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DS.I.1

DS.I.1. CSE 373: Data Structures and Algorithms. Autumn Quarter 2000. Instructor: Prof. Linda Shapiro Email: shapiro@cs.washington.edu Tas: Valentin Razmov Ann Li Karen Liu. Purpose: to study the fundamental data

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DS.I.1

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  1. DS.I.1 CSE 373: Data Structures and Algorithms Autumn Quarter 2000 Instructor: Prof. Linda Shapiro Email: shapiro@cs.washington.edu Tas: Valentin Razmov Ann Li Karen Liu Purpose: to study the fundamental data structures and algorithms used in computer science. • Data Structures • Algorithm Analysis • Applications

  2. DS.I.2 Text: Data Structures and Algorithm Analysis in C Topics to be Covered • Introduction (Ch 1) • Algorithm Analysis (Ch 2) • Review of Lists, Stacks, Queues (Ch 3) • Trees and Search Trees (Ch 4) • Hashing (Ch 5) • Priority Queues: Heaps (Ch 6) • Disjoint Sets: Union-Find (Ch 8) • Graph Algorithms (Ch 9) Applications Computer Graphics Computer Vision Artificial Intelligence Databases

  3. DS.I.3 INTRODUCTION Chapter 1 Overview • Motivation for Good Algorithm Design • Math Review (for use in reading proofs) • Proofs by Induction, Counterexample, Contradiction • Recursion What is an algorithm ? What would be a good algorithm for achieving tulips in my garden next spring?

  4. DS.I.4 An algorithm to achieve X is a procedure that: 1. Halts 2. Correctly achieves X Procedure for computing N / 5 for integer N count = 0; do { count = count + 1; N = N - 5; } until (N <= 0) Does it always halt? Does it correctly compute N / 5?

  5. DS.I.5 What is a data structure ? Informal Definition: a method of organizing data Examples? Formal Definition: an abstract data type (ADT) In C: • structs • functions that operate on them In C++: • classes • methods

  6. DS.I.6 Example: VectorArray Conceptual: Size (of array) NumElements Data 0 1 Size-1 What abstract operations are needed?

  7. DS.I.7 Iteration, Recursion, Induction 1. Write an iterative function to find the sum of the first num elements of a VectorArray stored in array v. int sumit ( int v[ ], int num) { int sum; sum = 0; for ( ) sum = ; return sum; }

  8. DS.I.8 2. Write a recursive function to find the sum of the first num elements of a VectorArray stored in array v. int sumit ( int v[ ], int num) { } Strengths of recursive approach: - simplifies code - can be proven correct Weaknesses: - slower than iteration - uses more memory for the stack

  9. DS.I.9 Principle of Mathematical Induction Let P(c) be true for small constant c >= 0. Suppose whenever P(k) is true, so is P(k+1). Then P(n) is true for all n >= 0. Ex. 1.10a Prove by induction that N  (2i - 1) = N . i=1 2 Basis: N=1 Inductive Hypothesis: Induction Step:

  10. DS.I.10 int sumit (int v[ ], int num) { if (num == 0) return 0; else return v[num-1] + sumit(v,num-1); } Prove by induction that sumit(v,n) correctly returns the sum of the first n elements of array v, n0. Basis: If n=0, Inductive Hypothesis: Assume sumit(v,k) ... Inductive Step: sumit(v,k+1)

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