You are currently browsing the category archive for the ‘Uncategorized’ category.

Informally speaking, Impagliazzo’s hardcore lemma says that if a boolean function is “hard to compute on average” by small circuits, then there exists a set of inputs on which the same function is “extremely hard to compute on average” by slightly smaller circuits.

In this post, I am going to explain how I understand the proof of the hardcore lemma presented in the Arora-Barak complexity book (here). Because the formal proof can be found in the book I intend to write in an informal way. I think some subtleties are involved in turning the context of the lemma into a suitable two-player zero-sum game. Doing so enables one to use von Neumann’s minimax theorem to effectively “exchange the quantifiers” in the contrapositive statement of the lemma. Although the Arora-Barak proof mentions these subtleties, I am going to explore these in more detail and in a more accessible way for a beginner like me.

Read the rest of this entry »

I changed my mind before entering the blue doors of the weight room in my gym. There was the soothing sound of water trickling  down from the fountain. I felt more thirsty as I waited.

— “Hey, Saad!” exclaimed a low voice. I looked to my right. He just came out of that blue door of the weight room.

— “You used to be my TA !” he said. It always feels good to meet your past students, especially ones that still remember you. I looked hard. Remembering my students’ names used to be my forte, but probably not anymore. Read the rest of this entry »

You must have known it already, but I can’t stop myself from showing off the interplay of const with pointers in C/C++.

Here goes the code. It’s simple.


void main() {

	int arr[]={1,2,3,4}; // data

	const int *p1 = &arr[0]; // non-const ptr to const data
	int const *p2 = &arr[0]; // non-const ptr to const data
	int* const p3 = &arr[0]; // const ptr to non-const data
	const int* const p4 = &arr[0]; // const ptr to const data

	p1++; // ok
	p1[0]++; // compile error: modifying const data

	p2++; // ok
	p2[0]++; // compile error: modifying const data

	p3++; // compile error: modifying const ptr
	p3[0]++; // ok 

	p4++; // compile error: modifying const ptr
	p4[0]++; // compile error: modifying const data

}

Read the rest of this entry »

We will show that the eigenvalues of the n\times n Laplacian matrix L of the complete graph K_n are \{0,1\} where  the eigenvalue 0 has (algebraic) multiplicity 1 and the eigenvalue n has multiplicity n-1.

Read the rest of this entry »

Welcome to WordPress.com. This is your first post. Edit or delete it and start blogging!