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Chapter 9: Graph Algorithms. Graph ADT. Topological Sort. Shortest Path. Maximum Flow. Minimum Spanning Tree. Depth-First Search. Undecidability. NP-Completeness. CS 340. Page 143. 4. 3. 6. 6. 5. 2. 2. 3. 3. 4. 5. The Graph Abstract Data Type.

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Presentation Transcript
slide1

Chapter 9: Graph Algorithms

  • Graph ADT
  • Topological Sort
  • Shortest Path
  • Maximum Flow
  • Minimum Spanning Tree
  • Depth-First Search
  • Undecidability
  • NP-Completeness

CS 340

Page 143

slide2

4

3

6

6

5

2

2

3

3

4

5

The Graph Abstract Data Type

A graph G = (V, E) consists of a set of vertices, V, and a set of edges, E, each of which is a pair of vertices. If the edges are ordered pairs of vertices, then the graph is directed.

Directed, unweighted, cyclic, strongly connected, loopless graph

(length of longest simple cycle: 7)

Undirected, unweighted, unconnected, loopless graph

(length of longest simple path: 2)

Directed, unweighted, acyclic, weakly connected, loopless graph

(length of longest simple path: 6)

Undirected, unweighted, connected graph, with loops

(length of longest simple path: 4)

Directed,

weighted, cyclic, weakly connected, loopless graph

(weight of longest simple cycle: 27)

CS 340

Page 144

slide3

C

C

C

4

3

B

B

B

A

A

A

6

6

5

E

E

E

D

D

D

2

2

3

F

F

F

3

H

H

H

4

5

G

G

G

Graph Representation Option 1: Adjacency Matrix

The Problem: Most graphs are sparse (i.e., most vertex pairs are not edges), so the memory requirement is excessive: (V2).

CS 340

Page 145

slide4

C

C

C

4

3

B

B

B

A

A

A

6

6

5

E

E

E

D

D

D

2

2

3

F

F

F

3

H

H

H

4

5

G

G

G

Graph Representation Option 2: Adjacency Lists

A

A

B

G

A

B

D

A

B

3

D

6

E

6

B

A

C

D

B

D

B

C

4

C

B

C

B

C

E

F

B

5

E

2

D

B

D

G

D

G

3

E

E

G

E

C

E

H

3

F

G

F

A

F

G

G

4

G

A

E

F

H

G

G

H

A

2

H

G

H

E

H

G

5

CS 340

Page 146

slide5

Topological Sort

A topological sort of an acyclic directed graph orders the vertices so that if there is a path from vertex u to vertex v, then vertex v appears after vertex u in the ordering.

MATH 120

One topological sort of the course prerequisite graph at left:

CS 111, MATH 120, CS 140, MATH 125, CS 150, MATH 224, CS 234, CS 240, ECE 282, CS 312, CS 325, MATH 150, MATH 152, STAT 380, CS 321, MATH 250, MATH 321, CS 314, MATH 423, CS 340, CS 425, ECE 381, CS 434, ECE 482, CS 330, CS 382, CS 423, CS 438, CS 454, CS 447, CS 499, CS 482, CS 456, ECE 483

MATH 125

CS 140

MATH 150

MATH 224

ECE 282

CS 111

CS 150

ECE 381

MATH 152

MATH 423

ECE 482

MATH 250

CS 234

CS 240

STAT 380

ECE 483

MATH 321

CS 312

CS 321

CS 325

CS 482

CS 340

CS 434

CS 382

CS 314

CS 330

CS 425

CS 447

CS 423

CS 499

CS 438

CS 456

CS 454

CS 340

Page 147

slide6

Indegree of A:

0

B

C

Indegree of B:

1

0

0

0

Indegree of C:

3

3

2

2

1

0

0

Indegree of D:

1

1

1

0

0

A

E

F

D

Indegree of E:

1

0

Indegree of F:

4

4

3

2

2

1

1

1

Indegree of G:

1

0

0

0

0

0

Indegree of H:

3

3

2

1

1

1

0

0

G

H

I

Indegree of I:

0

0

0

Output:

A

Topological Sort Algorithm

  • Place all indegree-zero vertices in a list.
  • While the list is non-empty:
    • Output an element v in the indegree-zero list.
    • For each vertex w with (v,w) in the edge set E:
      • Decrement the indegree of w by one.
      • Place w in the indegree-zero list if its new indegree is 0.

