  double red_[VALUES_IN_BYTE];
  double blue_[VALUES_IN_BYTE];
  double distance_[NUM_DISTANCE_CODES];
  double* literal_;
} CostModel;

static int BackwardReferencesTraceBackwards(
    int xsize, int ysize, const uint32_t* const argb, int quality,
    int cache_bits, const VP8LHashChain* const hash_chain,
    VP8LBackwardRefs* const refs);

static void ConvertPopulationCountTableToBitEstimates(
    int num_symbols, const uint32_t population_counts[], double output[]) {
  uint32_t sum = 0;
  int nonzeros = 0;
  int i;
  for (i = 0; i < num_symbols; ++i) {
    sum += population_counts[i];
    if (population_counts[i] > 0) {
      ++nonzeros;
    }
  }
  if (nonzeros <= 1) {
    memset(output, 0, num_symbols * sizeof(*output));
  } else {
    const double logsum = VP8LFastLog2(sum);
    for (i = 0; i < num_symbols; ++i) {
      output[i] = logsum - VP8LFastLog2(population_counts[i]);
    }
  }
}

static int CostModelBuild(CostModel* const m, int cache_bits,
                          VP8LBackwardRefs* const refs) {
  int ok = 0;
  VP8LHistogram* const histo = VP8LAllocateHistogram(cache_bits);
  if (histo == NULL) goto Error;

  VP8LHistogramCreate(histo, refs, cache_bits);

  ConvertPopulationCountTableToBitEstimates(
      VP8LHistogramNumCodes(histo->palette_code_bits_),
      histo->literal_, m->literal_);
  ConvertPopulationCountTableToBitEstimates(
      VALUES_IN_BYTE, histo->red_, m->red_);
  ConvertPopulationCountTableToBitEstimates(
      VALUES_IN_BYTE, histo->blue_, m->blue_);
  ConvertPopulationCountTableToBitEstimates(
      VALUES_IN_BYTE, histo->alpha_, m->alpha_);
  ConvertPopulationCountTableToBitEstimates(
      NUM_DISTANCE_CODES, histo->distance_, m->distance_);
  ok = 1;

 Error:
  VP8LFreeHistogram(histo);
  return ok;
}

static WEBP_INLINE double GetLiteralCost(const CostModel* const m, uint32_t v) {
  return m->alpha_[v >> 24] +
         m->red_[(v >> 16) & 0xff] +
         m->literal_[(v >> 8) & 0xff] +
         m->blue_[v & 0xff];
}

static WEBP_INLINE double GetCacheCost(const CostModel* const m, uint32_t idx) {
  const int literal_idx = VALUES_IN_BYTE + NUM_LENGTH_CODES + idx;
  return m->literal_[literal_idx];
}

static WEBP_INLINE double GetLengthCost(const CostModel* const m,
                                        uint32_t length) {
  int code, extra_bits;
  VP8LPrefixEncodeBits(length, &code, &extra_bits);
  return m->literal_[VALUES_IN_BYTE + code] + extra_bits;
}

static WEBP_INLINE double GetDistanceCost(const CostModel* const m,
                                          uint32_t distance) {
  int code, extra_bits;
  VP8LPrefixEncodeBits(distance, &code, &extra_bits);
  return m->distance_[code] + extra_bits;
}

static void AddSingleLiteralWithCostModel(const uint32_t* const argb,
                                          VP8LColorCache* const hashers,
                                          const CostModel* const cost_model,
                                          int idx, int use_color_cache,
                                          double prev_cost, float* const cost,
                                          uint16_t* const dist_array) {
  double cost_val = prev_cost;
  const uint32_t color = argb[0];
  if (use_color_cache && VP8LColorCacheContains(hashers, color)) {
    const double mul0 = 0.68;
    const int ix = VP8LColorCacheGetIndex(hashers, color);
    cost_val += GetCacheCost(cost_model, ix) * mul0;
  } else {
    const double mul1 = 0.82;
    if (use_color_cache) VP8LColorCacheInsert(hashers, color);
    cost_val += GetLiteralCost(cost_model, color) * mul1;
  }
  if (cost[idx] > cost_val) {
    cost[idx] = (float)cost_val;
    dist_array[idx] = 1;  // only one is inserted.
  }
}

// -----------------------------------------------------------------------------
// CostManager and interval handling

// Empirical value to avoid high memory consumption but good for performance.
#define COST_CACHE_INTERVAL_SIZE_MAX 100

// To perform backward reference every pixel at index index_ is considered and
// the cost for the MAX_LENGTH following pixels computed. Those following pixels
// at index index_ + k (k from 0 to MAX_LENGTH) have a cost of:
//     distance_cost_ at index_ + GetLengthCost(cost_model, k)
//            (named cost)            (named cached cost)
// and the minimum value is kept. GetLengthCost(cost_model, k) is cached in an
// array of size MAX_LENGTH.
// Instead of performing MAX_LENGTH comparisons per pixel, we keep track of the
// minimal values using intervals, for which lower_ and upper_ bounds are kept.
// An interval is defined by the index_ of the pixel that generated it and
// is only useful in a range of indices from start_ to end_ (exclusive), i.e.
// it contains the minimum value for pixels between start_ and end_.
// Intervals are stored in a linked list and ordered by start_. When a new
// interval has a better minimum, old intervals are split or removed.
typedef struct CostInterval CostInterval;
struct CostInterval {
  double lower_;
  double upper_;
  int start_;
  int end_;
  double distance_cost_;
  int index_;
  CostInterval* previous_;
  CostInterval* next_;
};

// The GetLengthCost(cost_model, k) part of the costs is also bounded for
// efficiency in a set of intervals of a different type.
// If those intervals are small enough, they are not used for comparison and
// written into the costs right away.
typedef struct {
  double lower_;  // Lower bound of the interval.
  double upper_;  // Upper bound of the interval.
  int start_;
  int end_;       // Exclusive.
  int do_write_;  // If !=0, the interval is saved to cost instead of being kept
                  // for comparison.
} CostCacheInterval;

