shortest paths definitions single source algorithms –bellman ford –dag shortest path algorithm...
Post on 20-Dec-2015
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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• Uses adjacency list representation
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
• Task– 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
• Task– Compute a shortest path from all nodes in V to
all nodes in V
Comments
• 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 Algorithm
• 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
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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)– In order for links[v] = k+1, then links[vk] = k.– 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 e k+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 shortest path algorithm
• 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);
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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’s Algorithm
• 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);
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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?
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 vOnly 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.
s u vOnly nodes in S
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)
Two adjacency matrix based algorithms
• 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)?
Conceptual pictures
i j
Shortest path using at most 2m edges
Shortest path using at most m edges Shortest path using at most m edges
kTry all possible nodes k
ki j
Shortest path using nodes 1 through k
Shortest path using nodes 1 through k-1
Shortest path using nodes 1 through k-1
Shortest path using nodes 1 through k-1
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)
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
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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]