Recitation 5

Recitation notes for the week of September 25, 2024.

Homework 3, Question 1

A seating

A seating (as you might expect) is an assignment of the $p$ ADFs to the $p^2$ chairs in the theater so that each ADF gets his/her own chair.

Only 1 ADF will be in the first row, but having $p^2$ seats means that there is enough space in each row to hold all ADFs, and there are enough rows that each ADF could have their own row.

Admissible seating

A seating is called admissible if

  1. for every pair of friends $(b_1,b_2)$, they can talk to each other during the movies
    They sit 1 or 0 rows apart
  2. some ADF is assigned a seat in the first row
    Self explanatory, someone sits in the front row
  3. the seating satisfies the distance compatible property.
    Each ADF is friendship distance from the ADF in the front row. If they are direct friends of the front row ADF, they sit in the second row. If they are friends of friends, they sit in the third row and so on.

Example 1

An example seatings with explanation of why they uphold the properties:

Friendships = { (A, B), (B, C) }
            +--+---+--+
            |  | A |  |
            +--+---+--+
            |  | B |  |
            +--+---+--+
            |  | C |  |
            +--+---+--+

            +---+---+---+
            |   | B |   |
            +---+---+---+
            | A |   | C |
            +---+---+---+
            |   |   |   |
            +---+---+---+

These are both valid seatings for the ADFs.

How the first example upholds the properties

  1. A can talk with B, B can talk with C. All friends can talk to each other since they are at most 1 row apart.
  2. ADF A is seated in the first row
  3. A is first level friends with B, because they are directly friends. A is second level friends with C because A is friends with B who is friends with C. Thus, the distance compatible property is upheld because B is 1 row from A and C is 2 rows away.

How the second example upholds the properties

  1. B can talk with A, and B can talk with C. All friends can talk to each other.
  2. ADF B is seated in the first row.
  3. A and C are both direct friends of B, and they are 1 row.

    But why are A and C not far apart?

    Note that A and C have friendship distance of $2$ but they are sitting in the same row. However, this does not violate the distance compatibility property since neither A nor C are in the first row.

Example 2


                        Friendships = { (A,D),(C,B),(A,C),(B,E) }
                        +---+---+---+--+--+
                        |   | A |   |  |  |
                        +---+---+---+--+--+
                        | C |   | D |  |  |
                        +---+---+---+--+--+
                        |   | B |   |  |  |
                        +---+---+---+--+--+
                        |   | E |   |  |  |
                        +---+---+---+--+--+


                        +---+---+---+--+--+
                        |   | C |   |  |  |
                        +---+---+---+--+--+
                        | A |   | B |  |  |
                        +---+---+---+--+--+
                        |   | D | E |  |  |
                        +---+---+---+--+--+

These are both valid seatings for the ADFs.
Note: Not all seats are shown, but there are 5x5 seats available.

How the first example upholds the properties

  1. A is friends with C and D, and can talk to both of them. C is friends with B, they can talk together. B is friends with E and they can talk together.
  2. ADF A is seated in the first row
  3. A is direct friends with C and D, and they are correctly 1 row behind A. C is friends with B, which makes B two levels away from A so the third row is a correct placement. B is friends with E, which makes E three levels away from A so the fourth row is a correct placement.

How the second example upholds the properties

  1. C is friends with A and B, and they can talk since they are 1 row apart or in the same row. A is friends with D, and they are 1 row apart. B is friends with E and they are 1 row apart. All friends can talk.
  2. ADF C is seated in the first row
  3. C is friends with both A and B, and they are 1 row behind C, which is the correct friendship distance. A is friends with D, which makes D two rows away from C which is correct. B is friends with E, which makes two rows away from C the correct placement for E.