// This structure is in charge of managing intervals and costs.
// It caches the different CostCacheInterval, caches the different
// GetLengthCost(cost_model, k) in cost_cache_ and the CostInterval's (whose
// count_ is limited by COST_CACHE_INTERVAL_SIZE_MAX).
#define COST_MANAGER_MAX_FREE_LIST 10
typedef struct {
  CostInterval* head_;
  int count_;  // The number of stored intervals.
  CostCacheInterval* cache_intervals_;
  size_t cache_intervals_size_;
  double cost_cache_[MAX_LENGTH];  // Contains the GetLengthCost(cost_model, k).
  double min_cost_cache_;          // The minimum value in cost_cache_[1:].
  double max_cost_cache_;          // The maximum value in cost_cache_[1:].
  float* costs_;
  uint16_t* dist_array_;
  // Most of the time, we only need few intervals -> use a free-list, to avoid
  // fragmentation with small allocs in most common cases.
  CostInterval intervals_[COST_MANAGER_MAX_FREE_LIST];
  CostInterval* free_intervals_;
  // These are regularly malloc'd remains. This list can't grow larger than than
  // size COST_CACHE_INTERVAL_SIZE_MAX - COST_MANAGER_MAX_FREE_LIST, note.
  CostInterval* recycled_intervals_;
  // Buffer used in BackwardReferencesHashChainDistanceOnly to store the ends
  // of the intervals that can have impacted the cost at a pixel.
  int* interval_ends_;
  int interval_ends_size_;
} CostManager;

static int IsCostCacheIntervalWritable(int start, int end) {
  // 100 is the length for which we consider an interval for comparison, and not
  // for writing.
  // The first intervals are very small and go in increasing size. This constant
  // helps merging them into one big interval (up to index 150/200 usually from
  // which intervals start getting much bigger).
  // This value is empirical.
  return (end - start + 1 < 100);
}

static void CostIntervalAddToFreeList(CostManager* const manager,
                                      CostInterval* const interval) {
  interval->next_ = manager->free_intervals_;
  manager->free_intervals_ = interval;
}

static int CostIntervalIsInFreeList(const CostManager* const manager,
                                    const CostInterval* const interval) {
  return (interval >= &manager->intervals_[0] &&
          interval <= &manager->intervals_[COST_MANAGER_MAX_FREE_LIST - 1]);
}

static void CostManagerInitFreeList(CostManager* const manager) {
  int i;
  manager->free_intervals_ = NULL;
  for (i = 0; i < COST_MANAGER_MAX_FREE_LIST; ++i) {
    CostIntervalAddToFreeList(manager, &manager->intervals_[i]);
  }
}

static void DeleteIntervalList(CostManager* const manager,
                               const CostInterval* interval) {
  while (interval != NULL) {
    const CostInterval* const next = interval->next_;
    if (!CostIntervalIsInFreeList(manager, interval)) {
      WebPSafeFree((void*)interval);
    }  // else: do nothing
    interval = next;
  }
}

static void CostManagerClear(CostManager* const manager) {
  if (manager == NULL) return;

  WebPSafeFree(manager->costs_);
  WebPSafeFree(manager->cache_intervals_);
  WebPSafeFree(manager->interval_ends_);

  // Clear the interval lists.
  DeleteIntervalList(manager, manager->head_);
  manager->head_ = NULL;
  DeleteIntervalList(manager, manager->recycled_intervals_);
  manager->recycled_intervals_ = NULL;

  // Reset pointers, count_ and cache_intervals_size_.
  memset(manager, 0, sizeof(*manager));
  CostManagerInitFreeList(manager);
}

static int CostManagerInit(CostManager* const manager,
                           uint16_t* const dist_array, int pix_count,
                           const CostModel* const cost_model) {
  int i;
  const int cost_cache_size = (pix_count > MAX_LENGTH) ? MAX_LENGTH : pix_count;
  // This constant is tied to the cost_model we use.
  // Empirically, differences between intervals is usually of more than 1.
  const double min_cost_diff = 0.1;

  manager->costs_ = NULL;
  manager->cache_intervals_ = NULL;
  manager->interval_ends_ = NULL;
  manager->head_ = NULL;
  manager->recycled_intervals_ = NULL;
  manager->count_ = 0;
  manager->dist_array_ = dist_array;
  CostManagerInitFreeList(manager);

  // Fill in the cost_cache_.
  manager->cache_intervals_size_ = 1;
  manager->cost_cache_[0] = 0;
  for (i = 1; i < cost_cache_size; ++i) {
    manager->cost_cache_[i] = GetLengthCost(cost_model, i);
    // Get an approximation of the number of bound intervals.
    if (fabs(manager->cost_cache_[i] - manager->cost_cache_[i - 1]) >
        min_cost_diff) {
      ++manager->cache_intervals_size_;
    }
    // Compute the minimum of cost_cache_.
    if (i == 1) {
      manager->min_cost_cache_ = manager->cost_cache_[1];
      manager->max_cost_cache_ = manager->cost_cache_[1];
    } else if (manager->cost_cache_[i] < manager->min_cost_cache_) {
      manager->min_cost_cache_ = manager->cost_cache_[i];
    } else if (manager->cost_cache_[i] > manager->max_cost_cache_) {
      manager->max_cost_cache_ = manager->cost_cache_[i];
    }
  }

  // With the current cost models, we have 15 intervals, so we are safe by
  // setting a maximum of COST_CACHE_INTERVAL_SIZE_MAX.
  if (manager->cache_intervals_size_ > COST_CACHE_INTERVAL_SIZE_MAX) {
    manager->cache_intervals_size_ = COST_CACHE_INTERVAL_SIZE_MAX;
  }
  manager->cache_intervals_ = (CostCacheInterval*)WebPSafeMalloc(
      manager->cache_intervals_size_, sizeof(*manager->cache_intervals_));
  if (manager->cache_intervals_ == NULL) {
    CostManagerClear(manager);
    return 0;
  }

