Algorithms To Live By

Optimal Stopping

Look and Leap

The Secretary Problem

Problem Definition

Optimal Stopping Simulation Using Core Python - 3 Secretaries - 1,000,000 runs

import random

random.seed(42)

iteration_count = 1000000
right_choice_made = 0

for i in range(iteration_count):
    assigned_nums = []

    while len(assigned_nums) != 3:
        rand_ordinal = random.randint(0, 2)
        if rand_ordinal in assigned_nums:
            continue
        assigned_nums.append(rand_ordinal)

    if assigned_nums[2] > assigned_nums[1]:
        right_choice_made += 1

print(right_choice_made / iteration_count) # 0.500469

Optimal Stopping Simulation Using Pandas - 100 Secretaries - 1,000,000 runs

Do not hire any candidate from the first 37 percentile, store the value of best in this phase. Starting from 38 percentile, hire the first candidate encountered where the candidate is better compared to best observed in first 37 percentile. Following this strategy will lead to hiring the best candidate 37% of the time, the best you can have.

import numpy as np
import pandas as pd

rng = np.random.RandomState(42)

numOfSecretaries = 100
numOfTries = 1000000
optimal_stopping_location = int(numOfSecretaries * 0.37)

secretaries = pd.DataFrame(
    rng.choice(
        np.arange(0, 100000), 
        size=[numOfSecretaries, numOfTries]
    )
)

training = secretaries[0:optimal_stopping_location]
threshold = training.max()

decision = secretaries[optimal_stopping_location:].reset_index(drop=True)

df = decision[decision > threshold]
df = df.idxmax() == df.apply(pd.Series.first_valid_index).values
print(np.sum(df) / numOfTries) # 0.370514

Walkthrough An explanation of what is going on in the above implementation with a smaller set of data: 15 candidates, 5 runs.

import numpy as np
import pandas as pd

rng = np.random.RandomState(42)

numOfSecretaries = 15
numOfTries = 5
optimal_stopping_location = int(numOfSecretaries * 0.37)

secretaries = pd.DataFrame(
    rng.choice(
        np.arange(0, 100), 
        size=[numOfSecretaries, numOfTries]
    )
)

# Initial DataFrame representing secretary points.
print(secretaries)
#      0   1   2   3   4
# 0   51  92  14  71  60
# 1   20  82  86  74  74
# 2   87  99  23   2  21
# 3   52   1  87  29  37
# 4    1  63  59  20  32
# 5   75  57  21  88  48
# 6   90  58  41  91  59
# 7   79  14  61  61  46
# 8   61  50  54  63   2
# 9   50   6  20  72  38
# 10  17   3  88  59  13
# 11   8  89  52   1  83
# 12  91  59  70  43   7
# 13  46  34  77  80  35
# 14  49   3   1   5  53

training = secretaries[0:optimal_stopping_location]
# DataFrame we will be using to adjust our threshold value.
print(training)
#     0   1   2   3   4
# 0  51  92  14  71  60
# 1  20  82  86  74  74
# 2  87  99  23   2  21
# 3  52   1  87  29  37
# 4   1  63  59  20  32

threshold = training.max()

# Our threshold value for each run.
print(threshold)
# 0    87
# 1    99
# 2    87
# 3    74
# 4    74
# dtype: int64

decision = secretaries[optimal_stopping_location:].reset_index(drop=True)

# DataFrame where we will be picking from. 
# Remember: Pick the first value greater than threshold.
print(decision)
#     0   1   2   3   4
# 0  75  57  21  88  48
# 1  90  58  41  91  59
# 2  79  14  61  61  46
# 3  61  50  54  63   2
# 4  50   6  20  72  38
# 5  17   3  88  59  13
# 6   8  89  52   1  83
# 7  91  59  70  43   7
# 8  46  34  77  80  35
# 9  49   3   1   5  53

# Figure out the first value > threshold.
# masked is a DataFrame where values lower than threshold are NaN
masked = decision[decision > threshold] # type: pd.DataFrame
print(masked)
#       0   1     2     3     4
# 0   NaN NaN   NaN  88.0   NaN
# 1  90.0 NaN   NaN  91.0   NaN
# 2   NaN NaN   NaN   NaN   NaN
# 3   NaN NaN   NaN   NaN   NaN
# 4   NaN NaN   NaN   NaN   NaN
# 5   NaN NaN  88.0   NaN   NaN
# 6   NaN NaN   NaN   NaN  83.0
# 7  91.0 NaN   NaN   NaN   NaN
# 8   NaN NaN   NaN  80.0   NaN
# 9   NaN NaN   NaN   NaN   NaN

# Now that we have `masked`, we will actually be picking the first !NaN value.
# For example for 0th run we will be picking 90, since that is the first value 
# greater than our threshold.
# However, in this case, we are not actually picking the best candidate we can..
# There is a better candidate at index 7 with a value of 91! Tough luck..

