EECS 311: Chapter 2 Notes
Chris Riesbeck
EECS 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
5 209 13
6
Triple-Loop Algorithm
For every start
point
For
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 +
Save thenext item
biggest
Algorithm 2 Run-Times
Divide and Conquer Algorithm
Sum for max
Sumright
for 1
subsequence
element
startingsubsequence
right of
center
Return
largest of
the 3 sums Sums for max
subsequences
in left and
right halves
Sum for max left
subsequence
ending on center
Analysis of Algorithm 3
To derive T(N), time to solve a
problem with N items
… O(N) + …
Base case
T(1) = O(1)
Recursive case
T(N) = 2T(N/2) +…
… 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
If it does,
Does this
how long
algorithm
does it take,
always
in the worst
stop?
case?
If it does, does
it always return
the right
answer?
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


Assume x not in a[].
Assume a[] has 1 element.
 Binary search example:


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
If it does,
Does this
how long
algorithm
does it take,
always
in the worst
stop?
case?
If it does, does
it always return
the right
answer?
Exponentiation
If it does,
Does this
how long
algorithm
does it take,
always
in the worst
stop?
case?
If it does, does
it always return
the right
answer?
Hailstone Numbers
void printHailStones(int n) {
cout << n << ":";
If it does,
Does this
while (n > 1) {
how long
algorithm
cout << " " << n;
does it take,
always
in the worst
if ( n % 2 == 0 )
stop?
case?
n /= 2;
else
If it does, does
it always return
n = 3 * n + 1;
the right
}
answer?
cout << " " << n << endl;
}
http://en.wikipedia.org/wiki/Collatz_conjecture