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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 20% • Two programming assignments 5+5% • Midterm 30% • Final 40%

  5. Home Work • Problem sets • weekly • handed in/out Mondays • Due next Monday • 1.4-1 p.17 • 1.4-2 p.17

  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] & 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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