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Programming Practicum Day 3: Problem Solving with Graphs Aaron Tan NUS School of Computing
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2 Contents Review of Day 2 problems Graphs [Programming Practicum, December 2009]
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3 Day 2 Ex 1: Max. Subseq. Sum (1/3) Given list 3 [Programming Practicum, December 2009] -271-939 Answer = 11
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4 Day 2 Ex 1: Max. Subseq. Sum (2/3) An algorithm 4 [Programming Practicum, December 2009] 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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5 Day 2 Ex 1: Max. Subseq. Sum (3/3) Another algorithm 5 [Programming Practicum, December 2009] 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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6 Day 2 Ex 3: Finding k th 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. 6 [Programming Practicum, December 2009]
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7 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.) [Programming Practicum, December 2009]
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8 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. [Programming Practicum, December 2009]
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9 Graphs: Example A road network, where nodes represent cities and edges represent costs (distance, or time). [Programming Practicum, December 2009] A C D B EF 5 51 2 3 1 3 1 4
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10 Graphs: Representation One simple representation is the 2-dimensional array, known as adjacency matrix. [Programming Practicum, December 2009] A C D B EF 5 51 2 3 1 3 1 4 ABCDEF A5 B113 C351 D E2 F4 (There are other graph representations that can give rise to faster algorithms. We introduce adjacency matrix for its simplicity.)
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11 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? [Programming Practicum, December 2009] A C D B EF 5 51 2 3 1 3 1 4 ABCDEF A5 B113 C351 D E2 F4
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12 Matrix Multiplication (1/2) Multiplication on square matrices (2-dimensional arrays) To compute C = A B, where A, B, C are matrices c i,j = (a i,0 b 0,j ) + (a i,1 b 1,j ) +... + (a i,n-1 b n-1,j ) c i,j is sum of terms produced by multiplying the elements of A’s row i with B’s column j. [Programming Practicum, December 2009] 2301 1213 0210 3122 0210 0010 2421 3210 3660 111281 2441 1018102 =
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13 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. [Programming Practicum, December 2009]
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14 THE END
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