about this book
This book is designed to be easy to follow. I avoid big leaps of thought. Any time a new concept is introduced, I explain it right away or tell you when I’ll explain it. Core concepts are reinforced with exercises and multiple explanations so that you can check your assumptions and make sure you’re following along.
I lead with examples. Instead of writing symbol soup, my goal is to make it easy for you to visualize these concepts. I also think we learn best by being able to recall something we already know, and examples make recall easier. So when you’re trying to remember the difference between arrays and linked lists (explained in chapter 2), you can just think about getting seated for a movie. Also, at the risk of stating the obvious, I’m a visual learner. This book is chock-full of images. The contents of the book are carefully curated. There’s no need to write a book that covers every sorting algorithm—that’s why we have Wikipedia and Khan Academy. All the algorithms I’ve included are practical. I’ve found them useful in my job as a software engineer, and they provide a good foundation for more complex topics.
Happy reading!
Roadmap
The first three chapters of this book lay the foundations:
• Chapter 1—You’ll learn your first practical algorithm: binary search. You also learn to analyze the speed of an algorithm using Big O notation. Big O notation is used throughout the book to analyze how slow or fast an algorithm is.
• Chapter 2—You’ll learn about two fundamental data structures: arrays and linked lists. These data structures are used throughout the book, and they’re used to make more advanced data structures like hash tables (chapter 5).
• Chapter 3—You’ll learn about recursion, a handy technique used by many algorithms (such as quicksort, covered in chapter 4).
In my experience, Big O notation and recursion are challenging topics for beginners. So I’ve slowed down and spent extra time on these sections.
The rest of the book presents algorithms with broad applications:
• Problem-solving techniques—Covered in chapters 4, 8, and 9. If you come across a problem and aren’t sure how to solve it efficiently, try divide and conquer (chapter 4) or dynamic programming (chapter 9). Or you may realize there’s no efficient solution, and get an approximate answer using a greedy algorithm instead (chapter 8).
• Hash tables—Covered in chapter 5. A hash table is a very useful data structure. It contains sets of key and value pairs, like a person’s name and their email address, or a username and the associated password. It’s hard to overstate hash tables’ usefulness. When I want to solve a problem, the two plans of attack I start with are “Can I use a hash table?” and “Can I model this as a graph?”
• Graph algorithms—Covered in chapters 6 and 7. Graphs are a way to model a network: a social network, or a network of roads, or neurons, or any other set of connections. Breadth-first search (chapter 6) and Dijkstra’s algorithm (chapter 7) are ways to find the shortest distance between two points in a network: you can use this approach to calculate the degrees of separation between two people or the shortest route to a destination.
• K-nearest neighbors (KNN)—Covered in chapter 10. This is a simple machine-learning algorithm. You can use KNN to build a recommendations system, an OCR engine, a system to predict stock values—anything that involves predicting a value (“We think Adit will rate this movie 4 stars”) or classifying an object (“That letter is a Q”).
• Next steps—Chapter 11 goes over 10 algorithms that would make good further reading.
about this book
How to use this book
The order and contents of this book have been carefully designed. If you’re interested in a topic, feel free to jump ahead. Otherwise, read the chapters in order—they build on each other.
I strongly recommend executing the code for the examples yourself. I can’t stress this part enough. Just type out my code samples verbatim (or download them from www.manning.com/books/grokkingalgorithms or https://github.com/egonschiele/grokking_algorithms), and execute them. You’ll retain a lot more if you do.
I also recommend doing the exercises in this book. The exercises are short—usually just a minute or two, sometimes 5 to 10 minutes. They will help you check your thinking, so you’ll know when you’re off track before you’ve gone too far.
Who should read this book
This book is aimed at anyone who knows the basics of coding and wants to understand algorithms. Maybe you already have a coding problem and are trying to find an algorithmic solution. Or maybe you want to understand what algorithms are useful for. Here’s a short, incomplete list of people who will probably find this book useful:
• Hobbyist coders
• Coding boot camp students
• Computer science grads looking for a refresher
• Physics/math/other grads who are interested in programming
Code conventions and downloads
All the code examples in this book use Python 2.7. All code in the book is presented in a fixed-width font like this to separate it from ordinary text. Code annotations accompany some of the listings, highlighting important concepts.
You can download the code for the examples in the book from the publisher’s website at www.manning.com/books/grokking-algorithms or from https://github.com/egonschiele/grokking_algorithms.
I believe you learn best when you really enjoy learning—so have fun, and run the code samples!
About the author
Aditya Bhargava is a software engineer at Etsy, an online marketplace for handmade goods. He has a master’s degree in computer science from the University of Chicago. He also runs a popular illustrated tech blog at adit.io.
Author Online
Purchase of Grokking Algorithms includes free access to a private web forum run by Manning Publications where you can make comments about the book, ask technical questions, and receive help from the author and from other users. To access the forum and subscribe to it, point your web browser to www.manning.com/books/grokkingalgorithms. This page provides information on how to get on the forum once you are registered, what kind of help is available, and the rules of conduct on the forum.
Manning’s commitment to our readers is to provide a venue where a meaningful dialog between individual readers and between readers and the author can take place. It isn’t a commitment to any specific amount of participation on the part of the author, whose contribution to Author Online remains voluntary (and unpaid). We suggest you try asking the author some challenging questions lest his interest stray! The Author Online forum and the archives of previous discussions will be accessible from the publisher’s website as long as the book is in print.
In this chapter
introduction to algorithms
• You get a foundation for the rest of the book.
• You write your first search algorithm (binary search).
• You learn how to talk about the running time of an algorithm (Big O notation).
• You’re introduced to a common technique for designing algorithms (recursion).
