Shortest Paths

1 / 28

# Shortest Paths - PowerPoint PPT Presentation

Shortest Paths. Definitions Single Source Algorithms Bellman Ford DAG shortest path algorithm Dijkstra All Pairs Algorithms Using Single Source Algorithms Matrix multiplication Floyd-Warshall Both of above use adjacency matrix representation and dynamic programming Johnson’s algorithm

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## Shortest Paths

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Shortest Paths
• Definitions
• Single Source Algorithms
• Bellman Ford
• DAG shortest path algorithm
• Dijkstra
• All Pairs Algorithms
• Using Single Source Algorithms
• Matrix multiplication
• Floyd-Warshall
• Both of above use adjacency matrix representation and dynamic programming
• Johnson’s algorithm
Single Source Definition
• Input
• Weighted, connected directed graph G=(V,E)
• Weight (length) function w on each edge e in E
• Source node s in V
• Compute a shortest path from s to all nodes in V
All Pairs Definition
• Input
• Weighted, connected directed graph G=(V,E)
• Weight (length) function w on each edge e in E
• We will typically assume w is represented as a matrix
• Compute a shortest path from all nodes in V to all nodes in V
• If edges are not weighted, then BFS works for single source problem
• Optimal substructure
• A shortest path between s and t contains other shortest paths within it
• No known algorithm is better at finding a shortest path from s to a specific destination node t in G than finding the shortest path from s to all nodes in V
Negative weight edges
• Negative weight edges can be allowed as long as there are no negative weight cycles
• If there are negative weight cycles, then there cannot be a shortest path from s to any node t (why?)
• If we disallow negative weight cycles, then there always is a shortest path that contains no cycles
Relaxation technique
• For each vertex v, we maintain an upper bound d[v] on the length of shortest path from s to v
• d[v] initialized to infinity
• Relaxing an edge (u,v)
• Can we shorten the path to v by going through u?
• If d[v] > d[u] + w(u,v), d[v] = d[u] + w(u,v)
• This can be done in O(1) time
Bellman-Ford (G, w, s)

Initialize-Single-Source(G,s)

for (i=1 to V-1)

for each edge (u,v) in E

relax(u,v);

for each edge (u,v) in E

if d[v] > d[u] + w(u,v)

return NEGATIVE WEIGHT CYCLE

-5

-5

3

3

5

5

-1

7

s

s

1

1

3

3

2

2

Bellman-Ford Algorithm

G1

G2

Running Time
• for (i=1 to V-1)
• for each edge (u,v) in E
• relax(u,v);
• The above takes (V-1)O(E) = O(VE) time
• for each edge (u,v) in E
• if d[v] > d[u] + w(u,v)
• return NEGATIVE WEIGHT CYCLE
• The above takes O(E) time
Proof of Correctness
• Theorem: If there is a shortest path from s to any node v, then d[v] will have this weight at end of Bellman-Ford algorithm
• Theorem restated:
• Define links[v] to be the minimum number of edges on a shortest path from s to v
• After i iterations of the Bellman-Ford for loop, nodes v with links[v] ≤ i will have their d[v] value set correctly
• Prove this by induction on links[v]
• Base case: links[v] = 0 which means v is s
• d[s] is initialized to 0 which is correct unless there is a negative weight cycle from s to s in which case there is no shortest path from s to s.
• Induction hypothesis: After k iterations for k ≥ 0, d[v] must be correct if links[v] ≤ k
• Inductive step: Show after k+1 iterations, d[v] must be correct if links[v] ≤ k+1
• If links[v] ≤ k, then by IH, d[v] is correctly set after kth iteration
• If links[v] = k+1, let p = (e1, e2, …, ek+1) = (v0, v1, v2, …, vk+1) be a shortest path from s to v
• s = v0, v = vk+1, ei = (vi-1, vi)
• By the inductive hypothesis, d[vk] will be correctly set after the kth iteration.
• During the k+1st iteration, we relax all edges including edge ek+1 = (vk,v)
• Thus, at end of the k+1st iteration, d[v] will be correct
Negative weight cycle
• for each edge (u,v) in E
• if d[v] > d[u] + w(u,v)
• return NEGATIVE WEIGHT CYCLE
• If no neg weight cycle, d[v] ≤d[u] + w(u,v) for all (u,v)
• If there is a negative weight cycle C, for some edge (u,v) on C, it must be the case that d[v] > d[u] + w(u,v).
• Suppose this is not true for some neg. weight cycle C
• sum these (d[v] ≤ d[u] + w(u,v)) all the way around C
• We end up with Σv in C d[v] ≤ (Σu in C d[u]) + weight(C)
• This is impossible unless weight(C) = 0
• But weight(C) is negative, so this cannot happen
• Thus for some (u,v) on C, d[v] > d[u] + w(u,v)
DAG-SP (G, w, s)

Initialize-Single-Source(G,s)

Topologically sort vertices in G

for each vertex u, taken in sorted order

for each edge (u,v) in E

relax(u,v);

DAG shortest path algorithm

5

3

4

-4

s

5

1

3

8

Running time Improvement
• O(V+E) for the topological sorting
• We only do 1 relaxation for each edge: O(E) time
• for each vertex u, taken in sorted order
• for each edge (u,v) in E
• relax(u,v);
• Overall running time: O(V+E)
Proof of Correctness
• If there is a shortest path from s to any node v, then d[v] will have this weight at end
• Let p = (e1, e2, …, ek) = (v1, v2, v3, …, vk+1) be a shortest path from s to v
• s = v1, v = vk+1, ei = (vi, vi+1)
• Since we sort edges in topological order, we will process node vi (and edge ei) before processing later edges in the path.
Dijkstra (G, w, s)

