1. CSE 326: Data Structures Lecture #24 The Algorhythmics
Steve Wolfman
Winter Quarter 2000
Today we’ll look at a pair of interesting and important graph algorithms that also happen to make good use of data structures!
The algorithms were developed by Dijkstra and Kruskal.

2. Today’s Outline Greedy Divide & Conquer Dynamic Programming Randomized
Backtracking
Since I’m getting filmed today, I need you to sign a release.
The form just says that you don’t mind being filmed.
Then, we can finish off some graph properties from Wednesday. It’s important to understand what these terms mean!
Then, we’ll hit the two algorithms.

3. Greedy Algorithms Repeat until problem is solved:
Measure options according to marginal value
Commit to maximum
Greedy algorithms are normally fast and simple.
Sometimes appropriate as a heuristic solution or to approximate the optimal solution.

4. Hill-Climbing
Global Maximum
Local Maximum

5. Greed in Action Best First Search A* Search Huffman Encodings
Kruskal’s Algorithm
Dijkstra’s Algorithm
Prim’s Algorithm
Scheduling

6. Scheduling Problem Given:
a group of tasks {T1, …, Tn}
each with a duration {d1, …, dn}
a single processor without interrupts
Select an order for the tasks that minimizes average completion time 1
1.5
0.5
0.3 2
1.4 T1 T2 T3 T4 T5 T6
Average time to completion:

7. Greedy Solution

8. Proof of Correctness
Common technique for proving correctness of greedy algorithms is by contradiction:
assume there is a non-greedy best way
show that making that way more like greedy solution improves the supposedly best way: contradiction!
Proof of correctness for greedy scheduling:

9. Divide & Conquer Divide problem into multiple smaller parts
Solve smaller parts
Solve base cases directly
Otherwise, solve subproblems recursively
Merge solutions together (Conquer!)
Often leads to elegant and simple recursive implementations.

10. Divide & Conquer in Action
Mergesort
Quicksort
buildHeap
buildTree
Closest points

11. Closest Points Problem
Given:
a group of points {(x1, y1), …, (xn, yn)}
Return the distance between the closest pair of points
0.75

12. Closest Points Algorithm
Closest pair is:
closest pair on left or
closest pair on right or
closest pair spanning the middle
runtime:

13. Closest Points Algorithm Refined
Closest pair is:
closest pair on left or
closest pair on right or
closest pair in middle strip within one  of each other vertically
runtime:

14. Memoizing/ Dynamic Programming
Define problem in terms of smaller subproblems
Solve and record solution for base cases
Build solutions for subproblems up from solutions to smaller subproblems
Can improve runtime of divide & conquer algorithms that have shared subproblems with optimal substructure.
Usually involves a table of subproblem solutions.

15. Dynamic Programming in Action
Sequence Alignment
Fibonacci numbers
All pairs shortest path
Optimal Binary Search Tree

16. Fibonacci Numbers F(n) = F(n - 1) + F(n - 2) F(0) = 1 F(1) = 1
int fib(int n) {
if (n <= 1)
return 1;
else
return fib(n - 1) +
fib(n - 2); }
“Divide” & Conquer
int fib(int n) {
static vector<int> fibs;
if (n <= 1)
return 1;
if (fibs[n] == 0)
fibs[n] = fib(n - 1) +
fib(n - 2);
return fibs[n]; }
Memoized

17. Optimal Binary Search Tree Problem
Given:
a set of words {w1, …, wn}
probabilities of each word’s occurrence {p1, …, pn}
Produce a Binary Search Tree which includes all the words and has the lowest expected cost:
Expected cost = (where di is the depth of word i in the tree)

18. The Optimal BST Optimal Substructure Shared Subproblems Falcon
Millenium
Falcon
And to
Ever
Millenium to
Zaphod
And to
Ever
First to
Meet
Can an optimal solution possibly
have suboptimal subtrees?

19. Optimal BST Cost
Let CLeft,Right be the cost of the optimal subtree between wLeft and wRight. Then, CLeft,Right is:
Let’s maintain a 2-D table to store values of Ci,j

20. Optimal BST Algorithm …a …am …and …egg …if …the …two a… am… and… egg…

21. To Do Finish Project IV!!!!!! Read Chapter 10 (algorithmic techniques)
Work on final review (remember you can and should work with others)
That’s it. You should finish off project IV this weekend. Tell us if you’re having trouble!
Read chapter 9 on graphs and start reading chapter 10 on algorithmic techniques.
We’ll talk about chapter 10 Monday and Wednesday of next week.
Finally, start working on that final review. It’s not homework, but it will help you get ready for the final.
We’ll have the official final review next week.

22. Coming Up Course Wrap-up Project IV due (March 7th; that’s tomorrow!)
Final Exam (March 13th)