0

E

I

B

D

G

C

H

F

CS 340

Page 148

slide7

Shortest Paths

The shortest path between two locations is dependent on the function that is used to measure a path’s “cost”.

Does the path “cost” less if the amount of boring scenery is minimized?

Does the path “cost” less if the total distance traveled is minimized?

Does the path “cost” less if the amount of traffic encountered is minimized?

In general, then, how do you find the shortest path from a specified vertex to every other vertex in a directed graph?

CS 340

Page 149

slide8

(1,A)

(1,A)

(2,B)

B

B

B

B

B

C

C

C

C

C

(1,A)

(1,A)

(2,E)

(0,-)

(0,-)

(0,-)

A

A

A

A

A

E

E

E

E

E

F

F

F

F

F

D

D

D

D

D

G

G

G

G

G

H

H

H

H

H

I

I

I

I

I

(1,A)

(1,A)

(2,E)

(1,A)

(2,B)

(1,A)

(2,B)

(1,A)

(2,E)

(1,A)

(2,E)

(0,-)

(3,F)

(0,-)

(3,F)

(1,A)

(1,A)

(4,D)

(2,E)

(2,E)

Shortest Path Algorithm: Unweighted Graphs

Use a breadth-first search, i.e., starting at the specified vertex, mark each adjacent vertex with its distance and its predecessor, until all vertices are marked. Algorithm’s time complexity: O(E+V).

CS 340

Page 150

slide9

Shortest Path Algorithm:

Weighted Graphs With No Negative Weights

(7,A)

(27,E)

(7,A)

(,-)

(,-)

(,-)

19

19

19

19

19

19

B

B

B

B

B

B

C

C

C

C

C

C

(,-)

7

7

7

7

7

7

(6,A)

(19,E)

(,-)

21

21

21

21

21

21

5

5

5

5

5

5

(6,A)

(,-)

(,-)

(,-)

(,-)

(0,-)

(0,-)

(0,-)

6

6

6

6

6

6

13

13

13

13

13

13

2

2

2

2

2

2

A

A

A

A

A

A

E

E

E

E

E

E

F

F

F

F

F

F

D

D

D

D

D

D

8

8

8

8

8

8

16

16

16

16

16

16

2

2

2

2

2

2

4

4

4

4

4

4

3

3

3

3

3

3

(8,A)

(,-)

G

G

G

G

G

G

H

H

H

H

H

H

I

I

I

I

I

I

(8,A)

(,-)

(,-)

(,-)

8

8

8

8

8

8

3

3

3

3

3

3

(22,E)

(,-)

(,-)

(7,A)

(26,B)

(7,A)

(7,A)

(26,B)

(26,B)

(,-)

(6,A)

(18,H)

(,-)

(,-)

(6,A)

(19,E)

(6,A)

(19,E)

(0,-)

(0,-)

(0,-)

(8,A)

(,-)

(8,A)

(,-)

(8,A)

(,-)

(16,G)

(16,G)

(22,E)

Use Dijkstra’s Algorithm, i.e., starting at the specified vertex, finalize the unfinalized vertex whose current cost is minimal, and update each vertex adjacent to the finalized vertex with its (possibly revised) cost and predecessor, until all vertices are finalized. Algorithm’s time complexity: O(E+V2) = O(V2).

CS 340

Page 151

slide10

(7,A)

(7,A)

(7,A)

(25,D)

(25,D)

(26,B)

(20,F)

(20,F)

(20,F)

(6,A)

(18,H)

(6,A)

(18,H)

(6,A)

(18,H)

(0,-)

(0,-)

(0,-)

(8,A)

(23,D)

(8,A)

(23,D)

(8,A)

(,-)

19

19

19

(16,G)

(16,G)

(16,G)

B

B

B

B

C

C

C

C

7

7

7

21

21

21

5

5

5

6

6

6

13

13

13

2

2

2

A

A

A

A

E

E

E

E

F

F

F

F

D

D

D

D

(7,A)

(25,D)

19

8

8

8

16

16

16

2

2

2

4

4

4

3

3

3

7

G

G

G

G

H

H

H

H

I

I

I

I

21

5

(20,F)

(6,A)

(18,H)

8

8

8

3

3

3

(0,-)

6

13

2

8

16

2

4

3

(8,A)

(23,D)

8

3

(16,G)

Note that this algorithm would not work for graphs with negative weights, since a vertex cannot be finalized when there might be some negative weight in the graph which would reduce a particular path’s cost.

