david luebke 1 9/15/2015 cs 332: algorithms topological sort minimum spanning trees
TRANSCRIPT
David Luebke 1 04/19/23
CS 332: Algorithms
Topological Sort
Minimum Spanning Trees
David Luebke 2 04/19/23
Review: Breadth-First Search
BFS(G, s) {
initialize vertices;
Q = {s}; // Q is a queue (duh); initialize to s
while (Q not empty) {
u = RemoveTop(Q);
for each v u->adj { if (v->color == WHITE)
v->color = GREY;
v->d = u->d + 1;
v->p = u;
Enqueue(Q, v);
}
u->color = BLACK;
}
}
v->p represents parent in tree
v->d represents depth in tree
David Luebke 3 04/19/23
Review: DFS Code
DFS(G)
{
for each vertex u G->V {
u->color = WHITE;
}
time = 0;
for each vertex u G->V {
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = YELLOW;
time = time+1;
u->d = time;
for each v u->Adj[] {
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
David Luebke 4 04/19/23
Review: DFS Example
sourcevertex
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Review: DFS Example
1 | | |
| | |
| |
sourcevertex
d f
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Review: DFS Example
1 | | |
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2 | |
sourcevertex
d f
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Review: DFS Example
1 | | |
| | 3 |
2 | |
sourcevertex
d f
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Review: DFS Example
1 | | |
| | 3 | 4
2 | |
sourcevertex
d f
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Review: DFS Example
1 | | |
| 5 | 3 | 4
2 | |
sourcevertex
d f
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Review: DFS Example
1 | | |
| 5 | 63 | 4
2 | |
sourcevertex
d f
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Review: DFS Example
1 | 8 | |
| 5 | 63 | 4
2 | 7 |
sourcevertex
d f
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Review: DFS Example
1 | 8 | |
| 5 | 63 | 4
2 | 7 |
sourcevertex
d f
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Review: DFS Example
1 | 8 | |
| 5 | 63 | 4
2 | 7 9 |
sourcevertex
d f
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Review: DFS Example
1 | 8 | |
| 5 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
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Review: DFS Example
1 | 8 |11 |
| 5 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
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Review: DFS Example
1 |12 8 |11 |
| 5 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
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Review: DFS Example
1 |12 8 |11 13|
| 5 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
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Review: DFS Example
1 |12 8 |11 13|
14| 5 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
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Review: DFS Example
1 |12 8 |11 13|
14|155 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
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Review: DFS Example
1 |12 8 |11 13|16
14|155 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
David Luebke 21 04/19/23
Review: Kinds Of Edges
● Thm: If G is undirected, a DFS produces only tree and back edges
● Thm: An undirected graph is acyclic iff a DFS yields no back edges
● Thus, can run DFS to find cycles
David Luebke 22 04/19/23
1 |12 8 |11 13|16
14|155 | 63 | 4
2 | 7 9 |10
sourcevertex
d f
Tree edges Back edges Forward edges Cross edges
Review: Kinds of Edges
David Luebke 23 04/19/23
DFS And Cycles
● Running time: O(V+E)● We can actually determine if cycles exist in
O(V) time:■ In an undirected acyclic forest, |E| |V| - 1 ■ So count the edges: if ever see |V| distinct edges,
must have seen a back edge along the way■ Why not just test if |E| <|V| and answer the
question in constant time?
