/**
* Determines whether the edge-weighted digraph {@code G} has a topological
* order and, if so, finds such an order.
* @param G the edge-weighted digraph
*/
public Topological(EdgeWeightedDigraph G) {
EdgeWeightedDirectedCycle finder = new EdgeWeightedDirectedCycle(G);
if (!finder.hasCycle()) {
DepthFirstOrder dfs = new DepthFirstOrder(G);
order = dfs.reversePost();
}
}
private IndexMinPQ<Double> pq; // priority queue of vertices
/**
* Computes a shortest-paths tree from the source vertex {@code s} to every other
* vertex in the edge-weighted digraph {@code G}.
*
* @param G the edge-weighted digraph
* @param s the source vertex
* @throws IllegalArgumentException if an edge weight is negative
* @throws IllegalArgumentException unless {@code 0 <= s < V}
*/
public DijkstraSP(EdgeWeightedDigraph G, int s) {
for (DirectedEdge e : G.edges()) {
if (e.weight() < 0)
throw new IllegalArgumentException("edge " + e + " has negative weight");
}
distTo = new double[G.V()];
edgeTo = new DirectedEdge[G.V()];
validateVertex(s);
for (int v = 0; v < G.V(); v++)
distTo[v] = Double.POSITIVE_INFINITY;
distTo[s] = 0.0;
// relax vertices in order of distance from s
pq = new IndexMinPQ<Double>(G.V());
pq.insert(s, distTo[s]);
while (!pq.isEmpty()) {
int v = pq.delMin();
for (DirectedEdge e : G.adj(v))
relax(e);
}
// check optimality conditions
assert check(G, s);
}
/**
* Reads the currency exchange table from standard input and
* prints an arbitrage opportunity to standard output (if one exists).
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
// V currencies
int V = StdIn.readInt();
String[] name = new String[V];
// create complete network
EdgeWeightedDigraph G = new EdgeWeightedDigraph(V);
for (int v = 0; v < V; v++) {
name[v] = StdIn.readString();
for (int w = 0; w < V; w++) {
double rate = StdIn.readDouble();
DirectedEdge e = new DirectedEdge(v, w, -Math.log(rate));
G.addEdge(e);
}
}
// find negative cycle
BellmanFordSP spt = new BellmanFordSP(G, 0);
if (spt.hasNegativeCycle()) {
double stake = 1000.0;
for (DirectedEdge e : spt.negativeCycle()) {
StdOut.printf("%10.5f %s ", stake, name[e.from()]);
stake *= Math.exp(-e.weight());
StdOut.printf("= %10.5f %s\n", stake, name[e.to()]);
}
}
else {
StdOut.println("No arbitrage opportunity");
}
}
// check optimality conditions: either
// (i) there exists a negative cycle reacheable from s
// or
// (ii) for all edges e = v->w: distTo[w] <= distTo[v] + e.weight()
// (ii') for all edges e = v->w on the SPT: distTo[w] == distTo[v] + e.weight()
private boolean check(EdgeWeightedDigraph G, int s) {
// has a negative cycle
if (hasNegativeCycle()) {
double weight = 0.0;
for (DirectedEdge e : negativeCycle()) {
weight += e.weight();
}
if (weight >= 0.0) {
System.err.println("error: weight of negative cycle = " + weight);
return false;
}
}
// no negative cycle reachable from source
else {
// check that distTo[v] and edgeTo[v] are consistent
if (distTo[s] != 0.0 || edgeTo[s] != null) {
System.err.println("distanceTo[s] and edgeTo[s] inconsistent");
return false;
}
for (int v = 0; v < G.V(); v++) {
if (v == s) continue;
if (edgeTo[v] == null && distTo[v] != Double.POSITIVE_INFINITY) {
System.err.println("distTo[] and edgeTo[] inconsistent");
return false;
}
}
// check that all edges e = v->w satisfy distTo[w] <= distTo[v] + e.weight()
for (int v = 0; v < G.V(); v++) {
for (DirectedEdge e : G.adj(v)) {
int w = e.to();
if (distTo[v] + e.weight() < distTo[w]) {
System.err.println("edge " + e + " not relaxed");
return false;
}
}
}
// check that all edges e = v->w on SPT satisfy distTo[w] == distTo[v] + e.weight()
for (int w = 0; w < G.V(); w++) {
if (edgeTo[w] == null) continue;
DirectedEdge e = edgeTo[w];
int v = e.from();
if (w != e.to()) return false;
if (distTo[v] + e.weight() != distTo[w]) {
System.err.println("edge " + e + " on shortest path not tight");
return false;
}
}
}
StdOut.println("Satisfies optimality conditions");
StdOut.println();
return true;
}
/**
* Determines a depth-first order for the edge-weighted digraph {@code G}.
