Algorithm Problems
Plurality Problem
Now let's take our new knowledge of algorithms and create a program that runs a plurality election that will yield the following output:
$ ./plurality Alice Bob Charlie
Number of voters: 4
Vote: Alice
Vote: Bob
Vote: Charlie
Vote: Alice
Alice
Background
Elections come in all shapes and sizes. In the UK, the Prime Minister is officially appointed by the monarch, who generally chooses the leader of the political party that wins the most seats in the House of Commons. The United States uses a multi-step Electoral College process where citizens vote on how each state should allocate Electors who then elect the President.
Perhaps the simplest way to hold an election, though, is via a method commonly known as the “plurality vote” (also known as “first-past-the-post” or “winner take all”). In the plurality vote, every voter gets to vote for one candidate. At the end of the election, whichever candidate has the greatest number of votes is declared the winner of the election.
Getting Started
We are going to take the following code and complete the vote and print_winner functions:
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78 | #include <cs50.h>
#include <stdio.h>
#include <string.h>
// Max number of candidates
#define MAX 9
// Candidates have name and vote count
typedef struct
{
string name;
int votes;
}
candidate;
// Array of candidates
candidate candidates[MAX];
// Number of candidates
int candidate_count;
// Function prototypes
bool vote(string name);
void print_winner(void);
int main(int argc, string argv[])
{
// Check for invalid usage
if (argc < 2)
{
printf("Usage: plurality [candidate ...]\n");
return 1;
}
// Populate array of candidates
candidate_count = argc - 1;
if (candidate_count > MAX)
{
printf("Maximum number of candidates is %i\n", MAX);
return 2;
}
for (int i = 0; i < candidate_count; i++)
{
candidates[i].name = argv[i + 1];
candidates[i].votes = 0;
}
int voter_count = get_int("Number of voters: ");
// Loop over all voters
for (int i = 0; i < voter_count; i++)
{
string name = get_string("Vote: ");
// Check for invalid vote
if (!vote(name))
{
printf("Invalid vote.\n");
}
}
// Display winner of election
print_winner();
}
// Update vote totals given a new vote
bool vote(string name)
{
// TODO
return false;
}
// Print the winner (or winners) of the election
void print_winner(void)
{
// TODO
return;
}
|
We can make up the following code for the vote and print_winner functions that solves the problem:
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107 | #include <cs50.h>
#include <stdio.h>
#include <string.h>
// Max number of candidates
#define MAX 9
// Candidates have name and vote count
typedef struct
{
string name;
int votes;
}
candidate;
// Array of candidates
candidate candidates[MAX];
// Number of candidates
int candidate_count;
// Function prototypes
bool vote(string name);
void print_winner(void);
int main(int argc, string argv[])
{
// Check for invalid usage
if (argc < 2)
{
printf("Usage: plurality [candidate ...]\n");
return 1;
}
// Populate array of candidates
candidate_count = argc - 1;
if (candidate_count > MAX)
{
printf("Maximum number of candidates is %i\n", MAX);
return 2;
}
for (int i = 0; i < candidate_count; i++)
{
candidates[i].name = argv[i + 1];
candidates[i].votes = 0;
}
int voter_count = get_int("Number of voters: ");
// Loop over all voters
for (int i = 0; i < voter_count; i++)
{
string name = get_string("Vote: ");
// Check for invalid vote
if (!vote(name))
{
printf("Invalid vote.\n");
}
}
// Display winner of election
print_winner();
}
int get_index(string name)
{
for (int i = 0; i < candidate_count; i++)
{
if (strcmp(name, candidates[i].name) == 0)
return i;
}
return -1;
}
// Update vote totals given a new vote
bool vote(string name)
{
int candidate_index = get_index(name); // gets the index for name user inputs
if (candidate_index != -1)
{
candidates[candidate_index].votes++; // increments vote count for each candidate
return true;
}
return false;
}
int get_max(void)
{
int max_votes = candidates[0].votes;
for (int i = 1; i < candidate_count; i++)
if (candidates[i].votes > max_votes)
max_votes = candidates[i].votes;
return max_votes;
}
// Print the winner (or winners) of the election
void print_winner(void)
{
int max_votes = get_max(); // see who has the most voites
for (int i = 0; i < candidate_count; i++) // prints name of candidate with most votes
{
if (candidates[i].votes == max_votes)
printf("%s\n", candidates[i].name);
}
}
|
Runoff
Now let's take the previous problem and make it a little more intricate.