Introduction
An algorithm is a set of instructions for accomplishing a task. Every piece of code could be called an algorithm, but this book covers the more interesting bits. I chose the algorithms in this book for inclusion because they’re fast, or they solve interesting problems, or both. Here are some highlights:
• Chapter 1 talks about binary search and shows how an algorithm can speed up your code. In one example, the number of steps needed goes from 4 billion down to 32!
• A GPS device uses graph algorithms (as you’ll learn in chapters 6, 7, and 8) to calculate the shortest route to your destination.
• You can use dynamic programming (discussed in chapter 9) to write an AI algorithm that plays checkers.
In each case, I’ll describe the algorithm and give you an example. Then I’ll talk about the running time of the algorithm in Big O notation. Finally, I’ll explore what other types of problems could be solved by the same algorithm.
What you’ll learn about performance
The good news is, an implementation of every algorithm in this book is probably available in your favorite language, so you don’t have to write each algorithm yourself! But those implementations are useless if you don’t understand the trade-offs. In this book, you’ll learn to compare trade-offs between different algorithms: Should you use merge sort or quicksort? Should you use an array or a list? Just using a different data structure can make a big difference.
What you’ll learn about solving problems
You’ll learn techniques for solving problems that might have been out of your grasp until now. For example:
• If you like making video games, you can write an AI system that follows the user around using graph algorithms.
• You’ll learn to make a recommendations system using k-nearest neighbors.
• Some problems aren’t solvable in a timely manner! The part of this book that talks about NP-complete problems shows you how to identify those problems and come up with an algorithm that gives you an approximate answer.
More generally, by the end of this book, you’ll know some of the most widely applicable algorithms. You can then use your new knowledge to learn about more specific algorithms for AI, databases, and so on. Or you can take on bigger challenges at work.
What you need to know
You’ll need to know basic algebra before starting this book. In particular, take this function: f(x) = x × 2. What is f(5)? If you answered 10, you’re set.
Additionally, this chapter (and this book) will be easier to follow if you’re familiar with one programming language. All the examples in this book are in Python. If you don’t know any programming languages and want to learn one, choose Python—it’s great for beginners. If you know another language, like Ruby, you’ll be fine.
Binary search
Suppose you’re searching for a person in the phone book (what an oldfashioned sentence!). Their name starts with K. You could start at the beginning and keep flipping pages until you get to the Ks. But you’re more likely to start at a page in the middle, because you know the Ks are going to be near the middle of the phone book. Or suppose you’re searching for a word in a dictionary, and it starts with O. Again, you’ll start near the middle.
Now suppose you log on to Facebook. When you do, Facebook has to verify that you have an account on the site. So, it needs to search for your username in its database. Suppose your username is karlmageddon. Facebook could start from the As and search for your name—but it makes more sense for it to begin somewhere in the middle.
This is a search problem. And all these cases use the same algorithm to solve the problem: binary search.
Binary search is an algorithm; its input is a sorted list of elements (I’ll explain later why it needs to be sorted). If an element you’re looking for is in that list, binary search returns the position where it’s located. Otherwise, binary search returns null.
For example:
Here’s an example of how binary search works. I’m thinking of a number between 1 and 100.
You have to try to guess my number in the fewest tries possible. With every guess, I’ll tell you if your guess is too low, too high, or correct. Suppose you start guessing like this: 1, 2, 3, 4 …. Here’s how it would go.
Looking for companies in a phone book with binary search
This is simple search (maybe stupid search would be a better term). With each guess, you’re eliminating only one number. If my number was 99, it could take you 99 guesses to get there!
A better way to search
Here’s a better technique. Start with 50.
A bad approach to number guessing
Too low, but you just eliminated half the numbers! Now you know that 1–50 are all too low. Next guess: 75.
Too high, but again you cut down half the remaining numbers! With binary search, you guess the middle number and eliminate half the remaining numbers every time. Next is 63 (halfway between 50 and 75).
This is binary search. You just learned your first algorithm! Here’s how many numbers you can eliminate every time.
Whatever number I’m thinking of, you can guess in a maximum of seven guesses—because you eliminate so many numbers with every guess!
Suppose you’re looking for a word in the dictionary. The dictionary has 240,000 words. In the worst case, how many steps do you think each search will take?
Simple search could take 240,000 steps if the word you’re looking for is the very last one in the book. With each step of binary search, you cut the number of words in half until you’re left with only one word.
Eliminate half the numbers every time with binary search.
So binary search will take 18 steps—a big difference! In general, for any list of n, binary search will take log2 n steps to run in the worst case, whereas simple search will take n steps.
Logarithms
You may not remember what logarithms are, but you probably know what exponentials are. log10 100 is like asking, “How many 10s do we multiply together to get 100?” The answer is 2: 10 × 10. So log10 100 = 2. Logs are the flip of exponentials.
Logs are the flip of exponentials.
In this book, when I talk about running time in Big O notation (explained a little later), log always means log 2 . When you search for an element using simple search, in the worst case you might have to look at every single element. So for a list of 8 numbers, you’d have to check 8 numbers at most. For binary search, you have to check log n elements in the worst case. For a list of 8 elements, log 8 == 3, because 23 == 8. So for a list of 8 numbers, you would have to check 3 numbers at most. For a list of 1,024 elements, log 1,024 = 10, because 210 == 1,024. So for a list of 1,024 numbers, you’d have to check 10 numbers at most.
Note
I’ll talk about log time a lot in this book, so you should understand the concept of logarithms. If you don’t, Khan Academy (khanacademy.org) has a nice video that makes it clear.
Note
Binary search only works when your list is in sorted order. For example, the names in a phone book are sorted in alphabetical order, so you can use binary search to look for a name. What would happen if the names weren’t sorted?
