Embed Size (px)
DESCRIPTION
CS2420: Lecture 40. Vladimir Kulyukin Computer Science Department Utah State University. Outline. Graph Algorithms (Chapter 9). Spanning Trees. An edge e in G is redundant if, after the removal of e , G remains connected. - PowerPoint PPT Presentation
Citation preview
CS2420: Lecture 40
Vladimir Kulyukin
Computer Science Department
Utah State University
Outline
• Graph Algorithms (Chapter 9)
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 G’s connectivity without any redundant edges.
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 graph’s connectivity.
• In a weighted connected graph, the weight of a spanning tree is defined as the sum of the tree’s edges.
• A minimal (minimum) spanning tree of a weighted connected graph is a spanning tree of the smallest weight.
Spanning Tree: Example
A
B C
D
A
B C
D
Spanning Trees: Examples
A B
C D
5
1
2
3
Original Graph
A B
DC
1
2
3
Tree T1
Weight(T1) = 6
A B
C D
1
53
Tree T2
Weight(T2) = 9
A B
C D
1
25
Tree T3
Weight(T3) = 8
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.
Min Spanning Tree Construction: Example
A B
C D
5
1
2
3
Original Graph
A B
D
C
1
2Step 1
A B1
Step 2
A B
D
1
2
3
Step 3
Min Spanning Tree Construction: Prim’s Algorithm
• G = (V, E) is a weighted connected graph
• Input: G = (V, E).
• Output: ET, the set of edges composing a minimum spanning tree.
Prim’s Algorithm: Insight
v2
v3
v4
u*
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
w2
w3
w4
Prim’s 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 v’s current weight label, update v’s weight label (V.Wght) by the weight of (u, v) and set v’s previous label (V.Prev) to u.
Prim’s Algorithm: ExampleB C
A DF
E
3
1
6
86 2
445 5
Tree Remaining VerticesNULL <A, _, INF>, <B, _, INF>,
<C, _, INF>, <D, _, INF>,<E, _, INF>, <F, _, INF>
<A, _, 0> <B, A, 3> , <C,_,INF>,<D, _, INF>, <E, A, 6> ,<F, A, 5>
<B, A, 3> <C, B, 1>, <D, _, INF>,<E, A, 6>, <F, B, 4>
<C, B, 1> <D, C, 6>, <E, A, 6>,<F, B, 4>
<F, B, 4> <D, F, 5> , <E, F, 2>
<E, F, 2> <D, F, 5>
<D, F, 5>
Prim’s Algorithm: Example
B C
A DF
E
3
1
6
86 2
44
5 5
Prim’s Algorithm: Example
B C
A DF
E
3
1
6
86 2
44
5 5
Prim’s Algorithm: Example
B C
A DF
E
3
1
6
86 2
44
5 5
Prim’s Algorithm: Example
B C
A DF
E
3
1
6
86 2
44
5 5
Prim’s Algorithm: Example
B C
A DF
E
3
1
6
86 2
44
5 5
Prim’s Algorithm: Result Spanning Tree
B C
A DF
E
3
1
2
4
5
Prim’s Algorithm: Implementation
• A vertex v has four labels:– 1) The vertex’s 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.
Prim’s Algorithm: Implementation
• We assume that we have implemented a dynamic priority queue data structure (DynamicPriorityQueue).
• 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.
