Lecture 16. Shortest Path Algorithms The single-source shortest path problem is the following: given...
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Transcript of Lecture 16. Shortest Path Algorithms The single-source shortest path problem is the following: given...
Lecture 16. Shortest Path Algorithms The single-source shortest path problem is the following: given a
source vertex s, and a sink vertex v, we'd like to find the shortest path from s to v. Here shortest path means a sequence of directed edges from s to v with the smallest total weight.
There are some subtleties here. Should we allow negative edges? Of course, there are no negative distances; nevertheless, there are actually some cases where negative edges make logical sense. But then there may not be a shortest path, because if there is a cycle with negative weight, we could simply go around that cycle as many times as we want and reduce the cost of the path as much as we like. To avoid this, we might want to detect negative cycles. This can be done in the algorithm itself.
Non-negative cycles aren't helpful, either. Suppose our shortest path contains a cycle of non-negative weight. Then by cutting it out we get a path with the same weight or less, so we might as well cut it out.
Weighted Graphs In a weighted graph, each edge has an associated numerical value,
called the weight of the edge Edge weights may represent distances, costs, etc. Example:
In a flight route graph, the weight of an edge represents the distance in miles between the endpoint airports
ORD PVD
MIADFW
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Shortest Path Problem Given a weighted graph and two vertices u and v, we want to find a path
of minimum total weight between u and v. Length of a path is the sum of the weights of its edges.
Example: Shortest path between Providence and Honolulu
Applications Internet packet routing Flight reservations Driving directions
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Shortest Path PropertiesProperty 1:
A subpath of a shortest path is itself a shortest path
Property 2:There is a tree of shortest paths from a start vertex to all the other vertices
Example:Tree of shortest paths from Providence
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Dijkstra’s Algorithm The distance of a vertex v
from a vertex s is the length of a shortest path between s and v
Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex s
Assumptions: the graph is connected the edges are
undirected the edge weights are
nonnegative
We grow a “cloud” of vertices, beginning with s and eventually covering all the vertices
We store with each vertex v a label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices
At each step We add to the cloud the vertex
u outside the cloud with the smallest distance label, d(u)
We update the labels of the vertices adjacent to u
Edge Relaxation Consider an edge e (u,z) such
that u is the vertex most recently
added to the cloud z is not in the cloud
The relaxation of edge e updates distance d(z) as follows:
d(z) min{d(z),d(u) weight(e)}
d(z) 75
d(u) 5010
zsu
d(z) 60
d(u) 5010
zsu
e
e
Example
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Dijkstra’s Algorithm A priority queue stores the
vertices outside the cloud Key: distance Element: vertex
Locator-based methods insert(k,e) returns a
locator replaceKey(l,k) changes
the key (distance) of an item
We store two labels with each vertex: Distance (d(v) label) locator in priority queue
Algorithm DijkstraDistances(G, s)Q new heap-based priority queuefor all v G.vertices()
if v ssetDistance(v, 0)
else setDistance(v, )
l Q.insert(getDistance(v), v)setLocator(v,l)
while Q.isEmpty()u Q.removeMin() for all e G.incidentEdges(u)
{ relax edge e }z G.opposite(u,e)r getDistance(u) weight(e)if r getDistance(z)
setDistance(z,r) Q.replaceKey(getLocator(z),r)
Analysis Graph operations
Method incidentEdges is called once for each vertex Label operations
We set/get the distance and locator labels of vertex z O(deg(z)) times Setting/getting a label takes O(1) time
Priority queue operations Each vertex is inserted once into and removed once from the priority
queue, where each insertion or removal takes O(log n) time The key of a vertex in the priority queue is modified at most deg(w)
times, where each key change takes O(log n) time Dijkstra’s algorithm runs in O((n m) log n) time provided the graph is
represented by the adjacency list structure Recall that v deg(v) 2m
The running time can also be expressed as O(m log n) since the graph is connected
Extension Using the template
method pattern, we can extend Dijkstra’s algorithm to return a tree of shortest paths from the start vertex to all other vertices
We store with each vertex a third label: parent edge in the
shortest path tree In the edge relaxation
step, we update the parent label
Algorithm DijkstraShortestPathsTree(G, s)
…
for all v G.vertices()…
setParent(v, )…
for all e G.incidentEdges(u){ relax edge e }z G.opposite(u,e)r getDistance(u) weight(e)if r getDistance(z)
setDistance(z,r)setParent(z,e) Q.replaceKey(getLocator(z),r)
Why Dijkstra’s Algorithm Works Dijkstra’s algorithm is based on the greedy method. It
adds vertices by increasing distance.
