MST Algorithm Review

1
MST Algorithm Review
2
Minimal Cost Spanning Trees

• Minimal Cost Spanning Tree (MST) Problem
• Input An undirected, connected graph G.
• Output The subgraph of G that
• 1) has minimum total cost as measured by summing
  the weight of all the edges in the subset, and
• 2) keeps the vertices connected.

3
Partition Property

• Partition Property
• Consider a partition of the vertices of G into
  subsets U and V
• Let e be an edge of minimum weight across the
  partition
• There is a minimum spanning tree of G containing
  edge e

U
V
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f
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9
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e
7
4
Prims MST Algorithm

• As with Dijkstras algorithm, uses a priority
  queue.
• Running time is identical to Dijkstras algorithm
  implementations.

B
5
A
7
C
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9
D
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Prims MST Algorithm
6
Prims MST Algorithm
(?)
(?)
B
Step 1
5
A
7
C
(?)
6
1
2
9
D
2
(?)
F
E
(0)
1
(?)
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Prims MST Algorithm
(6,F)
(?)
B
Step 2
5
A
7
C
(2,F)
6
1
2
9
D
2
(2,F)
F
E
(0)
1
(1,F)
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Prims MST Algorithm
(6,F)
(?)
B
Step 3
5
A
7
C
(2,F)
6
1
2
9
D
2
(2,F)
F
E
(0)
1
(1,F)
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Prims MST Algorithm
(6,F)
(9,E)
B
Step 3a
5
A
7
C
(2,F)
6
1
2
9
D
2
(2,F)
F
E
(0)
1
(1,F)
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Prims MST Algorithm
(6,F)
(9,E)
B
Step 4
5
A
7
C
(2,F)
6
1
2
9
D
2
(2,F)
F
E
(0)
1
(1,F)
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Prims MST Algorithm
(5,C)
(7,C)
B
Step 4a
5
A
7
C
(2,F)
6
1
2
9
D
2
(1,C)
F
E
(0)
1
(1,F)
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Prims MST Algorithm
(5,C)
(7,C)
B
Step 5
5
A
7
C
(2,F)
6
1
2
9
D
2
(1,C)
F
E
(0)
1
(1,F)
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Prims MST Algorithm
(5,C)
(7,C)
B
Step 5a
5
A
7
C
(2,F)
6
1
2
9
D
2
(1,C)
F
E
(0)
1
(1,F)
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Prims MST Algorithm
(5,C)
(7,C)
B
Step 6
5
A
7
C
(2,F)
6
1
2
9
D
2
(1,C)
F
E
(0)
1
(1,F)
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Prims MST Algorithm
(5,C)
(7,C)
B
Step 6a
5
A
7
C
(2,F)
6
1
2
9
D
2
(1,C)
F
E
(0)
1
(1,F)
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Prims MST Algorithm
(5,C)
(7,C)
B
Step 7
5
A
7
C
(2,F)
6
1
2
9
D
2
(1,C)
F
E
(0)
1
(1,F)
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Kruskals MST Algorithm
B
5
A
7
C
6
1
2
9
D
2
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1

• Initially, each vertex is in its own MST.
• Merge two MSTs that have the shortest edge
  between them.
• Use a priority queue to order the unprocessed
  edges. Grab the minimum one at each step.
• How to tell if an edge connects two vertices
  already in the same MST?
• Use parent-pointer representation.

18
Kruskals MST Algorithm
Step 1
E
D
F
B
C
A
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Kruskals MST Algorithm
Step 2
B
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A
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D
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B
C
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Kruskals MST Algorithm
Step 3
B
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B
C
A
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Kruskals MST Algorithm
Step 4
B
5
A
E
D
F
B
C
A
7
C
6
1
2
9
D
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E
1
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Kruskals MST Algorithm
Step 5
B
5
A
E
D
F
B
C
A
7
C
6
1
2
9
D
2
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E
1
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Kruskals MST Algorithm
Step 6
B
5
A
E
D
F
B
C
A
7
C
6
1
2
9
D
2
F
E
1
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Kruskals MST Algorithm
Step 7
B
5
A
E
D
F
B
C
A
7
C
6
1
2
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D
2
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E
1
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Kruskals MST Algorithm

• Cost is dominated by the time to remove edges
  from the heap.
• Can stop processing edges once all vertices are
  in the same MST
• Total cost Q((n m) log m).

26
Baruvkas MST Algorithm
B
5
A
7
C
6
1
2
9
D
2
F
E
1

• Create a forest of n trees
• Loop while (there is gt 1 tree in the forest)
• For each tree Ti in the forest
• Find the smallest edge e (u,v), in the edge
  list with u in Ti and v in Tj ?Ti
• connects 2 trees from the forest in to one.
• end loop

27
Baruvkas MST Algorithm
Step 1
E
D
F
B
C
A
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Baruvkas MST Algorithm
Step 2
B
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A
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D
F
B
C
A
7
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1
2
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D
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1
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Baruvkas MST Algorithm
Step 3
B
5
A
E
D
F
B
C
A
7
C
6
1
2
9
D
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1
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Baruvkas MST Algorithm
Step 4
B
5
A
E
D
F
B
C
A
7
C
6
1
2
9
D
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1
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Baruvkas MST Algorithm
Step 5
B
5
A
E
D
F
B
C
A
7
C
6
1
2
9
D
2
F
E
1
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Baruvkas MST Algorithm
Step 6
B
5
A
E
D
F
B
C
A
7
C
6
1
2
9
D
2
F
E
1
33
Baruvkas Algorithm

• Similar to Kruskals Algorithm, but no Priority
  queueS.
• Each iteration of the while-loop halves the
  number of connected compontents in T.
• The running time is O(m log n).