  // Fill in the cache_intervals_.
  {
    double cost_prev = -1e38f;  // unprobably low initial value
    CostCacheInterval* prev = NULL;
    CostCacheInterval* cur = manager->cache_intervals_;
    const CostCacheInterval* const end =
        manager->cache_intervals_ + manager->cache_intervals_size_;

    // Consecutive values in cost_cache_ are compared and if a big enough
    // difference is found, a new interval is created and bounded.
    for (i = 0; i < cost_cache_size; ++i) {
      const double cost_val = manager->cost_cache_[i];
      if (i == 0 ||
          (fabs(cost_val - cost_prev) > min_cost_diff && cur + 1 < end)) {
        if (i > 1) {
          const int is_writable =
              IsCostCacheIntervalWritable(cur->start_, cur->end_);
          // Merge with the previous interval if both are writable.
          if (is_writable && cur != manager->cache_intervals_ &&
              prev->do_write_) {
            // Update the previous interval.
            prev->end_ = cur->end_;
            if (cur->lower_ < prev->lower_) {
              prev->lower_ = cur->lower_;
            } else if (cur->upper_ > prev->upper_) {
              prev->upper_ = cur->upper_;
            }
          } else {
            cur->do_write_ = is_writable;
            prev = cur;
            ++cur;
          }
        }
        // Initialize an interval.
        cur->start_ = i;
        cur->do_write_ = 0;
        cur->lower_ = cost_val;
        cur->upper_ = cost_val;
      } else {
        // Update the current interval bounds.
        if (cost_val < cur->lower_) {
          cur->lower_ = cost_val;
        } else if (cost_val > cur->upper_) {
          cur->upper_ = cost_val;
        }
      }
      cur->end_ = i + 1;
      cost_prev = cost_val;
    }
    manager->cache_intervals_size_ = cur + 1 - manager->cache_intervals_;
  }

  manager->costs_ = (float*)WebPSafeMalloc(pix_count, sizeof(*manager->costs_));
  if (manager->costs_ == NULL) {
    CostManagerClear(manager);
    return 0;
  }
  // Set the initial costs_ high for every pixel as we will keep the minimum.
  for (i = 0; i < pix_count; ++i) manager->costs_[i] = 1e38f;

  // The cost at pixel is influenced by the cost intervals from previous pixels.
  // Let us take the specific case where the offset is the same (which actually
  // happens a lot in case of uniform regions).
  // pixel i contributes to j>i a cost of: offset cost + cost_cache_[j-i]
  // pixel i+1 contributes to j>i a cost of: 2*offset cost + cost_cache_[j-i-1]
  // pixel i+2 contributes to j>i a cost of: 3*offset cost + cost_cache_[j-i-2]
  // and so on.
  // A pixel i influences the following length(j) < MAX_LENGTH pixels. What is
  // the value of j such that pixel i + j cannot influence any of those pixels?
  // This value is such that:
  //               max of cost_cache_ < j*offset cost + min of cost_cache_
  // (pixel i + j 's cost cannot beat the worst cost given by pixel i).
  // This value will be used to optimize the cost computation in
  // BackwardReferencesHashChainDistanceOnly.
  {
    // The offset cost is computed in GetDistanceCost and has a minimum value of
    // the minimum in cost_model->distance_. The case where the offset cost is 0
    // will be dealt with differently later so we are only interested in the
    // minimum non-zero offset cost.
    double offset_cost_min = 0.;
    int size;
    for (i = 0; i < NUM_DISTANCE_CODES; ++i) {
      if (cost_model->distance_[i] != 0) {
        if (offset_cost_min == 0.) {
          offset_cost_min = cost_model->distance_[i];
        } else if (cost_model->distance_[i] < offset_cost_min) {
          offset_cost_min = cost_model->distance_[i];
        }
      }
    }
    // In case all the cost_model->distance_ is 0, the next non-zero cost we
    // can have is from the extra bit in GetDistanceCost, hence 1.
    if (offset_cost_min < 1.) offset_cost_min = 1.;

    size = 1 + (int)ceil((manager->max_cost_cache_ - manager->min_cost_cache_) /
                         offset_cost_min);
    // Empirically, we usually end up with a value below 100.
    if (size > MAX_LENGTH) size = MAX_LENGTH;

    manager->interval_ends_ =
        (int*)WebPSafeMalloc(size, sizeof(*manager->interval_ends_));
    if (manager->interval_ends_ == NULL) {
      CostManagerClear(manager);
      return 0;
    }
    manager->interval_ends_size_ = size;
  }

  return 1;
}

// Given the distance_cost for pixel 'index', update the cost at pixel 'i' if it
// is smaller than the previously computed value.
static WEBP_INLINE void UpdateCost(CostManager* const manager, int i, int index,
                                   double distance_cost) {
  int k = i - index;
  double cost_tmp;
  assert(k >= 0 && k < MAX_LENGTH);
  cost_tmp = distance_cost + manager->cost_cache_[k];

  if (manager->costs_[i] > cost_tmp) {
    manager->costs_[i] = (float)cost_tmp;
    manager->dist_array_[i] = k + 1;
  }
}

// Given the distance_cost for pixel 'index', update the cost for all the pixels
// between 'start' and 'end' excluded.
static WEBP_INLINE void UpdateCostPerInterval(CostManager* const manager,
                                              int start, int end, int index,
                                              double distance_cost) {
  int i;
  for (i = start; i < end; ++i) UpdateCost(manager, i, index, distance_cost);
}

// Given two intervals, make 'prev' be the previous one of 'next' in 'manager'.
static WEBP_INLINE void ConnectIntervals(CostManager* const manager,
                                         CostInterval* const prev,
                                         CostInterval* const next) {
  if (prev != NULL) {
    prev->next_ = next;
  } else {
    manager->head_ = next;
  }

  if (next != NULL) next->previous_ = prev;
}

// Pop an interval in the manager.
static WEBP_INLINE void PopInterval(CostManager* const manager,
                                    CostInterval* const interval) {
  CostInterval* const next = interval->next_;

  if (interval == NULL) return;

