[ Announcements |
Administrative Basics |
Scribe Notes |
Course Journal ]
In this course we shall advanced topics in the area of computer
algorithms. Since Algorithms as a field of research is extremely vast,
any choice of topics will necessarily cover only a tiny fraction of the
field. The choices made for this course will lean towards relatively
recent basic research work in algorithms in order to give computer
science graduate students (the intended audience) a feel for what
working in algorithms is like.
A strong mathematical background and a love of mathematical reasoning
are expected and may be required frequently.
- [Mar 9] Some solutions to
HW2 have been posted. You are free to refer to them, and to
everything else linked from this page, while working on your final
- [Mar 7, 6pm] The final exam is here. Please read it and
think a little about each problem before class on Wednesday. The
exam is due Mar 14, 5pm sharp.
- [Mar 4] Play around with the applet at this
website to gain some intuition for the KMP algorithm.
- [Feb 21, 6pm] A change has been made to clarify problem 3 in
HW2. Please reload the homework in your browser.
- [Feb 19] Some solutions to
HW1 have been posted. Please read and compare with your own
- [Feb 16] The second and final homework,
Homework 2, is up on
the website and is due Feb 28, before class.
- [Feb 3] Problem 2 in the homework has been spelled out to take
care of different versions of Problem 11-4 in different printings of
[CLRS]. Please make sure you're solving the correect problem in your
- [Jan 26] Homework 1 is up on
the website and is due Feb 7, before class.
- [Jan 3] You will be required to use LaTeX for your work in this
course. Homework 0 tests your proficiency with LaTeX; please take it
seriously and do it ASAP.
Sudikoff 214 | 10 hour | MWF 10:00-11:05, X-hr Th 12:00-12:50
Amit Chakrabarti | Sudikoff 107 | 6-1710 |
Office hours: MW 11:30-12:30 and 15:00-16:00; also by appointment
- Required: none
- "Introduction to Algorithms", by Cormen, Leiserson, Rivest
and Stein, MIT Press.
- "Randomized Algorithms", by Motwani and Raghavan, Cambridge
CS 25, CS 39
Scribing and preparing lecture notes for one week.
Class presentation on one of the topics in the list, or
a related topic of the student's choosing.
One take-home final exam, due around March 15 (date to be
Handouts are accessible only from within Dartmouth, for
"Storing a Sparse Table with O(1) Worst Case Access Time".
Fredman, Komlos, Szemeredi. JACM 31(3), 1984, pp. 538-544.
"Self-Adjusting Binary Search Trees".
Sleator, Tarjan. JACM 32(3), 1985, pp. 652-686.
"A Simple Min-Cut Algorithm".
Stoer, Wagner. JACM 44(4), 1997, pp. 585-591.
"A Randomized Linear-Time Algorithm to Find Minimum Spannning Trees".
Karger, Klein, Tarjan. JACM 42(2), 1995, pp. 321-328.
"Linear-Time Pointer-Machine Algorithms for Least Common
Ancestors, MST Verification, and Dominators".
Buchsbaum, Kaplan, Rogers, Westbrook. STOC 1998.
- Two sets of notes on maximum matching algorithms: Vempala's notes (detailed)
and Tarjan's notes (very
- Khanh Do Ba's lecture
notes for his lecture on the shortest superstring problem.
Here are the unedited scribe notes. I will try to edit and collate these
some time soon.
Course Journal, Homeworks
- Homework Zero: Learn to typeset mathematical writing using
LaTeX, by the first day of class (i.e., January 5). Then, typeset at
least one page of one of these two PDF documents: Doc 1 or Doc 2. Submit your ".tex" and
".pdf" files by the second day of class (i.e., January 7).
- We reviewed the basics of random variables and expectation,
then defined the concept of a family of universal hash functions. We
bounded the expected search time, under a universal hashing scheme
with linear probing to resolve collisions, to be 1 + (load factor).
- We showed how to construct a simple number-theory-based family
of universal hash functions and proved the universality of this
family. We then discussed perfect hashing and showed how an
easy-to-compute perfect hash function could be constructed if only
we were allowed a quadratic-sized hash table.
- Using two-level hashing with universal hash functions, we
showed how to build an overall O(n)-sized hash structure for
storing n keys. The construction was randomized. We showed that it
is expected to succeed within O(n) time.
- Sara Sinclair showed how to improve upon the above construction
and get the storage space down to (n + o(n)), for
- We discussed splay trees, described the splaying operation and
the potential function used in the amortized analysis of the cost of
- We completed the proof of the access lemma, which bounds the
amortized cost of splaying. Using it, we proved the Balance Theorem
(total access time bound) and the Static Optimality Theorem for
splay trees. We had to skip the Static Finger Theorem for lack of
- Chien-Chung proved the Static Finger Theorem and the Long Splay
Theorem. He described how to perform updates
(join/split/insert/delete) on splay trees and also presented
(without proof) the Snapshot Theorem. There are many more things
about splay trees discussed in the paper but unfortunately our
course is too short to do them all justice, so we must move on.
