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Analysis and Design of Algorithms

Analysis and Design of Algorithms. An algorithm is a method of solving problem (on a computer) Problem example: given a set of points on the plane find the closest pair Algorithm: find distance between all pairs Can we do it faster?. Combinatorial Problems. Closest pair

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Analysis and Design of Algorithms

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  1. Analysis and Design of Algorithms • An algorithm is a method of solving problem (on a computer) • Problem example: • given a set of points on the plane • find the closest pair • Algorithm: • find distance between all pairs • Can we do it faster?

  2. Combinatorial Problems • Closest pair • O(n^2) algorithm • TSP • O(n!) algorithm • too slow • difficult problem

  3. Course Overview • General algorithmic methods • divide and conquer, greedy algorithms, dynamic programming • Data structures • hashing, priority queues, binary search trees, binomial heaps • Combinatorial problems • MST, TSP, Vertex/Set Cover, Matrix • Computational Complexity • NP-completeness, reducibility, approximation • Cormen-Leiserson-Rivest Introduction to Algorithms

  4. Grading • Home work 1/3 • problems from Cormen ... • two programming assignments • 4 Quizes 1/3 • 4520 Final 1/3 • 6520 Project 1/3

  5. Home Work • Problem sets • weekly • handed in/out Tuesdays (usually) • Extra-credit problems! • Due next Tuesday • 1.4-1 p.17 / 1.2-2 p.13 • 1.4-2 p.17 / 1.2-2 p.13

  6. Sorting • Input: sequence of numbers Output: a sorted sequence • Insertion-Sort for j = 2 to n do current=A[j] i = j - 1 while i > 0 & A[i] > current do A[i + 1] = A[i] i = i - 1 A[i + 1] = current

  7. How it works • Insertion-Sort for j = 2 to n do current = A[j] next current i = j - 1 go left while i > 0 A[i] & A[i] > current do find place for current A[i + 1] = A[i] shift sorted right i = i - 1 go left A[i + 1] = current put current in place

  8. Running Time • Depends on • input size • input quality (partially ordered) • Kinds of analysis • Worst case (standard) • Average case (sometimes) • Best case (never)

  9. Asymptotic Analysis • Ignore machine dependent constants • Look at growth of T(n) while n   • O - notation • O(n^3)>O(n^2)

  10. Insertion Sort Analysis • Worst Case O(n^2) • Average Case O(n^2) • Can we do better? • New paradigms

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