  ConnectIntervals(manager, interval->previous_, next);
  if (CostIntervalIsInFreeList(manager, interval)) {
    CostIntervalAddToFreeList(manager, interval);
  } else {  // recycle regularly malloc'd intervals too
    interval->next_ = manager->recycled_intervals_;
    manager->recycled_intervals_ = interval;
  }
  --manager->count_;
  assert(manager->count_ >= 0);
}

// Update the cost at index i by going over all the stored intervals that
// overlap with i.
static WEBP_INLINE void UpdateCostPerIndex(CostManager* const manager, int i) {
  CostInterval* current = manager->head_;

  while (current != NULL && current->start_ <= i) {
    if (current->end_ <= i) {
      // We have an outdated interval, remove it.
      CostInterval* next = current->next_;
      PopInterval(manager, current);
      current = next;
    } else {
      UpdateCost(manager, i, current->index_, current->distance_cost_);
      current = current->next_;
    }
  }
}

// Given a current orphan interval and its previous interval, before
// it was orphaned (which can be NULL), set it at the right place in the list
// of intervals using the start_ ordering and the previous interval as a hint.
static WEBP_INLINE void PositionOrphanInterval(CostManager* const manager,
                                               CostInterval* const current,
                                               CostInterval* previous) {
  assert(current != NULL);

  if (previous == NULL) previous = manager->head_;
  while (previous != NULL && current->start_ < previous->start_) {
    previous = previous->previous_;
  }
  while (previous != NULL && previous->next_ != NULL &&
         previous->next_->start_ < current->start_) {
    previous = previous->next_;
  }

  if (previous != NULL) {
    ConnectIntervals(manager, current, previous->next_);
  } else {
    ConnectIntervals(manager, current, manager->head_);
  }
  ConnectIntervals(manager, previous, current);
}

// Insert an interval in the list contained in the manager by starting at
// interval_in as a hint. The intervals are sorted by start_ value.
static WEBP_INLINE void InsertInterval(CostManager* const manager,
                                       CostInterval* const interval_in,
                                       double distance_cost, double lower,
                                       double upper, int index, int start,
                                       int end) {
  CostInterval* interval_new;

  if (IsCostCacheIntervalWritable(start, end) ||
      manager->count_ >= COST_CACHE_INTERVAL_SIZE_MAX) {
    // Write down the interval if it is too small.
    UpdateCostPerInterval(manager, start, end, index, distance_cost);
    return;
  }
  if (manager->free_intervals_ != NULL) {
    interval_new = manager->free_intervals_;
    manager->free_intervals_ = interval_new->next_;
  } else if (manager->recycled_intervals_ != NULL) {
    interval_new = manager->recycled_intervals_;
    manager->recycled_intervals_ = interval_new->next_;
  } else {   // malloc for good
    interval_new = (CostInterval*)WebPSafeMalloc(1, sizeof(*interval_new));
    if (interval_new == NULL) {
      // Write down the interval if we cannot create it.
      UpdateCostPerInterval(manager, start, end, index, distance_cost);
      return;
    }
  }

  interval_new->distance_cost_ = distance_cost;
  interval_new->lower_ = lower;
  interval_new->upper_ = upper;
  interval_new->index_ = index;
  interval_new->start_ = start;
  interval_new->end_ = end;
  PositionOrphanInterval(manager, interval_new, interval_in);

  ++manager->count_;
}

// When an interval has its start_ or end_ modified, it needs to be
// repositioned in the linked list.
static WEBP_INLINE void RepositionInterval(CostManager* const manager,
                                           CostInterval* const interval) {
  if (IsCostCacheIntervalWritable(interval->start_, interval->end_)) {
    // Maybe interval has been resized and is small enough to be removed.
    UpdateCostPerInterval(manager, interval->start_, interval->end_,
                          interval->index_, interval->distance_cost_);
    PopInterval(manager, interval);
    return;
  }

  // Early exit if interval is at the right spot.
  if ((interval->previous_ == NULL ||
       interval->previous_->start_ <= interval->start_) &&
      (interval->next_ == NULL ||
       interval->start_ <= interval->next_->start_)) {
    return;
  }

  ConnectIntervals(manager, interval->previous_, interval->next_);
  PositionOrphanInterval(manager, interval, interval->previous_);
}

// Given a new cost interval defined by its start at index, its last value and
// distance_cost, add its contributions to the previous intervals and costs.
// If handling the interval or one of its subintervals becomes to heavy, its
// contribution is added to the costs right away.
static WEBP_INLINE void PushInterval(CostManager* const manager,
                                     double distance_cost, int index,
                                     int last) {
  size_t i;
  CostInterval* interval = manager->head_;
  CostInterval* interval_next;
  const CostCacheInterval* const cost_cache_intervals =
      manager->cache_intervals_;

  for (i = 0; i < manager->cache_intervals_size_ &&
              cost_cache_intervals[i].start_ < last;
       ++i) {
    // Define the intersection of the ith interval with the new one.
    int start = index + cost_cache_intervals[i].start_;
    const int end = index + (cost_cache_intervals[i].end_ > last
                                 ? last
                                 : cost_cache_intervals[i].end_);
    const double lower_in = cost_cache_intervals[i].lower_;
    const double upper_in = cost_cache_intervals[i].upper_;
    const double lower_full_in = distance_cost + lower_in;
    const double upper_full_in = distance_cost + upper_in;

    if (cost_cache_intervals[i].do_write_) {
      UpdateCostPerInterval(manager, start, end, index, distance_cost);
      continue;
    }

    for (; interval != NULL && interval->start_ < end && start < end;
         interval = interval_next) {
      const double lower_full_interval =
          interval->distance_cost_ + interval->lower_;
      const double upper_full_interval =
          interval->distance_cost_ + interval->upper_;

      interval_next = interval->next_;

      // Make sure we have some overlap
      if (start >= interval->end_) continue;

      if (lower_full_in >= upper_full_interval) {
        // When intervals are represented, the lower, the better.
        // [**********************************************************]
        // start                                                    end
        //                   [----------------------------------]
        //                   interval->start_       interval->end_
        // If we are worse than what we already have, add whatever we have so
        // far up to interval.
        const int start_new = interval->end_;
        InsertInterval(manager, interval, distance_cost, lower_in, upper_in,
                       index, start, interval->start_);
        start = start_new;
        continue;
      }

      // We know the two intervals intersect.
      if (upper_full_in >= lower_full_interval) {
        // There is no clear cut on which is best, so let's keep both.
        // [*********[*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*]***********]
        // start     interval->start_     interval->end_         end
        // OR
        // [*********[*-*-*-*-*-*-*-*-*-*-*-]----------------------]
        // start     interval->start_     end          interval->end_
        const int end_new = (interval->end_ <= end) ? interval->end_ : end;
        InsertInterval(manager, interval, distance_cost, lower_in, upper_in,
                       index, start, end_new);
        start = end_new;
      } else if (start <= interval->start_ && interval->end_ <= end) {
        //                   [----------------------------------]
        //                   interval->start_       interval->end_
        // [**************************************************************]
        // start                                                        end
        // We can safely remove the old interval as it is fully included.
        PopInterval(manager, interval);
      } else {
        if (interval->start_ <= start && end <= interval->end_) {
          // [--------------------------------------------------------------]
          // interval->start_                                  interval->end_
          //                     [*****************************]
          //                     start                       end
          // We have to split the old interval as it fully contains the new one.
          const int end_original = interval->end_;
          interval->end_ = start;
          InsertInterval(manager, interval, interval->distance_cost_,
                         interval->lower_, interval->upper_, interval->index_,
                         end, end_original);
        } else if (interval->start_ < start) {
          // [------------------------------------]
          // interval->start_        interval->end_
          //                     [*****************************]
          //                     start                       end
          interval->end_ = start;
        } else {
          //              [------------------------------------]
          //              interval->start_        interval->end_
          // [*****************************]
          // start                       end
          interval->start_ = end;
        }

        // The interval has been modified, we need to reposition it or write it.
        RepositionInterval(manager, interval);
      }
    }
    // Insert the remaining interval from start to end.
    InsertInterval(manager, interval, distance_cost, lower_in, upper_in, index,
                   start, end);
  }
}

static int BackwardReferencesHashChainDistanceOnly(
    int xsize, int ysize, const uint32_t* const argb, int quality,
    int cache_bits, const VP8LHashChain* const hash_chain,
    VP8LBackwardRefs* const refs, uint16_t* const dist_array) {
  int i;
  int ok = 0;
  int cc_init = 0;
  const int pix_count = xsize * ysize;
  const int use_color_cache = (cache_bits > 0);
  const size_t literal_array_size = sizeof(double) *
      (NUM_LITERAL_CODES + NUM_LENGTH_CODES +
       ((cache_bits > 0) ? (1 << cache_bits) : 0));
  const size_t cost_model_size = sizeof(CostModel) + literal_array_size;
  CostModel* const cost_model =
      (CostModel*)WebPSafeCalloc(1ULL, cost_model_size);
  VP8LColorCache hashers;
  const int skip_length = 32 + quality;
  const int skip_min_distance_code = 2;
  CostManager* cost_manager =
      (CostManager*)WebPSafeMalloc(1ULL, sizeof(*cost_manager));

  if (cost_model == NULL || cost_manager == NULL) goto Error;

  cost_model->literal_ = (double*)(cost_model + 1);
  if (use_color_cache) {
    cc_init = VP8LColorCacheInit(&hashers, cache_bits);
    if (!cc_init) goto Error;
  }

  if (!CostModelBuild(cost_model, cache_bits, refs)) {
    goto Error;
  }

  if (!CostManagerInit(cost_manager, dist_array, pix_count, cost_model)) {
    goto Error;
  }

  // We loop one pixel at a time, but store all currently best points to
  // non-processed locations from this point.
  dist_array[0] = 0;
  // Add first pixel as literal.
  AddSingleLiteralWithCostModel(argb + 0, &hashers, cost_model, 0,
                                use_color_cache, 0.0, cost_manager->costs_,
                                dist_array);

  for (i = 1; i < pix_count - 1; ++i) {
    int offset = 0, len = 0;
    double prev_cost = cost_manager->costs_[i - 1];
    HashChainFindCopy(hash_chain, i, &offset, &len);
    if (len >= MIN_LENGTH) {
      const int code = DistanceToPlaneCode(xsize, offset);
      const double offset_cost = GetDistanceCost(cost_model, code);
      const int first_i = i;
      int j_max = 0, interval_ends_index = 0;
      const int is_offset_zero = (offset_cost == 0.);

      if (!is_offset_zero) {
        j_max = (int)ceil(
            (cost_manager->max_cost_cache_ - cost_manager->min_cost_cache_) /
            offset_cost);
        if (j_max < 1) {
          j_max = 1;
        } else if (j_max > cost_manager->interval_ends_size_ - 1) {
          // This could only happen in the case of MAX_LENGTH.
          j_max = cost_manager->interval_ends_size_ - 1;
        }
      }  // else j_max is unused anyway.

