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Algorithms, 4th Edition · Robert Sedgewick and Kevin Wayne · Copyright © 2002–2010 · February 6, 2011 4:59:40 PM
4.4 Shortest Paths
‣ edge-weighted digraph API‣ shortest-paths properties‣ Dijkstra's algorithm‣ edge-weighted DAGs‣ negative weights
Google maps
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Continental U.S. routes (August 2010)
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5:00MountainStandardTime
6:00CentralStandardTime
7:00EasternStandardTime
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Continental RouteFuture ServiceSeasonal ServiceTime Zone BoundaryContinental/Continental Express/Continental Connection DestinationSelected Airline Partner Destination (For a complete list of airline partners, see page 78.)
Domestic Routes8/10
7:00Eastern Standard Time
ATLANTICOCEAN
Philadelphia
Wilmington
Stamford
NewHaven
Boston
Newark(Liberty)
Washington, D.C.
New York(Penn Station)
Train RoutesCodeshare/OnePass ServiceOnePass Eligible Service
take flight/u.s. route map system
http://www.continental.com/web/en-US/content/travel/routes
4
Shortest outgoing routes on the Internet from Lumeta headquarters
map by Lumeta Corporation, March 8, 2006
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Given an edge-weighted digraph, find the shortest (directed) path from s to t.
5
Shortest paths in a weighted digraph
An edge-weighted digraph and a shortest path
4->5 0.35 5->4 0.35 4->7 0.37 5->7 0.28 7->5 0.28 5->1 0.32 0->4 0.380->2 0.26 7->3 0.39 1->3 0.29 2->7 0.346->2 0.40 3->6 0.526->0 0.586->4 0.93
0->2 0.262->7 0.347->3 0.393->6 0.52
edge-weighted digraph
shortest path from 0 to 6
Shortest path variants
Which vertices?
• Source-sink: from one vertex to another.
• Single source: from one vertex to every other.
• All pairs: between all pairs of vertices.
Restrictions on edge weights?
• Nonnegative weights.
• Arbitrary weights.
• Euclidean weights.
Cycles?
• No cycles.
• No "negative cycles."
Simplifying assumption. There exists a shortest path from s to each vertex v.6
• Map routing.
• Robot navigation.
• Texture mapping.
• Typesetting in TeX.
• Urban traffic planning.
• Optimal pipelining of VLSI chip.
• Telemarketer operator scheduling.
• Subroutine in advanced algorithms.
• Routing of telecommunications messages.
• Approximating piecewise linear functions.
• Network routing protocols (OSPF, BGP, RIP).
• Exploiting arbitrage opportunities in currency exchange.
• Optimal truck routing through given traffic congestion pattern.
7
Reference: Network Flows: Theory, Algorithms, and Applications, R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Prentice Hall, 1993.
Shortest path applications
8
‣ edge-weighted digraph API‣ shortest-paths properties‣ Dijkstra's algorithm‣ edge-weighted DAGs‣ negative weights
9
Weighted directed edge API
Idiom for processing an edge e: int v = e.from(), w = e.to();
vweight
w
public class DirectedEdge public class DirectedEdge
DirectedEdge(int v, int w, double weight) weighted edge v!w
int from() vertex v
int to() vertex w
double weight() weight of this edge
String toString() string representation
10
Weighted directed edge: implementation in Java
Similar to Edge for undirected graphs, but a bit simpler.
public class DirectedEdge{ private final int v, w; private final double weight;
public DirectedEdge(int v, int w, double weight) { this.v = v; this.w = w; this.weight = weight; }
public int from() { return v; }
public int to() { return w; }
public int weight() { return weight; }}
from() and to() replace
either() and other()
11
Edge-weighted digraph API
Conventions. Allow self-loops and parallel edges.