- After recalling the definition of a min cut and considering how
it can be computed using (n – 1) "parallel" max flow
computations, we described Karger's contraction-based randomized
algorithm. We analysed it to obtain a running time of
O(n4 log n) and a success probability of (1
– n–c) for any positive constant
- We described a recursive contraction-based algorithm for min
cut that achieves a run time of O(n2
log3n) and a success probability of (1 –
n–c) for any positive constant c.
This, therefore, beats the best known max flow time bound as well as
the best known deterministic time bound (to be described next).
- Anne Loomis described what is currently the fastest (and
simplest) known deterministic min cut algorithm, due to Stoer and
Wagner. She analyzed its running time, proved its correctness and
showed us an example run.
- Homework One: Here it is.
This is due on Mon Feb 7, before class. Please read all the
instructions carefully before you start. Have fun solving!
- Minimum Spanning Tree (MST) was our new topic. We studied basic
graph-theoretic properties of spanning trees, leading up to what
we're calling the "Switch Theorem". We described the Red Rule
and the Blue Rule for MSTs and proved the correctness of the
- We proved the correctness of the Blue Rule. After considering
the colourful history of MST algorithms, we described the three
classic algorithms: Kruskal's, Prim's and Boruvka's. We proved
their correctness using the Red Rule, Blue Rule and Blue Rule
- We defined the concept of F-heavy edges in a graph, with
respect to a sub-forest F. Then we described the Karger-Klein-Tarjan
randomized MST (actually MSF, minimum spanning forest) algorithm and
proved its correctness using the Red and Blue rules and induction.
- Elizabeth Moseman analyzed the worst case and the exected
running times of the Karger-Klein-Tarjan algorithm. The latter turns
out to be linear, provided we assume that MST verification can be
done in linear time. Lizz also gave intuition for why the algorithm
finishes in linear time with very high probability.
- We considered the MST verification problem, which makes up the
last and most complicated piece of the Karger-Klein-Tarjan MST
algorithm. We discussed how to perform m least common
ancestor (LCA) computations in a rooted n-vertex
l-leaf tree in time O(mα(m,l) + n). We then
introduced the microtree technique and gave intuition for why
it might help reduce the running time to O(m + n). We shall
eventually use LCA computations to do MST verification.
- Rajendra Magar gave some details of the microtree technique. He
then described an O(mα(m,l) + n)-time MST verification
algorithm very similar to last time's LCA algorithm. Finally, he
gave a brief outline of how to reduce the running time to O(m +
n) using the microtree technique and further tricks.
- We considered the maximum matching problem. We defined
augmenting paths and proved Berge's Theorem which gave us the
nucleus of a polynomial time maximum matching algorithm.
- The carnival holiday ate up one class.
- We considered a forest-growing algorithm for finding an
augmenting path. It turned out that to make it work we had to assume
that the input graph was bipartite. This led to a maximum matching
algorithm for bipartite graphs.
- Paritosh Kavathekar proved a cycle-shrinking lemma which helped
us modify the previous algorithm suitably to get an algorithm that
worked on general graphs. This was, in essence, Edmond's famous
"shrinking blossoms" algorithm.
- Homework Two: Here it is.
This is due on Mon Feb 28, before class. As before, please read
all the instructions carefully before you start. Do not
forget to have fun solving these!
- We defined the notion of approximation algorithms (for
minimization problems) and gave a simple 2-approximation algorithm
for the NP-hard minimum vertex cover problem.
- We considered approximation algorithms for the traveling
salesman problem (TSP). We showed that finding a k-approximation,
for any constant k, is NP-hard. Then we considered the more
specialized metric TSP for which we gave a 2-approximation
- Valika Wan introduced the set cover problem and gave a greedy
algorithm with approximation ratio H(dmax),
where dmax is the size of the largest set in the
instance and H(k) is the kth harmonic
number. On a universe of size n, this is therefore a (log
n + O(1))-approximation algorithm.
- Khanh Do Ba introduced the shortest superstring problem and
used a set cover approximation algorithm as a subroutine to obtain a
(2Hn)-approximation. In fact he needed an
approximation algorithm for weighted set cover, but noted
that the algorithm presented by Valika handles that too.
- Linear programs are an extremely important tool in designing
approximation algorithms. We introduced linear programs (LPs) and
noted that there are known algorithms that solve LPs in polynomial
(but not strongly polynomial) time. We formulated the vertex cover
problem as in integer program (IP), relaxed this IP to an LP, and
"rounded" the fractional optimal solution to the LP to a solution to
the IP to get a 2-approximation.
- We obtained an O(log n)-approximation for weighted
set cover using the LP rounding technique. This time our rounding
algorithm was randomized.
- Soumendra Nanda presented a PTAS for the subset sum problem.
- Our final problem is string matching. We saw the linear time
Knuth-Morris-Pratt algorithm, which maintains a left pointer and a
right pointer into the text to be searched and cleverly updates
them to skip over unnecessary comparisons. The algorithm requires a
preprocessing step which is itself a computationally interesting
- Ivelin Georgiev described the preprocessing step used in the
KMP algorithm. This step computes, for every prefix p of the
pattern, the longest proper prefix of p that is also a
suffix of p. Ivelin showed how to do the whole thing in
linear time. The proof of correctness is quite complicated.
- Here is the final exam.
- Final remarks.