      // Instead of considering all contributions from a pixel i by calling:
      //         PushInterval(cost_manager, prev_cost + offset_cost, i, len);
      // we optimize these contributions in case offset_cost stays the same for
      // consecutive pixels. This describes a set of pixels similar to a
      // previous set (e.g. constant color regions).
      for (; i < pix_count - 1; ++i) {
        int offset_next, len_next;
        prev_cost = cost_manager->costs_[i - 1];

        if (is_offset_zero) {
          // No optimization can be made so we just push all of the
          // contributions from i.
          PushInterval(cost_manager, prev_cost, i, len);
        } else {
          // j_max is chosen as the smallest j such that:
          //       max of cost_cache_ < j*offset cost + min of cost_cache_
          // Therefore, the pixel influenced by i-j_max, cannot be influenced
          // by i. Only the costs after the end of what i contributed need to be
          // updated. cost_manager->interval_ends_ is a circular buffer that
          // stores those ends.
          const double distance_cost = prev_cost + offset_cost;
          int j = cost_manager->interval_ends_[interval_ends_index];
          if (i - first_i <= j_max ||
              !IsCostCacheIntervalWritable(j, i + len)) {
            PushInterval(cost_manager, distance_cost, i, len);
          } else {
            for (; j < i + len; ++j) {
              UpdateCost(cost_manager, j, i, distance_cost);
            }
          }
          // Store the new end in the circular buffer.
          assert(interval_ends_index < cost_manager->interval_ends_size_);
          cost_manager->interval_ends_[interval_ends_index] = i + len;
          if (++interval_ends_index > j_max) interval_ends_index = 0;
        }

        // Check whether i is the last pixel to consider, as it is handled
        // differently.
        if (i + 1 >= pix_count - 1) break;
        HashChainFindCopy(hash_chain, i + 1, &offset_next, &len_next);
        if (offset_next != offset) break;
        len = len_next;
        UpdateCostPerIndex(cost_manager, i);
        AddSingleLiteralWithCostModel(argb + i, &hashers, cost_model, i,
                                      use_color_cache, prev_cost,
                                      cost_manager->costs_, dist_array);
      }
      // Submit the last pixel.
      UpdateCostPerIndex(cost_manager, i + 1);

      // This if is for speedup only. It roughly doubles the speed, and
      // makes compression worse by .1 %.
      if (len >= skip_length && code <= skip_min_distance_code) {
        // Long copy for short distances, let's skip the middle
        // lookups for better copies.
        // 1) insert the hashes.
        if (use_color_cache) {
          int k;
          for (k = 0; k < len; ++k) {
            VP8LColorCacheInsert(&hashers, argb[i + k]);
          }
        }
        // 2) jump.
        {
          const int i_next = i + len - 1;  // for loop does ++i, thus -1 here.
          for (; i <= i_next; ++i) UpdateCostPerIndex(cost_manager, i + 1);
          i = i_next;
        }
        goto next_symbol;
      }
      if (len > MIN_LENGTH) {
        int code_min_length;
        double cost_total;
        offset = HashChainFindOffset(hash_chain, i);
        code_min_length = DistanceToPlaneCode(xsize, offset);
        cost_total = prev_cost +
            GetDistanceCost(cost_model, code_min_length) +
            GetLengthCost(cost_model, 1);
        if (cost_manager->costs_[i + 1] > cost_total) {
          cost_manager->costs_[i + 1] = (float)cost_total;
          dist_array[i + 1] = 2;
        }
      }
    } else {    // len < MIN_LENGTH
      UpdateCostPerIndex(cost_manager, i + 1);
    }

    AddSingleLiteralWithCostModel(argb + i, &hashers, cost_model, i,
                                  use_color_cache, prev_cost,
                                  cost_manager->costs_, dist_array);

 next_symbol: ;
  }
  // Handle the last pixel.
  if (i == (pix_count - 1)) {
    AddSingleLiteralWithCostModel(
        argb + i, &hashers, cost_model, i, use_color_cache,
        cost_manager->costs_[pix_count - 2], cost_manager->costs_, dist_array);
  }

  ok = !refs->error_;
 Error:
  if (cc_init) VP8LColorCacheClear(&hashers);
  CostManagerClear(cost_manager);
  WebPSafeFree(cost_model);
  WebPSafeFree(cost_manager);
  return ok;
}

// We pack the path at the end of *dist_array and return
// a pointer to this part of the array. Example:
// dist_array = [1x2xx3x2] => packed [1x2x1232], chosen_path = [1232]
static void TraceBackwards(uint16_t* const dist_array,
                           int dist_array_size,
                           uint16_t** const chosen_path,
                           int* const chosen_path_size) {
  uint16_t* path = dist_array + dist_array_size;
  uint16_t* cur = dist_array + dist_array_size - 1;
  while (cur >= dist_array) {
    const int k = *cur;
    --path;
    *path = k;
    cur -= k;
  }
  *chosen_path = path;
  *chosen_path_size = (int)(dist_array + dist_array_size - path);
}

static int BackwardReferencesHashChainFollowChosenPath(
    const uint32_t* const argb, int cache_bits,
    const uint16_t* const chosen_path, int chosen_path_size,
    const VP8LHashChain* const hash_chain, VP8LBackwardRefs* const refs) {
  const int use_color_cache = (cache_bits > 0);
  int ix;
  int i = 0;
  int ok = 0;
  int cc_init = 0;
  VP8LColorCache hashers;

  if (use_color_cache) {
    cc_init = VP8LColorCacheInit(&hashers, cache_bits);
    if (!cc_init) goto Error;
  }

  ClearBackwardRefs(refs);
  for (ix = 0; ix < chosen_path_size; ++ix) {
    const int len = chosen_path[ix];
    if (len != 1) {
      int k;
      const int offset = HashChainFindOffset(hash_chain, i);
      BackwardRefsCursorAdd(refs, PixOrCopyCreateCopy(offset, len));
      if (use_color_cache) {
        for (k = 0; k < len; ++k) {
          VP8LColorCacheInsert(&hashers, argb[i + k]);
        }
      }
      i += len;
    } else {
      PixOrCopy v;
      if (use_color_cache && VP8LColorCacheContains(&hashers, argb[i])) {
        // push pixel as a color cache index
        const int idx = VP8LColorCacheGetIndex(&hashers, argb[i]);
        v = PixOrCopyCreateCacheIdx(idx);
      } else {
        if (use_color_cache) VP8LColorCacheInsert(&hashers, argb[i]);
        v = PixOrCopyCreateLiteral(argb[i]);
      }
      BackwardRefsCursorAdd(refs, v);
      ++i;
    }
  }
  ok = !refs->error_;
 Error:
  if (cc_init) VP8LColorCacheClear(&hashers);
  return ok;
}