You already know about plurality elections, which follow a very simple algorithm for determining the winner of an election: every voter gets one vote, and the candidate with the most votes wins.
But the plurality vote does have some disadvantages. What happens, for instance, in an election with three candidates, and the ballots below are cast?
| Ballot | Ballot | Ballot | Ballot | Ballot |
| Alice | Alice | Bob | Bob | Charlie |
A plurality vote would here declare a tie between Alice and Bob, since each has two votes. But is that the right outcome?
There’s another kind of voting system known as a ranked-choice voting system. In a ranked-choice system, voters can vote for more than one candidate. Instead of just voting for their top choice, they can rank the candidates in order of preference. The resulting ballots might therefore look like the below.
| Ballot | Ballot | Ballot | Ballot | Ballot |
| 1.Alice | 1.Alice | 1.Bob | 1.Bob | 1.Charlie |
| 2.Bob | 2.Charlie | 2.Alice | 2.Alice | 2.Alice |
| 3.Charlie | 3.Bob | 3.Charlie | 3.Charlie | 3.Bob |
Here, each voter, in addition to specifying their first preference candidate, has also indicated their second and third choices. And now, what was previously a tied election could now have a winner. The race was originally tied between Alice and Bob, so Charlie was out of the running. But the voter who chose Charlie preferred Alice over Bob, so Alice could here be declared the winner.
One such ranked choice voting system is the instant runoff system. In an instant runoff election, voters can rank as many candidates as they wish. If any candidate has a majority (more than 50%) of the first preference votes, that candidate is declared the winner of the election.
Let's look at the following code and see how we can implement this type of voting system. We will need to fill in the correct code for the following functions: vote, tabulate, print_winner, find_min, is_tie, and eliminate.
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167 | #include <cs50.h>
#include <stdio.h>
// Max voters and candidates
#define MAX_VOTERS 100
#define MAX_CANDIDATES 9
// preferences[i][j] is jth preference for voter i
int preferences[MAX_VOTERS][MAX_CANDIDATES];
// Candidates have name, vote count, eliminated status
typedef struct
{
string name;
int votes;
bool eliminated;
}
candidate;
// Array of candidates
candidate candidates[MAX_CANDIDATES];
// Numbers of voters and candidates
int voter_count;
int candidate_count;
// Function prototypes
bool vote(int voter, int rank, string name);
void tabulate(void);
bool print_winner(void);
int find_min(void);
bool is_tie(int min);
void eliminate(int min);
int main(int argc, string argv[])
{
// Check for invalid usage
if (argc < 2)
{
printf("Usage: runoff [candidate ...]\n");
return 1;
}
// Populate array of candidates
candidate_count = argc - 1;
if (candidate_count > MAX_CANDIDATES)
{
printf("Maximum number of candidates is %i\n", MAX_CANDIDATES);
return 2;
}
for (int i = 0; i < candidate_count; i++)
{
candidates[i].name = argv[i + 1];
candidates[i].votes = 0;
candidates[i].eliminated = false;
}
voter_count = get_int("Number of voters: ");
if (voter_count > MAX_VOTERS)
{
printf("Maximum number of voters is %i\n", MAX_VOTERS);
return 3;
}
// Keep querying for votes
