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S. J. Shyu Chap. 1 Introduction 1 The Design and Analysis of Algorithms Chapter 1 Introduction S. J. Shyu

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Chap. 1 Introduction 2 Why do we need to study algorithms? We study algorithms so that we can learn strategies to design efficient algorithms. Besides, we study algorithms to understand the difficulty of designing good algorithms for some problems, namely NP-complete problems.

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S. J. Shyu Chap. 1 Introduction 3 Consider the sorting problem. Sorting problem: To sort a set of elements into increasing or decreasing order. 11, 7, 14, 1, 5, 9, 10 ↓sort 1, 5, 7, 9, 10, 11, 14 Insertion sort Quick sort

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S. J. Shyu Chap. 1 Introduction 4 Comparison of these two algorithms implemented on two computers : A bad algorithm implemented on a fast computer does not perform as well as a good algorithm implemented on a slow computer. Insertion Sort on IBM SP2 Quick Sort on PC486 When n >400, PC performs better than SP2.

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S. J. Shyu Chap. 1 Introduction 5 Insertion Sort on IBM SP2 vs. Quick Sort on PC

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S. J. Shyu Chap. 1 Introduction 6 High Speed Computer High Speed Computation (if no idea about algorithms) A good knowledge of algorithms makes your computer effective and efficient.

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S. J. Shyu Chap. 1 Introduction 7 Analysis of algorithms Measure the goodness of algorithms efficiency asymptotic notations: e.g. O(n 2 ) worst case average case amortized Measure the difficulty of problems NP-complete undecidable lower bound Is an algorithm optimal?

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S. J. Shyu Chap. 1 Introduction 8 Algorithms vs. Problems How fast can an algorithm be? (in solving a problem) Complexity Theory How hard can a problem be? (for any problem solving it) Computability Theory

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S. J. Shyu Chap. 1 Introduction 9 0/1 Knapsack Problem Given a set of n items where each item P i has a value V i, weight W i and a limit M of the total weights, we want to select a subset of items such that the total weight does not exceed M and the total value is maximized.

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S. J. Shyu Chap. 1 Introduction 10 0/1 Knapsack problem M (weight limit)=14 best solution: P 1, P 2, P 3, P 5 (optimal) 10 +5 +1 +3 = 19 This problem is NP-complete. P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 P7P7 P8P8 Value10519341117 Weight73310192215

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S. J. Shyu Chap. 1 Introduction 11 Traveling salesperson problem Given: A set of n planar points Find: A closed tour which includes all points exactly once such that its total length is minimized. This problem is NP-complete.

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S. J. Shyu Chap. 1 Introduction 12 Partition problem Given: A set of positive integers S Find: S 1 and S 2 such that S 1 S 2 = , S 1 S 2 = S, and (Partition into S 1 and S 2 such that the sum of S 1 is equal to that of S 2 ) e.g. S={1, 7, 10, 4, 6, 8, 13} S 1 ={1, 10, 4, 8, 3} S 2 ={7, 6, 13} This problem is NP-complete.

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S. J. Shyu Chap. 1 Introduction 13 Given: an art gallery Determine: min # of guards and their placements such that the entire art gallery can be monitored. NP-complete Art gallery problem

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S. J. Shyu Chap. 1 Introduction 14 Set Cover Problem Given: some subsets from a set S Determine: min # of subsets whose union covers S NP-complete 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 ABCDABCD a b c d e f

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S. J. Shyu Chap. 1 Introduction 15 Knapsack problem P 5, P 2, P 1, P 8, W 1 + 3 + 7=11 +15 > 14 (= M ) X 1 1 1 3/15 V 3 + 5 + 10 + (3/15) 17 = 21.4 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 P7P7 P8P8 Value10519341117 Weight73310192215 V/WV/W 10/75/31/39/103/14/911/2217/15 P5P5 P2P2 P1P1 P8P8 P4P4 P7P7 P6P6 P3P3 Value35101791141 Weight13715102293 V/WV/W 31.671.431.130.90.50.440.33

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S. J. Shyu Chap. 1 Introduction 16 Minimal Spanning Trees Given a weighted graph G, a spanning tree T is a tree where all vertices of G are vertices of T and if an edge of T connects V i and V j, its weight is the weight of e ( V i, V j ) in G. A minimal spanning tree of G is a spanning tree of G whose total weight is minimized.

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S. J. Shyu Chap. 1 Introduction 17 Minimum Spanning Trees solved by greedy method # of possible spanning trees for n points: n n -2 n =10→10 8, n =100→10 196

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S. J. Shyu Chap. 1 Introduction 18 Convex hull Given a set of planar points, find a smallest convex polygon which contains all points. It is not obvious to find a convex hull by examining all possible solutions divide-and-conquer

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S. J. Shyu Chap. 1 Introduction 19 One-center problem Given a set of planar points, find a smallest circle which contains all points. Prune-and-search

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S. J. Shyu Chap. 1 Introduction 20 Easy Problems vs. Difficult Problems NP problems NP-complete problems P problems Sorting, Minimal Spanning Tree, Shortest Path, Convex Hall, Knapsack, Longest Common Subsequence, … Traveling salesperson problem, Gallery Guards, 0/1 Knapsack, Multiple Sequence Alignment, …

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S. J. Shyu Chap. 1 Introduction 21 Many strategies, such as the greedy approach divide-and-conquer approach branch and bound prune and search dynamic programming and so on will be introduced in this course.

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S. J. Shyu Chap. 1 Introduction 22 Strategies for P problems Greedy Dynamic Programming Divide and Conquer Prune and Search …

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S. J. Shyu Chap. 1 Introduction 23 Strategies for NP-hard problems Optimization Branch and bound Dynamic programming Exhausted search … Approximation Simple heuristics with a guaranteed error ratio (e.g. Greedy, Random, …) Sophisticated heuristics (e.g. Greedy, 2-opt, k-opt, …) Meta-heuristics

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