David Luebke 24 04/19/23
Directed Acyclic Graphs
● A directed acyclic graph or DAG is a directed graph with no directed cycles:
David Luebke 25 04/19/23
DFS and DAGs
● Argue that a directed graph G is acyclic iff a DFS of G yields no back edges:■ Forward: if G is acyclic, will be no back edges
○ Trivial: a back edge implies a cycle
■ Backward: if no back edges, G is acyclic○ Argue contrapositive: G has a cycle a back edge
Let v be the vertex on the cycle first discovered, and u be the predecessor of v on the cycle
When v discovered, whole cycle is white Must visit everything reachable from v before returning from DFS-Visit() So path from uv is yellowyellow, thus (u, v) is a back edge
David Luebke 26 04/19/23
Topological Sort
● Topological sort of a DAG:■ Linear ordering of all vertices in graph G such that
vertex u comes before vertex v if edge (u, v) G
● Real-world example: getting dressed
David Luebke 27 04/19/23
Getting Dressed
Underwear Socks
ShoesPants
Belt
Shirt
Watch
Tie
Jacket
David Luebke 28 04/19/23
Getting Dressed
Underwear Socks
ShoesPants
Belt
Shirt
Watch
Tie
Jacket
Socks Underwear Pants Shoes Watch Shirt Belt Tie Jacket
David Luebke 29 04/19/23
Topological Sort Algorithm
Topological-Sort()
{
Run DFS
When a vertex is finished, output it
Vertices are output in reverse topological order
}
● Time: O(V+E)● Correctness: Want to prove that
(u,v) G uf > vf
David Luebke 30 04/19/23
Correctness of Topological Sort
● Claim: (u,v) G uf > vf■ When (u,v) is explored, u is yellow
○ v = yellow (u,v) is back edge. Contradiction (Why?)○ v = white v becomes descendent of u vf < uf
(since must finish v before backtracking and finishing u)
○ v = black v already finished vf < uf
David Luebke 31 04/19/23
Minimum Spanning Tree
● Problem: given a connected, undirected, weighted graph:
1410
3
6 4
5
2
9
15
8
David Luebke 32 04/19/23
Minimum Spanning Tree
● Problem: given a connected, undirected, weighted graph, find a spanning tree using edges that minimize the total weight
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6 4
5
2
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8
David Luebke 33 04/19/23
Minimum Spanning Tree
● Which edges form the minimum spanning tree (MST) of the below graph?
H B C
G E D
F
A
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6 4
5
2
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8
David Luebke 34 04/19/23
Minimum Spanning Tree
● Answer:
H B C
G E D
F
A
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6 4
5
2
9
15
8
David Luebke 35 04/19/23
Minimum Spanning Tree
● MSTs satisfy the optimal substructure property: an optimal tree is composed of optimal subtrees■ Let T be an MST of G with an edge (u,v) in the middle
■ Removing (u,v) partitions T into two trees T1 and T2
■ Claim: T1 is an MST of G1 = (V1,E1), and T2 is an MST of G2 = (V2,E2) (Do V1 and V2 share vertices? Why?)
■ Proof: w(T) = w(u,v) + w(T1) + w(T2)(There can’t be a better tree than T1 or T2, or T would be suboptimal)
David Luebke 36 04/19/23
Minimum Spanning Tree
● Thm: ■ Let T be MST of G, and let A T be subtree of T■ Let (u,v) be min-weight edge connecting A to V-A■ Then (u,v) T
David Luebke 37 04/19/23
Minimum Spanning Tree
● Thm: ■ Let T be MST of G, and let A T be subtree of T■ Let (u,v) be min-weight edge connecting A to V-A■ Then (u,v) T
● Proof: in book (see Thm 24.1)
David Luebke 38 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);
David Luebke 39 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);
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3
6 45
2
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15
8
Run on example graph
David Luebke 40 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);
1410
3
6 45
2
9
15
8
Run on example graph
David Luebke 41 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);
0
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3
6 45
2
9
15
8
Pick a start vertex r
r
David Luebke 42 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);
0
1410
3
6 45
2
9
15
8
Red vertices have been removed from Q
u
David Luebke 43 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);
0
3
1410
3
6 45
2
9
15
8
Red arrows indicate parent pointers
u
David Luebke 44 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);
14
0
3
1410
3
6 45
2
9
15
8
u
David Luebke 45 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);
14
0
3
1410
3
6 45
2
9
15
8u
David Luebke 46 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);
14
0 8
3
1410
3
6 45
2
9
15
8u
David Luebke 47 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
0 8
3
1410
3
6 45
2
9
15
8u
David Luebke 48 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
0 8
3
1410
3
6 45
2
9
15
8u
David Luebke 49 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
3
1410
3
6 45
2
9
15
8u
David Luebke 50 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
8u
David Luebke 51 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 52 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 9
0 8 15
3
1410
3
6 45
2
9
15
8
u
David Luebke 53 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 9
0 8 15
3
4
1410
3
6 45
2
9
15
8
u
David Luebke 54 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);
5 2 9
0 8 15
3
4
1410
3
6 45
2
9
15
8
u
David Luebke 55 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);
5 2 9
0 8 15
3
4
1410
3
6 45
2
9
15
8
u
David Luebke 56 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);
5 2 9
0 8 15
3
4
1410
3
6 45
2
9
15
8
u
David Luebke 57 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);
5 2 9
0 8 15
3
4
1410
3
6 45
2
9
15
8
u
David Luebke 58 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);
5 2 9
0 8 15
3
4
1410
3
6 45
2
9
15
8
u
David Luebke 59 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 60 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 61 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 62 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 63 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?