/* Assumption: all edge weights non-negative */

Initialize-Single-Source(G,s)

Completed = {};

ToBeCompleted = V;

While ToBeCompleted is not empty

u =EXTRACT-MIN(ToBeCompleted);

Completed += {u};

for each edge (u,v) relax(u,v);

4

15

5

7

s

1

3

2

Dijkstra’s Algorithm
Running Time Analysis
• While ToBeCompleted is not empty
• u =EXTRACT-MIN(ToBeCompleted);
• Completed += {u};
• for each edge (u,v) relax(u,v);
• Each edge relaxed at most once: O(E)
• Need to decrease-key potentially once per edge
• Need to extract-min once per node
• The node’s d[v] is then complete
Running Time Analysis cont’d
• Priority Queue operations
• O(E) decrease key operations
• O(V) extract-min operations
• Three implementations of priority queues
• Array: O(V2) time
• decrease-key is O(1) and extract-min is O(V)
• Binary heap: O(E log V) time assuming E ≥ V
• decrease-key and extract-min are O(log V)
• Fibonacci heap: O(V log V + E) time
• decrease-key is O(1) amortized and extract-min is O(log V)
Compare Dijkstra’s algorithm to Prim’s algorithm for MST
• Dijsktra
• Priority Queue operations
• O(E) decrease key operations
• O(V) extract-min operations
• Prim
• Priority Queue operations
• O(E) decrease-key operations
• O(V) extract-min operations
• Is this a coincidence or is there something more here?

s

u

v

Only nodes in S

Proof of Correctness
• Assume that Dijkstra’s algorithm fails to compute length of all shortest paths from s
• Let v be the first node whose shortest path length is computed incorrectly
• Let S be the set of nodes whose shortest paths were computed correctly by Dijkstra prior to adding v to the processed set of nodes.
• Dijkstra’s algorithm has used the shortest path from s to v using only nodes in S when it added v to S.
• The shortest path to v must include one node not in S
• Let u be the first such node.

s

u

v

Only nodes in S

Proof of Correctness
• The length of the shortest path to u must be at least that of the length of the path computed to v.
• Why?
• The length of the path from u to v must be < 0.
• Why?
• No path can have negative length since all edge weights are non-negative, and thus we have a contradiction.
Computing paths (not just distance)
• Maintain for each node v a predecessor node p(v)
• p(v) is initialized to be null
• Whenever an edge (u,v) is relaxed such that d(v) improves, then p(v) can be set to be u
• Paths can be generated from this data structure
All pairs algorithms using single source algorithms
• Call a single source algorithm from each vertex s in V
• O(V X) where X is the running time of the given algorithm
• Dijkstra linear array: O(V3)
• Dijkstra binary heap: O(VE log V)
• Dijkstra Fibonacci heap: O(V2 log V + VE)
• Bellman-Ford: O(V2 E) (negative weight edges)
• Matrix-multiplication based algorithm
• Let Lm(i,j) denote the length of the shortest path from node i to node j using at most m edges
• What is our desired result in terms of Lm(i,j)?
• What is a recurrence relation for Lm(i,j)?
• Floyd-Warshall algorithm
• Let Lk(i,j) denote the length of the shortest path from node i to node j using only nodes within {1, …, k} as internal nodes.
• What is our desired result in terms of Lk(i,j)?
• What is a recurrence relation for Lk(i,j)?

Shortest path using at most 2m edges

Try all possible nodes k

i

k

j

Shortest path using at most m edges

Shortest path using at most m edges

Shortest path using nodes 1 through k

i

j

Shortest path using nodes 1 through k-1

Shortest path using nodes 1 through k-1

Conceptual pictures

Shortest path using nodes 1 through k-1

k

OR

Running Times
• Matrix-multiplication based algorithm
• O(V3 log V)
• log V executions of “matrix-matrix” multiplication
• Not quite matrix-matrix multiplication but same running time
• Floyd-Warshall algorithm
• O(V3)
• V iterations of an O(V2) update loop
• The constant is very small, so this is a “fast” O(V3)
Johnson’s Algorithm
• Key ideas
• Reweight edge weights to eliminate negative weight edges AND preserve shortest paths
• Use Bellman-Ford and Dijkstra’s algorithms as subroutines
• Running time: O(V2 log V + VE)
• Better than earlier algorithms for sparse graphs
Reweighting
• Original edge weight is w(u,v)
• New edge weight:
• w’(u,v) = w(u,v) + h(u) – h(v)
• h(v) is a function mapping vertices to real numbers
• Key observation:
• Let p be any path from node u to node v
• w’(p) = w(p) + h(u) – h(v)

-5

3

4

7

1

3

-7

Computing vertex weights h(v)
• Create a new graph G’ = (V’, E’) by
• adding a new vertex s to V
• adding edges (s,v) for all v in V with w(s,v) = 0
• Set h(v) to be the length of the shortest path from this new node s to node v
• This is well-defined if G’ does not contain negative weight cycles
• Note that h(v) ≤ h(u) + w(u,v) for all (u,v) in E’
• Thus, w’(u,v) = w(u,v) + h(u) – h(v) ≥ 0
Algorithm implementation
• Run Bellman-Ford on G’ from new node s
• If no negative weight cycle, then use h(v) values from Bellman-Ford
• Now compute w’(u,v) for each edge (u,v) in E
• Now run Dijkstra’s algorithm using w’
• Use each node as source node
• Modify d[u,v] at end by adding h(v) and subtracting h(u) to get true path weight
• Running time:
• O(VE) [from one run of Bellman-Ford] +
• O(V2 log V + VE) [from V runs of Dijkstra]