Let’s see how to write binary search in Python. The code sample here uses arrays. If you don’t know how arrays work, don’t worry; they’re covered in the next chapter. You just need to know that you can store a sequence of elements in a row of consecutive buckets called an array. The buckets are numbered starting with 0: the first bucket is at position #0, the second is #1, the third is #2, and so on.
The binary_search function takes a sorted array and an item. If the item is in the array, the function returns its position. You’ll keep track of what part of the array you have to search through. At the beginning, this is the entire array:
low = 0
high = len(list) - 1
Each time, you check the middle element:
mid = (low + high) / 2 guess = list[mid]
mid is rounded down by Python automatically if (low + high) isn’t an even number.
If the guess is too low, you update low accordingly: if guess < item: low = mid + 1
And if the guess is too high, you update high . Here’s the full code:
def binary_search(list, item): low = 0 high = len(list)—1 while low <= high: mid = (low + high) guess = list[mid] if guess == item: return mid if guess > item: high = mid - 1 else: low = mid + 1 return None
my_list = [1, 3, 5, 7, 9]
low and high keep track of which part of the list you’ll search in.
While you haven’t narrowed it down to one element …
… check the middle element.
Found the item.
The guess was too high. The guess was too low. The item doesn’t exist.
Let’s test it!
print binary_search(my_list, 3) # => 1
print binary_search(my_list, -1) # => None
EXERCISES
Remember, lists start at 0. The second slot has index 1.
“None” means nil in Python. It indicates that the item wasn’t found.
1.1 Suppose you have a sorted list of 128 names, and you’re searching through it using binary search. What’s the maximum number of steps it would take?
1.2 Suppose you double the size of the list. What’s the maximum number of steps now?
Running time
Any time I talk about an algorithm, I’ll discuss its running time. Generally you want to choose the most efficient algorithm— whether you’re trying to optimize for time or space.
Back to binary search. How much time do you save by using it? Well, the first approach was to check each number, one by one. If this is a list of 100 numbers, it takes up to 100 guesses. If it’s a list of 4 billion numbers, it takes up to 4 billion guesses. So the maximum number of guesses is the same as the size of the list. This is called linear time.
Binary search is different. If the list is 100 items long, it takes at most 7 guesses. If the list is 4 billion items, it takes at most 32 guesses. Powerful, eh? Binary search runs in logarithmic time (or log time, as the natives call it). Here’s a table summarizing our findings today.
Big O notation
Big O notation is special notation that tells you how fast an algorithm is. Who cares? Well, it turns out that you’ll use other people’s algorithms often—and when you do, it’s nice to understand how fast or slow they are. In this section, I’ll explain what Big O notation is and give you a list of the most common running times for algorithms using it.
Run times for search algorithms
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"Sticking together," was Master Henly's concise and descriptive answer.
"Glad to hear it," said Willard, repressing a strong inclination to laugh. "Is Mrs. Tom within?"
"She was when I left the house," said Carl, who seemed determined not to commit himself.
"Any one with her?" again inquired the young gentleman, looking as indifferent as possible.
"No, nobody," was the unexpected answer.
"What!" exclaimed Willard, surprised. "I thought I saw a young lady enter a moment ago!"
"Oh, Christie—she's nobody," said the gallant Mr. Henly.
"Christie—Mrs. Tom's niece—I thought she was away!" exclaimed Willard.
"So she was, but I went for her this morning; couldn't be bothered doing her work and my own both any longer," said Carl.
"I suppose I may go in?" said Willard, feeling a sudden thrill of pleasure at the knowledge that this radiant girl was an inhabitant of the island.
"Yes, I suppose you may, if you like," said Carl, in a tone of the utmost unconcern.
Thus kindly permitted, Willard advanced and rapped at the door. It was opened by Mrs. Tom, whose surprise was only equaled by her delight at being honored by this unexpected visit.
Near the window that overlooked the lodge, stood the golden-haired vision of the beach. She turned round with a quick, shy glance, and blushed most enchantingly beneath the deep, dark eyes of the stranger.
"My niece, Christie, Mr. Drummond," said Mrs. Tom, directing his attention to her with a wave of her hand. "She got back this mornin'. I allers find it powerful lonesome here without Christie."
"I have no doubt of it," said Mr. Drummond, seating himself. "But I have had the pleasure of seeing Miss Christie before."
"Where?" asked Christie, opening her blue eyes in wonder.
"Down on the beach, a few moments ago."
"Oh, yes."
And again Christie blushed vividly, as she recollected how she had been caught singing.
"Where's Miss Sibyl and Master Guy?" inquired Mrs. Tom.
"Miss Sibyl has gone to N—— with the clergyman's family, and will not return for a week; and Captain Campbell has gone to Westbrook, where his vessel is undergoing repairs. So I am left all alone, and came to pay my respects to you."
"Then you'll stay and spend the evenin'?" said Mrs. Tom, smiling complacently.
Mr. Drummond professed his willingness; and the little widow, delighted at the condescension, set about preparing tea instantly, assisted by Christie, whose wild, shy glances were bent on his face whenever she fancied herself unobserved. Half pleased, half afraid of him at first, she was reserved and timid; but as this wore off, he drew her into conversation, and, to his surprise, found her intelligent and well-educated. This Mrs. Tom accounted for, by saying she had gone to school for the last five years at Westbrook, residing there with the friend she had now been visiting.
The evening passed away with the rapidity of magic. Christie, after much solicitation, consented to sing for him; and if anything was needed to fairly enchant him, that sweet, clear voice would have done it. Then, too,
Carl added to the general hilarity, by drawing out a rusty Jew's-harp, and playing a favorite tune of his own composition. Not once during the evening did he think of Sibyl; her dark, resplendant face, and wild fierce, black eyes, were forgotten for the golden locks and sweet, fair face of blueeyed Christie—this dainty island Peri.