public class EdgeWeightedDigraph public class EdgeWeightedDigraph
EdgeWeightedDigraph(int V) edge-weighted digraph with V vertices
EdgeWeightedDigraph(In in) edge-weighted digraph from input stream
void addEdge(DirectedEdge e) add weighted directed edge e
Iterable<DirectedEdge> adj(int v) edges adjacent from v
int V() number of vertices
int E() number of edges
Iterable<DirectedEdge> edges() all edges in this digraph
String toString() string representation
12
Edge-weighted digraph: adjacency-lists representation
Edge-weighted digraph representation
adj0
1
2
3
4
5
6
7
0 2 .26 0 4 .38
Bag objects
reference to aDirectedEdge
object
8154 5 0.35 5 4 0.35 4 7 0.37 5 7 0.28 7 5 0.28 5 1 0.32 0 4 0.380 2 0.26 7 3 0.39 1 3 0.29 2 7 0.346 2 0.40 3 6 0.526 0 0.586 4 0.93
1 3 .29
2 7 .34
3 6 .52
4 7 .37 4 5 .35
5 1 .32 5 7 .28 5 4 .35
6 4 .93 6 0 .58 6 2 .40
7 3 .39 7 5 .28
tinyEWD.txtV
E
13
Edge-weighted digraph: adjacency-lists implementation in Java
Same as EdgeWeightedGraph except replace Graph with Digraph.
public class EdgeWeightedDigraph{ private final int V; private final Bag<Edge>[] adj;
public EdgeWeightedDigraph(int V) { this.V = V; adj = (Bag<DirectedEdge>[]) new Bag[V]; for (int v = 0; v < V; v++) adj[v] = new Bag<DirectedEdge>(); }
public void addEdge(DirectedEdge e) { int v = e.from(); adj[v].add(e); }
public Iterable<DirectedEdge> adj(int v) { return adj[v]; }}
similar to edge-weighted
undirected graph, but only
add edge to v's adjacency list
14
Single-source shortest paths API
Goal. Find the shortest path from s to every other vertex.
SP sp = new SP(G, s);
for (int v = 0; v < G.V(); v++)
{
StdOut.printf("%d to %d (%.2f): ", s, v, sp.distTo(v));
for (DirectedEdge e : sp.pathTo(v))
StdOut.print(e + " ");
StdOut.println();
}
public class SP public class SP public class SP
SP(EdgeWeightedDigraph G, int s) shortest paths from s in graph G
double distTo(int v) length of shortest path from s to v
Iterable <DirectedEdge> pathTo(int v) shortest path from s to v
boolean hasPathTo(int v) is there a path from s to v?
15
Single-source shortest paths API
Goal. Find the shortest path from s to every other vertex.
% java SP tinyEWD.txt 0
0 to 0 (0.00):
0 to 1 (1.05): 0->4 0.38 4->5 0.35 5->1 0.32
0 to 2 (0.26): 0->2 0.26
0 to 3 (0.99): 0->2 0.26 2->7 0.34 7->3 0.39
0 to 4 (0.38): 0->4 0.38
0 to 5 (0.73): 0->4 0.38 4->5 0.35
0 to 6 (1.51): 0->2 0.26 2->7 0.34 7->3 0.39 3->6 0.52
0 to 7 (0.60): 0->2 0.26 2->7 0.34
public class SP public class SP public class SP
SP(EdgeWeightedDigraph G, int s) shortest paths from s in graph G
double distTo(int v) length of shortest path from s to v
Iterable <DirectedEdge> pathTo(int v) shortest path from s to v
boolean hasPathTo(int v) is there a path from s to v?
16
‣ edge-weighted digraph API‣ shortest-paths properties‣ Dijkstra's algorithm‣ edge-weighted DAGs‣ negative weights
Goal. Find the shortest path from s to every other vertex.
Observation. A shortest path tree (SPT) solution exists. Why?
Consequence. Can represent the SPT with two vertex-indexed arrays:
• distTo[v] is length of shortest path from s to v.
• edgeTo[v] is last edge on shortest path from s to v.
17
Data structures for single-source shortest paths
shortest path tree from 0
Shortest paths data structures
edgeTo[] distTo[] 0 null 0 1 5->1 0.32 1.05 2 0->2 0.26 0.26 3 7->3 0.37 0.97 4 0->4 0.38 0.38 5 4->5 0.35 0.73 6 3->6 0.52 1.49 7 2->7 0.34 0.60
Goal. Find the shortest path from s to every other vertex.
Observation. A shortest path tree (SPT) solution exists. Why?
Consequence. Can represent the SPT with two vertex-indexed arrays:
• distTo[v] is length of shortest path from s to v.
• edgeTo[v] is last edge on shortest path from s to v.
18
Data structures for single-source shortest paths
public double distTo(int v)
{ return distTo[v]; }
public Iterable<DirectedEdge> pathTo(int v)
{
Stack<DirectedEdge> path = new Stack<DirectedEdge>();
for (DirectedEdge e = edgeTo[v]; e != null; e = edgeTo[e.from()])
path.push(e);
return path;
}
19
Edge relaxation
Relax edge e = v!w.
• distTo[v] is length of shortest known path from s to v.
• distTo[w] is length of shortest known path from s to w.
• edgeTo[w] is last edge on shortest known path from s to w.
• If e = v!w gives shorter path to w through v, update distTo[w] and edgeTo[w].
private void relax(DirectedEdge e) { int v = e.from(), w = e.to(); if (distTo[w] > distTo[v] + e.weight()) { distTo[w] = distTo[v] + e.weight(); edgeTo[w] = e; } }
Edge relaxation (two cases)
v->w is ineligible v->w successfully relaxes
s
v
wblack edges
are in edgeTo[]
s
w
s
v
w
no changes
s
w
edgeTo[w]
distTo[v]
distTo[w]
weight of v->w is 1.3v
v
3.1
3.3
3.1
3.1
7.2
4.4
black edgesare in edgeTo[]
distTo[w]
distTo[v]
weight of v->w is 1.3
20
Shortest-paths optimality conditions
Proposition. Let G be an edge-weighted digraph.Then distTo[] are the shortest path distances from s iff:
• For each vertex v, distTo[v] is the length of some path from s to v.
• For each edge e = v!w, distTo[w] ! distTo[v] + e.weight().
Pf. " [ necessary ]
• Suppose that distTo[w] > distTo[v] + e.weight() for some edge e = v!w.
• Then, e gives a path from s to w (through v) of length less than distTo[w].
s
w
v 3.1
7.2
distTo[w] > distTo[v] + e.weight()
distTo[w]
distTo[v]
weight of v->w is 1.3
21
Shortest-paths optimality conditions
Proposition. Let G be an edge-weighted digraph.Then distTo[] are the shortest path distances from s iff:
• For each vertex v, distTo[v] is the length of some path from s to v.
• For each edge e = v!w, distTo[w] ! distTo[v] + e.weight().
Pf. # [ sufficient ]
• Suppose that s = v0 ! v1 ! v2 ! … ! vk = w is a shortest path from s to w.
• Then,
• Collapsing these inequalities and eliminate distTo[v0] = distTo[s] = 0:
distTo[w] = distTo[vk] ! ek.weight() + ek-1.weight() + … + e1.weight()
• Thus, distTo[w] is the weight of shortest path to w. !
distTo[vk] ! distTo[vk-1] + ek.weight()
distTo[vk-1] ! distTo[vk-2] + ek-1.weight()
...
distTo[v1] ! distTo[v0] + e1.weight()
weight of shortest path from s to wweight of some path from s to w
ei = ith edge on shortest
path from s to w
Proposition. Generic algorithm computes SPT from s.Pf sketch.
• Throughout algorithm, distTo[v] is the length of a simple path from s to v and edgeTo[v] is last edge on path.
• Each successful relaxation decreases distTo[v] for some v.
• The entry distTo[v] can decrease at most a finite number of times. !
22
Generic shortest-paths algorithm
Initialize distTo[s] = 0 and distTo[v] = $ for all other vertices.
Repeat until optimality conditions are satisfied:
- Relax any edge.
