Knapsack.H

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/*
                          Aleph_w
 
  Data structures & Algorithms
  version 2.0.0b
  https://github.com/lrleon/Aleph-w
 
  This file is part of Aleph-w library
 
  Copyright (c) 2002-2026 Leandro Rabindranath Leon
 
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*/
 
#ifndef KNAPSACK_H
#define KNAPSACK_H
 
#include <algorithm>
#include <concepts>
#include <cstddef>
#include <limits>
#include <type_traits>
#include <utility>
 
#include <ah-errors.H>
#include <tpl_array.H>
 
namespace Aleph {
template <typename W, typename V>
struct Knapsack_Item
{
  W weight; 
  V value;  
};
 
template <typename V>
struct Knapsack_Result
{
  V optimal_value = V{};        
  Array<size_t> selected_items; 
};
 
namespace knapsack_detail {
template <std::integral W>
[[nodiscard]] inline size_t to_size_checked(const W value, const char *fn_name, const char *field_name)
{
  if constexpr (std::is_signed_v<W>)
    ah_domain_error_if(value < W{}) << fn_name << ": " << field_name << " must be non-negative";
 
  using UW = std::make_unsigned_t<W>;
  const UW uvalue = static_cast<UW>(value);
  if constexpr (sizeof(UW) > sizeof(size_t))
    ah_out_of_range_error_if(uvalue > static_cast<UW>(std::numeric_limits<size_t>::max()))
      << fn_name << ": " << field_name << " is too large for size_t";
 
  return static_cast<size_t>(uvalue);
}
 
template <std::integral W, typename V>
[[nodiscard]] inline Array<size_t> extract_weights_checked(const Array<Knapsack_Item<W, V>> &items,
                                                           const char *fn_name)
{
  Array<size_t> weights = Array<size_t>::create(items.size());
  for (size_t i = 0; i < items.size(); ++i)
    weights(i) = to_size_checked(items[i].weight, fn_name, "item weight");
  return weights;
}
// Safe DP addition: clamps to numeric_limits<V>::max() for integral V to
// prevent signed overflow UB and unsigned wrap-around.
template <typename V>
[[nodiscard]] inline V dp_add(V a, V b) noexcept
{
  if constexpr (std::is_integral_v<V>)
    if (b > V{} and a > std::numeric_limits<V>::max() - b)
      return std::numeric_limits<V>::max();
  return a + b;
}
}  // namespace knapsack_detail
 
template <typename V, std::integral W>
[[nodiscard]] Knapsack_Result<V> knapsack_01(const Array<Knapsack_Item<W, V>> &items, W capacity)
{
  const size_t n = items.size();
  const size_t C = knapsack_detail::to_size_checked(capacity, "knapsack_01", "capacity");
  const Array<size_t> weights = knapsack_detail::extract_weights_checked(items, "knapsack_01");
 
  if (n == 0)
    return Knapsack_Result<V>{V{}, Array<size_t>()};
 
  // dp[i][w] = best value using items 0..i-1 with capacity w
  Array<Array<V>> dp;
  dp.reserve(n + 1);
  for (size_t i = 0; i <= n; ++i)
    {
      Array<V> row = Array<V>::create(C + 1);
      for (size_t w = 0; w <= C; ++w)
        row(w) = V{};
      dp.append(std::move(row));
    }
 
  for (size_t i = 1; i <= n; ++i)
    {
      const size_t wi = weights[i - 1];
      const V vi = items[i - 1].value;
      for (size_t w = 0; w <= C; ++w)
        {
          dp[i][w] = dp[i - 1][w];
          if (wi <= w)
            {
              const V candidate = knapsack_detail::dp_add(dp[i - 1][w - wi], vi);
              if (candidate > dp[i - 1][w])
                dp[i][w] = candidate;
            }
        }
    }
 
  // reconstruct
  Array<size_t> sel;
  size_t w = C;
  for (size_t i = n; i > 0; --i)
    if (dp[i][w] != dp[i - 1][w])
      {
        sel.append(i - 1);
        w -= weights[i - 1];
      }
 
  // reverse to ascending order
  Array<size_t> selected;
  selected.reserve(sel.size());
  for (size_t k = sel.size(); k > 0; --k)
    selected.append(sel[k - 1]);
 
  return Knapsack_Result<V>{dp[n][C], std::move(selected)};
}
 
template <typename V, std::integral W>
[[nodiscard]] V knapsack_01_value(const Array<Knapsack_Item<W, V>> &items, W capacity)
{
  const size_t n = items.size();
  const size_t C = knapsack_detail::to_size_checked(capacity, "knapsack_01_value", "capacity");
  const Array<size_t> weights = knapsack_detail::extract_weights_checked(items, "knapsack_01_value");
 
  if (n == 0)
    return V{};
 
  Array<V> dp = Array<V>::create(C + 1);
  for (size_t w = 0; w <= C; ++w)
    dp(w) = V{};
 
  for (size_t i = 0; i < n; ++i)
    {
      const size_t wi = weights[i];
      const V vi = items[i].value;
      // w != static_cast<size_t>(-1) guards against unsigned wrap-around when wi == 0
      for (size_t w = C; w >= wi and w != static_cast<size_t>(-1); --w)
        {
          const V candidate = knapsack_detail::dp_add(dp[w - wi], vi);
          if (candidate > dp[w])
            dp(w) = candidate;
        }
    }
 
  return dp[C];
}
 
template <typename V, std::integral W>
[[nodiscard]] Knapsack_Result<V> knapsack_unbounded(const Array<Knapsack_Item<W, V>> &items,
                                                    W capacity)
{
  const size_t n = items.size();
  const size_t C = knapsack_detail::to_size_checked(capacity, "knapsack_unbounded", "capacity");
  const Array<size_t> weights
    = knapsack_detail::extract_weights_checked(items, "knapsack_unbounded");
 
  if (n == 0)
    return Knapsack_Result<V>{V{}, Array<size_t>()};
 
  for (size_t i = 0; i < n; ++i)
    ah_domain_error_if(weights[i] == 0 and items[i].value > V{})
      << "knapsack_unbounded: zero-weight item with positive value " << "leads to unbounded optimum";
 