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Suppose it didn’t find all shortest distances. Let F be the first wrong vertex the algorithm processed.
When the previous node, D, on the true shortest path was considered, its distance was correct.
But the edge (D,F) was relaxed at that time!
Thus, so long as d(F)>d(D), F’s distance cannot be wrong. That is, there is no wrong vertex.
Why It Doesn’t Work for Negative-Weight Edges
If a node with a negative incident edge were to be added late to the cloud, it could mess up distances for vertices already in the cloud.
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Dijkstra’s algorithm is based on the greedy method. It adds vertices by increasing distance.
C’s true distance is 1, but it is already in the
cloud with d(C)=5!
Bellman-Ford Algorithm Works even with negative-
weight edges Must assume directed edges
(for otherwise we would have negative-weight cycles)
Iteration i finds all shortest paths of length i .
Running time: O(nm). Can be extended to detect a
negative-weight cycle if it exists.
Algorithm BellmanFord(G, s)for all v G.vertices()
if v ssetDistance(v, 0)
else setDistance(v, )
for i 1 to n-1 do (*)for each directed edge e = u z
{ relax edge e }r getDistance(u) weight(e)if r getDistance(z)
setDistance(z,r)
Correctness
Lemma. After i repetitions of the "for" loop in BELLMAN-FORD, if there is a path from s to u with at most i edges, then d[u] is at most the length of the shortest path from s to u with at most i edges.
Proof. By induction on i. The base case is i=0. Trivial.
For the induction step, consider the shortest path from s to u with at most i edges. Let v be the last vertex before u on this path. Then the part of the path from s to v is the shortest path from s to v with at most i-1 edges. By the inductive hypothesis, d[v] after i-1 executions of the (*) "for" loop is at most the length of this path. Therefore, d[v] + w(u,v) is at most the length of the path from s to u, via v (as i-1st node). In the i'th iteration, d[u] gets compared with d[v] + w(u,v), and is set equal to it if d[v] + w(u,v) is smaller.
Therefore, after i iterations of the (*) "for" loop, d[u] is at most the length of the shortest path from s to u that uses at most i edges. So the lemma holds.
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Bellman-Ford Example
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Nodes are labeled with their d(v) values
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DAG-based Algorithm
Works even with negative-weight edges
Uses topological order Doesn’t use any fancy data
structures Is much faster than
Dijkstra’s algorithm Running time: O(n+m).
Algorithm DagDistances(G, s)for all v G.vertices()
if v ssetDistance(v, 0)
else setDistance(v, )
Perform a topological sort of the verticesfor u 1 to n do {in topological order}
for each e G.outEdges(u){ relax edge e }z G.opposite(u,e)r getDistance(u) weight(e)if r getDistance(z)
setDistance(z,r)
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DAG Example
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Nodes are labeled with their d(v) values
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(two steps)
All-Pairs Shortest Paths: Dynamic Programming Find the distance between
every pair of vertices in a weighted directed graph G.
We can make n calls to Dijkstra’s algorithm (if no negative edges), which takes O(nmlog n) time.
Likewise, n calls to Bellman-Ford would take O(n2m) time.
We can achieve O(n3) time using dynamic programming (Floyd-Warshall algorithm).
Algorithm AllPair(G) {assumes vertices 1,…,n} for all vertex pairs (i,j)
if i jD0[i,i] 0
else if (i,j) is an edge in GD0[i,j] weight of edge (i,j)
elseD0[i,j] +
for k 1 to n do for i 1 to n do for j 1 to n do
Dk[i,j] min{Dk-1[i,j], Dk-1[i,k]+Dk-1[k,j]} return Dn
k
j
i
Uses only verticesnumbered 1,…,k-1 Uses only vertices
numbered 1,…,k-1
Uses only vertices numbered 1,…,k(compute weight of this edge)