The hour for leaving came all too soon. As he reluctantly rose to go, he pressed the hand Christie extended, to his lips, with such passionate ardor that the blood flushed to her very temples, but not with pleasure. Ere he left, Mrs. Tom cordially invited him to visit her house while he remained on the island—an invitation he was not loth in accepting.
Christie stood at the window, watching his tall, elegant form as he walked toward the castle in the bright, clear moonlight.
"I like him, Cousin Christie; don't you?" said Carl, when he had gone.
But Cousin Christie turned away without reply, longing to lay her burning cheek on the pillow, and muse over the new and delicious joy that was thrilling her whole heart, and in her slumber to lie dreaming "Love's young dream."
And Willard Drummond, forgetting his vows, forgetting Sibyl, forgetting honor, forgetting all but this lovely island-maiden, sought his couch with but one name on his heart and lips—
"Christie, Christie!"
CHAPTER VII.
"She loves, but knows not whom she loves, Nor what his race, nor whence he came; Like one who meets in Indian groves
Some beauteous bird without a name, Brought by the last ambrosial breeze
To show his plumage for a day
To wondering eyes, then wing away " LALLA ROOKH
Pale, feverish, and unrefreshed, after a night of restless dreams, Willard Drummond arose from a vision of Christie to hail a new day.
Passion and principle were at war already. Bound by every pledge of love, by every vow, to Sibyl, his whole soul was steeped in this new, allabsorbing passion that had taken possession of him. He had fancied he loved her, until he beheld radiant, dazzling, bewildering Christie and from that moment he could have yielded heaven for her. Every feeling of his inmost heart and soul was up, in arms. Every feeling of honor bade him fly from this intoxicating siren, whose power he felt growing stronger each moment over him; but the voice of passion cried: "Remain! love her, if you will! What right has Sibyl to stand between you and the heaven of your dreams?" And, like all who allow the struggle between right and wrong to wage its warfare in their bosom, Willard Drummond was lost. For, with his hot, fervid, Southern nature, worldly considerations, former vows, reason, principle, justice, even honor, were swept away, like a wall of smoke, before the fierce impetuosity of passion.
With a head throbbing, and pulse quick and feverish with the inward conflict, Willard descended to breakfast.
Captain Campbell stood in the sitting-room, awaiting his coming. With a courteous "good morning," he advanced to meet him; but started back in surprise at beholding his extreme pallor.
"My dear fellow," he exclaimed, in a tone of solicitude, "you are ill— very ill, I am afraid. What in the world is the matter?"
"Nothing. I had bad dreams, and did not sleep well," said Drummond, with a forced smile. "A cup of Aunt Moll's excellent coffee will set me all right again."
"I don't know about that," said Captain Campbell, with his eyes fixed anxiously on his face, "You are looking terribly feverish, and you were complaining of a headache yesterday. I hope you are not going to be ill."
"I assure you it is nothing," said Willard, in a tone of slight impatience. "You are needlessly alarming yourself. A bad night's rest is the cause of all."
"Well, if it is not, I will have to call up Mrs. Tom to nurse you till Sibyl comes. And, by the way, I regret exceedingly that I shall be obliged to leave you solitary and alone for some days. Important business, that cannot be postponed, demands my immediate attention."
Willard's heart suddenly bounded—he would scarcely have acknowledged to himself the reason—at the words.
"It seems hardly courteous or hospitable to leave you thus," continued the young captain; "but I know you will excuse me, my dear fellow, when I tell you it cannot possibly be helped."
"Oh, certainly, certainly!" interrupted Drummond, cordially. "Go, by all means. I will get along well enough in your absence. When do you leave?"
"Immediately after breakfast. It is an affair that cannot be postponed. In fact, I will not have time even to go and see Sibyl; but, as you will probably be there during the day, you can tell her. Perhaps you will come over to the mainland with me?"
"No, I think not," said Willard, with affected carelessness. "I may go during the course of the day."
"But how? I will take the boat."
"Oh, with Carl Henly. He has one, I believe."
"Well, suit yourself. And now I'm off. Take care of yourself, my boy; and au revoir!"
"Good-by!" said Willard, accompanying him to the door. "Aunt Moll and I will keep bachelor's hall till you come back."
Captain Guy laughed, and hurried down to the beach. And when he was gone, Willard arranged his slightly disordered dress and disheveled locks, and sauntering out, almost mechanically took the road to the cottage.
It came in sight at last—this little, quaint, old house, that held all of heaven to him now.
"Shall I enter—shall I thrust myself into temptation?" was his thought. "If I look again on this fairy sylph I am lost!"
He thought of Sibyl, and her dark, bright, menacing eyes arose before him, as if to warn him back.
"For your honor's sake—for your life's sake—for your soul's sake—go not there!" said the threatening voice of conscience.
"And have I not a right to love whom I please? Why should I offer violence to myself in leaving this bright enchantress, for that dark, wild Amazon? Go, go, and be happy," said passion.
And, as if to overthrow his last good resolution, the image of Christie, radiant, dazzling, and beautiful, as he had beheld her first, in the bright flush of the fading sun light, arose before him, and once again passion conquered.
He approached and entered the cottage.
Mrs. Tom sat near the window, spinning and singing to herself. Willard's eyes wandered around in search of another; but bright Christie was not to be seen.
The widow arose, smiling, to welcome her guest, and placed a chair for him near herself. And still Willard's eyes went wandering round the room.
