david luebke 1 9/10/2015 cs 332: algorithms single-source shortest path
Post on 27-Dec-2015
233 Views
Preview:
TRANSCRIPT
David Luebke 2 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
David Luebke 3 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
1410
3
6 45
2
9
15
8
Run on example graph
David Luebke 4 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
1410
3
6 45
2
9
15
8
Run on example graph
David Luebke 5 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
0
1410
3
6 45
2
9
15
8
Pick a start vertex r
r
David Luebke 6 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
0
1410
3
6 45
2
9
15
8
Red vertices have been removed from Q
u
David Luebke 7 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
0
3
1410
3
6 45
2
9
15
8
Red arrows indicate parent pointers
u
David Luebke 8 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
14
0
3
1410
3
6 45
2
9
15
8
u
David Luebke 9 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
14
0
3
1410
3
6 45
2
9
15
8u
David Luebke 10 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
14
0 8
3
1410
3
6 45
2
9
15
8u
David Luebke 11 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
10
0 8
3
1410
3
6 45
2
9
15
8u
David Luebke 12 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
10
0 8
3
1410
3
6 45
2
9
15
8u
David Luebke 13 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
10 2
0 8
3
1410
3
6 45
2
9
15
8u
David Luebke 14 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
10 2
0 8 15
3
1410
3
6 45
2
9
15
8u
David Luebke 15 04/19/23
Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
10 2
0 8 15
3
1410
3
6 45
2
9
15
8
u
David Luebke 16 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
10 2 9
0 8 15
3
1410
3
6 45
2
9
15
8
u
David Luebke 17 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
10 2 9
0 8 15
3
4
1410
3
6 45
2
9
15
8
u
David Luebke 18 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
5 2 9
0 8 15
3
4
1410
3
6 45
2
9
15
8
u
David Luebke 19 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
5 2 9
0 8 15
3
4
1410
3
6 45
2
9
15
8
u
David Luebke 20 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
5 2 9
0 8 15
3
4
1410
3
6 45
2
9
15
8
u
David Luebke 21 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
5 2 9
0 8 15
3
4
1410
3
6 45
2
9
15
8
u
David Luebke 22 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
5 2 9
0 8 15
3
4
1410
3
6 45
2
9
15
8
u
David Luebke 23 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
What is the hidden cost in this code?
David Luebke 24 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
DecreaseKey(v, w(u,v));
David Luebke 25 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
DecreaseKey(v, w(u,v));
How often is ExtractMin() called?How often is DecreaseKey() called?
David Luebke 26 04/19/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
What will be the running time?A: Depends on queue binary heap: O(E lg V) Fibonacci heap: O(V lg V + E)
David Luebke 27 04/19/23
Single-Source Shortest Path
● Problem: given a weighted directed graph G, find the minimum-weight path from a given source vertex s to another vertex v■ “Shortest-path” = minimum weight ■ Weight of path is sum of edges■ E.g., a road map: what is the shortest path from
Chapel Hill to Charlottesville?
David Luebke 28 04/19/23
Shortest Path Properties
● Again, we have optimal substructure: the shortest path consists of shortest subpaths:
■ Proof: suppose some subpath is not a shortest path○ There must then exist a shorter subpath ○ Could substitute the shorter subpath for a shorter path○ But then overall path is not shortest path. Contradiction
David Luebke 29 04/19/23
Shortest Path Properties
● Define (u,v) to be the weight of the shortest path from u to v
● Shortest paths satisfy the triangle inequality: (u,v) (u,x) + (x,v)
● “Proof”: x
u v
This path is no longer than any other path
David Luebke 30 04/19/23
Shortest Path Properties
● In graphs with negative weight cycles, some shortest paths will not exist (Why?):
< 0
David Luebke 31 04/19/23
Relaxation
● A key technique in shortest path algorithms is relaxation■ Idea: for all v, maintain upper bound d[v] on (s,v)Relax(u,v,w) {
if (d[v] > d[u]+w) then d[v]=d[u]+w;
}
952
752
Relax
652
652
Relax
David Luebke 32 04/19/23
Bellman-Ford Algorithm
BellmanFord()
for each v V d[v] = ; d[s] = 0;
for i=1 to |V|-1
for each edge (u,v) E Relax(u,v, w(u,v));
for each edge (u,v) E if (d[v] > d[u] + w(u,v))
return “no solution”;
Relax(u,v,w): if (d[v] > d[u]+w) then d[v]=d[u]+w
Initialize d[], whichwill converge to shortest-path value
Relaxation: Make |V|-1 passes, relaxing each edge
Test for solution Under what conditiondo we get a solution?