* @param G the edge-weighted digraph
*/
public DepthFirstOrder(EdgeWeightedDigraph G) {
pre = new int[G.V()];
post = new int[G.V()];
postorder = new Queue<Integer>();
preorder = new Queue<Integer>();
marked = new boolean[G.V()];
for (int v = 0; v < G.V(); v++)
if (!marked[v]) dfs(G, v);
}
/**
* Unit tests the {@code TopologicalX} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
// create random DAG with V vertices and E edges; then add F random edges
int V = Integer.parseInt(args[0]);
int E = Integer.parseInt(args[1]);
int F = Integer.parseInt(args[2]);
Digraph G1 = DigraphGenerator.dag(V, E);
// corresponding edge-weighted digraph
EdgeWeightedDigraph G2 = new EdgeWeightedDigraph(V);
for (int v = 0; v < G1.V(); v++)
for (int w : G1.adj(v))
G2.addEdge(new DirectedEdge(v, w, 0.0));
// add F extra edges
for (int i = 0; i < F; i++) {
int v = StdRandom.uniform(V);
int w = StdRandom.uniform(V);
G1.addEdge(v, w);
G2.addEdge(new DirectedEdge(v, w, 0.0));
}
StdOut.println(G1);
StdOut.println();
StdOut.println(G2);
// find a directed cycle
TopologicalX topological1 = new TopologicalX(G1);
if (!topological1.hasOrder()) {
StdOut.println("Not a DAG");
}
// or give topologial sort
else {
StdOut.print("Topological order: ");
for (int v : topological1.order()) {
StdOut.print(v + " ");
}
StdOut.println();
}
// find a directed cycle
TopologicalX topological2 = new TopologicalX(G2);
if (!topological2.hasOrder()) {
StdOut.println("Not a DAG");
}
// or give topologial sort
else {
StdOut.print("Topological order: ");
for (int v : topological2.order()) {
StdOut.print(v + " ");
}
StdOut.println();
}
}
// by finding a cycle in predecessor graph
private void findNegativeCycle() {
int V = edgeTo.length;
EdgeWeightedDigraph spt = new EdgeWeightedDigraph(V);
for (int v = 0; v < V; v++)
if (edgeTo[v] != null)
spt.addEdge(edgeTo[v]);
EdgeWeightedDirectedCycle finder = new EdgeWeightedDirectedCycle(spt);
cycle = finder.cycle();
}
private Stack<DirectedEdge> cycle; // directed cycle (or null if no such cycle)
/**
* Determines whether the edge-weighted digraph {@code G} has a directed cycle and,
* if so, finds such a cycle.
* @param G the edge-weighted digraph
*/
public EdgeWeightedDirectedCycle(EdgeWeightedDigraph G) {
marked = new boolean[G.V()];
onStack = new boolean[G.V()];
edgeTo = new DirectedEdge[G.V()];
for (int v = 0; v < G.V(); v++)
if (!marked[v]) dfs(G, v);
// check that digraph has a cycle
assert check();
}
// run DFS in edge-weighted digraph G from vertex v and compute preorder/postorder
private void dfs(EdgeWeightedDigraph G, int v) {
marked[v] = true;
pre[v] = preCounter++;
preorder.enqueue(v);
for (DirectedEdge e : G.adj(v)) {
int w = e.to();
if (!marked[w]) {
dfs(G, w);
}
}
postorder.enqueue(v);
post[v] = postCounter++;
}
/**
* Returns a negative cycle, or {@code null} if there is no such cycle.
