1 week 9-10 graphs ii unweighted shortest paths dijkstra’s algorithm graphs with negative costs...

1

WEEK 9-10 Graphs II

Unweighted Shortest PathsDijkstra’s Algorithm

Graphs with negative costsAcyclic Graphs

Izmir University of Economics

2

Shortest-Path Algorithms


• The input is a weighted graph: associated with each edge (vi, vj) is a cost ci,j.

• The cost of a path v1v2...vN is ∑ci,i+1 for i in [1..N-1] weighted path length

• The unweighted path length is merely the number of edges on the path, namely, N-1.

3

Shortest-Path Algorithms


Single-Source Shortest-Path Problem:

Given as input a weighted graph G=(V, E), and a distinguished vertex, s, find the shortest weighted path from s to every other vertex in G.

Negative Cost Cycles

Izmir University of Economics 4

• In the graph to the left, the shortest path from v1 to v6 has a cost of 6 and the path itself is v1v4v7v6. The shortest unweighted path has 2 edges.

• In the graph to the right, we have a negative cost. The path from v5 to v4 has cost 1, but a shorter path exists by following the loop v5v4v2v5v4 which has cost -5. This path is still not the shortest, because we could stay in the loop arbitrarily long.

Shortest Path Lenght: ProblemsWe will examine 4 algorithms to solve four versions of

the problem

1.Unweighted shortest path O(|E|+|V|)

2.Weighted shortest path without negative edges O(|E|log|V|) using queues

3.Weighted shortest path with negative edges O(|E| . |V|)

4.Weighted shortest path of acyclic graphs linear time


Unweighted Shortest Paths• Using some vertex, s, which is an input

parameter, find the shortest path from s to all other vertices in an unweighted graph. Assume s=v3.


Unweighted Shortest Paths

• Algorithm: find vertices that are at distance 1, 2, ... N-1 by processing vertices in layers (breadth-first search)


Unweighted Shortest Paths-ImplementationFor each vertex v

1. The distance from s in the entry dv

2. The bookkeeping variable (path) which will allow us to print the actual path pv

3. A boolean variable to check is the vertex is processed or not known

9

class Vertex {

List adj;

bool known;

int dist;

Vertex path; }


• Complexity O(|V|2)Izmir University of Economics 10

Unweighted Shortest Paths - Improvement

11

Unweighted Shortest Paths - Improvement

• At any point in time there are only two types of unknown vertices that have dv≠∞. Some have dv

= currDist and the rest have dv = currDist +1.

• We can make use of a queue data structure.

• O(|E|+|V|)12

Weighted Shortest Path Dijkstra’s Algorithm

• With weighted shortest path,distance dv is tentative. It turns out to be the shortest path length from s to v using only known vertices as intermediates.

• Greedy algorithm: proceeds in stages doing the best at each stage.

• Dijkstra’s algorithm selects a vertex v with smallest dv among all unknown vertices and declares it known. Remainder of the stage consists of updating the values dw for all edges (v, w).


Dijkstra’s Algorithm - Example


►

►

►

Dijkstra’s Algorithm - Example

• A proof by contradiction will show that this algorithm always works as long as no edge has a negative cost.


►

►►

►

Dijkstra’s Algorithm Example – Stages I


Dijkstra’s Algorithm Example – Stages II


Dijkstra’s Algorithm - Pseudocode• If the vertices are

sequentially scanned to find minimum dv, each phase will take O(|V|) to find the minimum, thus O(|V|2) over the course of the algorithm.

• The time for updates is constant and at most one update per edge for a total of O(|E|).

• Therefore the total time spent is O(|V|2+|E|).

• If the graph is dense, OPTIMAL.

18

Dijkstra’s Algorithm-What if the graph is sparse?

• If the graph is sparse |E|=θ(|V|), algorithm is too slow. The distances of vertices need to be kept in a priority queue.

• Selection of vertex with minimum distance via deleteMin, and updates via decreaseKey operation. Hence; O(|E|log|V|+|V|log|V|)

• find operations are not supported, so you need to be able to maintain locations of di in the heap and update them as they change.

• Alternative: insert w and dw with every update.Izmir University of Economics 19

Graphs with negative edge costs

• Dijkstra’s algorithm does not work with negative edge costs. Once a vertex u is known, it is possible that from some other unknown vertex v, there is a path back to u that is very negative.

• Algorithm: A combination of weighted and unweighted algorithms. Forget about the concept of known vertices.


Graphs with negative edge costs - I• O(|E|*|V|) • Each vertex can dequeue at

most O(|V|) times.

(Why? Algorithm computes shortest paths with at most 0, 1, ..., |V|-1 edges in this order). Hence, the result!

• If negative cost cycles, then each vertex should be checked to have been dequeued at most |V| times.


Acyclic Graphs• If the graph is known to be acyclic, the

order in which vertices are declared known, can be set to be the topological order.

• Running time = O(|V|+|E|)

• This selection rule works because when a vertex is selected, its distance can no longer be lowered, since by topological ordering rule it has no incoming edges emanating from unknown nodes.


Acyclic Graphs - Example


1 week 9-10 graphs ii unweighted shortest paths dijkstra’s algorithm graphs with negative costs...

Documents