4.7.2 Graph Coloring (Backtracking)

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/*

Given an undirected graph, assign a color to every node such that no pair of adjacent nodes have the
same color, and that the total number of colors used is minimized.

This implementation finds the chromatic number by backtracking: nodes (ordered by degree to prune
sooner) are colored one at a time, each tried with every color already in use plus one new color,
while the fewest colors seen in a complete coloring so far bounds the search and cuts off any branch
at least as costly.

- `color_graph()` populates `color` and returns minimum color count for a global, bidirectionally
  pre-populated adjacency matrix `adj` which must consist of nodes numbered [0, `n`), where `n` is
  `adj.size()`.

Time Complexity:
- Exponential on the number of nodes per call to `color_graph()`.

Space Complexity:
- O(n^2) for storage of the graph, where $n$ is the number of nodes.
- O(n) auxiliary stack and heap space for `color_graph()`.

*/

#include <algorithm>
#include <vector>

std::vector<std::vector<bool>> adj;
int min_colors;
std::vector<int> color, curr, id, degree;

void rec(int lo, int hi, int n, int used_colors) {
  if (used_colors >= min_colors) {
    return;
  }
  if (n == hi) {
    for (int i = lo; i < hi; i++) {
      color[id[i]] = curr[i];
    }
    min_colors = used_colors;
    return;
  }
  std::vector<bool> used(used_colors + 1);
  for (int i = 0; i < n; i++) {
    if (adj[id[n]][id[i]]) {
      used[curr[i]] = true;
    }
  }
  for (int i = 0; i <= used_colors; i++) {
    if (!used[i]) {
      int tmp = curr[n];
      curr[n] = i;
      rec(lo, hi, n + 1, std::max(used_colors, i + 1));
      curr[n] = tmp;
    }
  }
}

int color_graph() {
  int n = static_cast<int>(adj.size());
  if (n == 0) {
    color.clear();
    return 0;
  }
  color.assign(n, 0);
  curr.assign(n, 0);
  id.assign(n + 1, 0);
  degree.assign(n + 1, 0);
  for (int i = 0; i <= n; i++) {
    id[i] = i;
    degree[i] = 0;
  }
  int res = 1, lo = 0;
  for (int hi = 1; hi <= n; hi++) {
    int best = hi;
    for (int i = hi; i < n; i++) {
      if (adj[id[hi - 1]][id[i]]) {
        degree[id[i]]++;
      }
      if (degree[id[best]] < degree[id[i]]) {
        best = i;
      }
    }
    std::swap(id[hi], id[best]);
    if (degree[id[hi]] == 0) {
      min_colors = n + 1;
      curr.assign(n, 0);
      rec(lo, hi, lo, 0);
      lo = hi;
      res = std::max(res, min_colors);
    }
  }
  return res;
}

/*** Example Usage and Output:

Colored using 3 color(s):
Color 1: 0 3
Color 2: 1 2
Color 3: 4

***/

#include <cassert>
#include <iostream>
using namespace std;

void add_edge(int u, int v) {
  adj[u][v] = true;
  adj[v][u] = true;
}

int main() {
  int nodes = 5;
  adj.assign(nodes, std::vector<bool>(nodes));
  add_edge(0, 1);
  add_edge(0, 4);
  add_edge(1, 3);
  add_edge(1, 4);
  add_edge(2, 3);
  add_edge(2, 4);
  add_edge(3, 4);
  int colors = color_graph();
  cout << "Colored using " << colors << " color(s):" << endl;
  for (int i = 0; i < colors; i++) {
    cout << "Color " << i + 1 << ":";
    for (int j = 0; j < nodes; j++) {
      if (color[j] == i) {
        cout << " " << j;
      }
    }
    cout << endl;
  }
  return 0;
}