Annoucements

1. Office hours canceled today
2. Re-submission for assignments 1 and 2 open Wednesday to Friday
3. Assignment 06 can be done in pairs!
   • partner questionnaire today by Wednesday
   • Assignment 06 due 03/24

Outline

1. Convex Hulls
2. Activity: Finding the Convex Hull
3. Graham’s Scan Algorithm

Last Time

Depth-first Search: A Case Study in Visualizing Algorithms

• Building a graph interactively
• Stepping through an execution
• Visualizing each step

Today

Convex hulls:

• fundamental problem in computational geometry
• topic of Assignment 06

Convex Hulls, Two Views

Given a finite set of points in the plane:

• What is the smallest convex region that contains all of the points?
• What is the minimum perimeter of a polygon that contains the points?

Visualizing Convex Hulls

Which Points are on the Boundary?

Convex Hull Problem

Input:

• set of points in plane
  • $(x, y)$-coordinates of each point

Output:

• a sequence of points $(x_1, y_1), (x_2, y_2), \ldots, (x_k, y_k)$ that define the “boundary” of the set of points
  • path around $(x_1, y_1), (x_2, y_2), \ldots, (x_k, y_k)$ surrounds all points in the set in clockwise order
  • the bounded region is convex

Convex Hull Output Depiction

Activity!

Find the Convex Hull!

Graphed Points

With the Convex Hull

Observe

It is easy to find by hand once you’ve graphed the points.

Question. How to program a computer to find the convex hull?

Notation

• $X = {(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)}$ a set of points
• $CH(X)$ is the convex hull of $X$
  • more specifically, sequence of points in $X$ that are vertices of the convex hull in clockwise order
  • start from left-most boundary point

Initial Question

Given two points $A = (x_i, y_i)$, $B = (x_j, y_j)$ in $X$, how can we determine if $AB$ is a (directed) segment of $CH(X)$?

Sub-Question

Given three points, $A, B, C$, how to determine if $C$ lies “to the right” of $AB$?

Brute-force Solution

Check all ordered pairs of points $AB$ to see if $AB$ is a segment of the convex hull!

Efficiency?

Question. If there are $n$ points in $X$, what is the running time of the brute-force procedure?

Slight Optimization

Question. How could we ensure that we only ever check $A$ that are guaranteed to be on the convex hull?

• Hint: how can we find our first $A$ that is sure to be in $CH(X)$?

Optimization Illustration

Unfortunately

Even with the optimization, our code still takes $\Theta(n^3)$ time.

Another Approach

Try an incremental algorithm:

1. Start with a set $X$ containing only one element
   • in this case finding $CH(X)$ is easy!
2. Add points one at a time to $X$ and update $CH(X)$ accordingly

Graham’s Scan Algorithm

R. Graham, 1972

First Idea

Pre-process points by sorting them by $x$-coordinate:

• process the points from left to right

Starting at left-most point, scan to the right to find upper convex hull boundary

• repeat process from right to left to get lower convex hull

Graham Scan Idea, Illustrated

How to Formalize/Implement?

Assume $X$ is an array of points, sorted by $x$-coordinate

• How to keep track of points in $CH(X)$ (so far)?

• How to update in response to next point?

Graham’s Scan Pseudocode

• X sorted by $x$-coordinate
• stk a stack, initially storing first two points in X

For each remaining C in X:

• if stk.size() == 1, stk.push(C)
• otherwise
  • A and B are top two elements in stk
  • while ABC is not a right turn and stk.size() > 1
    • stk.pop(), update A, B
  • stk.push(C)

Claim

When Graham’s Scan completes, stk stores the points along the upper boundary of the convex hull of $X$.

Why?

Graham’s Scan Efficiency?

If there are $n$ points, what is the running time of Graham’s scan?

Finishing the Computation

How to find the lower boundary of $CH(X)$?

Assignment 06

Make an interactive visualization for Graham’s scan algorithm

• user can add points in the plane
• program steps through execution and illustrates each step
• returns convex hull of points