4.1.8 Eulerian Trails

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/*

A Eulerian trail is a path in a graph which contains every edge exactly once. A Eulerian cycle or
circuit is an Eulerian trail which begins and ends on the same node. A directed graph has a
Eulerian trail when all nonzero-degree nodes belong to one connected part of the underlying graph,
and either every node has equal in-degree and out-degree or exactly one node has one extra outgoing
edge and exactly one node has one extra incoming edge. An undirected graph has a Eulerian trail when
all nonzero-degree nodes are connected and either zero or two nodes have odd degree.

Hierholzer's algorithm walks unused edges until stuck, then backtracks to splice each closed detour
into the final trail. This implementation stores and returns edge IDs, which supports multigraphs:
parallel edges are distinct because each edge receives its own ID.

- `EulerianGraph(n, directed)` constructs a graph of `n` nodes numbered [0, `n`). The graph is
  directed if `directed` is true, or undirected otherwise.
- `add_edge(u, v)` adds an edge and returns its edge ID.
- `eulerian_path(start)` returns a trail using every edge exactly once, or a result with
  `start` $= -1$ if no such trail exists. If `start` $= -1$, a valid start is chosen automatically.
- `EulerianTrail::is_cycle()` returns whether the trail begins and ends on the same node.

Time Complexity:
- O(max(n, m)) per call to `eulerian_path()`, where $n$ is the number of nodes and $m$ is the
  number of edges.

Space Complexity:
- O(max(n, m)) for graph storage and auxiliary arrays.

*/

#include <algorithm>
#include <vector>

class EulerianGraph {
  std::vector<std::pair<int, int>> edges;
  std::vector<std::vector<int>> adj;
  bool directed;

  bool valid_degrees(int start) const {
    int n = static_cast<int>(adj.size());
    if (directed) {
      std::vector<int> indeg(n), outdeg(n);
      for (const auto &[eu, ev] : edges) {
        outdeg[eu]++;
        indeg[ev]++;
      }
      int source = -1, sink = -1;
      for (int u = 0; u < n; u++) {
        int diff = outdeg[u] - indeg[u];
        if (diff == 1) {
          if (source != -1) {
            return false;
          }
          source = u;
        } else if (diff == -1) {
          if (sink != -1) {
            return false;
          }
          sink = u;
        } else if (diff != 0) {
          return false;
        }
      }
      if ((source == -1) != (sink == -1)) {
        return false;
      }
      if (start != -1) {
        return source == -1 ? outdeg[start] > 0 : start == source;
      }
      return true;
    }
    std::vector<int> degree(n);
    for (const auto &[eu, ev] : edges) {
      degree[eu]++;
      degree[ev]++;
    }
    std::vector<int> odd;
    for (int u = 0; u < n; u++) {
      if (degree[u] % 2 == 1) {
        odd.push_back(u);
      }
    }
    if (!odd.empty() && odd.size() != 2) {
      return false;
    }
    if (start != -1) {
      return odd.empty() ? degree[start] > 0 : (start == odd[0] || start == odd[1]);
    }
    return true;
  }

  int choose_start() const {
    int n = static_cast<int>(adj.size());
    if (directed) {
      std::vector<int> indeg(n), outdeg(n);
      for (const auto &[eu, ev] : edges) {
        outdeg[eu]++;
        indeg[ev]++;
      }
      for (int u = 0; u < n; u++) {
        if (outdeg[u] - indeg[u] == 1) {
          return u;
        }
      }
      for (int u = 0; u < n; u++) {
        if (outdeg[u] > 0) {
          return u;
        }
      }
    } else {
      std::vector<int> degree(n);
      for (const auto &[eu, ev] : edges) {
        degree[eu]++;
        degree[ev]++;
      }
      for (int u = 0; u < n; u++) {
        if (degree[u] % 2 == 1) {
          return u;
        }
      }
      for (int u = 0; u < n; u++) {
        if (degree[u] > 0) {
          return u;
        }
      }
    }
    return 0;
  }

  std::vector<int> build_vertices(int start, const std::vector<int> &trail_edges) const {
    std::vector<int> vertices{start};
    int u = start;
    for (int id : trail_edges) {
      const auto &[eu, ev] = edges[id];
      u = directed ? ev : eu ^ ev ^ u;
      vertices.push_back(u);
    }
    return vertices;
  }

 public:
  struct EulerianTrail {
    int start = -1;
    std::vector<int> edges;
    std::vector<int> vertices;

    bool is_cycle() const {
      return start != -1 && vertices.size() > 1 && vertices.front() == vertices.back();
    }
  };