# Basically the first index that is actually a value..
index_of_candidates_we_picked = masked.apply(pd.Series.first_valid_index).values
print(index_of_candidates_we_picked)
# [ 1. nan  5.  0.  6.]

# index of the actual best candidate..
idxmax = masked.idxmax()
print(idxmax)
# 0    7.0
# 1    NaN
# 2    5.0
# 3    1.0
# 4    6.0
# dtype: float64

# Have we made the right choice?
have_me_made_the_right_choice = (idxmax == index_of_candidates_we_picked)
print(have_me_made_the_right_choice)
# 0    False
# 1    False
# 2     True
# 3    False
# 4     True

# Finding the number of times we made the best choice at this point is easy.
number_of_right_choices = np.sum(have_me_made_the_right_choice)
print(number_of_right_choices)
# 2

# And so is finding the percentage..
print(number_of_right_choices / numOfTries)
# 0.4

Sorting

How to Match Socks from a Laundry Bag?

Repeat the following until no socks left in the bag:

With just 10 different pair of socks, following this method will take on average 19 pulls merely to complete the first pair.

Finding Average Pulls Required

import random
from operator import add


def get_pair_of_socks(num_of_socks):
    return random.sample(range(num_of_socks), num_of_socks)


def index_to_pull_sock_from(bag_of_socks: list):
    return random.randint(a=0, b=len(bag_of_socks) - 1)


def attempt_counts_matching_socks(num_of_socks_to_consider):
    # Keep attempt counts in this list.
    attempt_counts = []

    # Generate pairs of random socks.
    socks = get_pair_of_socks(num_of_socks_to_consider)
    socks = socks + get_pair_of_socks(num_of_socks_to_consider)
    
    while len(socks) != 0:
        # Pick one pair from the bag..
        first_pair = socks.pop(index_to_pull_sock_from(socks))

        # Pick a second pair..
        random_pick = index_to_pull_sock_from(socks)
        second_pair = socks[random_pick]

        # We did an attempt..
        attempt_count = 1

        # If they matched, perfect. We will never enter this block.
        # Otherwise loop until you do find the match..
        while second_pair != first_pair:
            # Increment the attempt_count whenever you loop..
            attempt_count = attempt_count + 1
            random_pick = index_to_pull_sock_from(socks)
            second_pair = socks[random_pick]

        # Remove the second matching pair from the bag..
        socks.pop(random_pick)

        # Keep the number of attempts it took you to find the second pair..
        attempt_counts.append(attempt_count)

    return attempt_counts


num_of_iterations = 1000
pair_of_socks = 10

# Initalise a list full of zeros of length `pair_of_socks`
attempt_counts = [0] * pair_of_socks

for _ in range(num_of_iterations):
    # Get attempt counts for 1 iteration..
    attempt_counts_single_iter = attempt_counts_matching_socks(pair_of_socks)

    # Add the attempt counts aligned by index. 
    # We will be dividing by the total number of iterations later for averages.
    attempt_counts = list(map(add, attempt_counts, attempt_counts_single_iter))

average_takes = list(map(lambda x: x / num_of_iterations, attempt_counts))
print(average_takes)
# [18.205, 16.967, 14.659, 12.82, 11.686, 9.444, 7.238, 4.854, 2.984, 1.0]

Matching vs Sorting

But is matching socks from a laundry bag really identical to (or a good real life analogy of) sorting? Obviously you can not sort your socks but imagine there were numbers between 0 to 19 in the bag.

How would matching socks be identical to sorting?