for (int i = 0; i < voter_count; i++)
{
// Query for each rank
for (int j = 0; j < candidate_count; j++)
{
string name = get_string("Rank %i: ", j + 1);
// Record vote, unless it's invalid
if (!vote(i, j, name))
{
printf("Invalid vote.\n");
return 4;
}
}
printf("\n");
}
// Keep holding runoffs until winner exists
while (true)
{
// Calculate votes given remaining candidates
tabulate();
// Check if election has been won
bool won = print_winner();
if (won)
{
break;
}
// Eliminate last-place candidates
int min = find_min();
bool tie = is_tie(min);
// If tie, everyone wins
if (tie)
{
for (int i = 0; i < candidate_count; i++)
{
if (!candidates[i].eliminated)
{
printf("%s\n", candidates[i].name);
}
}
break;
}
// Eliminate anyone with minimum number of votes
eliminate(min);
// Reset vote counts back to zero
for (int i = 0; i < candidate_count; i++)
{
candidates[i].votes = 0;
}
}
return 0;
}
// Record preference if vote is valid
bool vote(int voter, int rank, string name)
{
// TODO
return false;
}
// Tabulate votes for non-eliminated candidates
void tabulate(void)
{
// TODO
return;
}
// Print the winner of the election, if there is one
bool print_winner(void)
{
// TODO
return false;
}
// Return the minimum number of votes any remaining candidate has
int find_min(void)
{
// TODO
return 0;
}
// Return true if the election is tied between all candidates, false otherwise
bool is_tie(int min)
{
// TODO
return false;
}
// Eliminate the candidate (or candidiates) in last place
void eliminate(int min)
{
// TODO
return;
}
|
Let's take a look at the vote, tabulate, print_winner, find_min, is_tie, and eliminate functions to see what exactly we need to do.
vote
-
The function takes arguments voter, rank, and name. If name is a match for the name of a valid candidate, then you should update the global preferences array to indicate that the voter voter has that candidate as their rank preference (where 0 is the first preference, 1 is the second preference, etc.).
-
If the preference is successfully recorded, the function should return true; the function should return false otherwise (if, for instance, name is not the name of one of the candidates).
-
You may assume that no two candidates will have the same name.
tabulate
-
The function should update the number of votes each candidate has at this stage in the runoff.
-
Recall that at each stage in the runoff, every voter effectively votes for their top-preferred candidate who has not already been eliminated.
print_winner
-
If any candidate has more than half of the vote, their name should be printed to stdout and the function should return true.
-
If nobody has won the election yet, the function should return false.
find_min
- The function should return the minimum vote total for any candidate who is still in the election.
is_tie
-
The function takes an argument min, which will be the minimum number of votes that anyone in the election currently has.
-
The function should return true if every candidate remaining in the election has the same number of votes, and should return false otherwise.
eliminate
-
The function takes an argument min, which will be the minimum number of votes that anyone in the election currently has.
-
The function should eliminate the candidate (or candidates) who have min number of votes.