"She will appear presently," he thought, not yet liking to inquire for her.
"What a venerable-looking affair your wheel is, Mrs. Tom," he said, surveying it, with its hard polished wood and bright brass rings.
"Yes, it's as old as the hills," said Mrs. Tom, resuming her work; "and has been in our family since the flood. I think I spun on that there wheel all the yarn that makes the socks, mittens, and comforters for half the county round; besides making sheets, blankets, and lots of other things for ourselves," said Mrs. Tom, with conscious pride.
"You deserve a premium for industry, Mrs. Tom," said Willard.
"Well, you may be jokin' now, and I dare say you are; but it is true, for all that. Many a true word is spoke in jest, you know," said Mrs. Tom, as her wheel went merrily round. "There ain't many women in this place o' my age and means, can do, or does do more work than me, though I say it as hadn't oughter. I knit, and spin, and sew, wash, brew, bake, sow, and reap, and fifty other things, too numerous to mention, besides. Carl, if I go out there I'll put an end to your lazin', you idle, good-for-nothin' vagabone, you!" she added, breaking off in sudden wrath, as she espied Carl, leaning on the spade with which he should have been digging in the garden.
"You should make Carl do these things, Mrs. Tom," said Willard, still impatiently watching the door and wondering why Christie did not come.
"Carl?" said Mrs. Tom, with a short laugh. "Lor'-a-massy! he ain't worth his salt; that there's the laziest, most worthlessest young scape-goat ever any living 'oman was plagued with. I hain't a minute's peace with him night nor day; and if scolding was a mite of good, the Lord knows he might have been a saint by this time, for he gets enough of it."
Willard laughed. And in such conversation the morning slipped away— very rapidly to Mrs. Tom, but each moment an age to our impatient lover. For Christie was absent still; and a strange reluctance, for which he could not account, still prevented Willard from asking for her. It was an inward sense of guilt that troubled him; for, feeling toward her as he did, he felt he had no right even to mention her name.
At last, in despair, he arose to go. Mrs. Tom relieved his mind by saying:
"Christie will be disappointed at not seeing you," said the old lady, following him out. "She went out berrying to the woods this morning, and hain't got home yet."
Willard started at the information; and, inwardly cursing the folly that had detained him so many hours talking to a foolish old woman, he darted off, with a rapidity that quite amazed Mrs. Tom, in the direction of the pine woods.
"What a confounded fool I have been," he exclaimed, savagely, "to stay there listening to the way to make butter, and flannel, and 'yarb tea,' as if the old beldame thought I was going to be somebody's housekeeper, or a female doctress; and all the time this enchanting little blue-eyed witch was wandering alone by herself. What an opportunity I have lost! and now I suppose I may search for an hour and not find her."
He turned an abrupt angle in the winding path, and stifled a sudden exclamation of surprise and delight; for there before him, reclining on the grass, with half-veiled eyes, and soft, musing smile, sat the object of all his thoughts, wishes, and desires.
He paused for a moment to contemplate the picture before him, for, if Christie had seemed beautiful when he first beheld her, oh, doubly lovely did she appear now in her attitude of unstudied grace.
Her dress was a loose, light muslin robe, fitting to perfection her rounded waist and swelling bust. Her straw hat lay on the ground beside her, and her golden, sunshiny hair floated, with all its wealth of rippling ringlets, round her ivory throat. How dazzlingly fair looked that smooth, snowy brow, contrasted with the full crimson lips and delicately flushed cheeks—how enchanting the long curved lashes, falling over the deep-blue eyes—how beautiful that faultless form, that soft, gentle, happy smile of guileless girlhood.
Willard Drummond's breath came and went, quick and short, as he gazed, and his dark eyes filled with a subdued fire.
He advanced toward her. His shadow, falling on the grass at her feet, was the first token she had of his coming. With a quick, startled cry, she sprang to her feet in terror; but when she saw who it was that stood before her, she stopped short, while the color flushed gloriously to her rounded cheeks. Her first impression was: He has read my thoughts in my face, and knows I was thinking of him.
"Have I disturbed you, bright Christie?" he asked, coming nearer.
"Oh, no!" she answered, blushingly. "I was only waiting to rest a little while before going home."
"And dreaming, I perceive," said Willard. "May I ask, of what—of whom?"
"I wasn't dreaming," said Christie, innocently. "I was wide awake all the time."
"Day-dreaming, I mean," said Drummond, with a smile. "Do you know, fairest Christie, I have been at your cottage all the morning, waiting to see you?"
"To see me?" said Christie, with another quick, glad blush.
"And not finding you there, I have come in search of you," he continued.
"And found me," she said, laughing. "If I had known you were coming I should have staid at home."
"Perhaps it is better as it is, bright one; for I have found you alone. It is very pleasant to have found so fair companion on this lonely isle."
"Yes, it is a lonely place," said Christie, musingly; "and yet I like it better than Westbrook, or any other place I have ever been in. Only I would like always to have a friend with me to talk to; and that, you know, I cannot
have here. Aunt Tom is always too busy to go out; and Carl don't care about the trouble of talking, much less walking, so I always have to go alone."
"And if he would go, I fancy Master Carl is hardly the kind of companion Miss Christie would select," said Willard.
"Not if I could find any better," said Christie, with a laugh; "but I have grown so accustomed to being alone now that I do not mind it at all, as I used to."
"And so you are perfectly happy here, fairest Christie, reigning queen of this fairy isle?"
"Ah, no! beautiful Miss Sibyl is queen of the isle. I am only her most loyal subject," said Christie, gayly; "you ought to know that, having paid her your allegiance."
"What if I should say that the subject is more lovely than the queen?" said Willard, in a low voice, and in a tone that brought the hot blood flushing to Christie's face.