* @return a negative cycle as an iterable of edges,
* or {@code null} if there is no such cycle
*/
public Iterable<DirectedEdge> negativeCycle() {
for (int v = 0; v < distTo.length; v++) {
// negative cycle in v's predecessor graph
if (distTo[v][v] < 0.0) {
int V = edgeTo.length;
EdgeWeightedDigraph spt = new EdgeWeightedDigraph(V);
for (int w = 0; w < V; w++)
if (edgeTo[v][w] != null)
spt.addEdge(edgeTo[v][w]);
EdgeWeightedDirectedCycle finder = new EdgeWeightedDirectedCycle(spt);
assert finder.hasCycle();
return finder.cycle();
}
}
return null;
}
/**
* Initializes a new edge-weighted digraph that is a deep copy of {@code G}.
*
* @param G the edge-weighted digraph to copy
*/
public EdgeWeightedDigraph(EdgeWeightedDigraph G) {
this(G.V());
this.E = G.E();
for (int v = 0; v < G.V(); v++)
this.indegree[v] = G.indegree(v);
for (int v = 0; v < G.V(); v++) {
// reverse so that adjacency list is in same order as original
Stack<DirectedEdge> reverse = new Stack<DirectedEdge>();
for (DirectedEdge e : G.adj[v]) {
reverse.push(e);
}
for (DirectedEdge e : reverse) {
adj[v].add(e);
}
}
}
// check optimality conditions:
// (i) for all edges e: distTo[e.to()] <= distTo[e.from()] + e.weight()
// (ii) for all edge e on the SPT: distTo[e.to()] == distTo[e.from()] + e.weight()
private boolean check(EdgeWeightedDigraph G, int s) {
// check that edge weights are nonnegative
for (DirectedEdge e : G.edges()) {
if (e.weight() < 0) {
System.err.println("negative edge weight detected");
return false;
}
}
// check that distTo[v] and edgeTo[v] are consistent
if (distTo[s] != 0.0 || edgeTo[s] != null) {
System.err.println("distTo[s] and edgeTo[s] inconsistent");
return false;
}
for (int v = 0; v < G.V(); v++) {
if (v == s) continue;
if (edgeTo[v] == null && distTo[v] != Double.POSITIVE_INFINITY) {
System.err.println("distTo[] and edgeTo[] inconsistent");
return false;
}
}
// check that all edges e = v->w satisfy distTo[w] <= distTo[v] + e.weight()
for (int v = 0; v < G.V(); v++) {
for (DirectedEdge e : G.adj(v)) {
int w = e.to();
if (distTo[v] + e.weight() < distTo[w]) {
System.err.println("edge " + e + " not relaxed");
return false;
}
}
}
// check that all edges e = v->w on SPT satisfy distTo[w] == distTo[v] + e.weight()
for (int w = 0; w < G.V(); w++) {
if (edgeTo[w] == null) continue;
DirectedEdge e = edgeTo[w];
int v = e.from();
if (w != e.to()) return false;
if (distTo[v] + e.weight() != distTo[w]) {
System.err.println("edge " + e + " on shortest path not tight");
return false;
}
}
return true;
}
// relax vertex v and put other endpoints on queue if changed
private void relax(EdgeWeightedDigraph G, int v) {
for (DirectedEdge e : G.adj(v)) {
int w = e.to();
if (distTo[w] > distTo[v] + e.weight()) {
distTo[w] = distTo[v] + e.weight();
edgeTo[w] = e;
if (!onQueue[w]) {
queue.enqueue(w);
onQueue[w] = true;
}
}
if (cost++ % G.V() == 0) {
findNegativeCycle();
if (hasNegativeCycle()) return; // found a negative cycle
}
}
}
/**
* Unit tests the {@code EdgeWeightedDirectedCycle} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
// create random DAG with V vertices and E edges; then add F random edges
int V = Integer.parseInt(args[0]);
int E = Integer.parseInt(args[1]);
int F = Integer.parseInt(args[2]);
EdgeWeightedDigraph G = new EdgeWeightedDigraph(V);
int[] vertices = new int[V];
for (int i = 0; i < V; i++)
vertices[i] = i;
StdRandom.shuffle(vertices);
for (int i = 0; i < E; i++) {
int v, w;
do {
v = StdRandom.uniform(V);
w = StdRandom.uniform(V);
} while (v >= w);
double weight = StdRandom.uniform();
G.addEdge(new DirectedEdge(v, w, weight));
}
// add F extra edges
for (int i = 0; i < F; i++) {
int v = StdRandom.uniform(V);
int w = StdRandom.uniform(V);
double weight = StdRandom.uniform(0.0, 1.0);
G.addEdge(new DirectedEdge(v, w, weight));
}
StdOut.println(G);
// find a directed cycle
EdgeWeightedDirectedCycle finder = new EdgeWeightedDirectedCycle(G);
if (finder.hasCycle()) {
StdOut.print("Cycle: ");
for (DirectedEdge e : finder.cycle()) {
StdOut.print(e + " ");
}
StdOut.println();
}
// or give topologial sort
else {
StdOut.println("No directed cycle");
}
}
private DirectedEdge[] edgeTo; // edgeTo[v] = last edge on shortest s->v path
/**
* Computes a shortest paths tree from {@code s} to every other vertex in
* the directed acyclic graph {@code G}.
