CS 8833 Algorithms Algorithms Shortest Path Problems
CS 8833 Algorithms
Algorithms
Shortest Path Problems
CS 8833 Algorithms
G = (V, E) weighted directed graphw: ER weight function
Weight of a path p = <v0, v1,. . ., vn>
Shortest path weight from u to v
Shortest path from u to v: Any path from u to v with w(p) = (u,v)
[v] predecessor of v on a path
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CS 8833 Algorithms
CS 8833 Algorithms
Variants Single-source shortest paths:
find shortest paths from source vertex to every other vertex
Single-destination shortest paths:find shortest paths to a destination from every vertex
Single-pair shortest-pathfind shortest path from u to v
All pairs shortest paths
CS 8833 Algorithms
Lemma 25.1 Subpaths of shortest paths are shortest
paths.Given G=(G,E) w: E R
Let p = p = <v1, v2,. . ., vk> be a shortest path from v1 to vk
For any i,j such that 1 i j k, let pij be a subpath from vi to vj.. Then pij is a shortest path from vi to vj.
CS 8833 Algorithms
1 i j k
p
CS 8833 Algorithms
Corollary 25.2Let G = (V,E) w: E R
Suppose shortest path p from a source s to vertex v can be decomposed into
p’
s u v
for vertex u and path p’.
Then weight of the shortest path from s to v is
(s,v) = (s,u) + w(u,v)
CS 8833 Algorithms
Lemma 25.3
Let G = (V,E) w: E R
Source vertex s
For all edges (u,v)E
(s,v) (s,u) + w(u,v)
CS 8833 Algorithms
s v
u1
u2
u4
u3
un
CS 8833 Algorithms
Relaxation Shortest path estimate
d[v] is an attribute of each vertex which is an upper bound on the weight of the shortest path from s to v
Relaxation is the process of incrementally reducing d[v] until it is an exact weight of the shortest path from s to v
CS 8833 Algorithms
INITIALIZE-SINGLE-SOURCE(G, s)
1. for each vertex v V(G)
2. do d[v]
3. [v] nil
4. d[s] 0
CS 8833 Algorithms
CS 8833 Algorithms
Relaxing an Edge (u,v) Question: Can we improve the shortest
path to v found so far by going through u?
If yes, update d[v] and [v]
CS 8833 Algorithms
RELAX(u,v,w)
1. if d[v] > d[u] + w(u,v)
2. then d[v] d[u] + w(u,v)
3. [v] u
CS 8833 Algorithms
EXAMPLE 1
s
u
v
s
u
v
Relax
CS 8833 Algorithms
EXAMPLE 2
s
u
v
s
u
v
Relax
CS 8833 Algorithms
Dijkstra’s Algorithm Problem:
– Solve the single source shortest-path problem on a weighted, directed graph G(V,E) for the cases in which edge weights are non-negative
CS 8833 Algorithms
Dijkstra’s Algorithm Approach
– maintain a set S of vertices whose final shortest path weights from the source s have been determined.
– repeat» select vertex from V-S with the minimum
shortest path estimate» insert u in S» relax all edges leaving u
CS 8833 Algorithms
DIJKSTRA(G,w,s)
1. INITIALIZE-SINGLE-SOURCE(G,s)
2. S
3. Q V[G]
4. while Q
5. do u EXTRACT-MIN(Q)
6. S S {u}
7. for each vertex v Adj[u]
8. do RELAX(u,v,w)
CS 8833 Algorithms
CS 8833 Algorithms
Analysis of Dijkstra’s Algorithm
Suppose priority Q is:– an ordered (by d) linked list
» Building the Q O(V lg V)» Each EXTRACT-MIN O(V)» This is done V times O(V2)» Each edge is relaxed one time O(E)» Total time O(V2 + E) = O(V2)
CS 8833 Algorithms
Analysis of Dijkstra’s Algorithm
Suppose priority Q is:– a binary heap
» BUILD-HEAP O(V)» Each EXTRACT-MIN O(lg V)» This is done V times O(V lg
V)» Each edge is relaxation O(lg V)» Each edge relaxed one time O(E lg
V)» Total time O(V lg V + E lg V))
CS 8833 Algorithms
Properties of Relaxation Lemma 25.4
G=(V,E) w: E R (u,v) E
After relaxing edge (u,v) by executing RELAX(u,v,w) we have
d[v] d[u] + w(u,v)
CS 8833 Algorithms
Lemma 25.5– Given:
G=(V,E) w: E R source s V
Graph initialized by
INITIALIZE-SINGLE-SOURCE(G,s)– then
d[v] (s,v) for all v V
and this invariant is maintained over all relaxation steps
Once d[v] achieves a lower bound (s,v), it never changes
CS 8833 Algorithms
Corollary 25.6– Given:
G=(V,E) w: E R source s V
No path connects s to given v– then
after initialization d[v] (s,v)
and this inequality is maintained over all relaxation steps.
CS 8833 Algorithms
Lemma 25.7– Given:
G=(V,E) w: E R source s V
Let s - - u v be the shortest path in G for all vertices u and v.
Suppose G initialized by INITIALIZE-SINGLE-SOURCE is followed by a sequence of relaxations including RELAX(u,v,w)
– Then d[u] = (s,u) prior to call implies that d[u] = (s,u) after the call
CS 8833 Algorithms
Bottom Line Therefore, relaxation causes the
shortest path estimates to descend monotonically toward the actual shortest-path weights.
