This document is a rough guide on writing up solutions
in CSE 202.
Writeups should not be excessively long.
Get to the point fast.
A typical problem will require a writeup of 1-2 pages,
but this is just an estimate, not a hard rule.
If you can argue your point in 1 paragraph,
that's even better.
The hard rule is
Of course if you have some amazing insight into the problem
that is not asked for, like an optimality or tightness proof,
feel free to share it.
You will not lose points as long as the part that is asked for
- Did I say everything I needed to say? (answer the question asked
and backup all claims)
- Did I say only what I needed to say? (discard unused claims and babble)
The key idea of the algorithm should be conveyed clearly.
Following should come algorithmic details such as
what data structures are involved.
(For example, "To remove the least weighted edge
in O(ln n) time, we maintain a minheap.")
Pseudocode should be provided, where appropriate.
The idea is that an algorithm should be conveyed several
times but in increasing level of detail.
Proof must always be provided at a level of detail
allowing a peer to be convinced.
This may involve equations, induction, combinatoric
reasoning, etc., and should be written in prose.
Almost any proof in the textbook serves as a good model.
State any extra assumptions that you need.
For example, "The graph G is represented as an array of
adjacency lists, and we assume that the adjacency lists are sorted."
In this class we will be primarily interested in 3 properties of algorithms:
- correctness (meets specifications)
- complexity (efficiency in time, space, or other resources)
- optimality with respect to a certain class of algorithms
Correctness of an algorithm is often (but not always)
proved in 2 parts:
first we show that it yields the correct output upon termination,
and then we show that it terminates.
This division often makes the proof easier.
For example, consider the following pseudocode.
INPUT: a satisfiable conjunctive normal form formula F in the variables x1,...,xn
OUTPUT: an assignment to x=(x1,...xn) such that F(x)=1
if F(0)==1, return 0
choose prime p>=2^n
for each prime q|p-1
x = 1
while F(x mod 2^n)==0
x = g * x mod p
return x mod 2^n
It is clear that the output is correct if the algorithm terminates,
but it is less clear that the algorithm terminates,
so the majority of the effort in proving correctness will be in
proving the latter.
In this example, x runs through a strange permutation
of all of the numbers in [1,p-1] and so will hit a solution
if there is one.
By contrast, in the Ford-Fulkerson method for network flow,
it is clear that the algorithm terminates but it is less clear that
it is partially correct.
Indeed, for most algorithms in this class, termination
will be clear.
The most typical measures of resource consumption for
an algorithm are time and space, though sometimes we care
about others, such as number of key comparisons for a sorting
We usually care about worst case resource consumption
and occasionally expected resource consumption.
(expected values taken over the random choices that the
Usually, we do not assume any particular distribution of the
input. We might as well assume that some adversary has access
to the code of our algorithm and gives an input designed to thwart
Big-Oh class is usually sufficient, but occasionally we will
want an exact quantity. All relevant parameters should be included
in the expression.
For example, Theta(m+n) time is required to add
an m-bit number to an n-bit number.
To see the lower bound, note that any algorithm A must
examine all of the bits.
To see the upper bound, use the grade school addition method
(say, on a RAM model).
If we know that m<=n, then we may simplify this to Theta(n).
We will be concerned with 3 only occasionally -
such results are hard to come by.
Give credit to any other
students who contributed key ideas.
Credit should be given when using ideas from research papers,
especially when obscure.
For example, "by a slight modification to
the Agrawal-Kayal-Saxena algorithm..."
You do not need to give credit to the professor,
the TA, nor the textbook (though possibly to the author
of an algorithm appearing in the textbook!).
Innocent citation mistakes will not be brutally penalized.
The following is a nonexhaustive list of mistakes that
may cause the loss of 0 to 1 points.
These may cause greater loss of credit.