David Luebke 33 04/19/23
Bellman-Ford Algorithm
BellmanFord()
for each v V d[v] = ; d[s] = 0;
for i=1 to |V|-1
for each edge (u,v) E Relax(u,v, w(u,v));
for each edge (u,v) E if (d[v] > d[u] + w(u,v))
return “no solution”;
Relax(u,v,w): if (d[v] > d[u]+w) then d[v]=d[u]+w
What will be the running time?
David Luebke 34 04/19/23
Bellman-Ford Algorithm
BellmanFord()
for each v V d[v] = ; d[s] = 0;
for i=1 to |V|-1
for each edge (u,v) E Relax(u,v, w(u,v));
for each edge (u,v) E if (d[v] > d[u] + w(u,v))
return “no solution”;
Relax(u,v,w): if (d[v] > d[u]+w) then d[v]=d[u]+w
What will be the running time?
A: O(VE)
David Luebke 35 04/19/23
Bellman-Ford Algorithm
BellmanFord()
for each v V d[v] = ; d[s] = 0;
for i=1 to |V|-1
for each edge (u,v) E Relax(u,v, w(u,v));
for each edge (u,v) E if (d[v] > d[u] + w(u,v))
return “no solution”;
Relax(u,v,w): if (d[v] > d[u]+w) then d[v]=d[u]+w
B
E
DC
A
-1 2
2
1-3
5
3
4
Ex: work on board
s
David Luebke 36 04/19/23
Bellman-Ford
● Note that order in which edges are processed affects how quickly it converges
● Correctness: show d[v] = (s,v) after |V|-1 passes■ Lemma: d[v] (s,v) always
○ Initially true○ Let v be first vertex for which d[v] < (s,v)○ Let u be the vertex that caused d[v] to change:
d[v] = d[u] + w(u,v)○ Then d[v] < (s,v)
(s,v) (s,u) + w(u,v) (Why?) (s,u) + w(u,v) d[u] + w(u,v) (Why?)
○ So d[v] < d[u] + w(u,v). Contradiction.
David Luebke 37 04/19/23
Bellman-Ford
● Prove: after |V|-1 passes, all d values correct■ Consider shortest path from s to v:
s v1 v2 v3 v4 v○ Initially, d[s] = 0 is correct, and doesn’t change (Why?)
○ After 1 pass through edges, d[v1] is correct (Why?) and doesn’t change
○ After 2 passes, d[v2] is correct and doesn’t change
○ …○ Terminates in |V| - 1 passes: (Why?) ○ What if it doesn’t?
David Luebke 38 04/19/23
DAG Shortest Paths
● Problem: finding shortest paths in DAG■ Bellman-Ford takes O(VE) time. ■ How can we do better?■ Idea: use topological sort
○ If were lucky and processes vertices on each shortest path from left to right, 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. What will be the running time?
David Luebke 39 04/19/23
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]
David Luebke 40 04/19/23
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);RelaxationStepNote: this
is really a call to Q->DecreaseKey()
B
C
DA
10
4 3
2
15
Ex: run the algorithm
David Luebke 41 04/19/23
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?
What will be the total running time?
David Luebke 42 04/19/23
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 QCan acheive O(V lg V + E) with Fibonacci heaps
David Luebke 43 04/19/23
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);Correctness: we must show that when u is removed from Q, it has already converged
David Luebke 44 04/19/23
Correctness Of Dijkstra's Algorithm
● Note that d[v] (s,v) v ● Let u be first vertex picked s.t. shorter path than d[u] d[u] > (s,u)● Let y be first vertex V-S on actual shortest path from su d[y] = (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
xy
up2
p2
David Luebke 45 04/19/23
Correctness Of Dijkstra's Algorithm
● Note that d[v] (s,v) v ● Let u be first vertex picked s.t. shorter path than d[u] d[u] > (s,u)● Let y be first vertex V-S on actual shortest path from su d[y] = (s,y)● d[u] > (s,u)
= (s,y) + (y,u) (Why?)= d[y] + (y,u) d[y] But if d[u] > d[y], wouldn't have chosen u. Contradiction.