Generic algorithm (to compute SPT from s)
assuming SPT exists
Efficient implementations. How to choose which edge to relax?Ex 1. Dijkstra's algorithm (nonnegative weights).Ex 2. Topological sort algorithm (no directed cycles).Ex 3. Bellman-Ford algorithm (no negative cycles).
23
Generic shortest-paths algorithm
Initialize distTo[s] = 0 and distTo[v] = $ for all other vertices.
Repeat until optimality conditions are satisfied:
- Relax any edge.
Generic algorithm (to compute SPT from s)
24
‣ edge-weighted digraph API‣ shortest-paths properties‣ Dijkstra's algorithm‣ edge-weighted DAGs‣ negative weights
25
Edsger W. Dijkstra: select quotes
Edsger W. Dijkstra
Turing award 1972
“ Do only what only you can do. ”
“ In their capacity as a tool, computers will be but a ripple on the surface of our culture. In their capacity as intellectual challenge, they are without precedent in the cultural history of mankind. ”
“ The use of COBOL cripples the mind; its teaching should, therefore, be regarded as a criminal offence. ”
“ It is practically impossible to teach good programming to students that have had a prior exposure to BASIC: as potential programmers they are mentally mutilated beyond hope of regeneration. ”
“ APL is a mistake, carried through to perfection. It is the language of the future for the programming techniques of the past: it creates a new generation of coding bums. ”
26
Edsger W. Dijkstra: select quotes
• Consider vertices in increasing order of distance from s(non-tree vertex with the lowest distTo[] value).
• Add vertex to tree and relax all edges incident from that vertex.
Dijkstra's algorithm
27
5
4
7
13
2
6
v distTo[v] edgeTo[v]0 0.00 -1 2 34 567
An edge-weighted digraph and a shortest path
4->5 0.35 5->4 0.35 4->7 0.37 5->7 0.28 7->5 0.28 5->1 0.32 0->4 0.380->2 0.26 7->3 0.39 1->3 0.29 2->7 0.346->2 0.40 3->6 0.526->0 0.586->4 0.93
0->2 0.262->7 0.347->3 0.393->6 0.52
edge-weighted digraph
shortest path from 0 to 60
Dijkstra’s algorithm visualization
28
Dijkstra’s algorithm visualization
29
• Consider vertices in increasing order of distance from s(non-tree vertex with the lowest distTo[] value).
• Add vertex to tree and relax all edges incident from that vertex.
30
Shortest path trees
50% 100%25%
Proposition. Dijkstra's algorithm computes SPT in any edge-weighted digraph with nonnegative weights.
Pf.
• Each edge e = v!w is relaxed exactly once (when v is relaxed),leaving distTo[w] ! distTo[v] + e.weight().
• Inequality holds until algorithm terminates because:- distTo[w] cannot increase - distTo[v] will not change
• Thus, upon termination, shortest-paths optimality conditions hold. !
Dijkstra's algorithm: correctness proof
31
distTo[] values are monotone decreasing
edge weights are nonnegative and we choose
lowest distTo[] value at each step
32
Dijkstra's algorithm: Java implementation
public class DijkstraSP{ private DirectedEdge[] edgeTo; private double[] distTo; private IndexMinPQ<Double> pq;
public DijkstraSP(EdgeWeightedDigraph G, int s) { edgeTo = new DirectedEdge[G.V()]; distTo = new double[G.V()]; pq = new IndexMinPQ<Double>(G.V());
for (int v = 0; v < G.V(); v++) distTo[v] = Double.POSITIVE_INFINITY; distTo[s] = 0.0;
pq.insert(s, 0.0); while (!pq.isEmpty()) { int v = pq.delMin(); for (DirectedEdge e : G.adj(v)) relax(e); } } }
relax vertices in order
of distance from s
33
Dijkstra's algorithm: Java implementation
private void relax(DirectedEdge e) { int v = e.from(), w = e.to(); if (distTo[w] > distTo[v] + e.weight()) { distTo[w] = distTo[v] + e.weight(); edgeTo[w] = e; if (pq.contains(w)) pq.decreaseKey(w, distTo[w]); else pq.insert (w, distTo[w]); } }
update PQ
34
Dijkstra's algorithm: which priority queue?
