Programming Practicum Day 3: Problem Solving with Graphs Aaron Tan NUS School of Computing.

Programming Practicum

Day 3: Problem Solving with GraphsAaron Tan

NUS School of Computing

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Contents Review of Day 2 problems Graphs

[Programming Practicum, December 2009]

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Day 2 Ex 1: Max. Subseq. Sum (1/3) Given list

3[Programming Practicum, December 2009]

-2 7 1 -9 3 -1 9 -1

Answer = 11

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Day 2 Ex 1: Max. Subseq. Sum (2/3) An algorithm


What is the time complexity?

public static int maxSubseqSum(int[] arr) { int sum, maxSum; maxSum = 0; for (int i=0; i<arr.length; i++) { sum = 0; for (int j=i; j<arr.length; j++) { sum += arr[j]; if (sum > maxSum) maxSum = sum; } } return maxSum;}

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Day 2 Ex 1: Max. Subseq. Sum (3/3) Another algorithm


What is the time complexity?

public static int maxSubseqSum(int[] arr) { int sum, maxSum; sum = maxSum = 0; for (int i=0; i<arr.length; i++) { sum += arr[i]; if (sum < 0) sum = 0; else if (sum > maxSum) maxSum = sum; } return maxSum;}

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Day 2 Ex 3: Finding kth Smallest Element Belongs to the classic selection problem Many algorithms available We adapt the heapsort:

Create min-heap instead of max-heap After heapify, instead of swapping the top elements

n-1 times (n = array size), we need only to swap it k-1 times and sift-down.

Heapify: O(n) Swapping k-1 times and sift-down: O(k lg n). If k <

n/(lg n), then this is O(n) as well.


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Graphs A graph consists of a set of nodes (vertices)

connected by edges. Undirected graphs: edges are undirected. Directed graphs: directed edges from a node to

another. Edges may be weighted, that is, each of them

contains a value (weight). Graph is a very important data structure that

supports many applications (Shortest-path, minimum spanning tree, etc.)


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Graphs: Degrees Directed graphs:

In-degree of a node: Number of edges pointing towards that node

Out-degree of a node: Number of edges pointing away from that node.

Undirected graphs: Degree of a node: Number of edges connected to that

node.


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Graphs: Example A road network, where nodes represent cities

and edges represent costs (distance, or time).


A

C

D

B

EF

5

51

2

3

1

31

4

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Graphs: Representation One simple representation is the 2-dimensional

array, known as adjacency matrix.


A

C

D

B

EF

5

51

2

3

1

31

4

A B C D E F

A 5

B 1 1 3

C 3 5 1

D

E 2

F 4

(There are other graph representations that can give rise to faster algorithms. We introduce adjacency matrix for its simplicity.)

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Graphs: Exploration How do you determine the in-degree and out-

degree of each node by using the adjacency matrix representation of the directed graph?


A

C

D

B

EF

5

51

2

3

1

31

4

A B C D E F

A 5

B 1 1 3

C 3 5 1

D

E 2

F 4

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Matrix Multiplication (1/2) Multiplication on square matrices (2-dimensional

arrays) To compute C = A B, where A, B, C are matrices

ci,j = (ai,0 b0,j ) + (ai,1 b1,j ) + . . . + (ai,n-1 bn-1,j ) ci,j is sum of terms produced by multiplying the elements of A’s row i with B’s column j.


2 3 0 1

1 2 1 3

0 2 1 0

3 1 2 2

0 2 1 0

0 0 1 0

2 4 2 1

3 2 1 0

3 6 6 0

11

12

8 1

2 4 4 1

10

18

10 2

=

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Matrix Multiplication (2/2) In CS1101, you were given Matrices.java and told

to complete the matrixProduct() method. Download Matrices.java from the Practicum

website.


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THE END

Programming Practicum Day 3: Problem Solving with Graphs Aaron Tan NUS School of Computing.

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