"I should say you were laughing at me, as of course you would be. Certainly no one would ever think of me while Miss Sibyl was near. Oh! how I wish she would always stay here, and then I would have a companion."
"Ah, bright one! if I were in her place, what would I not surrender for such a privilege!"
"Would you?" said Christie, looking at him in unfeigned surprise; "then why not stay? I am sure I should be glad to have you here always."
Her innocent words, her enticing beauty, her child-like candor, were a strong temptation. For one moment he was about to fall before her, to clasp her in his arms, to hold her there forever, while he breathed forth his mad, passionate love, and told her nothing on earth should ever part them now. But again rose before him the dark, warning face of Sibyl to allay the fever in his blood. It seemed to him he could see her black, fierce eyes gleaming
on them through the trees—he could almost hear her voice shouting "Traitor!"
All unconscious of the struggle raging in his breast, Christie stood leaning against a tree, her curved crimson lips half parted—her blue eyes fixed on a cloud drifting slowly over the sky, little dreaming of the far darker clouds gathering rapidly, now, over the horizon of her life.
And still in Willard's heart went on the struggle. He dared not look at her as she stood before him—-bright, radiant, bewildering—lest the last lingering remains of fidelity and honor should be swept away by the fierce impetuosity of passion in his unstable heart.
But his good angel was in the ascendant still, for at that moment the voice of Carl was heard calling loudly;
"Christie! Christie!"
"Here, Carl! Here I am," she answered; and in another instant honest Carl stood before them.
"Aunt Tom sent me looking for you," said the young gentleman, rather sulkily; "and I've been tramping through the woods this half-hour, while you were taking it easy here," said Carl, wiping the perspiration from his brow.
"It was all my fault, my good Carl," said Willard, as Christie hastily snatched up her hat and basket and fled, having a just terror of Mrs. Tom's sharp tongue. "Make my excuses to your good aunt, and here is something for yourself."
Carl's dull face brightened wonderfully as Willard drew a gold piece from his pocket and pressed it into his hand, and then turned his steps slowly in the direction of Campbell Castle, thinking all earthly happiness lay centered in the opposite direction.
Mrs. Tom's reproaches fell unheeded, for the first time, on Christie's ear that day. She heard not a word of the long lecture delivered with more than
the good widow's usual eloquence, for she was thinking of another voice, whose lowest tone had power already to thrill to the innermost recesses of her heart. She loved without knowing it, without wishing to define the new, delicious feeling filling her breast, only conscious she had never been so happy before in her life, and longing for the time when she should see him again. Ah, well had it been for her had they never met more.
CHAPTER VIII.
THE TRIUMPH OF PASSION.
"All other passions have their hour of thinking, And hear the voice of reason. This alone Sweeps the soul in tempests."
"Well," said Aunt Moll, to her son and heir, Lem, as he entered the long, high kitchen of Campbell's Lodge, "I would like ter know what dat ar Master Drummin's up ter? I doesn't understan' dese yer new-fangled young men 'tall. Fust he comes a courtin' of our Miss Sibyl, and jes' as soon as her back's turned he goes rite off an' takes up wid dat ar Miss Chrissy."
"'Tain't no business your'n, ole woman," said Lem, gruffly. "I 'spec's as how Marse Drummin' knows what he's about."
"Yes, honey; but 'pears to me I ought to tell Miss Sibyl 'bout it. Ef he is her beau, he oughtn't to be takin' up wid dat ar Miss Chrissy.'
"Better let Miss Sibyl look arter her own beau," replied her dutiful son. "How does ye know he's a courtin' Miss Chrissy?"
"'Cause I seed dem, chile—yes I did—las' night, down on de shore. De moon was shinin' jes' as bright as a new pin, an' I took dat ar litter o' kittens down to de shore to drown 'em, when I seed Marse Drummin' a walkin' along wid Miss Chrissy, and he had his head stooped down jes' so,"—and Aunt Moll ducked her woolly head to illustrate it—"an' was whisperin' soft stuff, jes' as folks do when dey're in love."
"Well, what den?" asked Lem, growing interested.
"Well dey come up ah' seed me, in course, an', Lor', Lor'! I jes' wish you seed de look Marse Drummin' give me. 'Peared as ef he'd a liked to knocked my ole head off. But I warn't afeared, 'deed I warn't, chile; so I jes' stood still an' drapped a courtesy, an' Miss Chrissy, she got red rite up to de roots ob her hair. 'Good-evenin', marse and young miss,' sez I; 'don't be skeered; I only wants to drown dese little kittens,' sez I, for I thought as I might be perlite jest as well as not.
"'Oh, how does yer do, Aunt Moll?' sez Miss Chrissy, a laughin' an' blushin'; 'how is Lem and yer rheumatiz, dese times?'
"'T'ank you, honey,' sez I, 'dey's pretty well, bof ob dem.'
"An' den Marse Drummin' he pulled her arm right troo his'n, and marched her off wid him; an' den I pitched de kittens right in de water an' come home."
"Well, dat ar warn't much," said the skeptical Lem. "Dey might be walkin' on de beach, but that ain't by no means courtin'. Marse Drummin' walk wid her, 'cause Miss Sibyl's gone, an' he ain't got nobody else to talk to."
"'Cisely so, chile; but dat ain't all," says Aunt Moll. "Dis berry mornin', as I was passin' troo de hall, de sittin'-room door was open, and I heered voices a talkin' dere; so I listened and peeked in, an' dar was Marse Drummin', rampin' up and down, a talking to hisself."
"Well, dat ain't nothin', eider," said the still contradictory Lem. "I've hearn dat ar Carl talk to hisself when Miss Tom sent him out to work; an' he
ain't in love wid no one."
