1 Summary of lectures 1.Introduction to Algorithm Analysis and Design (Chapter 1-3). Lecture SlidesLecture Slides 2.Recurrence and Master Theorem (Chapter 4). Lecture SlidesLecture Slides 3.Sorting and Order Statistics (Chapter 8-9). Lecture SlidesLecture Slides 4.Balanced Search Trees: red-back tree (chapter 13) and others Lecture SlidesLecture Slides 5.Augmenting Data Structure (Chapter 14) Lecture SlidesLecture Slides 6.Dynamic Programming (Chapter 15). Lecture SlidesLecture Slides 7.Greedy Algorithms (Chapter 16). Lecture SlidesLecture Slides 8.Amortized Analysis (chapter 17) Lecture slidesLecture slides 9.Disjoint Sets (Chapter 21). Lecture SlidesLecture Slides 10.NP-Completeness (Chapter 34). Lecture SlidesLecture Slides 11.Parallel Algorithms (Selected from Chapter 30, the First Edition). Lecture SlidesLecture Slides 12.String/Pattern Matching (Chapter 32 & handout). Lecture SlidesLecture Slides 13.Approximation Algorithms (Chapter 35). Lecture SlidesLecture Slides 14.Divide and Conquer--closest pair (Chapter 33.4) Lecture slidesLecture slides 15.Lower bound: decision tree & adversary argument (handout) Lecture SlidesLecture Slides

2 Introduction Algorithms design: data structures and algorithms (disjoint set, red-black tree, AVL, B-Tree, 2-3-4) Algorithm analysis: complexities-- space and time worst, best, average asymptotic notations: order of growth Analysis methods: loop and loop invariant recursive relation and equations Substitution, Recursion tree, Master theorem, Domain Transformation, Change of variables amortized analysis adversary argument, decision argument (worst case lower bound) Algorithms: serial vs. parallel regular vs. approximate deterministic vs. probabilistic Design methods: divide and conquer dynamic programming, memoization greedy algorithm prune and search specific methods: 7 in closest pair, 5 in ordered statistic,

3 Sorting and order statistic Sorting: –Comparison: Lower bound O(nlg n), decision tree. –Non-comparison: Bucket sort, counting sort, radix sort, (linear time). ith smallest elements: –First (minimum), last (Maximum), both (3  n/2  ). –Prune-and-search RANDOMIZED-SELECT :Expected linear time O(n) but worst- case running time O(n 2 ). SELECT: Linear worst-case running time O(n).

Lower bound 4 Decision Tree Adversary Argument

5 Dynamic programming Elements of DP: –Optimal substructures –Overlapping subproblems Four steps: –Find/prove Optimal Substructure –Find recursive solution –write DP program to compute optimal value –Construct optimal solution (path). Analysis of DP program Relations among: recursive algorithm, divide-and- conquer, Memoization. Auxiliary table.

6 Data structures Disjoint set –Definition and implementation Union-by-rank, path compression –Analysis Fast increasing function and its slow reverse Amortized analysis Proof of the running time. Red-black trees –Balance –Rotation –Augmenting Other trees: –AVL, B-tree, B+-tree, 2-3-4, Treap, Splay

7 Amortized analysis Find the average worse-case performance over a sequence of operations Three methods: –Aggregate analysis: Total cost of n operations/n, –Accounting method: Assign each type of operation an (different) amortized cost overcharge some operations, store the overcharge as credit on specific objects, then use the credit for compensation for some later operations. –Potential method: Same as accounting method But store the credit as “potential energy” and as a whole.

8 NP-completeness P and NP Poly reduction Proof of NP-completeness by reduction. –Belong to NP –Is NP-hard Reduce a (general) instance of known NPC problem to a (concrete) instance of need-to-proof problem Prove poly reduction and their equivalence.

9 Parallel algorithms PRAM models: –EREW, CREW, ERCW, CRCW Design parallel algorithms –Analysis Relation among models.

10 String matching Naïve solution KMP algorithm –Prefix function –Analysis: amortized method. Appropriate string matching.

11 Approximate algorithms Find near-optimal solution in poly time Ratio Question: given two NP-complete problems A and B, if A can be reduced to B in poly, how about their corresponding appropriate algorithms and the ratios?

12 Cross-topic reviews NP-complete problem –Proof –Some (poly) algorithms such as DP algorithm (they may look like poly solution, but in fact, not). –Graph related problems, schedule problems, number and set related problems, etc. Given a problem, –determine whether it is NP-complete, If yes, –find special cases, or find appropriate solution –Otherwise, design its data structures, its algorithms, and analyze its complexity. –DP solution for NP-complete problem, pseudo-poly. Space and time trade-off Pre-processing

13 Questions types NP-complete proof Parallel algorithm Design data structures Design algorithms (by different methods) Given algorithm, analysis of its (different techniques) functions and complexity. Problem-related specific questions: many!! Recursive and recurrence. Proof, computation, design, analysis.