Big-O Notation

Big-O Notation for Sorting

Bubble Sort

Bubble Sort Implementation in Python Note how comparison count increases roughly by 4 (6, 30, 132) as the length of the lists increase by 2 (3, 6, 12).

def bubble_sort(a_list):
    comparison_count = 0
    unsorted = True
    while unsorted:
        unsorted = False
        for i in range(len(a_list) - 1):
            comparison_count = comparison_count + 1
            if a_list[i] > a_list[i + 1]:
                unsorted = True
                a_list[i + 1], a_list[i] = a_list[i], a_list[i + 1]

    return a_list, comparison_count

print(bubble_sort([3, 2, 1]))
# ([1, 2, 3], 6)

print(bubble_sort([6, 5, 4, 3, 2, 1]))
# ([1, 2, 3, 4, 5, 6], 30)

print(bubble_sort([12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]))
# ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 132)

Insertion Sort

Insertion Sort Implementation in Python

def insertion_sort(a_list):
    i = 1
    while i < len(a_list):
        temp = a_list[i]
        j = i
        while j !=  0 and a_list[j - 1] > temp:
            a_list[j] = a_list[j - 1]
            j = j - 1
        a_list[j] = temp
        i = i + 1
    return a_list

print(insertion_sort([5, 4, 3, 2, 1]))
# [1, 2, 3, 4, 5]

Merge Sort

Merge Sort is as important in the history of sorting as sorting in the history of computing.

Merge Sort Implementation in Python

def sort_merge_step(list_01: list, list_02: list = None):
    if list_02 is None:
        return list_01

    merge_sorted = []
    i = 0
    j = 0

    while i < len(list_01) and j < len(list_02):
        if list_01[i] <= list_02[j]:
            merge_sorted.append(list_01[i])
            i = i + 1
        else:
            merge_sorted.append(list_02[j])
            j = j + 1

    while i < len(list_01):
        merge_sorted.append(list_01[i])
        i = i + 1

    while j < len(list_02):
        merge_sorted.append(list_02[j])
        j = j + 1

    return merge_sorted


def sort_merge(a_list: list):
    if len(a_list) < 2:
        return a_list

    mid = int(len(a_list) / 2)

    left_half = sort_merge(a_list[0: mid])
    right_half = sort_merge(a_list[mid:])

    return sort_merge_step(left_half, right_half)

print(sort_merge([5, 1, 3, 8, 11, 2]))
# [1, 2, 3, 5, 8, 11]

Quotes / Statements

Whether it’s finding the largest or the smallest, the most common or the rarest, tallying, indexing, flagging duplicates, or just plain looking for the thing you want, they all generally begin under the hood with a sort.

I do not agree with this statement, since either finding the largest or the smallest, the most common or the rarest can easily be done without sorting. For finding the largest or the smallest, sorting may be useful, but it is definetly not useful at all for the most common or the rarest.

With sorting, size is a recipe for disaster: perversely, as a sort grows larger, the unit cost of sorting, instead of falling, rises.

Well, apparently!

Sorting five shelves of books will take not five times as long as sorting a single shelf, but twenty-five times as long.

I think what is meant is “Sorting a shelf five times longer will take twenty-five times longer.”

References

Caching

The idea of keeping around pieces of information that you refer to frequently.

The goal of cache management is to minimize the number of times you can not find what you are looking for in the cache. Not being able to find what you are looking for in the cache is named as a page fault or a cache miss.

A big book is a big nuisance.

Callimachus

I am not sure how this quote is related to caching really. It reminds me the following quotes, which I also like:

A designer knows he has achieved perfection not when there is nothing left to add, but when there is nothing left to take away.

Antoine de Saint-Exupery

It also reminds me a quote from The Information: A History, a Theory, a Flood, which I can not exactly remember but goes something like..

Too much information is just as bad as no information.

Cache Eviction

Cache eviction is the process of deciding what to remove from the cache when it is capacity is full but a new item needs to be cached.

The optimal cache eviction policy is to evict the item we will need again the longest from now.

Cache Eviction Policies

Scheduling

How we spend our days is, of course, how we spend our lives.

Annie Dillard


We are what we repeatedly do.

Aristotle

Inconsistency in Time Management Best Sellers

Source Suggestion
Getting Things DoneImmediately do a task that would take 2 minutes or less
Eat That Frog!Begin with the most difficult task and move to easier ones
The Now HabitFirst schedule your social engagements, fill the gaps with work
William JamesThere is nothing so fatiguing as the eternal hanging on of an uncompleted task
Wait by Frank PartnoyDeliberately do not do things right away, wait on them

Multiple Machine Scheduling

Johnson’s Rule Algorithm

Johnson’s Rule Algorithm Implementation in Java

class Task {
    String name;
    double workHoursA, workHoursB; // first workcenter, second workcenter

