Problem 8.3

Sorting variable-length items

1. You are given an array of integers, where different integers may have different number of digits, but the total number of digits over all the integers in the array is $n$. Show how to sort the array in $\O(n)$ time.
2. You are given an array of strings, where different strings may have different numbers of characters, but the total number of characters over all the strings is $n$. Show how to sort the strings in $\O(n)$ time.
(Note that the desired order here is the standard alphabetical order; for example; $\mathrm{a} < \mathrm{ab} < \mathrm{b}$.)

The numbers

For the numbers, we can do this:

1. Group the numbers by number of digits and order the groups

Let's analyze the number of steps. Let $G_i$ be the group of numbers with $i$ digits and $c_i = |G_i|$. Thus:

$$T(n) = \sum_{i=1}{n}c_i \cdot i = n$$

The strings

For the strings, we can do this:

1. Groups the strings by length and order the groups
2. Starting $i$ on the maximum length and going down to 1, perform counting sort on the $i$th character. Make sure to include only groups that have an $i$th character.

If the groups are subsequent subarrays in the original array, we're performing counting sort on a subarray ending on the last index of the original array.

The implementation

The C code is a bit clever, but this is an exercise, not a production system.

C code

#include <math.h>
#include <string.h>

#define MAX_LENGTH 10

// --- Structs and typedefs ---------------------------------------------------

// In order to simplify everything, both numbers and strings are meshed in a
// single union called key_t. The key does not know whether it is a number or a

union key_t {
int number;
char string[MAX_LENGTH + 1];
};

typedef struct {
union key_t key;
int value;
} item;

typedef int (*key_f)(item, int);
typedef int (*dimension_f)(item);
typedef int (*compare_f)(item, item);

// --- Prototypes -------------------------------------------------------------

// Various sorting functinos

void partition(item *A, int size, int digits, int *groups, dimension_f dimension);
void radix_sort(item *A, int left, int right, int digits, key_f key);
void counting_sort(item *A, int left, int right, int dimension, key_f key, int key_index);

// Functions to work on numbers

int item_nth_digit(item i, int d);
int item_digits(item i);

// Functions to work on strings

int item_string_length(item i);
int item_nth_char(item i, int d);

// --- The solutions ----------------------------------------------------------

void sort_numbers(item *A, int size, int max_digits) {
int groups[max_digits + 1];

partition(A, size, max_digits, groups, item_digits);

for (int i = 1; i < max_digits + 1; i++) {
radix_sort(A, groups[i - 1], groups[i], i, item_nth_digit);
}
}

void sort_strings(item *A, int size, int max_length) {
int groups[max_length + 1];

partition(A, size, max_length, groups, item_string_length);

for (int len = max_length; len > 0; len--) {
counting_sort(A, groups[len - 1], size, 26, item_nth_char, len - 1);
}
}

// --- Auxiliary sorting functions --------------------------------------------

// Performs counting sort on a dimension (number of digits or string length)
// and populates a table (groups) with the position of each dimension.

void partition(item *A, int size, int max_dimension, int *groups, dimension_f dimension) {
int counts[max_dimension + 1];
item temp[size];

for (int i = 0; i < max_dimension + 1; i++) { groups[i] = 0; }
for (int i = 0; i < size;              i++) { groups[dimension(A[i])]++; }
for (int i = 1; i < max_dimension + 1; i++) { groups[i] += groups[i - 1]; }
for (int i = 0; i < max_dimension + 1; i++) { counts[i] = groups[i]; }
for (int i = 0; i < size;              i++) { temp[i] = A[i]; }

for (int i = size - 1; i >= 0; i--) {
int d = dimension(temp[i]);
int count = counts[d];

A[count - 1] = temp[i];
counts[d]--;
}
}

void radix_sort(item *A, int left, int right, int digits, key_f key) {
for (int i = 0; i < digits; i++) {
counting_sort(A, left, right, 10, key, i);
}
}

// A slightly generalized counting sort

void counting_sort(item *A, int left, int right, int dimension, key_f key, int key_index) {
int size = right - left;
int counts[dimension];
item temp[size];

for (int i = 0;    i < dimension; i++) { counts[i] = 0; }
for (int i = left; i < right;     i++) { counts[key(A[i], key_index)]++; }
for (int i = 1;    i < dimension; i++) { counts[i] += counts[i - 1]; }
for (int i = 0;    i < size;      i++) { temp[i] = A[left + i]; }

for (int i = size - 1; i >= 0; i--) {
int n = key(temp[i], key_index);
int count = counts[n];

A[left + count - 1] = temp[i];
counts[n]--;
}
}

// --- Key handling -----------------------------------------------------------

int count_digits(int n) {
if (n == 0) {
return 1;
} else {
return (int) log10(n) + 1;
}
}

int nth_digit(int n, int d) {
int magnitude = (int) pow(10, d);

return (n / magnitude) % 10;
}

int item_nth_digit(item i, int d) {
return nth_digit(i.key.number, d);
}

int item_digits(item i) {
return count_digits(i.key.number);
}

int item_string_length(item a) {
return strlen(a.key.string);
}

int item_nth_char(item a, int n) {
return a.key.string[n] - 'a';
}