"But listen, honey, and don't you be puttin' me out so, 'cause 'tain't 'spectful—'deed it ain't," said Aunt Moll, getting slightly indignant. "As I was sayin', I clapt my ear to de door, an' I heered him sayin' jes' as plain as nothin' 'tall;
"'Oh, dischanting, onwildering Chrissy! ef I had nebber met you, I might yet be happy!' Dar, what he say dat for ef he warn't in lub?"
This last was a settler. Lem felt that his mother had the best of the argument, and unwilling to seem defeated, he went out, leaving the old lady to enjoy her triumph.
Three days had passed since the departure of Sibyl, and certainly Willard's conduct seemed to justify Aunt Moll's suspicions. Unable to break the thrall which bound him, wishing, yet unable to fly from the spell of the enchantress, he lingered still by her side. There were shame, dishonor, sin, in remaining, but oh! there were death, misery, and desolation in going. All worldly considerations, her unknown birth, her obscure connections, her lowly rank, were swept away like walls of cobweb before the fierce torrent of passion that overwhelmed, conquered every other feeling in its impetuous tide.
And she loved him, this angel of beauty, this fairy princess of the isle; he could see it in the quick flush of joy at his approach, the quick, burning glances shot from her beautiful eyes, more quickly averted when they met his—her low, impassioned tones, her bright, beautiful blushes. There was joy, there was rapture in the thought; and yet, unless he forgot honor, vows, all that should have been sacred, what did this love avail?
And so, like a tempest-tossed bark on a tempest-tossed sea, he strove with passion and honor, love and remorse, right and wrong.
Once only, fearing lest her suspicions might be aroused by his absence, he had visited Sibyl, whose rapturous greeting and confiding love made him feel far more of a villain than ever. He looked forward with dread to the period of her return, fearing for the discovery of his falsity; but, more than
all, fearing for the effects of her fierce wrath on Christie, knowing well what must be the strength of Sibyl's passion when unchained.
And so, when Mrs. Brantwell proposed that Sibyl should remain with her another week, instead of returning to the dreary isle, instead of feeling irritated now, he backed the proposal, saying that perhaps it would be better for her to do so, more especially during her brother's absence.
And Sibyl, in her deep love and woman's trust, suspecting nothing, fearing nothing, consented, to the inward joy and sincere relief of her false lover.
Resolving to visit her frequently, and so allay any suspicions that his absence might give rise to, Willard Drummond returned to the island and to —Christie, yielding himself without further effort to the witching spell of her love.
Mrs. Tom suspected nothing of the contraband courting carried on under her very eyes. It was the most natural thing in the world, she thought, that, in the absence of Sibyl and her brother, the young man should spend whole days with them, for it was not pleasant having no one to talk to but a couple of negroes, as she very well knew. Then, it was not to be wondered at, that he preferred talking and walking with Christie to any of the rest, for she was "book-l'arned" like himself, which neither she nor Carl was. She did wonder a little sometimes, and said as much to Christie, why he should stay on the island at all, in the absence of the other.
"But, I suppose," was always her conclusion, "It's because it's Miss Sibyl's home, and, for her sake, he stays there until she comes."
But Christie, though she only blushed and was silent, was of a different opinion—one that she would scarcely own to her own heart. As to his being in love with Christie, Mrs. Tom would have scouted the idea with scorn and unbelief, had she heard it. Every circumstance was against such a conclusion. He was rich, highly connected, and proud as a prince of the blood; she was poor, unknown, and, compared with him, uneducated. Besides, in the good widow's opinion, she was a child in feeling, as she certainly was in years, scarcely knowing the meaning of the word love.
Ah! she had been till he came; and his fervid, impassioned words, his burning glances, his thrilling touch, had swept away the glamour of childhood and simplicity, and revealed to her the passionate woman's heart within her. His words, his looks, his tones, were all new revelations to the artless, island maiden, changing her, as if by magic, from a child to a woman. She revered him as the embodiment of all that was brave, generous, and noble; worshipped him as a god, and loved him with all the affection of her fresh, young heart, with all the ardor of a first, deep love.
As yet, she knew not whether that love was returned; for, unfaithful as he was in thought to Sibyl, passion had not yet so totally conquered his reason as to make him sin in words. He had never said, "Christie, I love you;" but, ah, how often had his eyes said this, and much more; and how long would this slight barrier stand before the fiery impetuosity of unstable youth?
And so that day passed, and the next, and the next, and the next, and with every passing hour the temptation grew stronger and harder to be resisted. Matters must come to a crisis now, or never. Sibyl, in a day or two, would be home, and this wild frenzy of his could be hidden no longer. If she should come, as matters stood now, all would be lost.
And thus, torn between conflicting emotions, Willard sought Christie, on the day before Sibyl was expected home, with the determination of bringing this struggle to an end, then and there.
It was a glorious August afternoon. The island wore its bright dress of green, and nestled in the blue shining river like an emerald set in sapphire. The birds in the deep pine forest were filling the air with their melody, and the odor of the wild roses came floating softly on the summer breeze.
But Willard Drummond was in no mood to admire the beauties of Nature. The morning had been spent in pacing up and down his room, hesitating, resolving, doubting, wishing, yet undecided still. For, when duty and principle would appear for a moment victorious, the waving golden hair, the beautiful blue eyes, the gentle, loving face of Christie would arise before him, scattering all his good resolutions to the winds. And, mingled with this, there was a sort of superstitious foreboding of evil to come. He
thought of his dream, and of the yellow-haired siren luring him on to destruction; and of Sibyl, fiery daughter of a fiery race, fierce, vindictive, and implacable in her wrongs.