s
xy
up2
p2
David Luebke 46 04/19/23
Disjoint-Set Union Problem
● Want a data structure to support disjoint sets ■ Collection of disjoint sets S = {Si}, Si ∩ Sj =
● Need to support following operations:■ MakeSet(x): S = S U {{x}}
■ Union(Si, Sj): S = S - {Si, Sj} U {Si U Sj}
■ FindSet(X): return Si S such that x Si
● Before discussing implementation details, we look at example application: MSTs
David Luebke 47 04/19/23
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));
}
David Luebke 48 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 49 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 50 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 51 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 52 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 53 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 54 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 55 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 56 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 57 04/19/23
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
1725
148?
21
Run the algorithm:
David Luebke 58 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 59 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 60 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 61 04/19/23
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?
1725
148
21
Run the algorithm:
David Luebke 62 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 63 04/19/23
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
1725
14?8
21
Run the algorithm:
David Luebke 64 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 65 04/19/23
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
148
21
Run the algorithm:
David Luebke 66 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 67 04/19/23
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
1725
148
21?
Run the algorithm:
David Luebke 68 04/19/23
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
1725?
148
21
Run the algorithm:
David Luebke 69 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 70 04/19/23
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
1725
148
21
Run the algorithm:
David Luebke 71 04/19/23
Correctness Of Kruskal’s Algorithm
● Sketch of a proof that this algorithm produces an MST for T:■ Assume algorithm is wrong: result is not an MST■ Then algorithm adds a wrong edge at some point■ If it adds a wrong edge, there must be a lower weight
edge (cut and paste argument)■ But algorithm chooses lowest weight edge at each step.
Contradiction
● Again, important to be comfortable with cut and paste arguments
David Luebke 72 04/19/23
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));
}
What will affect the running time?
David Luebke 73 04/19/23
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));
}
What will affect the running time? 1 Sort
O(V) MakeSet() callsO(E) FindSet() callsO(V) Union() calls
(Exactly how many Union()s?)
David Luebke 74 04/19/23
Kruskal’s Algorithm: Running Time
● To summarize: ■ Sort edges: O(E lg E) ■ O(V) MakeSet()’s■ O(E) FindSet()’s■ O(V) Union()’s
● Upshot: ■ Best disjoint-set union algorithm makes above 3
operations take O(E(E,V)), almost constant■ Overall thus O(E lg E), almost linear w/o sorting
David Luebke 75 04/19/23
Disjoint Set Union
● So how do we implement disjoint-set union?■ Naïve implementation: use a linked list to
represent each set:
○ MakeSet(): ??? time○ FindSet(): ??? time○ Union(A,B): “copy” elements of A into B: ??? time
David Luebke 76 04/19/23
Disjoint Set Union
● So how do we implement disjoint-set union?■ Naïve implementation: use a linked list to represent each
set:
○ MakeSet(): O(1) time○ FindSet(): O(1) time○ Union(A,B): “copy” elements of A into B: O(A) time
■ How long can a single Union() take?■ How long will n Union()’s take?
David Luebke 77 04/19/23
Disjoint Set Union: Analysis
● Worst-case analysis: O(n2) time for n Union’sUnion(S1, S2) “copy” 1 element
Union(S2, S3) “copy” 2 elements
…
Union(Sn-1, Sn) “copy” n-1 elements
O(n2)
● Improvement: always copy smaller into larger■ Why will this make things better?■ What is the worst-case time of Union()?
● But now n Union’s take only O(n lg n) time!
David Luebke 78 04/19/23
Amortized Analysis of Disjoint Sets
● Amortized analysis computes average times without using probability
● With our new Union(), any individual element is copied at most lg n times when forming the complete set from 1-element sets■ Worst case: Each time copied, element in smaller set
1st time resulting set size 2
2nd time 4
…
(lg n)th time n
David Luebke 79 04/19/23
Amortized Analysis of Disjoint Sets
● Since we have n elements each copied at most lg n times, n Union()’s takes O(n lg n) time
● We say that each Union() takes O(lg n) amortized time■ Financial term: imagine paying $(lg n) per Union■ At first we are overpaying; initial Union $O(1)■ But we accumulate enough $ in bank to pay for later
expensive O(n) operation. ■ Important: amount in bank never goes negative
top related