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Recursion is a method that defining a thing using the thing being defined in its definition. For example, we define an expression as a number, or a concatenation, in order, of an expression, an operand, and an another expression. The definition of an expression is recursive since the definition applies expressions. In Computer Science, "divide and conquer" is a very general programming strategy to solve problems. The idea of this strategy is based on simplifying a problem into small scaled same problems. For example, assuming we want to sort a list of books in a book store, we can partition the books to be sorted into multiple small groups of books. For each group, we can partition the books again, etc. When each small groups of books are sorted, we can combine the sorted groups of books together and eventually the whole book lot will be sorted. Recursion can be applied to write sorting algorithms based on the divide and conquer strategy. Indeed, the Quick Sort algori...

When studying mathematical proof techniques, I found that it is important to study with an open mind. To prove a statement, first it is important to clearly formulate a problem, and then to collect facts, and then to follow logic reasoning. For example, assuming we want to prove log(n!) is O(nlogn). The problem is clear. The fact we can use is the definition of Big-O in addition to the problem formulation. The definition of Big-O indicates that to prove a function f(n)is O(g(n))we need find constants c>0(real number) and n_0>0(integer) such that f(n) =n_0. To conclude the result, we need find c and n_0 in order to apply the definition of Big-O, which should be the goal during our reasoning. Here is the sequence. log(n!) = log(1 * 2 *...*n) = log1 + ... + logn = nlogn Therefore, there exist c =1, when n>=1, log(n!) based on the definition of Big-O, log(n!) = O(nlogn).

When developing code for a software application, we frequently need estimate how the application code consumes resources such as processor, time, and memory and whether the code scales well. It turns out asymptotic notations and analysis can help to answer these questions. The asymptotic notations include Big-O, little-o, Big-Omega, little-omega, and Big-Theta. Most algorithms handles input whose size affects the time and memory to execute the algorithms. If we use a function f(n), where n is standing for the size of the input, to indicate the cost to execute the algorithm, f(n) is a non-decreasing function, which grows as the input n does. Usually the larger the input n is, the longer the time or the memory is used to execute an algorithm. Asymptotic notations are usually discussed in theoretical CS courses. For a cost function f(n), Big-O notation is usually used to describe a tight-upper bound for the cost of an algorithm; little-o notation is used to describe an upper bound; Bi...

Abstraction is a purposeful process to focus on essentials, observable behavior, and ignore irrelevant details. Abstraction is a process frequently used in mathematics and computer science. In computer science, there is a complementary process called encapsulation for abstraction. Encapsulation applies information hiding to hide secrets so that observable behavior is abstracted. For example, assuming we want to reinvent a virtual pizza store, a chef in the store is responsible to make pizzas. A customer orders a cheese pizza and the order is delivered to the chef, who then cooks the pizza to fulfill the order. The customer and/or some waiter doesn't need to know how the chef take actions to make the cheese pizza. The chef, however, hides the 'secrets' to make the pizza. Abstraction and encapsulation work together to invent the automating agents in the pizza store. In mathematics, abstraction is important to produce a simple, elegant proof. Assuming that we want to prove ...

I was asked a question, "Can I find a data structure book writing for dummies?" I knew the boy was joking. But I think this is a good question. When we read a book, we are supposed to interact with the book like interacting with a friend. We are asking questions and the friend answers and guides our exploration. The unfortunate thing is that we have so many books that are dry to read like hanging with very boring friends. Sometimes the friend is so arrogant and monotoning. I personally think this is a big problem in Computer Science as well as Mathematics. There are so many terrific things happened in computing fields. And yet it is hard to find exciting textbooks for students to enjoy.

Buy at AllPosters.com My daughter loves to watch Christmas movies not only in Christmas season but the whole year long. I remember one movie she has watched a lot of times (because it is funny) is about a man chosen to be Santa Clause after the other Santa fell down from his roof. The movie was certainly very funny and we were laughing all the time when we watched it. When I am considering Object-Oriented (OO) programming, I am thinking about the movie and the role Santa Clause who plays in Christmas. Almost everybody loves Santa in Christmas because he delivers gifts, happiness, and hope to us. According to the movie, if one Santa falls, another one comes to play. I think this is really wonderful. In Object-Oriented programming, a program is a collection of interacting objects. A software application behaves and implements responsibilities. Objects in the application plays roles, performing corresponding responsibilities. For example, if we reinvent the North Pole machine...

An algorithm is a finite sequence of instructions to solve a problem. And a program is computational method implementing algorithms to solve problems. Algorithm is one of essential factors to guarantee the success of a software system. Quite often there are multiple algorithms to solve a same problem. Certainly we would like to choose good algorithms to solve problems. How to measure if an algorithm is a good one? There are several factors such as effectiveness, correctness , efficiency, complexity, etc. Intuitively, a good algorithm should correctly and effectively solve a problem we have. It should be efficient, running fast and consuming memory space economically. It should be simple and natural. When evaluating algorithms, most time we focus on time and space efficiency of the algorithms. When a programmer is designing and implementing a program, the construction requires to pay attention to not only essentials but also details and to make decisions when there a...