Algorithm BaruvkaMST(G) T ? V just the
vertices of G while T has fewer than n-1 edges
do for each connected component C in T
do Let edge e be the smallest-weight edge from
C to another component in T. if e is not
already in T then Add edge e to T return T
34
Network Flow Problem
35
Flow Network

• A network N consists of
• A weighted digraph G with nonnegative integer
  edge weights, where the weight of an edge e is
  called the capacity c(e) of e
• Two distinguished vertices, s and t of G, called
  the source and sink or target, respectively, such
  that s has no incoming edges and t has no
  outgoing edges.
• Example

36
Flow

• A flow f for a network N is an assignment of an
  value f(e) to each edge e that satisfies the
  following properties
• Capacity Rule For each edge e, 0 ? f (e) ? c(e)

37
Flow

• A flow f for a network N is an assignment of an
  value f(e) to each edge e that satisfies the
  following properties
• Capacity Rule For each edge e, 0 ? f (e) ? c(e)
• Conservation Rule For each vertex v ? s,t

38
Flow

• The value of a flow f , denoted f, is
• the total flow from the source
• which is the same as,
• the total flow into the sink

39
Maximum Flow

• A flow for a network N is said to be maximum if
  its value is the largest of all flows for N
• The maximum flow problem consists of finding a
  maximum flow for a given network N
• Applications
• Hydraulic systems
• Electrical circuits
• Traffic movements
• Freight transportation

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Maximum Flow
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Flow 2 3 3 1 3 4 8
41
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Flow Graph (Gf)
42
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
43
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
v
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6
1
t
7
3
s
w
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5
1
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2
44
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
v
3
6
1
t
7
3
s
w
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3
1
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45
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
v
3
6
1
t
7
3
s
w
9
3
1
3
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u
46
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
v
3
6
1
t
7
s
w
6
1
3
z
u
47
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
v
3
6
1
t
7
s
w
6
1
3
z
u
48
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
v
3
1
t
7
s
w
6
1
3
z
u
49
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
v
3
1
t
7
s
w
6
1
3
z
u
50
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
v
3
1
t
6
s
w
6
2
z
u
51
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
v
3
1
t
6
s
w
6
2
z
u
52
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
v
2
t
5
s
w
6
2
z
u
53
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
/2
2
w
v
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/2
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/3
3
s
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/4
4
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/1
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/2
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/3
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z
z
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/3
3
54
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
0/2
2
w
v
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v
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2
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3
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s
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4
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z
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55
A Simple Flow Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
0/2
2
w
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v
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z
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3
Algorithm terminates with an sub-optimal flow
solution
56
Ford-Fulkersons Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• Add a reversed path in the residual graph
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
0/2
2
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Ford-Fulkersons Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• Add a reversed path in the residual graph
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
0/2
2
w
v
w
v
0/2
2
3/3
-3
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-3
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3
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Ford-Fulkersons Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• Add a reversed path in the residual graph
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
0/2
2
w
v
w
v
0/2
2
3/3
-3
s
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3/4
1
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0/1
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-3
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2
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-3
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z
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0/3
3
59
Ford-Fulkersons Algorithm

• Initialize all edges of the flow graph with zero
  flow
• Loop while ? a path in Gr from s to t
• find a path in Gr from s to t (augmenting path)
• Add to the flow graph the minimum residual
  capacity
• from this path
• Reduce the residual capacity of the edges
• Add a reversed path in the residual graph
• end Loop

Residual Graph (Gr)
Flow Graph (Gf)
2/2
-2
w
v
w
v
2/2
-2
3/3
-3
s
s
1/4
3
t
0/1
t
1
-1
2/2
-2
3/3
-3
-2
z
z
u
u
2/3
1
60
Analysis

• Each augmenting path increases the flow value by
  at least 1.
• Let f is a maximum flow, then in the worst
  case, Ford-Fulkersons algorithm performs f
  flow augmentations.
• Finding an augmenting path and augmenting the
  flow takes O(n m) time
• The running time of Ford-Fulkersons algorithm is
  O(f(n m))

61
Analysis
v
50
50
Classic worst case example
s
1
t
50
50
u
Flow Graph (Gf)
Residual Graph (Gr)
v
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49
1/50
0/50
-1
s
s
1/1
t
t
-1
-1
49
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1/50
0/50
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62
Analysis
v
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Classic worst case example
s
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u
Flow Graph (Gf)
Residual Graph (Gr)
v
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49
1/50
1/50
-1
-1
s
s
1
0/1
t
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-1
-1
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49
1/50
1/50
u
u