CS 340

Page 152

slide11

Shortest Path Algorithm:

Weighted Graphs With Negative Weights

(,-)

(13,A)

(13,A)

(,-)

(,-)

(,-)

15

15

15

B

C

B

C

B

C

13

13

13

-9

-9

-9

7

(,-)

7

(,-)

7

(,-)

(,-)

(,-)

(25,A)

(,-)

(25,A)

(23,E)

(0,-)

25

-2

6

(0,-)

25

-2

6

(0,-)

25

-2

6

A

E

F

D

A

E

F

D

A

E

F

D

26

11

-4

8

26

11

-4

8

26

11

-4

8

-9

-9

-9

G

H

I

G

H

I

G

H

I

(,-)

(,-)

(26,A)

(,-)

(26,A)

(,-)

6

-3

6

-3

6

-3

(,-)

(,-)

(,-)

(13,A)

(13,A)

(13,A)

(28,B)

(28,B)

(28,B)

15

15

15

B

C

B

C

B

C

13

13

13

-9

-9

-9

7

(,-)

7

(,-)

7

(25,A)

(20,B)

(25,A)

(20,B)

(25,A)

(20,B)

(26,F)

(0,-)

25

-2

6

(0,-)

25

-2

6

(0,-)

25

-2

6

A

E

F

D

A

E

F

D

A

E

F

D

26

11

-4

8

26

11

-4

8

26

11

-4

8

-9

-9

-9

G

H

I

G

H

I

G

H

I

(26,A)

(,-)

(26,A)

(,-)

(26,A)

(,-)

6

-3

6

-3

6

-3

(,-)

(32,G)

(31,F)

Use a variation of Dijkstra’s Algorithm without using the concept of vertex finalization, i.e., starting at the specified vertex, update each vertex adjacent to the current vertex with its (possibly revised) cost and predecessor, placing each revised vertex in a queue. Continue until the queue is empty. Algorithm’s time complexity: O(EV).

CS 340

Page 153

slide12

(13,A)

(13,A)

(13,A)

(28,B)

(28,B)

(17,D)

15

15

15

B

C

B

C

B

C

13

13

13

-9

-9

-9

7

7

7

(25,A)

(20,B)

(26,F)

(22,H)

(20,B)

(26,F)

(22,H)

(20,B)

(26,F)

(0,-)

25

-2

6

(0,-)

25

-2

6

(0,-)

25

-2

6

A

E

F

D

A

E

F

D

A

E

F

D

26

11

-4

8

26

11

-4

8

26

11

-4

8

-9

-9

-9

G

H

I

G

H

I

G

H

I

(26,A)

(,-)

(26,A)

(,-)

(26,A)

(34,D)

6

-3

6

-3

6

-3

(31,F)

(31,F)

(31,F)

(13,A)

(13,A)

(13,A)

(17,D)

(17,D)

(17,D)

15

15

15

B

C

B

C

B

C

13

13

13

-9

-9

-9

7

7

7

(22,H)

(20,B)

(26,F)

(22,H)

(20,B)

(26,F)

(22,H)

(20,B)

(26,F)

(0,-)

25

-2

6

(0,-)

25

-2

6

(0,-)

25

-2

6

A

E

F

D

A

E

F

D

A

E

F

D

26

11

-4

8

26

11

-4

8

26

11

-4

8

-9

-9

-9

G

H

I

G

H

I

G

H

I

(26,A)

(34,D)

(26,A)

(34,D)

(26,A)

(34,D)

6

-3

6

-3

6

-3

(31,F)

(31,F)

(31,F)

Note that this algorithm would not work for graphs with negative-cost cycles. For example, if edge IH in the above example had cost -5 instead of cost -3, then the algorithm would loop indefinitely.

CS 340

Page 154

slide13

19

B

C

35

21

35

21

27

17

23

A

E

F

D

16

18

12

14

12

30

G

H

I

8

15

Maximum Flow

Assume that the directed graph G = (V, E) has edge capacities assigned to each edge, and that two vertices s and t have been designated the source and sink nodes, respectively.

We wish to maximize the “flow” from s to t by determining how much of each edge’s capacity can be used so that at each vertex, the total incoming flow equals the total outgoing flow.

This problem relates to such practical applications as Internet routing and automobile traffic control.

