Backtracking and recursion are both programming techniques used for solving problems, particularly those that can be broken down into smaller sub-problems. Understanding them requires starting from the basic principles of problem-solving in computer science.

Recursion

Fundamental Principle: Recursion is based on the idea that a problem can be solved by solving smaller instances of the same problem.

• Definition: A recursive function calls itself with a modified input, aiming to reach a base case.
• Base Case: This is a condition under which the recursion stops. It's necessary to prevent infinite loops.
• Divide and Conquer: Recursion is often used in divide-and-conquer algorithms where the problem is divided into smaller, more manageable sub-problems.

For example, consider calculating the factorial of a number n (denoted as n!). Factorial of n is defined as the product of all positive integers up to n. This can be expressed recursively as n! = n * (n-1)!, with the base case being 0! = 1.

Backtracking

Fundamental Principle: Backtracking is a refined form of brute force approach, where you try to build a solution incrementally and abandon a path as soon as it is determined that this path cannot possibly lead to a solution.

• Definition: It involves recursion, but with the added strategy of "tracking back" when a potential solution path proves wrong.
• State Space Tree: Backtracking is often visualized as a traversal over a state space tree, where each node represents a partial solution.
• Pruning: When a node (partial solution) is determined not to lead to a valid solution, the algorithm backtracks to explore other branches of the tree.

A classic example of backtracking is solving a maze. You start from a point, make a choice at each intersection, and if you reach a dead end (i.e., a point from which you can't proceed further towards the solution), you backtrack and try a different path.

Comparison

• Scope: Recursion is a broader concept. Backtracking is a specific use of recursion for solving optimization problems and searching.
• Performance: In backtracking, the performance gain is achieved by pruning, which eliminates the need to explore all possible solutions.
• Use Cases: While recursion is used in a variety of contexts (like tree traversal, sorting algorithms like quicksort, etc.), backtracking is specifically used in constraint satisfaction problems like puzzles, scheduling, and partitioning.

Both recursion and backtracking are critical for understanding how complex problems can be decomposed and solved in a step-wise and systematic manner in computer science.