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LECTURES in PPTLecture
1 |
Textbook: "Introduction to Algorithms" by Cormen et al (old edition/new edition) GRADUATE
PROJECTS: 1.
Dynamic Programming on a line for
paper 5 from 2003 2.
Minimum spanning tree via different algorithm 3.
Anything you suggest – should make appointment All graduate students should have their project topics chosen and acknowledged by March 17. ANNOUNCEMENTS/HOMEWORKS/ASSIGNMENTS Turning-in/grading policy:
(a)
1.2-2,
1.2-3, p. 13 (new edition) (b)
Extra
credit: Give an algorithm for exchanging 2 memory contents X and Y
without using any extra memory. Due
(a)
HW: 4-1
p.85, 4-6 p.87 (b)
Extra
credit: One asks another: How many kids do you have? – 2. At least one
of them is a boy? Yes. What is the probability that the other is a boy?
(a)
Given
lever scales (i.e., scales with two cups, after each measurement they show
either left or right cup content is greater or the contents are equal) and a
set of 12 coins in which exactly one coin is false and either lighter or
heavier than any other coin which is standard. In 3 tests find the false coin
and tell if it is heavier or lighter than a standard one. (b)
Describe the algorithm which in 7 comparisons
correctly sorts five unequal numbers (rocks) A1, A2, A3, A4, A5 (c)
There
are 25 horses and a 5 race lanes. If a horse A is faster than horse B, then
in any race A will finish earlier. In 7 races find the fastest horse, the
second fastest and the 3d fastest horses.
(a)
6-1 p.142 and 6-2 p.143 (b) Extra credit: Describe the algorithm which in 6 comparisons correctly finds median (the third) out of five unequal numbers (rocks) A1, A2, A3, A4, A5 (c)
Extra credit: Prove that a monk coming up down of a tower (in two different days)
should be in the same place in the same day time.
(a)
22-3
p.559 (b)
Give
an example of a pentagon where greedy algorithm does not give optimal
solution (c)
(extra credit) Give the worst-case example for
performance of the greedy algorithm for minimum length triangulation of a
convex polygon, i.e., describe an example of convex polygon, give greedy
algorithm solution, the optimum solution and the ratio of the greedy over
optimum length.
(a)
Prove
using Euler's formula that, 3 houses - 3 wells graph, K_{3,3}, is not planar (b)
For
Petersen graph (complement of line graph of K5): show how to draw on the
plane with two self-intersections and
(extra-credit) prove that it cannot be drawn with a single
self-intersection (c)
Prove
that the faces of the Eulerian planar graph G=(V,E) can be colored into two colors such that any two
adjacent faces (i.e., faced with common border edge) have different colors.
HINT. Show that in the dual graph G' (in which each node is a
"capital" of each face and two capitals are connected iff the faces are adjacent) does not contain odd cycles
(a)
A guy is coming to a train station in a random
time and takes the first train. Trains go west each 10 minutes and go east
each ten minutes. After several months he finds out that he goes west several
times more often than east. Why? (b)
Draw
10 “right” stars of the same size without taking hand from paper (c)
Put 4
empty bottles from
(a)
24-2 and
24-3 p.615
Graph Representation, Planar Graphs, Greedy algorithm, MST,
Depth/Breadth First Search, Bellman-Ford, Dijkstra
Dynamic Programming. All-pairs shortest paths: Matrix multiplication, Floyd-Warshall, Johnson's Convex
Closest pair (D&C) Voronoi graph/diagram Rectilinear & Octilinear metrics Zero-skew trees Steiner trees Minimum Disk covering n points NP-completeness
Max independent set (reduction from 3CNF) Max clique/Min vertex cover Approximation algorithms Vertex cover/Traveling Salesman
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