- Forgetting to prove a major claim
- Proof incorrect
- Proof too confusing to understand. Poor notation may
cause this. For example, suppose you need to distinguish
between the value of a variable x at the beginning of an algorithm
and at step i. Use a disambiguating notation in the proof
such as xi.
- Algorithm too suboptimal. Almost any problem can be
solved inefficiently by brute force. Strive for better.
- Too much slack in the analysis.
For example do not say that the algorithm takes time O(n^100)
if it is more obviously O(n^2).
- Algorithm unnecessarily complex
These can give extra points.
- Finding a better algorithm than that of the professor
or the TA. Better can mean significantly simpler with the
same asymptotic complexity or better asymptotic complexity.
This really can be done. Look for creative solutions!
- Finding tighter upper or lower bounds than that of the
professor or the TA. To find a lower bound on a specific
algorithm, you might have to find a specific input on which
the algorithm performs poorly. (but not necessarily!
it may be the case that all inputs yield the same behavior.)
To find an upper bound, you need to consider all possible
How to Write a Good Proof
The following suggestions need not be followed rigorously,
but may help you to improve clarity.
Notice how good proof writing practice mirrors
good programming practice.
- Develop the necessary notation and
be consistent with it.
Notation developed within a proof is usually considered
local to the proof.
If you need notation to have a more global scope,
declare it more globally.
(If G is a graph, let A(G) be its adjacency matrix.)
- State what you want to prove clearly and precisely.
(G is connected iff the largest and second largest
eigenvalues of A(G) are unequal)
- Write the word, "proof."
- Prove it.
- If you are going to use a proof by contradiction,
state the assumption which is supposed to cause the
contradiction and clearly indicate that it is not to
be believed outright, but that it will cause a contradiction
For example, "To show that the number of primes is infinite,
suppose indirectly that p_1< ...< p_n are the only primes.
(notice the use of the word "indirectly")
Then p_1*...*p_n+1 is divisible by one of them, say p_i.
But then 1 is divisible by p_i, a contradiction.
So the number of primes must be infinite."
- If you use induction and the induction assumption
is not obvious, state it. Sometimes, to prove a statement
A by induction, we need to prove a stronger statement B
or write the claim in a clever way to allow the induction to work.
This often occurs when proving loop invariants.
For example, "We prove by induction on the size n of the larger
of the 2 graphs that the algorithm uses only n comparisons."
- When using induction, prove the base cases.
If these are truly trivial, it
is permissible to state this, e.g. "The n=0,1,2 cases are
If a case requires checking nothing, for example if some assertion
is made about all of the elements of the empty set, then
it is permissible to state that it holds "vacuously".
But sometimes base cases are hard and require careful justification.
Recognize these times.
Also be careful of what constitutes a base case.
For example, if, exactly when n>=1, we replace n by floor(n/2)
and invoke the induction
hypothesis, then 0 is the base case.
But if, exactly when n>=100, we replace n by floor(n/2) and invoke
the induction hypothesis, then 50 through 99 are the base cases.
(and possibly 0 though 49 if we care about these)
- List the important calculations, but truly trivial
calculations can be omitted.
"x>=1 => x+1>= 2 => (x+1)/2>=1 => -(x+1)/2<=-1"
is verbose, "x>=1 => -(x+1)/2<=-1" is better.
- Approximate equality is fine for building intuition
and can even be stated in a proof, but exact inequalities should
be used to back it up. For example 1+x is approximately e^x
for small x, but it is useful to know that 1+x<=e^x for all x
and Taylor's theorem can be used to bound the gap if need be.
- For a very tricky proof, give the high-level intuition first.
- If an idea surfaces within a proof that could be used later
and is interesting in its own right, consider making it a lemma.
This is like code modularity and reuse.
- Indicate that the proof is over, e.g. by a
square or QED.
- Possibly state the significance of what was
just shown if it is not obvious.
(If all we cared about was whether G were connected,
we would use a simpler technique, such as Dijkstra's algorithm;
but the above is useful for analysis.)
Last modified 9/27/04