Depends on PQ implementation: V insert, V delete-min, E decrease-key.
Bottom line.
• Array implementation optimal for dense graphs.
• Binary heap much faster for sparse graphs.
• d-way heap worth the trouble in performance-critical situations.
• Fibonacci heap best in theory, but not worth implementing.
† amortized
PQ implementation insert delete-min decrease-key total
array 1 V 1 V 2
binary heap log V log V log V E log V
d-way heap
(Johnson 1975)d logd V d logd V logd V E log E/V V
Fibonacci heap
(Fredman-Tarjan 1984)1 † log V † 1 † E + V log V
35
Priority-first search
Insight. Four of our graph-search methods are the same algorithm!
• Maintain a set of explored vertices S.
• Grow S by exploring edges with exactly one endpoint leaving S.
DFS. Take edge from vertex which was discovered most recently.BFS. Take edge from vertex which was discovered least recently.Prim. Take edge of minimum weight.Dijkstra. Take edge to vertex that is closest to S.
Challenge. Express this insight in reusable Java code.
S
e
s
v
w
36
‣ edge-weighted digraph API‣ shortest-paths properties‣ Dijkstra's algorithm‣ edge-weighted DAGs‣ negative weights
Q. Suppose that an edge-weighted digraph has no directed cycles. Is it easier to find shortest paths than in a general digraph?
A. Yes!
37
Acyclic edge-weighted digraphs
5->4 0.35 4->7 0.37 5->7 0.28 5->1 0.32 4->0 0.380->2 0.26 3->7 0.39 1->3 0.29 7->2 0.346->2 0.40 3->6 0.526->0 0.586->4 0.93
source
Topological sort algorithm.
• Consider vertices in topologically order.
• Relax all edges incident from vertex.
38
Shortest paths in edge-weighted DAGs
topological order: 5 1 3 6 4 7 0 2
39
Shortest paths in edge-weighted DAGs
public class AcyclicSP{ private DirectedEdge[] edgeTo; private double[] distTo;
public AcyclicSP(EdgeWeightedDigraph G, int s) { edgeTo = new DirectedEdge[G.V()]; distTo = new double[G.V()];
for (int v = 0; v < G.V(); v++) distTo[v] = Double.POSITIVE_INFINITY; distTo[s] = 0.0;
Topological topological = new Topological(G); for (int v : topological.order()) for (DirectedEdge e : G.adj(v)) relax(e); } }
topological order
Topological sort algorithm.
• Consider vertices in topologically order.
• Relax all edges incident from vertex.
Proposition. Topological sort algorithm computes SPT in any edge-weighted DAG in time proportional to E + V.
Pf.
• Each edge e = v!w is relaxed exactly once (when v is relaxed),leaving distTo[w] ! distTo[v] + e.weight().
• Inequality holds until algorithm terminates because:- distTo[w] cannot increase - distTo[v] will not change
• Thus, upon termination, shortest-paths optimality conditions hold. !
40
Shortest paths in edge-weighted DAGs
distTo[] values are monotone decreasing
because of topological order, no edge pointing to v
will be relaxed after v is relaxed
Formulate as a shortest paths problem in edge-weighted DAGs.
• Negate all weights.
• Find shortest paths.
• Negate weights in result.
Key point. Topological sort algorithm works even with negative edge weights.41
Longest paths in edge-weighted DAGs
equivalent: reverse sense of equality in relax()
5->4 0.35 4->7 0.37 5->7 0.28 5->1 0.32 4->0 0.380->2 0.26 3->7 0.39 1->3 0.29 7->2 0.346->2 0.40 3->6 0.526->0 0.586->4 0.93
5->4 -0.35 4->7 -0.37 5->7 -0.28 5->1 -0.32 4->0 -0.380->2 -0.26 3->7 -0.39 1->3 -0.29 7->2 -0.346->2 -0.40 3->6 -0.526->0 -0.586->4 -0.93
longest paths input shortest paths input
5->4 -0.35 4->7 -0.37 5->7 -0.28 5->1 -0.32 4->0 -0.380->2 -0.26 3->7 -0.39 1->3 -0.29 7->2 -0.346->2 -0.40 3->6 -0.526->0 -0.586->4 -0.93
Longest paths in edge-weighted DAGs: application
Parallel job scheduling. Given a set of jobs with durations and precedence constraints, schedule the jobs (by finding a start time for each) so as to achieve the minimum completion time while respecting the constraints.