* @param G the acyclic digraph
* @param s the source vertex
* @throws IllegalArgumentException if the digraph is not acyclic
* @throws IllegalArgumentException unless {@code 0 <= s < V}
*/
public AcyclicSP(EdgeWeightedDigraph G, int s) {
distTo = new double[G.V()];
edgeTo = new DirectedEdge[G.V()];
validateVertex(s);
for (int v = 0; v < G.V(); v++)
distTo[v] = Double.POSITIVE_INFINITY;
distTo[s] = 0.0;
// visit vertices in topological order
Topological topological = new Topological(G);
if (!topological.hasOrder())
throw new IllegalArgumentException("Digraph is not acyclic.");
for (int v : topological.order()) {
for (DirectedEdge e : G.adj(v))
relax(e);
}
}
private DirectedEdge[] edgeTo; // edgeTo[v] = last edge on longest s->v path
/**
* Computes a longest paths tree from {@code s} to every other vertex in
* the directed acyclic graph {@code G}.
* @param G the acyclic digraph
* @param s the source vertex
* @throws IllegalArgumentException if the digraph is not acyclic
* @throws IllegalArgumentException unless {@code 0 <= s < V}
*/
public AcyclicLP(EdgeWeightedDigraph G, int s) {
distTo = new double[G.V()];
edgeTo = new DirectedEdge[G.V()];
validateVertex(s);
for (int v = 0; v < G.V(); v++)
distTo[v] = double.NegativeInfinity;// NEGATIVE_INFINITY;
distTo[s] = 0.0;
// relax vertices in topological order
Topological topological = new Topological(G);
if (!topological.hasOrder())
throw new ArgumentException("Digraph is not acyclic.");
foreach (int v in topological.order()) {
foreach ( DirectedEdge e in G.adj(v))
relax(e);
}
}
// certify that digraph is acyclic
private boolean check(EdgeWeightedDigraph G) {
// digraph is acyclic
if (hasOrder()) {
// check that ranks are a permutation of 0 to V-1
boolean[] found = new boolean[G.V()];
for (int i = 0; i < G.V(); i++) {
found[rank(i)] = true;
}
for (int i = 0; i < G.V(); i++) {
if (!found[i]) {
System.err.println("No vertex with rank " + i);
return false;
}
}
// check that ranks provide a valid topological order
for (int v = 0; v < G.V(); v++) {
for (DirectedEdge e : G.adj(v)) {
int w = e.to();
if (rank(v) > rank(w)) {
System.err.printf("%d-%d: rank(%d) = %d, rank(%d) = %d\n",
v, w, v, rank(v), w, rank(w));
return false;
}
}
}
// check that order() is consistent with rank()
int r = 0;
for (int v : order()) {
if (rank(v) != r) {
System.err.println("order() and rank() inconsistent");
return false;
}
r++;
}
}
return true;
}
private Iterable<DirectedEdge> cycle; // negative cycle (or null if no such cycle)
/**
* Computes a shortest paths tree from {@code s} to every other vertex in
* the edge-weighted digraph {@code G}.