In the graph at left, for instance, the total of the incoming capacities for node C is 40 while the total of the outgoing capacities for node C is 35.

Obviously, the maximum flow for this graph will have to “waste” some of the incoming capacity at node C.

Conversely, node B’s outgoing capacity exceeds its incoming capacity by 5, so some of its outgoing capacity will have to be wasted.

CS 340

Page 155

slide14

19

0

19

B

C

B

C

B

C

35

0

35

21

35

0

0

21

35

21

0

21

27

17

23

0

0

0

27

17

23

A

E

F

D

A

E

F

D

A

E

F

D

16

0

16

18

12

14

0

0

0

18

12

14

12

0

12

30

0

30

G

H

I

G

H

I

G

H

I

8

15

0

0

8

15

19

0

19

B

C

B

C

B

C

21

35

21

14

21

21

35

21

0

0

35

21

0

21

27

17

23

0

0

21

27

17

2

A

E

F

D

A

E

F

D

A

E

F

D

16

0

16

21

18

12

14

0

0

0

18

12

14

12

0

12

30

0

30

G

H

I

G

H

I

G

H

I

8

15

0

0

8

15

Maximum Flow Algorithm

To find a maximum flow, keep track of a flow graph and a residual graph, which keep track of which paths have been added to the flow.

Keep choosing paths which yield maximal increases to the flow; add these paths to the flow graph, subtract them from the residual graph, and add their reverse to the residual graph.

Original Graph

Flow Graph

Residual Graph

CS 340

Page 156

slide15

Original Graph

Flow Graph

Residual Graph

19

0

19

B

C

B

C

B

C

21

17

35

21

14

21

21

35

21

17

0

18

17

21

17

4

27

17

23

17

17

21

10

0

A

E

F

D

A

E

F

D

2

A

E

F

D

16

0

17

16

21

17

12

12

18

14

0

0

0

18

12

14

0

12

30

0

30

G

H

I

G

H

I

G

H

I

8

15

0

0

8

15

14

19

14

B

C

B

C

B

C

5

35

31

35

35

0

21

21

35

21

31

0

4

17

21

17

4

27

17

23

17

17

21

10

0

A

E

F

D

A

E

F

D

2

A

E

F

D

16

0

17

16

21

17

12

12

18

14

0

0

0

18

12

14

0

12

30

0

30

G

H

I

G

H

I

G

H

I

8

15

0

0

8

15

14

19

14

B

C

B

C

B

C

5

35

31

35

35

0

21

21

35

21

31

0

4

17

21

17

4

27

17

23

17

17

21

10

0

A

E

F

D

A

E

F

D

2

A

E

F

D

16

12

17

4

21

17

12

12

12

18

14

12

0

0

6

12

14

12

0

12

30

12

18

12

12

G

H

I

G

H

I

G

H

I

8

15

0

12

8

3

12

CS 340

Page 157

slide16

Original Graph

Flow Graph

Residual Graph

14

19

14

B

C

B

C

B

C

5

35

31

35

35

0

21

21

35

21

31

0

4

17

21

17

4

27

17

23

25

17

21

2

0

2

A

E

F

D

A

E

F

D

A

E

F

D

16

12

25

4

21

17

20

18

12

14

12

8

6

6

4

6

12

4

8

4

30

20

8

10

12

12

8

G

H

I

G

H

I

G

H

I

8

15

8

12

0

3

12

8

14

19

14

B

C

B

C

B

C

5

35

31

35

35

0

21

21

35

21

31

0

4

17

21

17

4

27

17

23

25

17

21

2

0

2

A

E

F

D

A

E

F

D

A

E

F

D

16

16

25

0

21

17

24

18

12

14

16

12

10

2

0

2

12

8

4

8

30

24

12

6

16

8

12

G

H

I

G

H

I

G

H

I

8

15

8

12

0

3

12

8

Thus, the maximum flow for the graph is 76.

CS 340

Page 158

slide17

B

C

B

C

D

E

H

3

D

E

H

A

A

F

G

F

G

6

B

B

B

B

B

C

C

C

C

C

5

5

4

4

4

3

3

3

3

3

D

D

D

D

D

E

E

E

E

E

H

H

H

H

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Minimum Spanning Tree: Kruskal’s Algorithm

When a large set of vertices must be interconnected (i.e., “spanning”) and there are several possible means of doing so, redundancy can be eliminated (i.e., “tree”) and costs can be reduced (i.e., “minimum”) by employing a minimum spanning tree.

