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What NOT to doI get soooooFrustrated!

Marking the SAMEwrong answer hundreds of times!

I will give a list of mistakes which I particularly hate marking& for which there is no excuse.

If you do any of these thingsyou will get -20%

on the question.

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Think about Algorithms Abstractly

Devastated by the midterm

Change your thinking now.

This course requires completely changing the way you think about algorithms.

Though I keep warning people, they tend not to get it until they are

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• Easy algorithmic problems testing whether you get the correct structure of iterative & recursive programs. – Get the structure correct, even if you can’t get the details of the

algorithm correct. – Tell me that you know what you need to do

and where you get stuck

Types of Questions

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In high school, I hated teachers who marked answers wrong if not done their way. Now I do the same. The method of thinking abstractly is important. 99% of other types of answers are wrong.

Types of Questions

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Types of Questions• Did you learn the basics of the algorithms taught in class?

Can you trace them? • Did you do the homework? Did you read the solution set?• Math: summing, recurrence relations, & , running time

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Give the exact time complexity (running time).No one gave the correct

answer!

Do not measure running time

using the value of the input.

Size = log n = s. n = 2s. Time = 2n2 + 3n + 4 = 2·22s + 3·2s + 4.

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T(n) = T(n/ 4) + T(n-5) + 3.

Not T(n) = 3 T(n/ 4) + T(n-5) + 3.Not Eg(n) = Eg(n/ 4) + Eg(n-5) + 3.

Give the exact time complexity (running time).

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T(n) = T(n/ 4) + T(n-5) + n + 3.

Not T(n) = T(n/ 4) + T(n-5) + T(n) + 3.

Give the exact time complexity (running time).

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Typical Loop InvariantIf the input consists of an array of objects

Each iteration considers the next input item.

Loop Invariants are picturesof current state.

Not actionsNot algorithms

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Typical Loop InvariantIf the input consists of an array of objects

We have considered the ith input element.

What about elements [1...i]?

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Typical Loop InvariantIf the input consists of an array of objects

We have considered input elements [1..i].

So?

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Typical Loop InvariantIf the input consists of an array of objects

We have a solution for theprefix consisting of elements [1..i].

plus some additional information

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Typical Loop InvariantIf the output consists of an array of objects

I have produced the first i objects.

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For all major algorithms covered. Learn pre and post conditions. Learn the Loop Invariant Learn how to make progress while maintaining the LI.

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• Trust your friends to solve subinstances. • The subinstance given must be smaller

and must be an instance to the same problem.• Combine solution given by friend to construct

your own solution for your instance. • Focus on one step. • Do not talk of their friends friends friends. • Solve small instances on your own.

I am obsessed with the Friends - Strong Induction View of Recursion.

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I am obsessed with the Friends - Strong Induction View of Recursion.

For binary trees, make sure that your program works for

generic generic generic

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Define pre & post conditions

Don't have inputs or outputs that are not explained!

Typical Test Answer

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Typical Test Answer

Call recursively

on the correct types of input

k,num,v

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Typical Test Answer

Call recursively

Save the results (or don't bother calling)

returning the correct types of output

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Typical Test Answer

Combine solutions given by friends

to construct your own solution.

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Typical Test Answer

Return things of the correct types.

Not an element

In every path through code

the root?

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Typical Test Answer

Subinstancesneed to be smaller.

“Size” is size of the tree

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Typical Test Answer

Subinstancesneed to be smaller.

When the instance sufficiently small

solve on your own.

This does not actas base case

because k is not getting smaller

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Typical Test Answer

Subinstancesneed to be smaller.

When the instance sufficiently small

solve on your own.

Code does not checkwhether the tree is empty.

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Typical Test Answer

Local variables (other than those holding and

combining solutions) are not usually needed.

Value of n?

Local variables don't change outside

of the current stack frame.

n = 2

No global variables allowed!

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Typical Test Answer

No global returns. Things returned

by your friends do not get returned

to your boss unless you do the returning.

My friend finds and returns the answer.

I drop his answer.

And return nothing!

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Typical Test Answer

Loops are rarely used in recursive programs.

on the same input

getting the same result each time!

Returns the root??

Called n times

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Typical Test Answer

Don’t expect your markerto be a compiler!