* @param G the acyclic digraph
* @param s the source vertex
* @throws IllegalArgumentException unless {@code 0 <= s < V}
*/
public BellmanFordSP(EdgeWeightedDigraph G, int s) {
distTo = new double[G.V()];
edgeTo = new DirectedEdge[G.V()];
onQueue = new boolean[G.V()];
for (int v = 0; v < G.V(); v++)
distTo[v] = Double.POSITIVE_INFINITY;
distTo[s] = 0.0;
// Bellman-Ford algorithm
queue = new Queue<Integer>();
queue.enqueue(s);
onQueue[s] = true;
while (!queue.isEmpty() && !hasNegativeCycle()) {
int v = queue.dequeue();
onQueue[v] = false;
relax(G, v);
}
assert check(G, s);
}
// find shortest augmenting path and upate
private void augment() {
// build residual graph
EdgeWeightedDigraph G = new EdgeWeightedDigraph(2*n+2);
int s = 2*n, t = 2*n+1;
for (int i = 0; i < n; i++) {
if (xy[i] == UNMATCHED)
G.addEdge(new DirectedEdge(s, i, 0.0));
}
for (int j = 0; j < n; j++) {
if (yx[j] == UNMATCHED)
G.addEdge(new DirectedEdge(n+j, t, py[j]));
}
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
if (xy[i] == j) G.addEdge(new DirectedEdge(n+j, i, 0.0));
else G.addEdge(new DirectedEdge(i, n+j, reducedCost(i, j)));
}
}
// compute shortest path from s to every other vertex
DijkstraSP spt = new DijkstraSP(G, s);
// augment along alternating path
for (DirectedEdge e : spt.pathTo(t)) {
int i = e.from(), j = e.to() - n;
if (i < n) {
xy[i] = j;
yx[j] = i;
}
}
// update dual variables
for (int i = 0; i < n; i++)
px[i] += spt.distTo(i);
for (int j = 0; j < n; j++)
py[j] += spt.distTo(n+j);
}
/**
* Determines whether the edge-weighted digraph {@code G} has a
* topological order and, if so, finds such a topological order.
* @param G the digraph
*/
public TopologicalX(EdgeWeightedDigraph G) {
// indegrees of remaining vertices
int[] indegree = new int[G.V()];
for (int v = 0; v < G.V(); v++) {
indegree[v] = G.indegree(v);
}
// initialize
ranks = new int[G.V()];
order = new Queue<Integer>();
int count = 0;
// initialize queue to contain all vertices with indegree = 0
Queue<Integer> queue = new Queue<Integer>();
for (int v = 0; v < G.V(); v++)
if (indegree[v] == 0) queue.enqueue(v);
while (!queue.isEmpty()) {
int v = queue.dequeue();
order.enqueue(v);
ranks[v] = count++;
for (DirectedEdge e : G.adj(v)) {
int w = e.to();
indegree[w]--;
if (indegree[w] == 0) queue.enqueue(w);
}
}
// there is a directed cycle in subgraph of vertices with indegree >= 1.
if (count != G.V()) {
order = null;
}
assert check(G);
}
/**
* Reads the precedence constraints from standard input
* and prints a feasible schedule to standard output.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
// number of jobs
int n = StdIn.readInt();
// source and sink
int source = 2*n;
int sink = 2*n + 1;
// build network
EdgeWeightedDigraph G = new EdgeWeightedDigraph(2*n + 2);
for (int i = 0; i < n; i++) {
double duration = StdIn.readDouble();
G.addEdge(new DirectedEdge(source, i, 0.0));
G.addEdge(new DirectedEdge(i+n, sink, 0.0));
G.addEdge(new DirectedEdge(i, i+n, duration));
// precedence constraints
int m = StdIn.readInt();
for (int j = 0; j < m; j++) {
int precedent = StdIn.readInt();
G.addEdge(new DirectedEdge(n+i, precedent, 0.0));
}
}
// compute longest path
AcyclicLP lp = new AcyclicLP(G, source);
// print results
StdOut.println(" job start finish");
StdOut.println("--------------------");
for (int i = 0; i < n; i++) {
StdOut.printf("%4d %7.1f %7.1f\n", i, lp.distTo(i), lp.distTo(i+n));
}
StdOut.printf("Finish time: %7.1f\n", lp.distTo(sink));
}
/**
* Computes a shortest paths tree from each vertex to to every other vertex in
* the edge-weighted digraph {@code G}.
* @param G the edge-weighted digraph
* @throws IllegalArgumentException if an edge weight is negative
* @throws IllegalArgumentException unless {@code 0 <= s < V}
*/
public DijkstraAllPairsSP(EdgeWeightedDigraph G) {
all = new DijkstraSP[G.V()];
for (int v = 0; v < G.V(); v++)
all[v] = new DijkstraSP(G, v);
}