42
Parallel job scheduling solution
0
4
3
5
9
7
6 8 2
1
41 700 91 123 173
A job scheduling problem
0 41.0 1 7 9 1 51.0 2 2 50.0 3 36.0 4 38.0 5 45.0 6 21.0 3 8 7 32.0 3 8 8 32.0 2 9 29.0 4 6
job duration must completebefore
CPM. To solve a parallel job-scheduling problem, create acyclic edge-weighted digraph:
• Source and sink vertices.
• Two vertices (begin and end) for each job.
• Three edges for each job.
- begin to end (weighted by duration)
- source to begin (0 weight)- end to sink (0 weight)
Critical path method
43Edge-weighted DAG representation of job scheduling
410 0
511 1
502 2
363 3
384 4
455 5
216 6
327 7
328 8
29 9 9
precedence constraint(zero weight)
job start job finish
duration
zero-weight edge to each
job start
zero-weight edge from each
job finish
A job scheduling problem
0 41.0 1 7 9 1 51.0 2 2 50.0 3 36.0 4 38.0 5 45.0 6 21.0 3 8 7 32.0 3 8 8 32.0 2 9 29.0 4 6
job duration must completebefore
Critical path method
44
CPM. Use longest path from the source to schedule each job.
Longest paths solution to job scheduling example
410 0
511 1
502 2
363 3
384 4
455 5
216 6
327 7
328 8
29 9 9
critical path
duration
Parallel job scheduling solution
0
4
3
5
9
7
6 8 2
1
41 700 91 123 173
Deadlines. Add extra constraints to the parallel job-scheduling problem.Ex. “Job 2 must start no later than 12 time units after job 4 starts.”
Consequences.
• Corresponding shortest-paths problem has cycles (and negative weights).
• Possibility of infeasible problem (negative cycles).
Deep water
45
zero-weight edge from each
job finish
zero-weight edge to each
job start
Edge-weighted digraph representation of parallel job scheduling with deadlines
410 0
511 1
502 2
363 3
38
-12
-70
-80
4 4
455 5
216 6
327 7
328 8
29 9 9
deadline
46
‣ edge-weighted digraph API‣ shortest-paths properties‣ Dijkstra's algorithm‣ edge-weighted DAGs‣ negative weights
Dijkstra. Doesn’t work with negative edge weights.
Re-weighting. Add a constant to every edge weight doesn’t work.
Bad news. Need a different algorithm.47
Shortest paths with negative weights: failed attempts
0
3
1
2
4
2-9
6
0
3
1
11
13
20
15
Dijkstra selects vertex 3 immediately after 0.
But shortest path from 0 to 3 is 0%1%2%3.
Adding 9 to each edge weight changes the
shortest path from 0%1%2%3 to 0%3.
Def. A negative cycle is a directed cycle whose sum of edge weights is negative.
Proposition. A SPT exists iff no negative cycles.
48
Negative cycles
An edge-weighted digraph with a negative cycle
4->5 0.35 5->4 -0.66 4->7 0.37 5->7 0.28 7->5 0.28 5->1 0.32 0->4 0.380->2 0.26 7->3 0.39 1->3 0.29 2->7 0.346->2 0.40 3->6 0.526->0 0.586->4 0.93
digraph
5->4->7->5 negative cycle (-0.66 + 0.37 + 0.28)
0->4->7->5->4->7->5...->1->3->6 shortest path from 0 to 6
assuming all vertices reachable from s
Proposition. Dynamic programming algorithm computes SPT in any edge-weighted digraph with no negative cycles in time proportional to E " V.Pf idea. After phase i, found shortest path containing at most i edges.
for (int i = 1; i <= G.V(); i++) for (int v = 0; v < G.V(); v++) for (DirectedEdge e : G.adj(v)) relax(e);
49
Shortest paths with negative weights: dynamic programming algorithm
phase i (relax each edge)
Initialize distTo[s] = 0 and distTo[v] = # for all other vertices.
