Introduction to Algorithms

A step-by-step procedure for solving a problem, consisting of a series of instructions that can be executed by a computer.

A problem that can be solved by applying a general problem-solving method, like finding the largest number in a sequence, sorting a list in ascending order, or finding the shortest path between points in a network.

A specific instance of a general problem. For example, finding the largest integer in the sequence 101, 12, 144, 212, 98 is a specific case of the general problem of finding the largest integer in any sequence.

Representations of problems using mathematical models, such as sets, sequences, graphs, and relations. They provide a framework for analyzing and solving problems.

A branch of mathematics that deals with the study of discrete objects, like sets, graphs, sequences, and functions, often used in computer science for problem-solving and algorithm development.

Data that is provided to an algorithm, like ingredients in a recipe.

The result produced by an algorithm after processing the input, like the finished cake.

Each step in an algorithm must be clearly defined and unambiguous, leaving no room for interpretation.

An algorithm should always produce the correct answer for any given input.

An algorithm must be able to complete in a finite number of steps (not infinite).

Each step in an algorithm can be performed in a limited amount of time by a computer.

An algorithm should work for all problems of a similar type, not just a single specific case.

A method for arranging items in a specific order, usually ascending or descending.

A way to sort a list by repeatedly comparing adjacent elements and swapping them if they're in the wrong order. It's simple but not very efficient.

A sorting algorithm that builds a sorted list by inserting elements one at a time into their correct positions.

A measure of how well an algorithm performs, often considering time taken and memory used.

The property of an algorithm that makes it easy to understand and implement.

The process of comparing two adjacent elements in a list and swapping them if they are in the wrong order.

A specific order in which elements are arranged, often from smallest to largest (ascending) or largest to smallest (descending).

The process of repeating a set of operations until a condition is met, such as sorting a list completely.

An algorithm that finds the largest element in a sequence of integers. It compares each element to the current maximum and updates the maximum if a larger element is found.

The property of an algorithm where each step is precisely defined and unambiguous. There's no room for interpretation.

The problem of locating a specific element within an ordered list, such as a dictionary or a sorted array.

A procedure where each step is clearly defined and can be executed by a computer.

A type of algorithm that involves repeatedly checking a condition and performing actions until the condition is met.

A specific step in an algorithm that compares a value to a condition and then chooses a path based on whether the condition is true or false.

A procedure that determines the sum of all integers in a list.

The insertion sort algorithm proceeds by comparing the current element (aj) with the elements already in the sorted sublist (a1, a2, ..., ai-1). If aj is greater than ai, the comparison continues with the next element (ai+1). Once aj is found to be less than or equal to ai, the elements from ai to aj-1 are shifted one position to the right, and aj is inserted into the position where the comparison stopped.

The insertion sort algorithm continues iterating until the last element in the list is correctly placed relative to the elements already arranged in the sorted portion of the list.

The insertion sort algorithm is a comparison-based sorting algorithm, meaning it compares elements to determine their relative positions.

In the insertion sort, the already sorted portion of the list is expanded by one element in each iteration.

The insertion sort algorithm is considered an in-place sorting algorithm because it modifies the original list directly without requiring additional memory for new data structures.

The insertion sort algorithm is generally efficient for small lists or lists that are already nearly sorted. However, for larger lists with a high degree of disorder, its performance can be significantly slower compared to other sorting algorithms like merge sort or quick sort.

The insertion sort algorithm has a time complexity of O(n^2) in the worst-case scenario, which occurs when the list is sorted in reverse order. In the best case, when the list is already sorted, its complexity is O(n).