Q1 Part (a)

The Problem

For part (a) you must formally define the problem in terms of properties of a graph

Formalizing the problem

  • How to represent the Ava DuVernay fans (ADFs) as a graph $G = (V, E)$?
    1. Fans are the vertices of the graph $G$
    2. If ADFs $i$ and $k$ are friends, then $(i, k)$ exists in the set of edges $E$
  • How to define a seating?
    • A seating is a one-to-one function $s:V\to [p]^2$, i.e. if $s(u)=(r,c)$, then it means that $u$ is seated in row $r$ and column $c$, for notational simplicity the rows and columns are indexed by $[p]$.
  • Defining an admissible seating in terms of properties of a graph $G$:
    1. If there exists an edge (friendship) between ADFs $i$ and $k$, then the absolute value of the difference between their assigned rows must be less than or equal to $1$, i.e. if $s(i)=(r_i,c_i)$ and $s(k)=(r_k,c_k)$, then $|r_i-r_k|\le 1$.
      This means that connected nodes must be placed 1 or 0 rows apart
    2. Some node (ADF) is placed in the first row, i.e. there exists some ADF $u\in V$ and $c\in [p]$ such that $s(u)=(1,c)$.
      This implies that you should start by seating a single ADF
    3. This third property, the distance compatible property, must be solved on your own for part (a).

Q1 Part (b)

For part (b) you should use the graph from part (a) to come up with an algorithm that will in time $O(p^2+f)$ (where $f=|E|$) seat the ADFs in an admissible seating. Note that the runtime $O(f+p^2)$ is always $O(p^2)$.

There might be a reduction to a problem that you have seen in class.

Homework 3, Question 2

We first recall the question:

The Problem

A 4-clique in a graph $G=(V,E)$ is a set of four distinct vertices $\{u,v,w,x\}$ such that $(u,v),(v,w),(w,x),(x,u),(u,w),(v,x)\in E$. (Note that $G$ is undirected.) In this problem you will design a series of algorithms that given a connected graph $G$ as input, lists all the $4$-cliques in $G$. (It is fine to list one $4$-clique more than once.) We call this the 4-clique listing problem (duh!). You can assume that as input you are given $G$ in both the adjacency matrix and adjacency list format. For this problem you can also assume that $G$ is connected.

Sample Input/Output pair

Question 2, part (a)

The Problem

Present an $O(n^4)$ time algorithm to solve the $4$-clique listing problem.

Algorithm Idea

Let us assume we have access to a function Check-if-4-clique that takes as input $G=(V,E)$ and four vertices $u,v,w,x\in V$ and outputs true if $\{u,v,w,x\}$ is a $4$-clique in $G$ and false otherwise.

Then the algorithm is simple: just go through all possible $n^4$ $4$-tuples $\{u,v,w,x\}$ and if Check-if-4-clique$(G,u,v,w,x)$ outputs true then we add $\{u,v,w,x\}$ to the output.

Duplicates

The above algorithm would output the same set $\{u,v,w,x\}$ $24$ times but that is OK for this problem. As long as you (1) only output $4$-cliques; (2) output all $4$-cliques and (3) maintain the given time bound, you can have duplicates.

Exercise

Show that Check-if-4-clique can be implemented in $O(1)$ time.

Algorithm Details

Even though this is not needed for your HW submission, we write down the details of the $O(n^4)$ time algorithm (assuming Check-if-4-clique as a blackbox of course): 

                                    Let GL and GM be the adjacency list and adjacency matrix representation of G (both are input).
                                    # GL is an array of pointers to linked list. The indices to vertices are in GL.keys()

                                    #Get the set of vertices
                                    V = GL.keys()

                                    L = null

                                    #Cycle through all 4-tuples
                                    for u in V:
                                        for v in V:
                                            for w in V:
                                                for x in V:
                                                    if Check-if-4-clique(GM,u,v,w,x) == true:
                                                        Add {u,v,w,x} to L

                                    return L

Question 2, part (b)

The Problem

Present an $O(m^2)$ time algorithm to solve the $4$-clique listing problem.

A Comment

We conclude with a comment:

Homework 3, Question 3

One property of BFS that is useful for Q3 on HW 3:

BFS and distance

Say we run BFS from node $s$ in graph $G=(V,E)$ and let the layers be denoted by $L_0,L_1,\dots$. If $u\in L_i$ for any $u\in V$ and $i\ge 0$, then we have that the distance between $s$ and $u$ is exactly $i$. This is (3.3) in the textbook.

Please see the textbook for a proof of this claim.

Further Reading