# Problem 10.2

## Mergeable heaps using linked lists

A mergeable heap supports the following operations: MAKE-HEAP (which creates an empty mergeable heap), INSERT, MINIMUM, EXTRACT-MIN, and UNION. Show how to implement mergeable heaps using linked lists in each of the following cases. Try to make each operation as efficient as possible. Analyze the running time of each operation in terms of the size of the dynamic set(s) being operated on.

1. Lists are sorted.
2. Lists are unsorted.
3. Lists are unsorted, and dynamic sets to be merged are disjoint.

There isn't a difference in my implementation between the sorted and unsorted lists. There will be a difference between (a) and (b) if we don't allow repetitions in the lists, but since this will make insertion linear, I find it better to let insert create duplicates and them remove them on EXTRACT-MIN. That way the latter is linear (even if with a larger constant), but UNION can be implemented constantly with doubly-linked lists.

I've chosen singly-linked lists for no good reason. Mostly because I did not realize UNION is going to take linear time with such an implementation (in order to find the last element of the first list). If we switch to doubly-linked, we can make it constant easily.

I find this problem too simple to bother further.

Otherwise, the running times are as follows

sorted unsorted
MAKE-HEAP constant constant
INSERT linear constant
MINIMUM constant linear
EXTRACT-MIN constant linear
UNION linear linear*

As I noted, UNION can be made constant if we keep track of the last element of the list.

### C code

#include <stdlib.h>
#include <stdarg.h>

/////////////////////////////////////////////////////////////////////////////
// List operations
/////////////////////////////////////////////////////////////////////////////

struct list_t {
int key;
struct list_t *next;
};

typedef struct list_t list_t;

list_t *insert_sorted(list_t *list, int key) {
list_t *new = malloc(sizeof(list_t));
new->key = key;

if (!list || key < list->key) {
new->next = list;
return new;
}

list_t *l = list;
while (l->next && l->next->key < key) {
l = l->next;
}

new->next = l->next;
l->next = new;

return list;
}

list_t *delete_key(list_t *list, int key) {
list_t *match;
while (list && list->key == key) {
match = list;
list = list->next;
free(match);
}

if (!list) {
return NULL;
}

list_t *node = list;
while (node->next) {
if (node->next->key == key) {
match = node->next;
node->next = match->next;
free(match);
} else {
node = node->next;
}
}

return list;
}

list_t *prepend(list_t *list, int key) {
list_t *new = malloc(sizeof(list_t));

new->key = key;
new->next = list;

return new;
}

int find_min(list_t *list) {
int min = list->key;
list = list->next;

while (list) {
if (list->key < min) {
min = list->key;
}
list = list->next;
}

return min;
}

list_t *link_together(list_t *a, list_t *b) {
list_t *result;

if (!a) {
result = b;
} else {
result = a;
while (a->next) {
a = a->next;
}
a->next = b;
}

return result;
}

list_t *merge_sorted(list_t *a, list_t *b) {
list_t dummy;
list_t *new = &dummy;

while (a && b) {
if (a->key < b->key) {
new->next = a;
a = a->next;
new = new->next;
} else {
new->next = b;
b = b->next;
new = new->next;
}
}

if (a) {
new->next = a;
} else {
new->next = b;
}

return dummy.next;
}

/////////////////////////////////////////////////////////////////////////////
// 1. Mergreable heaps with sorted list
/////////////////////////////////////////////////////////////////////////////

typedef struct {
} heap1;

heap1 *make_heap1() {
heap1 *result = malloc(sizeof(heap1));
return result;
}

void insert1(heap1 *heap, int key) {
}

int minimum1(heap1 *heap) {
}

int extract_min1(heap1 *heap) {
return result;
}

heap1 *union1(heap1 *ha, heap1 *hb) {
heap1 *result = malloc(sizeof(heap1));

free(ha);
free(hb);

return result;
}

/////////////////////////////////////////////////////////////////////////////
// 2. Mergreable heaps with unsorted lists
/////////////////////////////////////////////////////////////////////////////

typedef struct {
} heap2;

heap2 *make_heap2() {
heap2 *result = malloc(sizeof(heap2));
return result;
}

void insert2(heap2 *heap, int key) {
}

int minimum2(heap2 *heap) {
}

int extract_min2(heap2 *heap) {
int min = minimum2(heap);