    Task(String name, double workHoursA, double workHoursB) {
        this.name = name;
        this.workHoursA = workHoursA;
        this.workHoursB = workHoursB;
    }

    double minimumWork() {
        return workHoursA < workHoursB ? workHoursA : workHoursB;
    }

    public String toString() {
        return "Task{" + name +'}';
    }
}

class JohnsonRuleAlgorithm {
    void scheduleTasks() {
        // Schedule the following tasks
        List<Task> tasks = Arrays.asList(
            new Task("A", 3.2, 4.2),
            new Task("B", 4.7, 1.5),
            new Task("C", 2.2, 5.0),
            new Task("D", 5.8, 4.0),
            new Task("E", 3.1, 2.8)
        );
        
        // Sort tasks by minimum work needed. It can be either workA or workB.
        List<Task> collect = tasks.stream()
                                  .sorted(comparing(Task::minimumWork))
                                  .collect(toList());
        
        // Tasks to be scheduled first
        List<Task> scheduled = new ArrayList<>();
        
        // Use a first in last out to push the items to be scheduled last
        // Retrieve them by popping each later to scheduled
        ArrayDeque<Task> lastToDo = new ArrayDeque<>();
        
        collect.forEach(task -> {
            if (task.workHoursA < task.workHoursB) 
                scheduled.add(task);
            else 
                lastToDo.push(task);
        });
        
        while (!lastToDo.isEmpty()) scheduled.add(lastToDo.pop());
        // scheduled: [Task{C}, Task{A}, Task{D}, Task{E}, Task{B}]
    }
}

Single Machine Scheduling

If we have a list of tasks and only a single machine (unlike the example above), no matter how we order the tasks we can not optimize finishing running the all tasks in terms of shortest time. However, if every task has a deadline, we can

Earliest Due Date

Minimizes the cumulative delay.

Moore’s Algorithm

Minimizes the delayed task count.

Shortest Processing Time

Maximizes the finished task counts earlier.

Earliest Due Date vs Moore’s Algorithm Example

class Task {
    int requiredTime, deadLine;

    static Task of(int requiredTime, int deadLine) {
        Task task = new Task();
        task.requiredTime = requiredTime;
        task.deadLine = deadLine;
        return task;
    }

    // Sample data
    static List<Task> tasks() {
        return asList(of(2, 3), of(3, 4), of(6, 8), of(4, 10));
    }
}

class TaskExecution {
    Task task;
    int startTime, endTime;

    int delay() {
        return endTime < task.deadLine ? 0 : endTime - task.deadLine;
    }

    boolean isDelayed() {
        return delay() > 0;
    }
}

class TaskExecutor {
    void earliestDueDate(List<Task> tasks) {
        int time = 1;
        tasks.sort(Comparator.comparing(task -> task.deadLine));
        List<TaskExecution> taskExecutions = new ArrayList<>();

        while (!tasks.isEmpty()) {
            Task task = tasks.remove(0);
            TaskExecution taskExecution = executeTask(time, task);
            taskExecutions.add(taskExecution);
            time = taskExecution.endTime;
        }
    }

    void mooresAlgorithm(List<Task> tasks) {
        int time = 1;
        tasks.sort(Comparator.comparing(task -> task.deadLine));
        List<TaskExecution> taskExecutions = new ArrayList<>();
        List<Task> delayedTasks = new ArrayList<>();

        while (!tasks.isEmpty()) {
            Task task = tasks.remove(0);
            if (time + task.requiredTime > task.deadLine) {
                delayedTasks.add(task);
                continue;
            }
            TaskExecution taskExecution = executeTask(time, task);
            taskExecutions.add(taskExecution);
            time = taskExecution.endTime;
        }

        while (!delayedTasks.isEmpty()) {
            Task task = delayedTasks.remove(0);
            TaskExecution taskExecution = executeTask(time, task);
            taskExecutions.add(taskExecution);
            time = taskExecution.endTime;
        }
    }

    TaskExecution executeTask(int startTime, Task task) {
        TaskExecution taskExc = new TaskExecution();
        taskExc.task = task;
        taskExc.startTime = startTime;
        taskExc.endTime = startTime + task.requiredTime;
        return taskExc;
    }
}

Execution

TaskExecutor taskExecutor = new TaskExecutor();
taskExecutor.earliestDueDate(new LinkedList<>(Task.tasks()));
taskExecutor.mooresAlgorithm(new LinkedList<>(Task.tasks()));

Report

Moore’s Algorithm skips executing the 2nd and 3rd tasks in favor of getting the 4rd task on time and causes delay amounts of 6 and 8 compared to 2 and 4 on tasks 2 and 3.

