Groups will be assigned by either yourselves or me and will not change for the duration of the project.

By the next lab please submit a 1-2 page write up of how you are attacking the problem, progress, and pending work.

By the end of the project, groups should have working code, an
analysis of their algorithm, a discussion of the approach used
including test data, and a
discussion of things you might have liked to do but did not have time
for.

CPU scheduling

You have recently been hired by a large data center to develop a new
algorithm for scheduling jobs on a single processor. Clients can
submit jobs to the data center. Each job Sum of n dice

Most people are familiar with basic probabilities when rolling one die
or two dice. For example, what is the probability of a roll of two
random six sided dice summing to seven? But not much is typically known about Google Maps Spatial Queries

Google maps and other map search engines allow users to search using text queries that describe geographic locations. Another way to search for objects would be to allow searches to be describe by a geographic box, circle, or in the most general case, an arbitrary two dimensional simply connected planar region. How would you design a data structure to store and query two dimensional point data using such query regions. It suffices to focus on bounded rectangular axis-aligned queries. Can you design a structure to support queries and updates? Analyze the space and runtime complexity of your approach.
Keep 'em separated

Election fever is sweeping through campus and a variety of student organizations are planning to staff a variety of booths over the weekend to support their cause. Groups have place their rectangular booths across Cunningham field (the rugby teams were away that weekend) in a haphazard manner. With no football team and no ruggers on campus to break up anticipated disputes, the campus administration has decided it would be safest if they could set up a temporary partition or fence to separate the two main groups of booths: Democrats and Republicans. The partition must be a simple straight line. Depending on the configuration of booths, such a partition may not be possible. Design an algorithm that determines if two groups of booths can be separated by a single line and returns one possible line if a separation is possible, or informs users that no such separation is possible. What is the complexity of your algorithm. Will it work on all possible configurations of booths, or only under limiting assumptions? Explain.
Intersection monitoring

Traffic in Philadelphia has become a mess. City planners would like to remote traffic monitors to observe traffic patterns to eventually commission construction project to relieve the traffic. Suppose a traffic monitor can only be placed at intersections and can only observe traffic along any road meeting at that intersection until the next intersection along that road. (It is easier for me to draw a picture of this). Naturally, the city would like to monitor all roads in the city with the minimal number of monitors. Design an algorithm that can determine the minimal number of monitors needed and their location such that all roads segments can be seen by at least one monitor.
Union of Rectangles

Given a collection of Range Minimum Queries

What were the high and low values of the S&P 500 in the past day, week, month, year, or decade. More generally, given two calendar days, what were the high and low values of the market between the two given days. Since the cases are symmetric, we are interested in an algorithm for supporting such range minimum queries. Given an array Art Gallerys

A local museum is exhibiting a large collection of priceless art. Naturally they would not want the art to be stolen so they would like to set up a number of cameras to observe the room. Unfortunately with artists being creative types the room is not a rectangle which can be observed with a single camera. Suppose the room is decribed by a simple but non-convex polygon. We wish to place cameras at the vertices of the polygon such that the entire interior of the polygon is visible by the cameras. Suppose the each camera can see arbitrarily far in a straight line as long as no walls obstruct the view. Furthermore assume the cameras have a viewing angle of 360 degrees. How many cameras are needed to cover a polygonal art gallery with Line Segment Intersection

Given a collection of Image classification with limited memory

A common task in image processing is to identify the number, size, and shape of connected regions in the image that have the same (or similar) intensity values. For example, a traffic camera might detect cars as darker blobs against a lighter higway background. An infared camera might identify animals as brighter spots against a dark backround. Suppose I am given a grid of pixels where each pixel has an integer intensity value. I would like to assign a unique label to each connected component in the image where a connected component is a connected region in the image where all pixels in the component have the same intensity value and for which I can go from any pixel in the connected component to any other pixel in the component via a path that is completely within the component. One easy solution is to run a breadth first search on each unvisited pixel to identify its connected compenent. This works great if I can load the entire image into memory. Suppose I add the constraint that only a constant number of rows fit into memory. Design an algorithm that still solves the problem, but reduces the number of times I need to load a row into memory.