Shortest Path Problems

Section 29.15 in Carrano/Savitch

Section 4.2 in Neapolitan/Naimipour

Single source shortest path problems

• We want to find the shortest path from a given vertex to all the others

– The input is a graph (stored either as a adjacency matrix or list)

– The cost of a path is the sum of the cost of each edge in the path

• Two types

– Weighted shortest path

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

– Unweighted shortest path

– A special case of the problem above. Consider all the weights one.

Negative weights

• Suppose a graph G has a cycle whose total weight is negative.

• By repeatedly going through the negative cycle, any path length could become negative.

• So, even if there is a path from v to u in G, the distance may be undefined if a negative cycle exists in G.

• Therefore, the only graphs we will look at have all positive edges.

Unweighted shortest path

• Our algorithm will first find path lengths of length zero, then of length one, then two etc.

• First we choose the beginning vertex. The length from it to itself is zero

• Then we find all the paths adjacent to original vertex. They are path length one.

Unweighted shortest path

• We want to find the shortest path from s to all the other vertices.

• The algorithm keeps track of three things for each vertex

– Whether or not the vertex has been processed – a boolean known

• Initially all paths are unknown; it is set to true after a vertex is processed

_– The distance from the start vertex – call it dist

_• Initially, the distance to all vertices except s is infinity

– The path to vertex s—this will be the last vertex to cause a change to the dist Initially this is zero for all vertices

Draw this unweighted graph

-- means infinity; i.e. no edge between the vertices

	1	2	3	4	5	6	7
1	0	1	--	1	--	--	--
2	--	0	--	1	1	--	--
3	1	--	0	--	--	1	--
4	--		1	0	1	1	1
5	--	--	--	--	0	--	1
6	--	--	--	--	--	0	--
7	--	--	--	--	--	1	0

• Let V₃ be the start vertex

_• Mark as known all vertices that are path length 1 from v₃

_– The path will be v₃

• Now mark as know all the path lengths of 2

– These will be adjacent to the nodes whose dist =1

– The path will be the vertex at dist = 1

• Mark as known the vertices whose path length from

v₃ is 3, then 4 etc, until all the vertices are known

Weighted graph—shortest path

• The general method to solve this problem is called Dijkstra’s algorithm

• This is a classic example of a greedy algorithm

– Greedy algorithms usually solve a problem in stages by doing what is best at that stage

– The problem with greedy algorithms is that sometimes they do not give optimal solutions

• Dijkstra’s algorithm works much like the unweighted shortest path algorithm

– We keep track of the same things

Dijkstra’s algorithm to solve the single source shortest path problem

• Initial configuration

– known = false for all vertices except start

– dist is infinity for all vertices except s; it is 0 for s

– path is zero for all vertices

• Look at the graph on the overhead, we will find the shortest path from v1 to all the other vertices

Going through Dijkstra’s algorit6hm

• Mark V1 as known,

– the dist from v1 to v1 is 0; so is the path

• The two vertices adjacent to V1 are V2 and V4

– Mark their dist from V1

– Mark their path as V1

Select a vertex V which has the smallest dist among all the unknown vertices, mark it known

– Mark V4 known

_– put down the dist of V4’s adjacent vertices from v1 (the start)

• This involves adding the dist to v4 to v4’s adjacent vertices

– The path for all of V4’s adjacent vertices is through v4

Now find the smallest dist that is not marked known

repeat the above step, replacing the dist if the dist through that vertex is smaller than the dist marked down.

Draw this graph, then find the shortest path using Dijkstra’s algorithm. Show the table.

Let V1 be the start vertex

-- means infinity; i.e. no edge

	1	2	3	4	5
1	0	7	4	6	1
2	--	0	--	--	--
3	--	2	0	5	--
4	--	3	--	0	--
5	--	--	--	1	0

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