Best Bin First
Best Bin First (BBF) is a deterministic greedy approximation heuristic utilized in solving the NP-hard 1D Bin Packing Problem. By sorting items in descending order of size and placing each item into the tightest fitting active bin—the bin that leaves the minimum residual capacity greater than or equal to the item's size—BBF drastically minimizes fragmentation compared to Next Fit or First Fit strategies. When backed by self-balancing binary search trees or augmented priority queues, Best Bin First achieves optimal sorting-bound time complexity while maximizing global spatial utilization.
HOW-TO IMPLEMENT & CONFIGURE
- Modern C++ compiler supporting C++17 or higher (GCC 9+, Clang 10+, MSVC 2019+)
- Standard Template Library (STL) vector and multiset components
- CMake 3.15+ build system
Step 1
Define the primary data structures for items and bins. Utilize an ordered multiset to maintain active bin residual capacities in logarithmic time.
#include <vector>
#include <set>
#include <algorithm>
#include <iostream>
struct Item {
int id;
int size;
};
struct Bin {
int id;
int capacity;
int residual_capacity;
};Step 2
Implement the Best Fit Decreasing logic by sorting items descending and querying an ordered multiset for the tightest valid bin capacity.
std::vector<Bin> pack_bins(std::vector<Item>& items, int bin_capacity) {
// Sort items descending for optimal Best Fit Decreasing performance
std::sort(items.begin(), items.end(), [](const Item& a, const Item& b) {
return a.size > b.size;
});
std::vector<Bin> bins;
// Stores pairs of (residual_capacity, bin_index)
std::multiset<std::pair<int, int>> active_bins;
for (const auto& item : items) {
// Find the best bin: smallest residual capacity >= item.size
auto it = active_bins.lower_bound({item.size, 0});
if (it == active_bins.end()) {
// Open a new bin
int new_bin_idx = bins.size();
bins.push_back({new_bin_idx, bin_capacity, bin_capacity - item.size});
active_bins.insert({bin_capacity - item.size, new_bin_idx});
} else {
// Allocate to existing best bin
int bin_idx = it->second;
active_bins.erase(it);
bins[bin_idx].residual_capacity -= item.size;
active_bins.insert({bins[bin_idx].residual_capacity, bin_idx});
}
}
return bins;
}Step 3
Construct a verification suite to guarantee capacity constraints are never violated and allocation completeness is preserved.
void verify_packing(const std::vector<Bin>& bins, int capacity, int total_item_size) {
int calculated_total = 0;
for (const auto& bin : bins) {
if (bin.residual_capacity < 0 || bin.residual_capacity > capacity) {
throw std::runtime_error("Invariant violated: Invalid bin capacity boundary.");
}
calculated_total += (capacity - bin.residual_capacity);
}
if (calculated_total != total_item_size) {
template_error("Mass conservation failure during bin allocation.");
}
}Step 4
Ensure numerical stability under integer overflows, handle zero-sized items correctly, and isolate mutable state for thread safety.
// Ensure inputs are sanitized prior to execution
void sanitize_inputs(const std::vector<Item>& items, int bin_capacity) {
if (bin_capacity <= 0) throw std::invalid_argument("Bin capacity must be positive.");
for (const auto& item : items) {
if (item.size <= 0) throw std::invalid_argument("Item size must be positive.");
if (item.size > bin_capacity) throw std::out_of_range("Item exceeds maximum bin capacity.");
}
}Mathematical Formulation
Let U be a set of n items with individual integer sizes s(u) for each u in U, and let C be the uniform capacity of an unlimited supply of bins. The Bin Packing Problem seeks to partition U into the minimum number of subsets (bins) B_1, B_2, …, B_k such that the sum of sizes in each subset does not exceed C. The Best Bin First heuristic processes items sequentially after sorting them such that s(u_1) >= s(u_2) >= … >= s(u_n). For each item u_i, the algorithm queries the set of currently active bins to find a bin B_j minimizing C – sum(s(u)) subject to the constraint that C – sum(s(u)) >= s(u_i).
Complexity Analysis
The time complexity is dominated by two distinct phases: sorting the item collection, which requires O(n log n) operations, and querying/updating the active bin multiset. In the worst-case scenario where a new bin is opened for every item, the multiset contains up to n elements, making each insertion and deletion operation take O(log n) time. Executing this across n items yields an overall time complexity of O(n log n). Space complexity is strictly bounded at O(n) to store both the item arrays and the index mappings within the balanced binary search tree structures.
Failure Modes & Edge Cases
A primary production failure mode involves floating-point inaccuracy if continuous size allocations are attempted without fixed-point scaling. Developers must convert fractional weights into strict integer representations (e.g., scaling gigabytes to megabytes or bytes) to prevent rounding anomalies during capacity checks. Furthermore, unmonitored sorting degradation on pre-sorted reverse inputs can lead to pathological cache thrashing if memory layouts are non-contiguous.
📚 STANDARDS & ACADEMIC RFC CITATIONS
- Garey, M. R., & Johnson, D. S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman.
- Johnson, D. S. (1974). Near-Optimal Bin Packing Algorithms. Ph.D. dissertation, MIT.
- Korf, R. E. (2002). Optimal Bin Packing Algorithms. Proceedings of the National Conference on Artificial Intelligence (AAAI).