"Oh, that I had never met this dark, passionate girl!" he murmured, distractedly, "who now stands between me and the heaven of my dreams; or would that I had seen this beautiful, enchanting Christie first! Oh, for that angel as my wife! And but for those fatal vows once made to Sibyl, she might be mine. I was mad, crazed, to mistake my fancy for that dark, wildeyed girl for love! And now, for that one mistake, am I to be wretched for life? Shall I give up this beautiful, radiant creature, who loves me, for one I care for no longer? No; the struggle is past. Christie shall be my bride, and I will brave the worst that may follow!"
He set his teeth hard; and, as if fearing second reflection might make him change his mind, he left the house and hurried out to meet Christie.
Down on the shore, under the shade of an overhanging willow, he knew Christie had a favorite seat, where, on pleasant days, she used to take her work. Here he was sure of finding her, and in this direction he bent his steps.
She sat, sewing, under the shade of the drooping willow, singing softly to herself, and looking like some sylvan goddess of a sylvan scene; or some beautiful sea-nymph, just risen from her grotto of coral and chrystal.
Radiant and bewildering was the smile and blush with which she welcomed him—a smile and blush that might have been found too strong even for more potent principles than his.
He seated himself beside her, with a look of moody abstraction, all unusual with him, watching her covertly from under his eyelashes, as she bent smiling and happy over her work.
For a time, Christie chatted gayly on various commonplace matters; but, at last, catching her tone from his, she, too, grew silent and thoughtful. She bent lower over her work, wondering if she had offended him, and involuntarily sighed.
He heard it, and said:
"And wherefore that sigh, Christie! Are you unhappy?"
"No not unhappy; but troubled."
"And why should you be troubled, bright one? What can there be to grieve one so fair?"
"I—I—feared I had offended you," she answered, timidly. "You appear out of spirits."
"You offend me, gentle one—you who never offended any one in your life? No, no; it is not that."
"Then you are unhappy," she said, shyly.
"Yes, I am miserable—wretched!" he cried, vehemently. "I wish to Heaven I had never been born!"
"Oh, Mr. Drummond! what has happened!" she cried, laying her hand on his, and looking up wistfully in his face.
Her touch, her tone, her look swept away every remaining trace of fidelity. He forgot everything he should have remembered—his vows, his honor, his truth—and saw nothing but the bright, radiant, bewildering vision before him. In an instant he was on his knees at her feet, exclaiming, with impassioned vehemence:
"Christie! Christie! my life, my dream, my hope, I love you. See, I am at your feet, where my heart, my name, my fortune, long have been. With my whole heart, and soul, and life, I love you with a love stronger than death or the grave. All the devotion and hopes of my life I offer you, if you can only say you love me."
He was pale and panting; his eyes were fierce and burning; his tones low, thrilling, and passionate.
Trembling, shrinking, blushing, yet, with a deep, intense, fervent joy thrilling through her heart and being, Christie listened. The blood swept in torrents to her face, neck, and bosom, which rose and fell with her rapid breathing. She dare not look up to meet his ardent, burning, gaze.
"Christie, Christie! my love, my life! look up; speak—answer me—tell me that you love me!"
Still no reply, only those downcast eyes, deepest blushes, and quick, hurried breathing.
"Speak! speak! my beautiful love! only one word from those sweet lips; but one touch of your dear hand to tell me I may live," he cried, growing more wild and impassioned.
With a low, glad cry of intense joy, she buried her blushing face on his shoulder.
"Thanks! my heart's thanks for this sweetest, loveliest Christie!" he cried, with exultant joy, pressing her yielding form to his bounding heart. "My life, with all its hopes, energies, and ambitions, shall be devoted to but one purpose now—that of rewarding you for your priceless love."
"Oh, Mr. Drummond, your love is all the reward I ask!" she said, in the deep, earnest voice of perfect trust.
"Not Mr. Drummond now, sweetest Christie. I am Willard to you, now and forever. Let me always hear that name in music from your lips, and earth has no higher boon in store for me."
"But oh! can you love me thus—me, a poor, little, nameless, uncultured girl, while you are rich, distinguished, and highly connected. Oh, Willard, will you not, some day, repent this choice—you, who might win the highest and fairest in the land?"
"Repent! never—never! Perish my heart, if it ever admit of any love but thine; palsied be my arm, if it ever encircle any form but this; accursed be
my lips, if they ever perjure the words I have spoken now; lost forever be my soul, when it is false to thee!" he cried, with passionate vehemence.
"Oh, Willard! dearest, hush! I do not doubt you—Heaven forbid. I should die, if I thought you could be false to me."
"Speak not of death; it is not for such as you, bright, beautiful Christie. And now, only one thing is wanting, to make me the happiest of men."
She lifted her radiant face with a look of earnest inquiry.
"Christie, one little word from you, and ere the sun rises on a new day, my joy will be complete—my cup of earthly happiness will be filled to the brim."
Still the same earnest, anxious gaze.
"Dearest love, you will not refuse? It will be but a small matter to you, and will make me supremely blessed."
"And that?" she inquired, wonderingly.
"Brightest Christie, be my bride—my wife!" he cried, folding her closer in his arms, and speaking in a thrilling whisper.
Again the eloquent blood swept over her stainless neck and bosom, but she did not reply.
"You will not refuse me, my own Christie, this last, greatest favor? Comply now—to-day; for if the present opportunity passes, it may never occur again."
"But how—how can we be wedded here?" she said, shyly, lifting her eyes to his impassioned ones, and dropping them in brightest blushes.
"Christie, yonder lies a boat; it is three hours to sunset; long before that time we can reach Westport; there we can find a clergyman, and there you can become my own for life!"