// Returns 1 on success.
static int BackwardReferencesTraceBackwards(
    int xsize, int ysize, const uint32_t* const argb, int quality,
    int cache_bits, const VP8LHashChain* const hash_chain,
    VP8LBackwardRefs* const refs) {
  int ok = 0;
  const int dist_array_size = xsize * ysize;
  uint16_t* chosen_path = NULL;
  int chosen_path_size = 0;
  uint16_t* dist_array =
      (uint16_t*)WebPSafeMalloc(dist_array_size, sizeof(*dist_array));

  if (dist_array == NULL) goto Error;

  if (!BackwardReferencesHashChainDistanceOnly(
      xsize, ysize, argb, quality, cache_bits, hash_chain,
      refs, dist_array)) {
    goto Error;
  }
  TraceBackwards(dist_array, dist_array_size, &chosen_path, &chosen_path_size);
  if (!BackwardReferencesHashChainFollowChosenPath(
          argb, cache_bits, chosen_path, chosen_path_size, hash_chain, refs)) {
    goto Error;
  }
  ok = 1;
 Error:
  WebPSafeFree(dist_array);
  return ok;
}

static void BackwardReferences2DLocality(int xsize,
                                         const VP8LBackwardRefs* const refs) {
  VP8LRefsCursor c = VP8LRefsCursorInit(refs);
  while (VP8LRefsCursorOk(&c)) {
    if (PixOrCopyIsCopy(c.cur_pos)) {
      const int dist = c.cur_pos->argb_or_distance;
      const int transformed_dist = DistanceToPlaneCode(xsize, dist);
      c.cur_pos->argb_or_distance = transformed_dist;
    }
    VP8LRefsCursorNext(&c);
  }
}

// Returns entropy for the given cache bits.
static double ComputeCacheEntropy(const uint32_t* argb,
                                  const VP8LBackwardRefs* const refs,
                                  int cache_bits) {
  const int use_color_cache = (cache_bits > 0);
  int cc_init = 0;
  double entropy = MAX_ENTROPY;
  const double kSmallPenaltyForLargeCache = 4.0;
  VP8LColorCache hashers;
  VP8LRefsCursor c = VP8LRefsCursorInit(refs);
  VP8LHistogram* histo = VP8LAllocateHistogram(cache_bits);
  if (histo == NULL) goto Error;

  if (use_color_cache) {
    cc_init = VP8LColorCacheInit(&hashers, cache_bits);
    if (!cc_init) goto Error;
  }
  if (!use_color_cache) {
    while (VP8LRefsCursorOk(&c)) {
      VP8LHistogramAddSinglePixOrCopy(histo, c.cur_pos);
      VP8LRefsCursorNext(&c);
    }
  } else {
    while (VP8LRefsCursorOk(&c)) {
      const PixOrCopy* const v = c.cur_pos;
      if (PixOrCopyIsLiteral(v)) {
        const uint32_t pix = *argb++;
        const uint32_t key = VP8LColorCacheGetIndex(&hashers, pix);
        if (VP8LColorCacheLookup(&hashers, key) == pix) {
          ++histo->literal_[NUM_LITERAL_CODES + NUM_LENGTH_CODES + key];
        } else {
          VP8LColorCacheSet(&hashers, key, pix);
          ++histo->blue_[pix & 0xff];
          ++histo->literal_[(pix >> 8) & 0xff];
          ++histo->red_[(pix >> 16) & 0xff];
          ++histo->alpha_[pix >> 24];
        }
      } else {
        int len = PixOrCopyLength(v);
        int code, extra_bits;
        VP8LPrefixEncodeBits(len, &code, &extra_bits);
        ++histo->literal_[NUM_LITERAL_CODES + code];
        VP8LPrefixEncodeBits(PixOrCopyDistance(v), &code, &extra_bits);
        ++histo->distance_[code];
        do {
          VP8LColorCacheInsert(&hashers, *argb++);
        } while(--len != 0);
      }
      VP8LRefsCursorNext(&c);
    }
  }
  entropy = VP8LHistogramEstimateBits(histo) +
      kSmallPenaltyForLargeCache * cache_bits;
 Error:
  if (cc_init) VP8LColorCacheClear(&hashers);
  VP8LFreeHistogram(histo);
  return entropy;
}

// Evaluate optimal cache bits for the local color cache.
// The input *best_cache_bits sets the maximum cache bits to use (passing 0
// implies disabling the local color cache). The local color cache is also
// disabled for the lower (<= 25) quality.
// Returns 0 in case of memory error.
static int CalculateBestCacheSize(const uint32_t* const argb,
                                  int xsize, int ysize, int quality,
                                  const VP8LHashChain* const hash_chain,
                                  VP8LBackwardRefs* const refs,
                                  int* const lz77_computed,
                                  int* const best_cache_bits) {
  int eval_low = 1;
  int eval_high = 1;
  double entropy_low = MAX_ENTROPY;
  double entropy_high = MAX_ENTROPY;
  const double cost_mul = 5e-4;
  int cache_bits_low = 0;
  int cache_bits_high = (quality <= 25) ? 0 : *best_cache_bits;

  assert(cache_bits_high <= MAX_COLOR_CACHE_BITS);

  *lz77_computed = 0;
  if (cache_bits_high == 0) {
    *best_cache_bits = 0;
    // Local color cache is disabled.
    return 1;
  }
  if (!BackwardReferencesLz77(xsize, ysize, argb, cache_bits_low, hash_chain,
                              refs)) {
    return 0;
  }
  // Do a binary search to find the optimal entropy for cache_bits.
  while (eval_low || eval_high) {
    if (eval_low) {
      entropy_low = ComputeCacheEntropy(argb, refs, cache_bits_low);
      entropy_low += entropy_low * cache_bits_low * cost_mul;
      eval_low = 0;
    }
    if (eval_high) {
      entropy_high = ComputeCacheEntropy(argb, refs, cache_bits_high);
      entropy_high += entropy_high * cache_bits_high * cost_mul;
      eval_high = 0;
    }
    if (entropy_high < entropy_low) {
      const int prev_cache_bits_low = cache_bits_low;
      *best_cache_bits = cache_bits_high;
      cache_bits_low = (cache_bits_low + cache_bits_high) / 2;
      if (cache_bits_low != prev_cache_bits_low) eval_low = 1;
    } else {
      *best_cache_bits = cache_bits_low;
      cache_bits_high = (cache_bits_low + cache_bits_high) / 2;
      if (cache_bits_high != cache_bits_low) eval_high = 1;
    }
  }
  *lz77_computed = 1;
  return 1;
}

