EECS 311: Chapter 2 Notes

Chris RiesbeckEECS Northwestern

Unless otherwise noted, all tables, graphs and code from Mark Allen Weiss' Data Structures and Algorithm Analysis in C++, 3rd ed, copyright © 2006 by Pearson Education, Inc.

Maximal Subsequence Sum Problem

Given a sequence of N positive and negative integers A1, A2, A3, …, An

Find the subsequence Aj, … Ak with the largest sum

Example: -2, 11, -4, 13, -5, -2

65 9 1320

Triple-Loop Algorithm

For every start pointFor every

end point

Add up the subsequence

Save the biggest

Algorithm 1 Run-Times

Double-Loop Algorithm

For every start point

For every end point

New sum is old sum + next itemSave the

biggest

Algorithm 2 Run-Times

Divide and Conquer Algorithm

Sum for 1 element

subsequence

Sums for max subsequences

in left and right halves

Sum for max left subsequence

ending on center

Sum for max right subsequence

starting right of centerReturn largest of

the 3 sums

Analysis of Algorithm 3To derive T(N), time to solve a problem with N items

Base case T(1) = O(1)

Recursive case T(N) = 2T(N/2) +…

… O(N) + …

… O(N) + …

… O(1)

Analyzing Divide and Conquer

T(1) = O(1) T(N) = 2T(N/2) + N

T(2) = 2T(1) + 2 = 4 = 2*2 T(4) = 2T(2) + 4 = 12 = 4*3 T(8) = 2T(4) + 8 = 32 = 8*4 T(16) = 2T(8) + 16 = 80 = 16*5

T(2k) = 2k * (k+1) For general N, O(N log N)

Algorithm 3 Run-Times

Single-Loop Algorithm

For every start point

New sum is old sum + next item

If bigger, save;if negative, forget and start over

Algorithm 4 Run-Times

Algorithmic Analysis

Run-times for small N

Run-times for large N

Typical Growth Rates

Don't confuse with log log N

which is < log N

Binary Search

Does this algorithm

always stop?

If it does, does it always return

the right answer?

If it does, how long does it take, in the worst case?

Does it halt? Find a measure M such that

you can prove it monotonically decrease on every iteration

the algorithm halts when M passes some threshold, e.g., 0

Binary search example: M = high – low M decreases by at least 1 every iteration algorithm halts when M < 0

Does it give the right answer? Proof by cases

When it returns a value, is it correct (no false positives)?

When it returns not found, is it correct (no false negatives)?

When it returns a value, is it correct? Proof by contradiction:

Assume desired property is false. Prove contradiction results.

Binary search example: Assume k ≠ -1 is returned and a[k] ≠ x. k is returned on line 19. This means a[k] is neither > nor < than x. a[k] must be <, > or =. Contradiction.

When it returns not found, is it correct? Proof by induction:

Prove P is true for K , typically 1 or 2 Prove P is true for N+1 if it’s true for N Then P is true for all N ≥ K

Binary search example: Assume x not in a[]. Assume a[] has 1 element.

a[0] ≠ x and code correctly returns not found. Assume a[] has N + 1 elements.

Proof by cases: If a[mid] < x, search will look at a[mid] … a[high], which

has less than N elements. By assumption, that search returns correct answer. Similarly if a[mid] > x.

Ergo, binary search returns not found correctly for all N ≥ 1

Greatest Common Divisor

Does this algorithm

always stop?

If it does, does it always return

the right answer?

If it does, how long does it take, in the worst case?

Exponentiation

Does this algorithm

always stop?

If it does, does it always return

the right answer?

If it does, how long does it take, in the worst case?

Hailstone Numbersvoid printHailStones(int n) { cout << n << ":"; while (n > 1) { cout << " " << n; if ( n % 2 == 0 ) n /= 2; else n = 3 * n + 1; } cout << " " << n << endl;}http://en.wikipedia.org/wiki/Collatz_conjecture

Does this algorithm

always stop?

If it does, does it always return

the right answer?

If it does, how long does it take, in the worst case?