Analysis of Dijkstra's Algorithm and Kruskal's Algorithm with Disjoint Set Union, Slides of Computer Science

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Algorithms

Dijkstra’s Algorithm

Disjoint-Set Union

Homework 5

● Check web page this afternoon…

Review: DAG Shortest Paths

● Problem: finding shortest paths in DAG

■ Bellman-Ford takes O(VE) time.

■ Do better using topological sort.

○ Idea: if were lucky and processes vertices on each shortest path in order, B-F would be done in one pass ○ Every path in a dag is subsequence of topologically sorted vertex order, so processing verts in that order, we will do each path in forward order (will never relax edges out of vert before doing all edges into vert). ○ Thus: just one pass. Running time: O(V+E)

Review: Dijkstra’s Algorithm

● If no negative edge weights, we can beat BF

● Similar to breadth-first search

■ Grow a tree gradually, advancing from vertices

taken from a queue

● Also similar to Prim’s algorithm for MST

■ Use a priority queue keyed on d[v]

Dijkstra’s Algorithm

Dijkstra(G) for each v ∈ V d[v] = ∞; d[s] = 0; S = ∅; Q = V; while (Q ≠ ∅) u = ExtractMin(Q); S = S U {u}; for each v ∈ u->Adj[] if (d[v] > d[u]+w(u,v)) d[v] = d[u]+w(u,v);

How many times is ExtractMin() called?

How many times is DecreaseKey() called?

What will be the total running time?

Dijkstra’s Algorithm

Dijkstra(G) for each v ∈ V d[v] = ∞; d[s] = 0; S = ∅; Q = V; while (Q ≠ ∅) u = ExtractMin(Q); S = S U {u}; for each v ∈ u->Adj[] if (d[v] > d[u]+w(u,v)) d[v] = d[u]+w(u,v);

How many times is ExtractMin() called?

How many times is DecraseKey() called?

A: O(E lg V) using binary heap for Q Can acheive O(V lg V + E) with Fibonacci heapsDocsity.com

Correctness Of Dijkstra's Algorithm

● Note that d[v] ≥ δ(s,v) ∀v ● Let u be first vertex picked s.t. ∃ shorter path than d[u] ⇒d[u] > δ(s,u) ● Let y be first vertex ∈V-S on actual shortest path from s→u ⇒ d[y] = δ(s,y) ■ Because d[x] is set correctly for y's predecessor x ∈ S on the shortest path, and ■ When we put x into S, we relaxed (x,y), giving d[y] the correct value

s

x

y

u

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p (^2)

Correctness Of Dijkstra's Algorithm

● Note that d[v] ≥ δ(s,v) ∀v ● Let u be first vertex picked s.t. ∃ shorter path than d[u] ⇒d[u] > δ(s,u) ● Let y be first vertex ∈V-S on actual shortest path from s→u ⇒ d[y] = δ(s,y) ● d[u] > δ(s,u) = δ(s,y) + δ(y,u) ( Why? ) = d[y] + δ(y,u) ≥ d[y] But if d[u] > d[y], wouldn't have chosen u. Contradiction.

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x

y

u

p

p (^2)

Kruskal’s Algorithm

Kruskal()

{

T = ∅; for each v ∈ V MakeSet(v); sort E by increasing edge weight w for each (u,v) ∈ E (in sorted order) if FindSet(u) ≠ FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v));

}

Kruskal’s Algorithm

Kruskal()

{

T = ∅; for each v ∈ V MakeSet(v); sort E by increasing edge weight w for each (u,v) ∈ E (in sorted order) if FindSet(u) ≠ FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v));

}

(^2 ) 9

1

5 13

17 25

14 8

21

Run the algorithm:

Kruskal’s Algorithm

Kruskal()

{

T = ∅; for each v ∈ V MakeSet(v); sort E by increasing edge weight w for each (u,v) ∈ E (in sorted order) if FindSet(u) ≠ FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v));

}

(^2 ) 9

1

5 13

17 25

14 8

21

Run the algorithm:

Kruskal’s Algorithm

Kruskal()

{

T = ∅; for each v ∈ V MakeSet(v); sort E by increasing edge weight w for each (u,v) ∈ E (in sorted order) if FindSet(u) ≠ FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v));

}

(^2 ) 9

1?

5 13

17 25

14 8

21

Run the algorithm:

Kruskal’s Algorithm

Kruskal()

{

T = ∅; for each v ∈ V MakeSet(v); sort E by increasing edge weight w for each (u,v) ∈ E (in sorted order) if FindSet(u) ≠ FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v));

}

2? (^19) 9

1

5 13

17 25

14 8

21

Run the algorithm:

Kruskal’s Algorithm

Kruskal()

{

T = ∅; for each v ∈ V MakeSet(v); sort E by increasing edge weight w for each (u,v) ∈ E (in sorted order) if FindSet(u) ≠ FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v));

}

(^2 ) 9

1

5 13

17 25

14 8

21

Run the algorithm: