How do you modify your algorithm design for different constraints?

1 Divide and conquer

One technique that can help you reduce the time complexity of your algorithm is to divide the problem into smaller and simpler subproblems, and then combine the solutions of the subproblems to solve the original problem. This technique is called divide and conquer, and it can be applied to many problems, such as sorting, searching, or multiplying large numbers. For example, the merge sort algorithm uses divide and conquer to sort an array by recursively splitting it into two halves, sorting each half, and then merging them together. The time complexity of merge sort is O(n log n), which is better than O(n^2) for some other sorting algorithms.

2 Dynamic programming

Another technique that can help you optimize your algorithm is to use dynamic programming, which is a method of solving complex problems by breaking them down into overlapping subproblems, and then storing and reusing the solutions of the subproblems to avoid repeated computations. This technique can help you reduce both the time and space complexity of your algorithm, but it requires that the problem has two properties: optimal substructure and overlapping subproblems. For example, the Fibonacci sequence can be computed using dynamic programming by storing the previous two numbers in an array, and then using them to calculate the next number. The time and space complexity of this approach is O(n), which is better than O(2^n) for the recursive approach.

3 Greedy algorithm

A greedy algorithm is a technique that makes the locally optimal choice at each step, hoping to find the globally optimal solution. This technique can help you simplify your algorithm and find a fast and easy solution, but it does not guarantee that the solution is optimal or feasible for every problem. Therefore, you need to verify that the problem has a property called greedy choice property, which means that the optimal solution can be built by making the greedy choices. For example, the coin change problem can be solved using a greedy algorithm by choosing the largest coin value that does not exceed the remaining amount at each step, until the amount is zero. However, this algorithm may not work for some coin systems, such as {1, 3, 4}, where the greedy choice may lead to a suboptimal solution.

4 Heuristic algorithm

A heuristic algorithm is a technique that uses a rule of thumb or a shortcut to find an approximate or good-enough solution for a problem that is too hard or impractical to solve optimally. This technique can help you deal with complex or NP-hard problems, such as traveling salesman problem, knapsack problem, or sudoku, where finding the exact solution may take too long or require too much space. However, a heuristic algorithm does not guarantee that the solution is optimal or accurate, and it may depend on the quality of the heuristic function or the initial state. For example, the hill climbing algorithm is a heuristic algorithm that starts from a random state, and then moves to a better state if possible, until it reaches a local maximum or a plateau. However, this algorithm may get stuck in a suboptimal state or a cycle, and it may not explore the whole search space.

5 Randomized algorithm

A randomized algorithm is a technique that uses random numbers or probabilities to influence the decisions or outcomes of the algorithm. This technique can help you introduce some variability or diversity into your algorithm, and avoid some worst-case scenarios or biases. Randomized algorithms can be classified into two types: Las Vegas algorithms and Monte Carlo algorithms. Las Vegas algorithms always produce the correct solution, but the running time may vary depending on the random choices. Monte Carlo algorithms have a fixed running time, but the solution may be incorrect or approximate with some probability. For example, the quicksort algorithm is a randomized algorithm that uses a random pivot to partition the array, and then sorts the two subarrays recursively. The average time complexity of quicksort is O(n log n), but the worst-case time complexity is O(n^2) if the pivot is always the smallest or the largest element.

6 Trade-offs and compromises

When you modify your algorithm design for different constraints, you need to consider the trade-offs and compromises that you make. Sometimes, improving one aspect of your algorithm may worsen another aspect, such as speed, space, accuracy, or simplicity. Therefore, you need to balance the pros and cons of each technique, and choose the one that best suits your needs and preferences. For example, if you have limited memory, you may prefer a greedy algorithm over a dynamic programming algorithm, even if the latter has a better time complexity. Or, if you need a fast and approximate solution, you may prefer a heuristic algorithm over an exact algorithm, even if the former has a lower accuracy. The optimal algorithm design depends on the nature and context of the problem, and the resources and requirements that you have.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?