Shortest Paths

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 weight of shortest path from s to v

• d[v] initialized to infinity• Relaxing an edge (u,v)

– Can we improve the shortest 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

• 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, …, v,+1) be a shortest path from s to v– s = v1, v = vk+1, ei = (vi, vi+1)

• At beginning of ith iteration, during which we relax edge ei, d[vi] must be correct, so at end of ith iteration, d[vi+1] will e correct– Iteration is the full sweep of relaxing all edges– Proof by induction

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 sum(d[v]) ≤ sum(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, …, v,+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

Running Time Analysis cont’d

• O(E) edge relaxations• 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 time 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 k j

Shortest path using at most 2m edges

Shortest path using at most m edges Shortest path using at most m edges

Try all possible nodes k

i k 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]