Kruskal’s Algorithm accomplishes this by just selecting minimum-cost edges as long as they don’t form cycles.

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MINIMUM SPANNING TREE

ORIGINAL GRAPH

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MINIMUM SPANNING TREE

Minimum Spanning Tree: Prim’s Algorithm

An alternative to Kruskal’s Algorithm is Prim’s Algorithm, a variation of Dijkstra’sAlgorithm that starts with a minimum-cost edge, and builds the tree by adding minimum-cost edges as long as they don’t create cycles. Like Kruskal’s Algorithm, Prim’s Algorithm is O(ElogV)

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ORIGINAL GRAPH

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slide19

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Original Graph

Depth-First Search

(Solid lines are part of depth-first spanning tree; dashed lines are visits to previously marked vertices)

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Depth-First Spanning Tree

Depth-First Search

A convenient means to traverse a graph is to use a depth-first search, which recursively visits and marks the vertices until all of them have been traversed.

Such a traversal provides a means by which several significant features of a graph can be determined.

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slide20

Depth-First Search Application: Articulation Points

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Depth-First Search (with nodes numbered as they’re visited)

Original Graph

Nodes also marked with lowest-numbered vertex reachable via zero or more tree edges, followed by at most one back edge

A vertex v in a graph G is called an articulation point if its removal from G would cause the graph to be disconnected.

Such vertices would be considered “critical” in applications like networks, where articulation points are the only means of communication between different portions of the network.

A depth-first search can be used to find all of a graph’s articulation points.

The only articulation points are the root (if it has more than one child) and any other node v with a child whose “low” number is at least as large as v’s “visit” number. (In this example: nodes B, C, E, and G.)

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Depth-First Search Application: Euler Circuits

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Original Graph

After Removing First DFS Cycle: BCDFB

After Removing Second DFS Cycle: ABGHA

After Removing Third DFS Cycle: FHIF

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After Removing Fourth DFS Cycle: FEJF

An Euler circuit of a graph G is a cycle that visits every edge exactly once.

Such a cycle could be useful in applications like networks, where it could provide an efficient means of testing whether each network link is up.

A depth-first search can be used to find an Euler circuit of a graph, if one exists. (Note: An Euler circuit exists only if the graph is connected and every vertex has even degree.)

Splicing the first two cycles yields cycle A(BCDFB)GHA.

Splicing this cycle with the third cycle yields ABCD(FHIF)BGHA.

Splicing this cycle with the fourth cycle yields ABCD(FEJF)HIFBGHA

Note that this traversal takes O(E+V) time.

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Depth-First Search Application: Strong Components

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Third leg of depth-first search

Number the vertices according to a depth-first postorder traversal, and reverse the edges.

Depth-first search of the revised graph, always starting at the vertex with the highest number.

First leg of depth-first search

Second leg of depth-first search

Original Directed Graph

A subgraph of a directed graph is a strong component of the graph if there is a path in the subgraph from every vertex in the subgraph to every other vertex in the subgraph.

In network applications, such a subgraph could be used to ensure intercommunication.

A depth-first search can be used to find the strong components of a graph.

The trees in the final depth-first spanning forest form the set of strong components.

In this example, the strong components are: { C }, { B, F, I, E }, { A }, { D, G, K, J }, and { H }.

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slide23

Program P

Program H

YES (if P halts on I)

Input I

NO (if P loops forever on I)

Program X

YES

Program H

loop forever

Program P

NO

halt

Undecidable Problems

Some problems are difficult to solve on a computer, but some are impossible!

Example: The Halting Problem

We’d like to build a program H that will test any other program P and any input I and answer “yes” if P will terminate normally on input I and “no” if P will get stuck in an infinite loop on input I.

If this program H exists, then let’s make it the subroutine for program X, which takes any program P, and runs it through subroutine H as both the program and the input. If the result from the subroutine is “yes”, the program X enters an infinite loop; otherwise it halts.

Note that if we ran program X with itself as its input, then it would halt only if it loops forever, and it would loop forever only if it halts!?!?

This contradiction means that our assumption that we could build program H was false, i.e., that the halting problem is undecidable.

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slide24

P and NP Problems

A problem is said to be a P problem if it can be solved with a deterministic, polynomial-time algorithm. (Deterministic algorithms have each step clearly specified.)