Repeat V times: - Relax each edge.
Dynamic programming algorithm
50
Observation. If distTo[v] does not change during phase i,no need to relax any edge incident from v in phase i +1.
FIFO implementation. Maintain queue of vertices whose distTo[] changed.
Overall effect.
• The running time is still proportional to E " V in worst case.
• But much faster than that in practice.
Bellman-Ford algorithm
be careful to keep at most one copy
of each vertex on queue (why?)
51
Bellman-Ford algorithm trace
Trace of the Bellman-Ford algorithm
source
queue vertices foreach phase are in red
recolored edge
red: this pass
black: next pass
edgeTo[] 0 1 2 3 1->3 4 5 6 7
1 3
edgeTo[] 0 1 2 3 4 5 6 3->6 7
3 6
edgeTo[] 0 6->0 1 2 6->2 3 1->3 4 6->4 5 6 3->6 7
6 4 0 2
edgeTo[] 0 6->0 1 2 6->2 3 1->3 4 6->4 5 4->5 6 3->6 7 2->7
4 0 2 7 5
edgeTo[] 0 6->0 1 2 6->2 3 1->3 4 6->4 5 7->5 6 3->6 7 2->7
7 5
edgeTo[] 0 6->0 1 2 6->2 3 1->3 4 6->4 5 7->5 6 3->6 7 2->7
q
Trace of the Bellman-Ford algorithm
source
queue vertices foreach phase are in red
recolored edge
red: this pass
black: next pass
edgeTo[] 0 1 2 3 1->3 4 5 6 7
1 3
edgeTo[] 0 1 2 3 4 5 6 3->6 7
3 6
edgeTo[] 0 6->0 1 2 6->2 3 1->3 4 6->4 5 6 3->6 7
6 4 0 2
edgeTo[] 0 6->0 1 2 6->2 3 1->3 4 6->4 5 4->5 6 3->6 7 2->7
4 0 2 7 5
edgeTo[] 0 6->0 1 2 6->2 3 1->3 4 6->4 5 7->5 6 3->6 7 2->7
7 5
edgeTo[] 0 6->0 1 2 6->2 3 1->3 4 6->4 5 7->5 6 3->6 7 2->7
q
52
Bellman-Ford algorithm
public class BellmanFordSP{ private double[] distTo; private DirectedEdge[] edgeTo; private int[] onQ; private Queue<Integer> queue; public BellmanFordSPT(EdgeWeightedDigraph G, int s) { distTo = new double[G.V()]; edgeTo = new DirectedEdge[G.V()]; onq = new int[G.V()]; queue = new Queue<Integer>();
for (int v = 0; v < V; v++) distTo[v] = Double.POSITIVE_INFINITY; distTo[s] = 0.0;
queue.enqueue(s); while (!queue.isEmpty()) { int v = queue.dequeue(); onQ[v] = false; for (DirectedEdge e : G.adj(v)) relax(e); } }}
queue of vertices whose
distTo[] value changes
private void relax(DirectedEdge e){ int v = e.from(), w = e.to(); if (distTo[w] > distTo[v] + e.weight()) { distTo[w] = distTo[v] + e.weight(); edgeTo[w] = e; if (!onQ[w]) { queue.enqueue(w); onQ[w] = true; } } }
53
Bellman-Ford algorithm visualization
Bellman-Ford (250 vertices)
4 7 10
13 SPT
edges on queue in red phases
54
Single source shortest-paths implementation: cost summary
Remark 1. Directed cycles make the problem harder.Remark 2. Negative weights make the problem harder.Remark 3. Negative cycles makes the problem intractable.
algorithm restriction typical case worst case extra space
topological sortno directed
cyclesE + V E + V V
Dijkstra
(binary heap)
no negative
weightsE log V E log V V
dynamic programming
no negative
E V E V V
Bellman-Ford
cycles
E + V E V V
55
Finding a negative cycle
Negative cycle. Add two method to the API for SP.
boolean hasNegativeCycle() is there a negative cycle?
