[PDF] Dijkstras Algorithm Continued Dijkstras Algorithm: Pseudocode PDF lect22.pdf

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1 1

Dijkstra"s Algorithm Continued

E.W. Dijkstra (1930-2002)

Dijkstra"s Algorithm: Pseudocode

Initialize the cost of each node to ¥

Initialize the cost of the source to 0

While there are

unknownnodes left in the graph

Select an

unknownnode bwith the lowest cost

Mark bas

known

For each node a adjacent to b

a"s cost = min(a"s old cost, b"s cost + cost of (b, a)) 3 v3 v6 v1 v2v4 v5 v0s 12 2 2 76 1
1 35 6 10 pathDistKnownV v0 v6 v5 v4 v3 v2 v1 4 void Graph::dijkstra(Vertex s){

Vertex v,w;

Initialize s.dist = 0 and set dist of all other

vertices to infinity while (there exist unknown vertices, find the one bwith the smallest distance) b.known = true; for each a adjacent to b if (!a.known) if (b.dist + Cost_ba < a.dist){ decrease(a.dist to= b.dist + Cost_ba); a.path = b; 5

Dijkstra"s Alg: Implementation

Initialize the cost of each node to ¥

Initialize the cost of the source to 0

While there are unknown nodes left in the graph

Select the unknown node bwith the lowest cost

Mark bas known

For each node aadjacent to b

a"s cost = min(a"s old cost, b"s cost + cost of (b, a))

What data structures should we use?

Operations to be performed:

deleteMin() decreaseKey() 6

Dijkstra"s vs BFS

At each step:

1) Pick closest unknown vertex

2) Add it to finished vertices

3) Update distances

Dijkstra"s Algorithm

At each step:

1) Pick vertex from queue

2) Add it to visited vertices

3) Update queue with neighbors

Breadth-first Search

O(|V|2+ |E|) directly O(|V|+|E|)

O(|V|log|V|+|E|log|V|) heaps

2 7

Single-Source Shortest Path

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

All-Pairs Shortest Path:

• Find the shortest paths between all pairs of vertices in the graph. • How? 8

Analysis

• Total running time for Dijkstra"s:

O(|V|2+ |E|) (linear scan)

O(|V| log |V| + |E| log |V|) (heaps)

What if we want to find the shortest path from

each point to ALL other points? 9

Dynamic Programming

Algorithmic technique that systematically records

the answers to sub-problems in a table and re- usesthose recorded results (rather than re- computing them).

Simple Example: Calculating the Nth Fibonacci

number.

Fib(N) = Fib(N-1) + Fib(N-2)

Floyd-Warshall

for (intk= 1; k =< V; k++) for (int i = 1; i =< V; i++) for (int j = 1; j =< V; j++) if ( (

M[i][k]+ M[k][j] ) < M[i][j] )

M[i][j] =

M[i][k]+ M[k][j]

Invariant: After the kth iteration, the matrix includes the shortest paths for all pairs of vertices (i,j) containing only vertices 1..k as intermediate vertices

110----e40---d1-0--c31-20-b--4-20aedcba

b c de a -42 -2 1 3 1

4Initial state of the matrix:

M[i][j] =

min(M[i][j], M[i][k]+ M[k][j])120----e40---d1-0--c-11-20-b0-4020aedcba b c de a -42 -2 1 3 1

[PDF] [PDF] Dijkstras Algorithm Continued Dijkstras Algorithm: Pseudocode

Dijkstra"s Algorithm Continued

E.W. Dijkstra (1930-2002)

Dijkstra"s Algorithm: Pseudocode

Initialize the cost of each node to ¥

Initialize the cost of the source to 0

While there are

Select an

Mark bas

For each node a adjacent to b

Vertex v,w;

Initialize s.dist = 0 and set dist of all other

Dijkstra"s Alg: Implementation

Initialize the cost of each node to ¥

Initialize the cost of the source to 0

While there are unknown nodes left in the graph

Select the unknown node bwith the lowest cost

Mark bas known

For each node aadjacent to b

What data structures should we use?

Operations to be performed:

Dijkstra"s vs BFS

At each step:

1) Pick closest unknown vertex

2) Add it to finished vertices

3) Update distances

Dijkstra"s Algorithm

At each step:

1) Pick vertex from queue

2) Add it to visited vertices

3) Update queue with neighbors

Breadth-first Search

O(|V|2+ |E|) directly O(|V|+|E|)

O(|V|log|V|+|E|log|V|) heaps

Single-Source Shortest Path

All-Pairs Shortest Path:

Analysis

O(|V|2+ |E|) (linear scan)

O(|V| log |V| + |E| log |V|) (heaps)

What if we want to find the shortest path from

Dynamic Programming

Algorithmic technique that systematically records

Simple Example: Calculating the Nth Fibonacci

Fib(N) = Fib(N-1) + Fib(N-2)

Floyd-Warshall

M[i][k]+ M[k][j] ) < M[i][j] )

M[i][j] =

M[i][k]+ M[k][j]

110----e40---d1-0--c31-20-b--4-20aedcba

4Initial state of the matrix:

M[i][j] =

4Floyd-Warshall -for All-pairs shortest path

Final Matrix

Contents

Floyd-Warshall

Performance