Your program should behave per the example below:
./runoff Alice Bob Charlie
Number of voters: 5
Rank 1: Alice
Rank 2: Charlie
Rank 3: Bob
Rank 1: Alice
Rank 2: Charlie
Rank 3: Bob
Rank 1: Bob
Rank 2: Charlie
Rank 3: Alice
Rank 1: Bob
Rank 2: Charlie
Rank 3: Alice
Rank 1: Charlie
Rank 2: Alice
Rank 3: Bob
Alice
The correct code looks as follows:
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225 | #include <cs50.h>
#include <stdio.h>
#include <string.h>
// Max voters and candidates
#define MAX_VOTERS 100
#define MAX_CANDIDATES 9
// preferences[i][j] is jth preference for voter i
int preferences[MAX_VOTERS][MAX_CANDIDATES];
// Candidates have name, vote count, eliminated status
typedef struct
{
string name;
int votes;
bool eliminated;
}
candidate;
// Array of candidates
candidate candidates[MAX_CANDIDATES];
// Numbers of voters and candidates
int voter_count; // global variable
int candidate_count; // global variable
// Function prototypes
bool vote(int voter, int rank, string name);
void tabulate(void);
bool print_winner(void);
int find_min(void);
bool is_tie(int min);
void eliminate(int min);
int main(int argc, string argv[])
{
// Check for invalid usage
if (argc < 2)
{
printf("Usage: runoff [candidate ...]\n");
return 1;
}
// Populate array of candidates
candidate_count = argc - 1;
if (candidate_count > MAX_CANDIDATES)
{
printf("Maximum number of candidates is %i\n", MAX_CANDIDATES);
return 2;
}
for (int i = 0; i < candidate_count; i++)
{
candidates[i].name = argv[i + 1];
candidates[i].votes = 0;
candidates[i].eliminated = false;
}
voter_count = get_int("Number of voters: ");
if (voter_count > MAX_VOTERS)
{
printf("Maximum number of voters is %i\n", MAX_VOTERS);
return 3;
}
// Keep querying for votes
for (int i = 0; i < voter_count; i++)
{
// Query for each rank
for (int j = 0; j < candidate_count; j++)
{
string name = get_string("Rank %i: ", j + 1);
// Record vote, unless it's invalid
if (!vote(i, j, name))
{
printf("Invalid vote.\n");
return 4;
}
}
printf("\n");
}
// Keep holding runoffs until winner exists
while (true)
{
// Calculate votes given remaining candidates
tabulate();
// Check if election has been won
bool won = print_winner();
if (won)
{
break;
}
// Eliminate last-place candidates
int min = find_min();
bool tie = is_tie(min);
// If tie, everyone wins
if (tie)
{
for (int i = 0; i < candidate_count; i++)
{
if (!candidates[i].eliminated)
{
printf("%s\n", candidates[i].name);
}
}
break;
}
// Eliminate anyone with minimum number of votes
eliminate(min);
// Reset vote counts back to zero
for (int i = 0; i < candidate_count; i++)
{
candidates[i].votes = 0;
}
}
return 0;
}
// Record preference if vote is valid
bool vote(int voter, int rank, string name)
{
// TODO
for (int i = 0; i < candidate_count; i++)
{
if (strcmp(candidates[i].name, name) == 0)
{
preferences[voter][rank] = i;
return true;
}
}
return false;
}
// Tabulate votes for non-eliminated candidates
void tabulate(void)
{
// TODO
for (int i = 0; i < voter_count; i++)
{
for (int j = 0; j < candidate_count; j++) // j is rank
{
int candidate_index = preferences[i][j];
if (!candidates[candidate_index].eliminated)
{
candidates[candidate_index].votes++; // update vote count if candidate has not been eliminated
break;
}
}
}
}
// Print the winner of the election, if there is one
bool print_winner(void)
{
// TODO
for (int i = 0; i < candidate_count; i++)
{
if (candidates[i].votes > (voter_count / 2)) // checks to see if one candidate has over half the votes
{
printf("%s\n", candidates[i].name);
return true;
}
}
return false;
}
// Return the minimum number of votes any remaining candidate has
int find_min(void)
{
// TODO
int min = 0; // start from zero
bool find_first = false;
for (int i = 0; i < candidate_count; i++)
{
if (!candidates[i].eliminated) // see if candidate has been eliminated
{
if (!find_first)
{
min = candidates[i].votes;
find_first = true;
}
else if (candidates[i].votes < min)
{
min = candidates[i].votes;
}
}
}
return min;
}
// Return true if the election is tied between all candidates, false otherwise
bool is_tie(int min)
{
// TODO
for (int i = 0; i < candidate_count; i++)
{
if (!candidates[i].eliminated)
if (candidates[i].votes != min)
return false;
}
return true;
}
// Eliminate the candidate (or candidiates) in last place
void eliminate(int min)
{
// TODO
for (int i = 0; i < candidate_count; i++)
{
if (!candidates[i].eliminated)
{
if (candidates[i].votes == min)
candidates[i].eliminated = true;
}
}
}
|