DSA1 | Week 4

Interlude: A Card Trick & Asymptotics

The CARDSTACK data structure

...after the riffle shuffle.

The CardStack Data Structure

Introducing...

< maintain an ordered stack of cards >

The Data

how many cards in the stack?

Operations & Queries

pull out the top \(k\) cards

pull out the bottom \(k\) cards

cut shuffle at \(k\) cards

typedef struct s_card {
    int cardvalue;
    
    
    
    
} t_card;

typedef struct s_card {
    int cardvalue;
        
    struct s_card *next;
    struct s_card *prev;
    
} t_card;

last

first

typedef struct s_cardstack {

    struct s_card *first;
    struct s_card *last;
    
} t_cardstack;

t_cardstack* cardstackInit()
{

    t_cardstack* cardstack;
    cardstack = malloc(sizeof(t_cardstack));
    
    cardstack->first = NULL;
    cardstack->last = NULL;
    
    return cardstack;

}

The initialization is pretty straightforward.

Next up...

...let us write a simple helper function

to check if our stack of cards is empty.

int isEmpty(t_cardstack *cardstack)
{
    return !cardstack->first;
}

int isEmpty(t_cardstack *cardstack)
{
    return cardstack->first;
}

int isEmpty(t_cardstack *cardstack)
{
    return !cardstack->last;
}

int isEmpty(t_cardstack *cardstack)
{
    return cardstack->last;
}

Here is what the card stack looks like, schematically.

first

last

pushFront(cardstack, card)

first

last

first

last

first

last

first

last

first

last

first

last

first

last

first

last

void pushFront(t_cardstack* cardstack, int cardvalue)
{
 
    t_card *node = malloc(sizeof(t_card));
    
    node->cardvalue = cardvalue;
    
    node->prev = NULL;
    
    node->next = cardstack->first;
    
    if(cardstack->last) // the stack is NOT empty
        cardstack->first->prev = node;
    else
        cardstack->last = node;       
    
    cardstack->first = node;

}

popBack(cardstack)

first

last

first

last

first

last

first

last

first

last

int popBack(t_cardstack *cardstack)
{
    t_card *node;
    int cardvalue;
    
    if (isEmpty(cardstack))
        return -1;
    
    node = cardstack->last;    
    cardstack->last = node->prev;
    
    if (!cardstack->last)
        cardstack->first = NULL;
    else
        cardstack->last->next = NULL;
    
    cardvalue = node->cardvalue;
    free(node);
    return cardvalue;
}

View Implementation

	Front	Back
Add	allowed	allowed
Remove	allowed	allowed

DEQUES

	Front	Back
Add	allowed	❌
Remove	allowed	❌

STACKS

	Front	Back
Add	❌	allowed
Remove	❌	allowed

STACKS

	Front	Back
Add	❌	allowed
Remove	allowed	❌

QUEUES

	Front	Back
Add	allowed	❌
Remove	❌	allowed

QUEUES

	Adj. Matrix	Adj. List	Edge List
Adding a vertex	Painful	Tolerable	Decent
Deleting a vertex	Painful	Tolerable	Tolerable
Adding an edge	Brilliant	Brilliant	Brilliant
Deleting an edge	Brilliant	Decent	Tolerable
Finding degree(v)	Tolerable	Decent	Tolerable
Check if (u,v) is an edge	Brilliant	Decent	Tolerable

Recap: what we have seen so far

Graph Representations

Any number of cut shuffles are done.

Five people pull out the top five cards in order.

A deck is given to someone in class.

Magic with Cards

The ones with red cards stand up.

All cards are identified.

The Card Identifier

X Y A B C

00 - ♣️ | 01 - ♠️ | 10 - ♦️ | 11 - ♥️

Five bits correspond to a card

A B C - value of the card (000 interpreted as 8)

The Sequence