# Analyzing Non$-$Recursive Sorting Algorithms: A Comparative Study of Bubble Sort and Heap SortSorting algorithms play a crucial role in computer science, significantly influencing the efficiency of data organization and retrieval. Among the various sorting techniques employed by computer scientists, two prominent non$-$recursive algorithms are $**$Bubble Sort$**$ and $**$Heap Sort$**.$ Each of these algorithms offers distinct advantages and disadvantages, making them suitable for different types of data and scenarios.## Understanding the Algorithms$**$Bubble Sort$**$ is a straightforward sorting technique that functions through a series of comparisons. It iteratively traverses a list, comparing adjacent elements and swapping them if they are found to be in the incorrect order. This process is repeated until the entire list is sorted. Despite its simplicity, Bubble Sort is generally inefficient for large datasets due to its high time complexity.In contrast, $**$Heap Sort$**$ leverages a binary heap data structure, which allows it to sort elements more efficiently, especially as the size of the dataset increases. Heap Sort first builds a max heap from the input data, ensuring that the largest element is at the root. It then repeatedly extracts the maximum element from the heap, placing it at the end of the sorted array. This method is more sophisticated and performs better than Bubble Sort in most cases.The overarching goal of sorting algorithms is to arrange elements in a specified order, whether ascending or descending, based on numerical or other criteria.## Pseudocode Representation### Bubble Sort Pseudocode$```$function bubbleSort$($arr$)$: n $=$ length$($arr$)$ for i from $0$ to n$-1$: for j from $0$ to n$-$i$-2$: if arr$[$j$]>$ arr$[$j$+1]$: swap$($arr$[$j$],$ arr$[$j$+1])```$### Heap Sort Pseudocode$```$function heapSort$($arr$)$: n $=$ length$($arr$)$ buildMaxHeap$($arr$,$ n$)$ for i from n$-1$ down to $1$: swap$($arr$[0],$ arr$[$i$])$ heapify$($arr$,0,$ i$)$function buildMaxHeap$($arr$,$ n$)$: for i from floor$($n$/2)$ down to $0$: heapify$($arr$,$ i$,$ n$)```$## Analyzing Time Complexity### Bubble Sort ComplexityThe time complexity of Bubble Sort is characterized by its nested loops. The outer loop iterates $`$n$`$ times, while the inner loop runs $`$n $-$ i $-1`$ times for each pass of the outer loop. Consequently, the total number of comparisons in the worst$-$case scenario can be expressed as:$\backslash [$t$($n$)=\backslash$sum$\_\{$i$=0\}^\{$n$-1\}($n $-$ i $-1)=\backslash$frac$\{$n$($n$-1)\}\{2\}\backslash ]$This results in a time complexity of $\backslash ($O$($n$^2)\backslash ),$ which makes Bubble Sort impractical for large datasets.### Heap Sort ComplexityIn contrast, Heap Sort exhibits a better time complexity. The building of the max heap takes $\backslash ($O$($n$)\backslash )$ time, while the subsequent extraction of elements involves $\backslash ($O$(\backslash$log n$)\backslash )$ time for each of the $\backslash ($n$\backslash )$ elements. Thus, the overall time complexity for Heap Sort can be summarized as:$\backslash [$O$($n $\backslash$log n$)\backslash ]$This efficiency makes Heap Sort a more favorable option for larger datasets compared to Bubble Sort.## ConclusionIn summary, while both Bubble Sort and Heap Sort serve the fundamental purpose of sorting data, they differ significantly in their efficiency and operational mechanisms. Bubble Sort, with its simplicity, is best suited for small datasets or educational purposes, while Heap Sort's superior performance makes it a better choice for handling larger collections of data. Check amd reduce my similarity score