  Array<V> dp = Array<V>::create(C + 1);
  // choice[w] = which item was last added at capacity w
  Array<size_t> choice = Array<size_t>::create(C + 1);
  constexpr size_t NONE = std::numeric_limits<size_t>::max();
  for (size_t w = 0; w <= C; ++w)
    {
      dp(w) = V{};
      choice(w) = NONE;
    }
 
  for (size_t w = 1; w <= C; ++w)
    for (size_t i = 0; i < n; ++i)
      if (const size_t wi = weights[i]; wi <= w)
        {
          const V candidate = knapsack_detail::dp_add(dp[w - wi], items[i].value);
          if (candidate > dp[w])
            {
              dp(w) = candidate;
              choice(w) = i;
            }
        }
 
  // reconstruct
  Array<size_t> sel;
  size_t w = C;
  while (w > 0 and choice[w] != NONE)
    {
      const size_t idx = choice[w];
      sel.append(idx);
      w -= weights[idx];
    }
 
  return Knapsack_Result<V>{dp[C], std::move(sel)};
}
 
template <typename V, std::integral W>
[[nodiscard]] Knapsack_Result<V> knapsack_bounded(const Array<Knapsack_Item<W, V>> &items,
                                                  const Array<size_t> &counts,
                                                  W capacity)
{
  ah_invalid_argument_if(items.size() != counts.size())
    << "knapsack_bounded: items and counts must have same size";
 
  const size_t n = items.size();
  const size_t C = knapsack_detail::to_size_checked(capacity, "knapsack_bounded", "capacity");
  const Array<size_t> weights = knapsack_detail::extract_weights_checked(items, "knapsack_bounded");
 
  if (n == 0)
    return Knapsack_Result<V>{V{}, Array<size_t>()};
 
  // Binary decomposition: for item i with count c, create groups
  // of 1, 2, 4, ..., 2^k, remainder
  Array<Knapsack_Item<W, V>> expanded;
  Array<size_t> origin;      // maps expanded index -> original index
  Array<size_t> multiplier;  // how many copies this group represents
 
  for (size_t i = 0; i < n; ++i)
    {
      const size_t wi = weights[i];
      size_t rem = counts[i];
      size_t k = 1;
      while (rem > 0)
        {
          const size_t take = std::min(k, rem);
 
          bool value_ok;
          if constexpr (std::is_integral_v<V>)
            {
              if (items[i].value == V{0})
                value_ok = true;
              else if (items[i].value > V{0})
                {
                  // Positive: check take * value <= V::max() using UV to avoid
                  // truncation if V is wider than size_t.
                  using UV = std::make_unsigned_t<V>;
                  const UV value_uv = static_cast<UV>(items[i].value);
                  const UV maxV_uv  = static_cast<UV>(std::numeric_limits<V>::max());
                  value_ok = (static_cast<UV>(take) <= maxV_uv / value_uv);
                }
              else
                {
                  // Negative: check take * |value| <= |V::min()|
                  using UV = std::make_unsigned_t<V>;
                  const UV abs_value = static_cast<UV>(-(items[i].value + V{1})) + UV{1};
                  // Casting V::min() to its unsigned type yields |V::min()| in two's complement.
                  const UV abs_min = static_cast<UV>(std::numeric_limits<V>::min());
                  value_ok = (static_cast<UV>(take) <= abs_min / abs_value);
                }
            }
          else if constexpr (std::numeric_limits<V>::is_specialized)
            {
              const V maxV = std::numeric_limits<V>::max();
              if (items[i].value == V{0})
                value_ok = true;
              else
                {
                  // Use |value| so negative values are handled correctly.
                  const V absV = items[i].value > V{0} ? items[i].value : -items[i].value;
                  value_ok = (static_cast<V>(take) <= maxV / absV);
                }
            }
          else
            value_ok = true;
 
          if ((wi == 0 or take <= C / wi) and value_ok)
            {
              const V val = static_cast<V>(take) * items[i].value;
              expanded.append(Knapsack_Item<W, V>{static_cast<W>(wi * take), val});
              origin.append(i);
              multiplier.append(take);
            }
 
          rem -= take;
          if (rem == 0)
            break;
 
          if (k > std::numeric_limits<size_t>::max() / 2)
            k = rem;
          else
            k *= 2;
        }
    }
 
  // Solve as 0/1
  auto result = knapsack_01(expanded, capacity);
 
  // Map back to original indices
  Array<size_t> sel;
  for (size_t k = 0; k < result.selected_items.size(); ++k)
    {
      const size_t ei = result.selected_items[k];
      const size_t orig = origin[ei];
      const size_t mult = multiplier[ei];
      for (size_t j = 0; j < mult; ++j)
        sel.append(orig);
    }
 
  return Knapsack_Result<V>{result.optimal_value, std::move(sel)};
}
}  // namespace Aleph
 
#endif  // KNAPSACK_H