Question

Write an efficient algorithm that searches for a value in an \(n \times m\) table (two-dimensional array). This table is sorted along the rows and columns that is. \\[\begin{array}{l}\text { Table }[i][j] \leq \text { Table }[i][j+1] \\ \text { Table }[i][j] \leq \text { Table }[i+1][j]\end{array}\\]

Short answer

From the top right corner, move left if the current element is greater than the target, move down if it is less. Continue until the target is found or the current position is out of bounds. If the algorithm ends without finding the value, it is not in the array.

Step-by-step solution

Starting point

Start your search from the right-most top corner of the array. Let's call this element as current element.

Compare and Move

Compare the current element with the target value. If they are equal, you have found the value and you can exit. If the current element is less than the target, then move down one row and remain in the same column. If it's larger than the target, then move left one column and remain in the same row.

Recurse or return

Repeat step 2 until you find the target or the current position is out of the array boundaries. If you step outside of the array without finding the target value, then the value is not in the array. You should return an error or an indication that the value wasn't found.

Key concepts

Two-Dimensional Array Search

Searching for values within two-dimensional arrays, or 'tables', is a common operation in computer science. These arrays are essentially a grid, consisting of rows and columns, akin to a chessboard or a spreadsheet. When data is sorted in these structures—in this case, non-decreasingly from left to right across rows, and from top to bottom down columns—it empowers certain search algorithms to operate more effectively.

A strategic starting point is crucial. By starting at the top right cell, we leverage the sorted nature of the array. If the target value is less than the current element, we can confidently move leftwards, eliminating the entire column. Conversely, if the target is greater, moving down one row excludes that row from subsequent search. This method is akin to a game of 'hot and cold', where each move gets you closer. To enhance algorithm effectiveness:

• Utilize the array's sorted property to your advantage.
• Avoid unnecessary comparisons.
• Optimize movements to eliminate as many options as possible with each step.

Scanning row by row, despite them being sorted, would be far less efficient. The step-by-step approach detailed in our base solution exemplifies how to navigate such an array smartly and quickly.

Algorithm Efficiency

The efficiency of an algorithm is gauged by how fast and how economically it can solve a problem. Effective algorithms process data swiftly and with minimal resource consumption. Measuring this efficiency is crucial in computer science and is typically done in terms of time complexity and space complexity, which correspond to the time and memory an algorithm uses, respectively.

In the context of a two-dimensional array search, an efficient algorithm will find the target value using the least amount of comparisons and memory. The given step-by-step method demonstrates a time-efficient approach because it reduces the problem size significantly with each step—it's a case of ‘divide and conquer’ without explicitly dividing the array. Time complexity is often expressed using Big O notation, which provides a high-level understanding of the algorithm's scalability as the size of the input data grows.

To improve efficiency:

• Consider early termination once the target is found.
• Think about the best-case, average-case, and worst-case scenarios.
• Optimize the code to execute the fewest instructions possible.

Remember, the most brute-force method isn't always the most effective—smart, strategic steps count considerably more toward efficiency.

Sorted Data Structures

Sorted data structures are a cornerstone of efficient computing. When data is sorted, it provides inherent order, allowing algorithms to execute searches, insertions, and deletions more efficiently. In our exercise, the two-dimensional array represents a sorted data structure with guaranteed order in both rows and columns.

A sorted structure enables the use of binary search—often a far quicker alternative to linear search. However, in a two-dimensional plane, the algorithm needs adaptation. The process from the exercise improves upon a simple binary search by taking advantage of the array's two sorted dimensions for deterministic directional navigation.

Sorting data structures isn't free of cost, though. The sorting process itself takes time, so it is most beneficial when a data structure will be searched frequently. When improving search algorithms in sorted structures:

• Utilize the sort order to reduce search space.
• Understand the cost of maintaining sorted data, especially if frequent insertions or deletions occur.
• Choose the right sorted data structure for your specific use case—arrays for static data, binary search trees, or balanced trees (like AVL trees) when data is dynamic.

Ultimately, sorted data structures are powerful tools, magnifying the potency of the algorithms that work on them.