  EulerianGraph(int n, bool directed) : adj(n), directed(directed) {}

  int add_edge(int u, int v) {
    int id = static_cast<int>(edges.size());
    edges.emplace_back(u, v);
    adj[u].push_back(id);
    if (!directed) {
      adj[v].push_back(id);
    }
    return id;
  }

  EulerianTrail eulerian_path(int start = -1) const {
    int n = static_cast<int>(adj.size());
    int m = static_cast<int>(edges.size());
    if (m == 0) {
      int s = (start == -1 ? 0 : start);
      return EulerianTrail{s, {}, {s}};
    }
    if (!valid_degrees(start)) {
      return EulerianTrail{};
    }
    if (start == -1) {
      start = choose_start();
    }
    std::vector<bool> used(m, false);
    std::vector<int> ptr(n), vertex_stack{start}, edge_stack{-1}, trail_edges;
    while (!vertex_stack.empty()) {
      int u = vertex_stack.back();
      while (ptr[u] < static_cast<int>(adj[u].size()) && used[adj[u][ptr[u]]]) {
        ptr[u]++;
      }
      if (ptr[u] == static_cast<int>(adj[u].size())) {
        if (edge_stack.back() != -1) {
          trail_edges.push_back(edge_stack.back());
        }
        vertex_stack.pop_back();
        edge_stack.pop_back();
      } else {
        int id = adj[u][ptr[u]++];
        used[id] = true;
        const auto &[eu, ev] = edges[id];
        int v = directed ? ev : eu ^ ev ^ u;
        vertex_stack.push_back(v);
        edge_stack.push_back(id);
      }
    }
    if (static_cast<int>(trail_edges.size()) != m) {
      return EulerianTrail{};
    }
    std::reverse(trail_edges.begin(), trail_edges.end());
    return EulerianTrail{start, trail_edges, build_vertices(start, trail_edges)};
  }
};

/*** Example Usage ***/

#include <cassert>
using namespace std;

int main() {
  {
    EulerianGraph g(5, true);
    g.add_edge(0, 1);
    g.add_edge(1, 2);
    g.add_edge(2, 0);
    g.add_edge(1, 3);
    g.add_edge(3, 4);
    g.add_edge(4, 1);
    auto trail = g.eulerian_path(0);
    assert(trail.edges.size() == 6);
    assert(trail.vertices.front() == 0 && trail.vertices.back() == 0);
    assert(trail.is_cycle());
  }
  {
    EulerianGraph g(5, false);
    g.add_edge(0, 1);
    g.add_edge(1, 2);
    g.add_edge(2, 0);
    g.add_edge(1, 3);
    g.add_edge(3, 4);
    g.add_edge(4, 1);
    auto trail = g.eulerian_path();
    assert(trail.edges.size() == 6);
    assert(trail.vertices.front() == trail.vertices.back());
    assert(trail.is_cycle());
  }
  {
    EulerianGraph g(2, false);
    int a = g.add_edge(0, 1);
    int b = g.add_edge(0, 1);
    auto trail = g.eulerian_path(0);
    assert(trail.is_cycle());
    assert((trail.edges == vector<int>{a, b} || trail.edges == vector<int>{b, a}));
  }
  {
    EulerianGraph g(3, true);
    g.add_edge(0, 1);
    g.add_edge(1, 2);
    auto trail = g.eulerian_path();
    assert(trail.start == 0);
    assert((trail.vertices == vector<int>{0, 1, 2}));
    assert(!trail.is_cycle());
  }
  return 0;
}