ECE250: Algorithms and Data Structures Final Review Course

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1 ECE250: Algorithms and Data Structures Final Review Course Ladan Tahvildari, PEng, SMIEEE Professor Software Technologies Applied Research (STAR) Group Dept. of Elect. & Comp. Eng. University of Waterloo

2 The Course Review (1) v Analysis of Algorithms Ø Asymptotic Notations Ø Recurrences v Toolbox of Algorithmic Techniques Ø Divide and Conquer Merge sort, Quicksort, Binary search Ø Dynamic Programming Matrix Multiplication, LCS, Floyd-Warshall Ø Greedy Algorithms Prim-Jarnik, Kruskal, Dijkstra Lecture 34 ECE250 2

3 The Course Review (2) v Building algorithms - concept of ADT v Toolbox of Data Structures Ø Simple data structures and ADTs array, all sorts of linked lists, stacks, queues, heaps Ø Dictionaries hash tables binary search trees (unbalanced) AVL Trees B-Trees Lecture 34 ECE250 3

4 The Course Review (3) v Toolbox of Algorithms Ø Sorting insertion sort selection sort merge sort quick sort heap sort priority queues counting sort radix sort Lecture 34 ECE250 4

5 The Course Review (4) v Toolbox of Algorithms Ø Graphs graph traversal o breadth-first search o depth-first search o topological sort MST o Prim-Jarnik o Kruskal SSSP o Dijkstra o Bellman-Ford APSP o Floyd-Warshall Lecture 34 ECE250 5

6 Reading Material: Before Midterm Chapter CLRS 2 CLRS 3, A.1 CLRS 4 CLRS 10 CLRS 11 CLRS 12, B.5 CLRS 18 Weiss 4 Topics Algorithmic problems, insertion sort, merge sort Asymptotic notation, useful math Recurrences Arrays, lists, stacks, queues Hashing Binary Search Trees (BST) B-Trees AVL Trees Lecture 34 ECE250 6

7 Reading Material: After Midterm Chapter CLRS 6 CLRS 7 CLRS 8 CLRS 15 CLRS 16 CLRS 22, B.4 CLRS 21, 23 CLRS 24 CLRS 25 CLRS 34 Topics Heap, Heapsort, Priority Queue Quicksort Sorting in Linear Time Dynamic Programming Greedy Algorithms Elementary Graph Algorithms Minimum Spanning Trees (MST) Single-Source Shortest Paths (SSSP) All Pairs Shortest Paths (APSP) NP-Completeness Lecture 34 ECE250 7

8 Solving Recurrences v Repeated Substitution Procedure: Ø Substitute, expand, substitute, expand Ø Observe a pattern and write how your expression looks after the i-th substitution Ø Find out what the value of i (e.g., lgn or n-1) should be to get the base case of the recurrence (e.g., T(1)) Ø Insert the value of T(1) and the expression of i into your expression Ø Compute your recurrence for small values and check if your general solution gives the same results! Lecture 34 ECE250 8

9 Types of Recurrences v Number of substitutions i : Ø If T(n) = T(n - 1) and base case T(k) = then i = n k Ø If T(n) = T(n/b) and base case T(k) = then i = log b (n/k) Lecture 34 ECE250 9

10 Types of Recurrences v T(n) = at(n b) + c v T(n) = at(n/b) + cn + d Ø Geometric Series v T(n)=T(n b) + cn + d Ø Arithmetic Series v Prove it yourself! Lecture 34 ECE250 10

11 Graph Exercises v Types of graph exercises: Ø Decide what are the vertices and what are the edges Ø Un-weighted graphs Traverse and check(do) something => use DFS or BFS Find shortest paths => use BFS Schedule or order dependent activities or processes => use Topological sorting Lecture 34 ECE250 11

12 Graph Exercises v Weighted graphs Ø Decide what are the weights Ø Shortest paths, shortest times => use Dijkstra Ø Minimum (maximum) spanning tree => use Prim-Jarnik or Kruskal Ø Finding a critical path in a schedule (longest path in a DAG) => use Topological sorting on a DAG of activities Lecture 34 ECE250 12

13 Preparation for the Exam v First, concentrate on lecture notes v Second, studying CLRS/Weiss books wherever is needed (although you have to know the concepts!) v Third, solving exercises by yourself Ø When solving write your solutions down! v Good news: solutions to all exercises, assignments, and exams are on the course website J Lecture 34 ECE250 13

14 More Materials (115 Questions) v Study all solved exercises in the tutorials/lectures (24) v Study solutions for the assignments (12) v Previous ECE250 exams (62) v Study solutions for sample questions (17) Lecture 34 ECE250 14