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Typical Test Answer

Traverse nodes in in-order. Count the nodes. Output the

kth one.

Reasonable Sounding Alg

This is an iterative alg not a recursive alg.

Hard to write a recursive program that implements an iterative algorithm.

Fine (though asked for recursive)

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Greedy Algorithms Don’ts

What is the loop invariant of any greedy algorithm?

We prove that the algorithm's solution is …

The algorithm has made some commitments already, but this does not yet constitute

a valid solution.

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Greedy Algorithms Don’ts

What the algorithm has done so far is optimal.

What does this mean?

The “More of the Input” loop invariantdoes not work.

What is the loop invariant of any greedy algorithm?

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Greedy Algorithms Don’ts

“There exists an opt solution consistent with choices made so far.”

What is the loop invariant of any greedy algorithm?

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Greedy Algorithms Don’ts

We prove it is consistent, optimal, and valid.

Don't say "it" without saying what "it" is.

The loop invariant of any greedy algorithm is“There exists an opt solution consistent

with choices made so far.” How we prove that this loop invariant

is maintained?

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Greedy Algorithms Don’ts

We tell the fairy god mother to change her solution optSLI into optSours. We must prove optSours is consistent, optimal, and valid.

Great, but what does this mean?

The loop invariant of any greedy algorithm is“There exists an opt solution consistent

with choices made so far.” How we prove that this loop invariant

is maintained?

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Greedy Algorithms Don’ts

The Prover is unable to see the Fairy Godmother's optimal solution.

The Prover compares the Fairy Godmother's optimal solution to what the algorithm has done.

How do we prove optSours is consistent?

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Greedy Algorithms Don’ts

By the LI, optSLI is consistent with what the algorithm did in the first i steps.The Prover instructs the Fairy Godmother to change it to optSours to make it consistent with the i+1st step without missing up the earlier commitments.

How do we prove optSours is consistent?

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Greedy Algorithms Don’ts

It is true that is important.But it is not what we need here.

Show that the steps taken by the algorithm are valid.

How do we prove optSours is valid?

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Greedy Algorithms Don’ts

By the LI, optSLI is valid (ie does not contain conflicts within it.)The Prover instructs the Fairy Godmother to change it to optSours in a way that provably does not introduce conflicts.

How do we prove optSours is valid?

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Greedy Algorithms Don’ts

It is true that is important.But it is not what we need here.

Show that the steps taken by the algorithm are the best choice available.

How do we prove optSours is optimal?

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Greedy Algorithms Don’ts

By the LI, optSLI is optimal (ie there is not a valid solution worth more.)The Prover instructs the Fairy Godmother to change it to optSours in a way that provably does not decrease its worth.

How do we prove optSours is valid?

Good

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Optimization Problems• Don’t mix up the following

– What is an instance– What are the objects in an instance– What is a solution– What are the objects in a solution– What is the cost of a solution

• Greedy algorithm– What does the algorithm do & know– What does the prover do & know– What does the fairy god mother do & know

• Recursive Backtracking / Dynamic Programming– What does the algorithm do & know– What does the little bird do & know– What does the friend do & know

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Dynamic Programmingdon’ts

• Yes, the code has a basic structure that you should learn.• But don’t copy other code verbatim • Don’t say if(ai = bj)

(i.e. Longest Common Subsequence) when our problem has not bj

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Dynamic Programmingdon’ts

• When looping over the subinstances • be clear what the set of subinstances are • which is currently being solved,

i.e. which instance is cost(i,j)?• If you know that the set of subinstances are the

prefixes of the input, i.e. <a1,a2, …, ai>, then don’t have a two dimensional table. Table[1..n,1..n].

• Don’t loop over i and loop over j if j never gets mentioned again.

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Dynamic Programmingdon’ts

• .When trying all bird answers • be clear what the set of bird answers are,• which is currently being tried,• & what it says about the solution being looked for.

• When getting help from your friend,• be clear what the subinstance is that you are giving him• How do you use the current instance and the birds answer

to form his subinstance.• Don’t simply say cost(i-1,j-1)

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Dynamic Programmingdon’ts

• .Think about what the base cases should be.• Don’t make an instances a base cases

if they can be solved using the general method.• % is used to start a comment.

Don’t put it in front of code.

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End