// Update (in-place) backward references for specified cache_bits.
static int BackwardRefsWithLocalCache(const uint32_t* const argb,
                                      int cache_bits,
                                      VP8LBackwardRefs* const refs) {
  int pixel_index = 0;
  VP8LColorCache hashers;
  VP8LRefsCursor c = VP8LRefsCursorInit(refs);
  if (!VP8LColorCacheInit(&hashers, cache_bits)) return 0;

  while (VP8LRefsCursorOk(&c)) {
    PixOrCopy* const v = c.cur_pos;
    if (PixOrCopyIsLiteral(v)) {
      const uint32_t argb_literal = v->argb_or_distance;
      if (VP8LColorCacheContains(&hashers, argb_literal)) {
        const int ix = VP8LColorCacheGetIndex(&hashers, argb_literal);
        *v = PixOrCopyCreateCacheIdx(ix);
      } else {
        VP8LColorCacheInsert(&hashers, argb_literal);
      }
      ++pixel_index;
    } else {
      // refs was created without local cache, so it can not have cache indexes.
      int k;
      assert(PixOrCopyIsCopy(v));
      for (k = 0; k < v->len; ++k) {
        VP8LColorCacheInsert(&hashers, argb[pixel_index++]);
      }
    }
    VP8LRefsCursorNext(&c);
  }
  VP8LColorCacheClear(&hashers);
  return 1;
}

static VP8LBackwardRefs* GetBackwardReferencesLowEffort(
    int width, int height, const uint32_t* const argb,
    int* const cache_bits, const VP8LHashChain* const hash_chain,
    VP8LBackwardRefs refs_array[2]) {
  VP8LBackwardRefs* refs_lz77 = &refs_array[0];
  *cache_bits = 0;
  if (!BackwardReferencesLz77(width, height, argb, 0, hash_chain, refs_lz77)) {
    return NULL;
  }
  BackwardReferences2DLocality(width, refs_lz77);
  return refs_lz77;
}

static VP8LBackwardRefs* GetBackwardReferences(
    int width, int height, const uint32_t* const argb, int quality,
    int* const cache_bits, const VP8LHashChain* const hash_chain,
    VP8LBackwardRefs refs_array[2]) {
  int lz77_is_useful;
  int lz77_computed;
  double bit_cost_lz77, bit_cost_rle;
  VP8LBackwardRefs* best = NULL;
  VP8LBackwardRefs* refs_lz77 = &refs_array[0];
  VP8LBackwardRefs* refs_rle = &refs_array[1];
  VP8LHistogram* histo = NULL;

  if (!CalculateBestCacheSize(argb, width, height, quality, hash_chain,
                              refs_lz77, &lz77_computed, cache_bits)) {
    goto Error;
  }

  if (lz77_computed) {
    // Transform refs_lz77 for the optimized cache_bits.
    if (*cache_bits > 0) {
      if (!BackwardRefsWithLocalCache(argb, *cache_bits, refs_lz77)) {
        goto Error;
      }
    }
  } else {
    if (!BackwardReferencesLz77(width, height, argb, *cache_bits, hash_chain,
                                refs_lz77)) {
      goto Error;
    }
  }

  if (!BackwardReferencesRle(width, height, argb, *cache_bits, refs_rle)) {
    goto Error;
  }

  histo = VP8LAllocateHistogram(*cache_bits);
  if (histo == NULL) goto Error;

  {
    // Evaluate LZ77 coding.
    VP8LHistogramCreate(histo, refs_lz77, *cache_bits);
    bit_cost_lz77 = VP8LHistogramEstimateBits(histo);
    // Evaluate RLE coding.
    VP8LHistogramCreate(histo, refs_rle, *cache_bits);
    bit_cost_rle = VP8LHistogramEstimateBits(histo);
    // Decide if LZ77 is useful.
    lz77_is_useful = (bit_cost_lz77 < bit_cost_rle);
  }

  // Choose appropriate backward reference.
  if (lz77_is_useful) {
    // TraceBackwards is costly. Don't execute it at lower quality.
    const int try_lz77_trace_backwards = (quality >= 25);
    best = refs_lz77;   // default guess: lz77 is better
    if (try_lz77_trace_backwards) {
      VP8LBackwardRefs* const refs_trace = refs_rle;
      if (!VP8LBackwardRefsCopy(refs_lz77, refs_trace)) {
        best = NULL;
        goto Error;
      }
      if (BackwardReferencesTraceBackwards(width, height, argb, quality,
                                           *cache_bits, hash_chain,
                                           refs_trace)) {
        double bit_cost_trace;
        // Evaluate LZ77 coding.
        VP8LHistogramCreate(histo, refs_trace, *cache_bits);
        bit_cost_trace = VP8LHistogramEstimateBits(histo);
        if (bit_cost_trace < bit_cost_lz77) {
          best = refs_trace;
        }
      }
    }
  } else {
    best = refs_rle;
  }

  BackwardReferences2DLocality(width, best);

 Error:
  VP8LFreeHistogram(histo);
  return best;
}

VP8LBackwardRefs* VP8LGetBackwardReferences(
    int width, int height, const uint32_t* const argb, int quality,
    int low_effort, int* const cache_bits,
    const VP8LHashChain* const hash_chain, VP8LBackwardRefs refs_array[2]) {
  if (low_effort) {
    return GetBackwardReferencesLowEffort(width, height, argb, cache_bits,
                                          hash_chain, refs_array);
  } else {
    return GetBackwardReferences(width, height, argb, quality, cache_bits,
                                 hash_chain, refs_array);
  }
}