Algorithm Design and Time Analysis

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Cecil Tate
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1 Algorithm Design and Time Analysis CSE21 Winter 2017, Day 6 (B00), Day 4 (A00) January 23,

2 Today s Plan Analyzing algorithms that solve other problems (besides sorting and searching) Designing better algorithms pre-processing re-use of computation

3 Summing Triples: WHAT Given a list of real numbers a 1, a 2,..., a n look for three indices, i, j, k (each between 1 and n) such that a i + a j = a k Does the list 3,6,5,7,8 have a summing triple? A. Yes: 1,2,3 B. Yes: 1,3,5 C. No

4 Summing Triples: WHAT Given a list of real numbers a 1, a 2,..., a n look for three indices, i, j, k (each between 1 and n) such that a i + a j = a k Design an algorithm to look for summing triples

5 Summing Triples: HOW (1) What's the best-case runtime of this algorithm? A. O(1) B. O(n) C. O(n 2 ) D. O(n 3 ) E. None of the above

6 Summing Triples: HOW (1) Describe all best-case inputs?

7 Summing Triples: HOW (1) What's the worst-case runtime of this algorithm? A. O(1) B. O(n) C. O(n 2 ) D. O(n 3 ) E. None of the above

8 Summing Triples: HOW (1) Can we do better? How?

9 Summing Triples: HOW (2) Eliminate redundancy

10 Summing Triples: HOW (2) Eliminate redundancy What's the worst-case runtime of this algorithm? A. O(1) B. O(n) C. O(n 2 ) D. O(n 3 ) E. None of the above

11 Summing Triples: HOW (2) Eliminate redundancy Hmmmm Can we do better? How?

13 Reframing what we did: Summing Triples: HOW (2) For each candidate sum a i +a j, do linear search to find it Improvements??

14 Summing Triples: HOW (2) For each candidate sum a i +a j, do linear search to find it We have a faster search than linear search!

15 Summing Triples: HOW (3) For each candidate sum a i +a j, Worst-case runtime? A. O(n 3 ) B. O(n 2 ) C. O(n 2 log n) D. O(n log n) do binary search to find it

16 Summing Triples: HOW (3) For each candidate sum a i +a j, do binary search to find it Something is wrong!

17 Summing Triples: HOW (3) For each candidate sum a i +a j, do binary search to find it Does this algorithm really work?

18 Summing Triples: HOW (4) Preprocessing step This algorithm works! How long does it take? aka SortedSumTriples

19 Summing Triples: HOW (4) O(n 2 ) O(n 2 log n) SumTriples4 worst-case complexity is max of these: O(n 2 log n)

20 Summing Triples: HOW (4) O(n 2 ) O(n 2 log n) Max of these: O(n 2 log n) SumTriples4 does better than O(n 3 ). Using a faster sort won't help overall. Fastest known algorithm: O(n 2 )

21 Tight? To know that we've actually made improvements, need to make sure our original analysis was not overly pessimistic. A tight bound for runtime is a function g(n) so that the runtime is in Big-O: upper bound. Big-Ω: lower bound.

22 Summing Triples: WHEN (1) What's a lower bound on the worst-case runtime of this algorithm? A. B. C. D. E. None of the above

23 Summing Triples: WHEN (1) Strategy: work from the inside out

24 Summing Triples: WHEN (2) What's a lower bound on the worst-case runtime of this algorithm? A. B. C. D. E. None of the above

25 Summing Triples: WHEN (2) For at least n/2 values of i (1 n/2), we do inner for loop (k) at least n/2 times, each taking n steps

26 Summing Triples: WHEN (2) Observe: in both these examples, the product rule for calculating the nested loop runtime gave us tight upper bounds is that always the case?

runtime O(T 2 ) in the worst case and run at most O(T 1 )

27 When is the product rule for nested loops tight? Nested code: If Guard Condition is O(1) and body of the loop has runtime O(T 2 ) in the worst case and run at most O(T 1 ) iterations, then runtime is O(T 1 T 2 ) But what if many t k are much better than the worst case?

28 Intersecting sorted lists: WHAT Given two sorted lists a 1, a 2,..., a n and b 1, b 2,..., b n determine if there are indices i,j such that a i = b j Design an algorithm to look for indices of intersection

29 Intersecting sorted lists: HOW Given two sorted lists a 1, a 2,..., a n and b 1, b 2,..., b n determine if there are indices i,j such that a i = b j High-level description: Use linear search to see if b 1 is anywhere in first list, using early abort Since b 2 >b 1, start the search for b 2 where the search for b 1 left off And in general, start the search for b j where the search for b j-1 left off

30 Intersecting sorted lists: HOW

31 Intersecting sorted lists: WHY To practice: trace examples & generalize argument for correctness

32 Intersecting sorted lists: WHEN Using product rule O(n) O(1)

33 Intersecting sorted lists: WHEN Using product rule O(n) Total: O(n 2 )

34 Intersecting sorted lists: WHEN More careful analysis Every time the while loop condition is true, i is incremented. If i ever reaches n+1, the program terminates (returns)

35 More careful analysis Intersecting sorted lists: WHEN This executes O(n) times total (across all iterations of for loop)

36 More careful analysis Intersecting sorted lists: WHEN This executes O(n) times total (across all iterations of for loop) Total: O(n) Be careful: product rule isn't always tight!

37 Announcements HW2 Due tomorrow! (Tues 1/24, 11:59PM) Practice with Order Notation on Khan Academy e.g., Signup Is available!

Notes slides from before lecture. CSE 21, Winter 2017, Section A00. Lecture 4 Notes. Class URL:

Notes slides from before lecture CSE 21, Winter 2017, Section A00 Lecture 4 Notes Class URL: http://vlsicad.ucsd.edu/courses/cse21-w17/ Notes slides from before lecture Notes January 23 (1) HW2 due tomorrow