15 Office Hours Before Final Exam (April 8-24; 21 hours) Mon, April 8 Tue, April 9 Fri, April 12 Tue, April 23 Wed, April 24 Hari 10-11am DC-2550 Jenny 10-11am EIT-4143 Ladan 10-11am EIT-4136 Ladan 10-11am EIT-4136 Hari 11am-12pm DC-2550 Hari 11am-12pm DC-2550 Jenny 11am-12pm EIT-4143 Ladan 11am-12pm EIT-4136 Hari 11am-12pm DC-2550 Hari 1-2pm DC-2550 Jenny 1-2pm EIT-4143 Hari 1-2pm DC-2550 Ladan 1-2pm EIT-4136 Jenny 2-3pm EIT-4143 Jenny 2-3pm EIT-4143 Ladan 2-3pm EIT-4136 Hari 2-3pm DC-2550 Jenny 3-4pm EIT-4143 Ladan 3-4pm EIT-4136 Ladan 3-4pm EIT-4136 Jenny 3-4pm EIT-4143 Lecture 34 ECE250 15

16 ECE250 Final Exam v Date: Wednesday, April 24, 2019 v Time: 4:00-6:30pm v Locations: Ø PAC 1, PAC 2, and PAC 3 NOTE: Standard calculator allowed but no additional materials allowed. Lecture 34 ECE250 16

17 Potential Structure for Final Exam v Seven questions with at least two sub-problems 1. Algorithm Analysis (~20 marks) (CLRS 2, 3, 4, A.1, 10) 2. Hashing (~5 marks) (CLRS 11) 3. Trees and Tree Traversals (~10 marks) (CLRS 12, 18, B.5, Weiss 4) 4. Sorting Algorithms, Heaps, Heapsort, Priority Queues (~15 marks) (CLRS 6, 7, 8) 5. Graphs (~25 marks) (CLRS 22, 23, 24, 25, B.4) 6. Optimization Problems (~20 marks) (CLRS 15, 16) 7. NP-Completeness (~5 marks) (CLRS 34) Lecture 34 ECE250 17

18 Sample Questions

19 Question 1 Algorithm Analysis You have an unsorted array of size n. You need to search for k different values in the array. You consider two strategies: ü Method A: Do k linear searches, searching the entire array for each value. ü Method B: First sort the array using merge-sort, and then do k binary searches. How large should k be in such a way that Method B is faster than Method A? Show all your work and specify k as a function of n. Simplify your answer as much as possible. Lecture 34 ECE250 19

20 Question 2 Recurrences Give asymptotic upper and lower bounds for the following recurrence. Assume that is constant for sufficiently small n. T ( n) = 4T ( n / 2) + n 2 n Make your bounds as tight as possible, and justify your answer. Lecture 34 ECE250 20

21 Question 3 Hashing Suppose we have a hash table with M slots containing n keys. Assume the chaining technique is used when a collision occurs. Suppose that instead of a linked list, each slot is implemented as a B-Tree of degree 4. Give the worst and the best time complexity of adding an entry to this hash table. Justify your answers. Lecture 34 ECE250 21

22 Question 4 Trees Let B1 and B be two binary search trees that together store keys 2 k 1, k2,..., kn. Suppose we know that every key in B is smaller than every key in 1 B (according to some comparison operator). 2 Describe an algorithm that merges B 1 and B into a single BST in 2 O (min{ h 1, h2}) time, where h1 and h2 are the heights of B 1 andb 2, respectively. Pointers b 1 and b 2 to the roots of each tree are given. Lecture 34 ECE250 22

23 Question 5 Sorting Algorithms Write a Θ( nlgn) -time algorithm that, given a set of S of n integers and another integer x, determine whether or not there exists two elements in S whose sum is exactly x. Analyze the running time of your algorithm. Lecture 34 ECE250 23

24 Question 6 Graphs To get in shape, you have decided to start running to school. You want a route that goes entirely uphill and then entirely downhill so that you can work up a sweat going uphill and then get a nice breeze at the end of your run as you run faster downhill. Your run will start at home and end at school and you have a map detailing the roads with m road segments (any existing road between two intersections) and n intersections. Each road segment has a positive length, and each intersection has a distinct elevation. Assuming that every road segment is either uphill or downhill, write a O(m+n)-time algorithm to find the shortest route that meets your specifications. Analyze the running time of your algorithm. Lecture 34 ECE250 24

25 Question 7 Optimization Problems James Bond drives his car from Windsor to Toronto along Highway 401. His car s gas tank, when full, holds enough gas to travel n kilometers, his map gives the distances between gas stations on his route. James Bond wishes to make as few gas stops as possible along the way. Give an efficient algorithm by which James Bond can determine at which gas stations he should stop, and prove that your strategy yields an optimal solution. Lecture 34 ECE250 25

26 Question 8 NP Completeness Suppose that L 1, L 2 NP and that L1 p L2. For each of the following statements, determine whether it is true, false, or an open problem. Prove your answers. L L 1 2 L2 p L1 a) If and are both NP-complete, then. L 2 p L 1 L1 2 b) If, then both and L are NP-complete. c) Suppose there is a linear time algorithm that recognizes Then, there exists a linear time algorithm to recognize L 1. L 2 Lecture 34 ECE250 26

ECE250: Algorithms and Data Structures Midterm Review

ECE250: Algorithms and Data Structures Midterm Review Ladan Tahvildari, PEng, SMIEEE Associate Professor Software Technologies Applied Research (STAR) Group Dept. of Elect. & Comp. Eng. University of Waterloo