Earliest Due Date Algorithm Moore 's Algorithm
Delayed Task Count 3 2
Total Delay Amount 12 14
End Time 16 16

None of these tasks had weight (i.e. importance) associated with them in our examples. When tasks not only have deadline but also weight, things get complicated.

Priority Inheritance

If a low-priority task is found to be blocking a high-priority resource, the low-priority task should become the highest-priority. If you keep constantly thinking about the novel you are about to finish while studying for the exam you need to take, maybe it is better to finish the novel first, unblocking the high priority task at hand.

Context Switch

Context Switching helps us getting things done by pausing at a state of a task, getting other things done, and getting back to it.

Computers and people face the same challenge: The machine responsible for scheduling is the machine itself that will process the tasks. The scheduling task itself becomes a task in the to-do list which also must be scheduled.

Context Switching however is expensive, and may end up in asking the question: Now where was I?.

Better Scheduling

Bayes ‘s Rule

Rule of Succession

If we repeat an experiment that we know can result in a success or failure, n times independently, and get s successes, then what is the probability that the next repetition will succeed?

The more data we have, the less importance should be assigned to our prior information.

Source

Probabilities and Distributions

Normal Distribution

Power-Law Distribution

Erlang Distribution

Distribution Expectation
Normal Distribution The longer the incidents goes on, expect it to finish sooner.
Power-Law Distribution The longer the incidents goes on, expect it to go longer.
Erlang Distribution The longer the incidents goes on, assume it might finish any given time.

Predictions

Knowing what distribution you are up against makes all the difference when predicting the future. When .. discovered he had cancer, he found out half of the patients with his form of cancer dies within the eight months. But without the distribution, eight months did not tell him much. If it were a normal distribution, it would be normal for him to think his chances was going lower and lower as he lived every single day after the eight months. But if it were a power-law distribution, then he knew the more he lived, the more likely he would live even longer.

It turned out it was power-law distribution after all, and he lived twenty more years.

Randomness

A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the “average case” over all possible choices of random bits.

Source

Sampling

Buffon’s needle

import math
import random

iteration_count = 10000000
crossed = 0

needle_length = 1.0
gap_length = 2.0


for i in range(iteration_count):
    drop_point = random.uniform(-1, 1)
    # π / 2 = 157079632679
    drop_degree_rad = random.randint(0, 157079632679) / 100000000000

    tip = (math.sin(drop_degree_rad) * needle_length / 2) + drop_point
    bottom = drop_point - (math.sin(drop_degree_rad) * needle_length / 2)

    if math.fabs(tip) >= 1 or math.fabs(bottom) >= 1:
        crossed += 1

print(2 / crossed / iteration_count / 2)  # 3.140268574610112 Very close!

Prime Numbers and Sieve of Eratosthenes

Sieve of Eratosthenes Implementation in Java

int upto = 100;

Set<Integer> nonPrimeProcessed = new HashSet<>();
Set<Integer> primes = IntStream.rangeClosed(1, upto).boxed().collect(toSet());

for (int i = 2; i < upto; i++) {
    if (i * i > upto) {
        break;
    }
    if (nonPrimeProcessed.contains(i)) {
        continue;
    }
    int k = i;
    int mul = i * k;
    while (mul < upto + 1) {
        nonPrimeProcessed.add(mul);
        primes.remove(mul);
        k++;
        mul = i * k;
    }
}

Sieve of Eratosthenes Implementation in Python

def primes(up_to):
    if up_to < 2:
        return [False] * up_to

    # 0 indexed array hence the +1 so index is aligned with the integer value.
    prime_indices = [True] * (up_to + 1)

    # Special cases 0 and 1.. Not primes..
    prime_indices[0] = False
    prime_indices[1] = False

    for i in range(2, len(prime_indices)):
        if i * i >= len(prime_indices):
            break
        mul = i
        val = i * mul
        while val < len(prime_indices):
            prime_indices[mul * i] = False
            mul = mul + 1
            val = i * mul

    return prime_indices

Networking

The term connection has a wide variety of meanings. It can refer to a physical or logical path between two entities, it can refer to the flow over the path, it can inferentially refer to an action associated with the setting up of a path, or it can refer to an association between two or more entities, with or without regard to any path between them.

Vint Cerf and Bob Kahn, A Protocol for Packet Network Intercommunication

Circuit Switching vs Packet Switching

In Packet Switching, there are no connections. What you call a connection is a consensual illusion between two end points.

Stuart Cheshire

Acknowledgment

Exponential Backoff

Congestion

Buffer

Game Theory