Table of Contents

You are given a directed weighted graph $G$ with $N$ vertices and $M$ edges. Find any cycle of negative weight in it, if such a cycle exists.

In another formulation of the problem you have to find all pairs of vertices which have a path of arbitrarily small weight between them.

It is convenient to use different algorithms to solve these two variations of the problem, so we’ll discuss both of them here.

## 1. Using Bellman-Ford algorithm

Bellman-Ford algorithm allows you to check whether there exists a cycle of negative weight in the graph, and if it does, find one of these cycles.

The details of the algorithm are described in the article on the Bellman-Ford algorithm. Here we’ll describe only its application to this problem.

Do $N$ iterations of Bellman-Ford algorithm. If there were no changes on the last iteration, there is no cycle of negative weight in the graph. Otherwise take a vertex the distance to which has changed, and go from it via its ancestors until a cycle is found. This cycle will be the desired cycle of negative weight.

### 1.1. Implementation

struct Edge { int a, b, cost; }; int n, m; vector<Edge> edges; const int INF = 1000000000; void solve() { vector<int> d(n); vector<int> p(n, -1); int x; for (int i = 0; i < n; ++i) { x = -1; for (Edge e : edges) { if (d[e.a] + e.cost < d[e.b]) { d[e.b] = d[e.a] + e.cost; p[e.b] = e.a; x = e.b; } } } if (x == -1) { cout << "No negative cycle found."; } else { for (int i = 0; i < n; ++i) x = p[x]; vector<int> cycle; for (int v = x;; v = p[v]) { cycle.push_back(v); if (v == x && cycle.size() > 1) break; } reverse(cycle.begin(), cycle.end()); cout << "Negative cycle: "; for (int v : cycle) cout << v << ' '; cout << endl; } }

## 2. Using Floyd-Warshall algorithm

The Floyd-Warshall algorithm allows to solve the second variation of the problem – finding all pairs of vertices $(i, j)$ which don’t have a shortest path between them (i.e. a path of arbitrarily small weight exists).

Again, the details can be found in the **Floyd-Warshall** article, and here we describe only its application.

Run Floyd-Warshall algorithm on the graph. Initially $d[v][v] = 0$ for each $v$. But after running the algorithm $d[v][v]$ will be smaller than $0$ if there exists a negative length path from $v$ to $v$. We can use this to also find all pairs of vertices that don’t have a shortest path between them. We iterate over all pairs of vertices $(i, j)$ and for each pair we check whether they have a shortest path between them. To do this try all possibilities for an intermediate vertex $t$. $(i, j)$ doesn’t have a shortest path, if one of the intermediate vertices $t$ has $d[t][t] < 0$ (i.e. $t$ is part of a cycle of negative weight), $t$ can be reached from $i$ and $j$ can be reached from $t$. Then the path from $i$ to $j$ can have arbitrarily small weight. We will denote this with `-INF`

.

### 2.1. Implementation

for (int i = 0; i < n; ++i) { for (int j = 0; j < n; ++j) { for (int t = 0; t < n; ++t) { if (d[i][t] < INF && d[t][t] < 0 && d[t][j] < INF) d[i][j] = - INF; } } }