CS2420: Lecture 40

1
CS2420 Lecture 40

• Vladimir Kulyukin
• Computer Science Department
• Utah State University

2
Outline

• Graph Algorithms (Chapter 9)

3
Spanning Trees

• An edge e in G is redundant if, after the removal
  of e, G remains connected.
• A spanning tree of G is a tree (acyclic graph)
  that contains all vertices of G and preserves Gs
  connectivity without any redundant edges.

4
Spanning Tree

• A spanning tree of a connected graph is its
  connected acyclic sub-graph (a tree) that
  contains all vertices of the graph and preserves
  the graphs connectivity.
• In a weighted connected graph, the weight of a
  spanning tree is defined as the sum of the trees
  edges.
• A minimal (minimum) spanning tree of a weighted
  connected graph is a spanning tree of the
  smallest weight.

5
Spanning Tree Example
A
A
B
B
C
C
D
D
6
Spanning Trees Examples
1
1
1
A
B
A
B
A
B
2
2
5
5
3
3
3
D
C
C
D
C
D
Tree T1 Weight(T1) 6
Tree T2 Weight(T2) 9
Original Graph
1
A
B
2
5
C
D
Tree T3 Weight(T3) 8
7
Min Spanning Tree Construction

• Construct a minimum spanning tree by expanding it
  with one vertex at each iteration.
• The initial tree consists of one vertex, i.e. the
  root.
• On each iteration, the tree so far is expanded by
  attaching to it the nearest vertex.
• Stop when all the vertices are included.

8
Min Spanning Tree Construction Example
1
1
1
A
B
A
B
A
B
2
2
5
Step 1
3
D
C
D
Step 2
Original Graph
1
A
B
2
3
C
D
Step 3
9
Min Spanning Tree Construction Prims Algorithm

• G (V, E) is a weighted connected graph
• Input G (V, E).
• Output ET, the set of edges composing a minimum
  spanning tree.

10
Prims Algorithm Insight
u is the vertex we have reached next we choose
the vertex closest to u by comparing the
weights w1, w2, w3, and w4.
v1
w1
v2
w2
u
w3
v3
w4
v4
11
Prims Algorithm Update

• After v is reached, two operations must be
  performed
• Mark v as visited, i.e. move it into VT
• For each remaining vertex v not in VT and
  connected to u by a shorter edge than vs current
  weight label, update vs weight label (V.Wght) by
  the weight of (u, v) and set vs previous label
  (V.Prev) to u.

12
Prims Algorithm Example
Tree Remaining Vertices NULL ltA, _, INFgt, ltB,
_, INFgt, ltC, _, INFgt, ltD, _, INFgt, ltE, _,
INFgt, ltF, _, INFgt ltA, _, 0gt ltB, A, 3gt ,
ltC,_,INFgt, ltD, _, INFgt, ltE, A, 6gt , ltF, A,
5gt ltB, A, 3gt ltC, B, 1gt, ltD, _, INFgt, ltE, A,
6gt, ltF, B, 4gt ltC, B, 1gt ltD, C, 6gt, ltE, A,
6gt, ltF, B, 4gt ltF, B, 4gt ltD, F, 5gt , ltE, F,
2gt ltE, F, 2gt ltD, F, 5gt ltD, F, 5gt
1
B
C
3
6
4
4
5
5
A
D
F
2
6
8
E
13
Prims Algorithm Example
1
B
C
6
3
4
4
5
5
D
A
F
2
6
8
E
14
Prims Algorithm Example
1
B
C
6
3
4
4
5
5
D
A
F
2
6
8
E
15
Prims Algorithm Example
1
B
C
6
3
4
4
5
5
D
A
F
2
6
8
E
16
Prims Algorithm Example
1
B
C
6
3
4
4
5
5
D
A
F
2
6
8
E
17
Prims Algorithm Example
1
B
C
6
3
4
4
5
5
D
A
F
2
6
8
E
18
Prims Algorithm Result Spanning Tree
1
B
C
3
4
5
D
A
F
2
E
19
Prims Algorithm Implementation

• A vertex v has four labels
• 1) The vertexs name, i.e., V
• 2) The name of the vertex U from which V was
  reached (V.Prev)
• 3) The weight of (u, v) (V.Wght). Remember V.Wght
  weight(u,v)
• 4) Visitation status of V (V.Visisted) this is a
  boolean variable, initially false, and set to
  true when V is visited.
• Unreached vertices have NULL as the value of
  V.Prev and INF as the value of V.Wght.

20
Prims Algorithm Implementation

• We assume that we have implemented a dynamic
  priority queue data structure (DynamicPriorityQueu
  e).
• If Q is a dynamic priority queue, then we can
  perform the following operations on it
• Q.Insert(v) // inserts an element v into Q
• Q.UpdatePriority(v) // moves v up or down the
  minimal heap
• Q.Pop() // removes the minimal element
• Q.Push(v) // inserts v into Q.