Iterable <DirectedEdge> negativeCycle() negative cycle reachable from s
An edge-weighted digraph with a negative cycle
4->5 0.35 5->4 -0.66 4->7 0.37 5->7 0.28 7->5 0.28 5->1 0.32 0->4 0.380->2 0.26 7->3 0.39 1->3 0.29 2->7 0.346->2 0.40 3->6 0.526->0 0.586->4 0.93
digraph
5->4->7->5 negative cycle (-0.66 + 0.37 + 0.28)
0->4->7->5->4->7->5...->1->3->6 shortest path from 0 to 6
56
Finding a negative cycle
Observation. If there is a negative cycle, Bellman-Ford gets stuck in loop, updating distTo[] and edgeTo[] entries of vertices in the cycle.
Proposition. If any vertex v is updated in phase V, there exists a negative cycle (and can trace back edgeTo[v] entries to find it).
In practice. Check for negative cycles more frequently.
edgeTo[v]
s 3
v
2 6
1
4
5
Problem. Given table of exchange rates, is there an arbitrage opportunity?
Ex. $1,000 # 741 Euros # 1,012.206 Canadian dollars # $1,007.14497.
57
Negative cycle application: arbitrage detection
1000 × 0.741 × 1.366 × 0.995 = 1007.14497
USD EUR GBP CHF CAD
USD
EUR
GBP
CHF
CAD
1 0.741 0.657 1.061 1.011
1.350 1 0.888 1.433 1.366
1.521 1.126 1 1.614 1.538
0.943 0.698 0.620 1 0.953
0.995 0.732 0.650 1.049 1
Currency exchange graph.
• Vertex = currency.
• Edge = transaction, with weight equal to exchange rate.
• Find a directed cycle whose product of edge weights is > 1.
Challenge. Express as a negative cycle detection problem.58
Negative cycle application: arbitrage detection
An arbitrage opportunity
USD
0.741 1.
350
0.888
1.126
0.620
1.614
1.049
0.953
1.0110.995
0.650
1.538
0.732
1.366
0.657
1.5211.061
0.943
1.433
0.698
EUR
GBP
CHFCAD
0.741 * 1.366 * .995 = 1.00714497
Model as a negative cycle detection problem by taking logs.
• Let weight of edge v!w be - ln (exchange rate from currency v to w).
• Multiplication turns to addition; > 1 turns to < 0.
• Find a directed cycle whose sum of edge weights is < 0 (negative cycle).
Remark. Fastest algorithm is extraordinarily valuable!
A negative cycle that representsan arbitrage opportunity
USD
.2998 -.
3001
.1188
-.1187
.4780
-.4787
-.0478
.0481
-.0109.0050
.4308
-.4305
.3120
-.3119
.4201
-.4914-.0592
.0587
-.3598.3595
EUR
GBP
CHFCAD
replace eachweight w
with !ln(w)
.2998 - .3119 + .0050 = -.0071
-ln(.741) -ln(1.366) -ln(.995)
59
Negative cycle application: arbitrage detection Shortest paths summary
Dijkstra’s algorithm.
• Nearly linear-time when weights are nonnegative.
• Generalization encompasses DFS, BFS, and Prim.
Acyclic edge-weighted digraphs.
• Arise in applications.
• Faster than Dijkstra’s algorithm.
• Negative weights are no problem.
Negative weights and negative cycles.
• Arise in applications.
• If no negative cycles, can find shortest paths via Bellman-Ford.
• If negative cycles, can find one via Bellman-Ford.
Shortest-paths is a broadly useful problem-solving model.
60
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