A problem is said to be an NP problem if it can be solved with a nondeterministic, polynomial-time algorithm. In essence, at a critical point in the algorithm, a decision must be made, and it is assumed that a magical “choice” function always chooses correctly.

Example: Satisfiability

Given a set of n boolean variables b1, b2, …, bn and a boolean function f(b1, b2, …, bn).

Problem: Are there values that can be assigned to the variables so that the function would evaluate to TRUE?

For example, is the function:

(b1 OR b2) AND (b3 OR NOT b4) ever TRUE?

To try every combination takes exponential time, but a nondeterministic solution is polynomial-time:

for (i=1; i<=n; i++)

b[i] = choice(TRUE,FALSE);

if (f(b[1],b[2],…,b[n])==TRUE)

cout << “SATISFIABLE”;

else

cout << “UNSATISFIABLE”;

So Satisfiability is an NP problem.

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The Knapsack Problem

Given a set of n valuable jewels J1, J2, …, Jn with respective weights w1, w2, …, wn, and respective prices p1, p2, …, pn, as well as a knapsack capable of supporting a total weight of M.

Problem: Is there a way to pack at least T dollars worth of jewels, without exceeding the weight capacity of the knapsack?

(It’s not as easy as it sounds; three lightweight $1000 jewels might be preferable to one heavy $2500 jewel, for instance.)

A nondeterministic polynomial-time solution:

totalWorth = 0;

totalWeight = 0;

for (i=1; i<=n; i++)

{

b[i] = choice(TRUE,FALSE);

if (b[i]==TRUE)

{

totalWorth+= p[i];

totalWeight+= w[i];

}

}

if ((totalWorth >= T) && (totalWeight <= M))

cout << “YAHOO! I’M RICH!”;

else

cout << “@#$&%!”;

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NP-Complete Problems

Note that all P problems are automatically NP problems, but it’s unknown if the reverse is true.

The hardest NP problems are the NP-complete problems. An NP-complete problem is one to which every NP problem can be “polynomially reduced”. In other words, an instance of any NP problem can be transformed in polynomial time into an instance of the NP-complete problem in such a way that a solution to the first instance provides a solution to the second instance, and vice versa.

For example, consider the following two problems:

The Hamiltonian Circuit Problem: Given an undirected graph, does it contain a cycle that passes through every vertex in the graph exactly once?

The Spanning Tree Problem: Given an undirected graph and a positive integer k, does the graph have a spanning tree with exactly k leaf nodes?

Assume that the Hamiltonian Circuit Problem is known to be NP-complete. To prove that the Spanning Tree Problem is NP-complete, we can just polynomially reduce the Hamiltonian Circuit Problem to it.

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Let G = (V, E) be an instance of the Hamiltonian Circuit Problem.

Let x be some arbitrary vertex in V.

Define Ax = {y in V (x,y) is in E}.

Define an instance G = (V, E) of the Spanning Tree Problem as follows:

Let V = V  {u, v, w}, where u, v, and w are new vertices that are not in V.

Let E = E  {(u, x)}  {(y, v  y is inAx}  {(v, w)}.

Finally, let k = 2.

If G has a Hamiltonian circuit, then G has a spanning tree with exactly k leaf nodes.

Proof:G’s Hamiltonian circuit can be written as (x, y1, z1, z2, …, zm, y2, x), where V = {x, y1, y2, z1, z2, …, zm} and Ax contains y1 and y2.

Thus, (u, x, y1, z1, z2, …, zm, y2, v, w) forms a Hamiltonian path (i.e., a spanning tree with exactly two leaf nodes) of G.

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slide28

Conversely, if G has a spanning tree with exactly two leaf nodes, then G has a Hamiltonian circuit.

Proof: The two-leaf spanning tree of G must have u and w as its leaf nodes (since they’re the only vertices of degree one in G ).

Notice that the only vertex adjacent to w in G is v and the only vertex adjacent to u in G is x.

Also, the only vertices adjacent to v in G are those in Ax  {w}, and the only vertices adjacent to x in G are those in Ax  {u}.

So the Hamiltonian path in G must take the form (u, x, y1, z1, z2, …, zm, y2, v, w) where V = {x, y1, y2, z1, z2, …, zm} and Ax contains y1 and y2.

Thus, since y2 is adjacent to x, (x, y1, z1, z2, …, zm, y2, x) must be a Hamiltonian circuit in G.

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