CS 8833 Algorithms
Shortest-Paths Tree of G(V,E) The shortest-paths tree at S of G(V,E)
is a directed subgraph G’-(V’,E’), where V’ V, E’E, such that– V’ is the set of vertices reachable from S in
G– G’ forms a rooted tree with root s, and– for all v V’, the unique simple path from s
to v in G’ is a shortest path from s to v in G
CS 8833 Algorithms
Goal We want to show that successive
relaxations will yield a shortest-path tree
CS 8833 Algorithms
Lemma 25.8– Given:
G=(V,E) w: E R source s V
Assume that G contains no negative-weight cycles reachable from s.
– Then after the graph is initialized with INITIALIZE-SINGLE-SOURCE
• the predecessor subgraph G forms a rooted tree with root s, and
• any sequence of relaxation steps on edges in G maintains this property as an invariant.
CS 8833 Algorithms
Algorithms
Bellman-Ford Algorithm
Directed-Acyclic Graphs
All Pairs-Shortest Path Algorithm
CS 8833 Algorithms
Why does Dijkstra’s greedy algorithm work?
Because we know that when we add a node u to the set S, the value d is the length of the shortest path from s to u.
But, this only works if the edges of the graph are nonnegative.
CS 8833 Algorithms
a
b c
7 10
-4
Would Dikjstra’s Algorithm work with this graph?
CS 8833 Algorithms
a
b c
7 6
-14
What is the length of the shortest path from a to c in this graph?
CS 8833 Algorithms
Bellman-Ford Algorithm The Bellman-Ford algorithm can be
used to solve the general single source shortest path problem
Negative weights are allowed The algorithm detects negative cycles
and returns false if the graph contains one.
CS 8833 Algorithms
BELLMAN-FORD(G,w,s)
1 INITIALIZE-SINGLE-SOURCE(G,s)
2 for i 1 to |V[G]| -1
3 do for each edge (u,v) E[G]
4 do RELAX(u,v,w)
5 for each edge (u,v) E[G]
6 do if d[v] > d[u] + w(u,v)
7 then return false
8 return true
CS 8833 Algorithms
When there are no cycles of negative length, what is maximum number of edges in a shortest path when the graph has |V[G]| vertices and |E[G]| edges?
CS 8833 Algorithms
Dynamic Programming Formulation
The following recurrence shows how the Bellman Ford algorithm computes the d values for paths of length k.
)}},(][{min ],[min{][ 1i
1 uiwidistudud kkk
CS 8833 Algorithms
a
b c
7 10
-4
Processing order
1 2 3
(a,c)
(b,a)
(c,b)
(c,d)
(d,b)
d
-59
CS 8833 Algorithms
Complexity of the Bellman Ford Algorithm
Time complexity:
Performance can be improved by– adding a test to the loop to see if any d values
were updated on the previous iteration, or– maintain a queue of vertices whose d value
changed on the previous iteration-only process these on the next iteration
CS 8833 Algorithms
Single-source shortest paths in directed acyclic graphs
Topological sorting is the key to efficient algorithms for many DAG applications.
A topological sort of a DAG is a linear ordering of all of its vertices such that if G contains an edge (u,v), then u appears before v in the ordering.
CS 8833 Algorithms
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CS 8833 Algorithms
DAG-SHORTEST-PATHS(G,w,s)
1 topologically sort the vertices of G
2 INITIALIZE-SINGLE-SOURCE(G,s)
3 for each vertex u taken in topological order
4 do for each vertex v Adj[u]
5 do RELAX(u,v,w)
CS 8833 Algorithms
Topological Sorting
TOPOLOGICAL-SORT(G)
1 call DFS(G) to compute finishing times f[v] for each vertex v
2 as each vertex is finished, insert it onto the front of a linked list
3 return the linked list of vertices
CS 8833 Algorithms
Depth-first search Goal: search all edges in the graph one
time Strategy: Search deeper in the graph
whenever possible Edges are explored out of the most
recently discovered vertex v that still has unexplored edges leaving it.
Backtrack when a dead end is encountered
CS 8833 Algorithms
Predecessor subgraph The predecessor subgraph of a depth
first search– forms a depth-first forest– composed of depth-first trees
The edges in E are called tree edges
CS 8833 Algorithms
Vertex coloring scheme All vertices are initially white A vertex is colored gray when it is
discovered A vertex is colored black when it is
finished (all vertices adjacent to the vertex have been examined completely)
CS 8833 Algorithms
Time Stamps Each vertex v has two time-stamps
– d[v] records when v is first discovered (and grayed)
– f[v] records when the search finishes examining its adjacency list (and is blackened)
For every vertex u– d[u] < f[u]
CS 8833 Algorithms
Color and Time Stamp Summary
Vertex u is – white before d[u]– gray between d[u] and f[u]– black after f[u]
Time is a global variable in the pseudocode
CS 8833 Algorithms
DFS(G)
1 for each vertex u V[G]
2 do color[u] white
3 [u] nil
4 time 0
5 do for each vertex u Adj[u]
6 do if color[u] = white
7 then DFS-VISIT(u)
CS 8833 Algorithms
DFS-VISIT(u)
1 color[u] gray
2 d[u] time time time +1
3 for each vertex v Adj[u]
4 do if color[v] = white
5 then [v] u
6 DFS-VISIT(v)
7 color[u] black
8 f[u] time time time +1
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Running time of DFS lines 1-3 of DFS lines 5-7 of DFS lines 2-6 of DFS-VISIT
CS 8833 Algorithms
Running Time of Topological Sort
DFS Insertion in linked list
CS 8833 Algorithms
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CS 8833 Algorithms
Running Time for DAG-SHORTEST-PATHS
Topological sort Initialize-single source 3-5 each edge examined one time