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-**Insertion**: $O(1)$ for the end, $O(n)$ for beginning/middle
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-**Deletion**: $O(1)$ for the end, $O(n)$ for beginning/middle
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-**Append**: $O(1)$ amortized, $O(n)$ during resizing
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### Code Example: Basic Array Operations
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@@ -66,15 +66,15 @@ public class ArrayExample {
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### Advantages
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-**Speed**: Arrays provide \(O(1)\) access and append operations when appending at a known index (like the end).
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-**Speed**: Arrays provide $O(1)$ access and append operations when appending at a known index (like the end).
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-**Cache Performance**: Arrays, with their contiguous memory layout, are efficient for tasks involving sequential data access.
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### Disadvantages
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-**Size Limitations**: Arrays have a fixed size after allocation. Resizing means creating a new array, leading to potential memory overhead or data transfer costs.
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-**Mid-Array Changes**: Operations like insertions or deletions are \(O(n)\) due to necessary element shifting.
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-**Mid-Array Changes**: Operations like insertions or deletions are $O(n)$ due to necessary element shifting.
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### Considerations
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@@ -89,7 +89,7 @@ public class ArrayExample {
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### Answer
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**Indexing** refers to accessing specific elements in an array using unique indices, which range from 0 to \(n-1\) for an array of \(n\) elements.
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**Indexing** refers to accessing specific elements in an array using unique indices, which range from 0 to $n-1$ for an array of $n$ elements.
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### Key Concepts
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#### Memory Address Calculation
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The memory address of the \(i\)-th element is computed as:
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The memory address of the $i$-th element is computed as:
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\[
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$$
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\text{Memory Address}_{i} = P + (\text{Element Size}) \times i
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\]
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$$
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Here, \(P\) represents the pointer to the array's first element.
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Here, $P$ represents the pointer to the array's first element.
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### Code Example: Accessing Memory Address
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@@ -141,15 +141,15 @@ This ordering provides various benefits, such as **optimized search operations**
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### Advantages
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-**Efficient Searches**: Sorted arrays are optimized for search operations, especially when using algorithms like Binary Search, which has a \(O(\log n)\) time complexity.
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-**Efficient Searches**: Sorted arrays are optimized for search operations, especially when using algorithms like Binary Search, which has a $O(\log n)$ time complexity.
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-**Additional Query Types**: They support other specialized queries, like bisection to find the closest element and range queries to identify elements within a specified range.
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-**Cache Efficiency**: The contiguous memory layout improves cache utilization, which can lead to faster performance.
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### Disadvantages
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-**Slow Updates**: Insertions and deletions generally require shifting elements, leading to \(O(n)\) time complexity for these operations.
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-**Slow Updates**: Insertions and deletions generally require shifting elements, leading to $O(n)$ time complexity for these operations.
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-**Memory Overhead**: The need to maintain the sorted structure can require extra memory, especially during updates.
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@@ -163,11 +163,11 @@ This ordering provides various benefits, such as **optimized search operations**
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### Time Complexity of Basic Operations
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-**Access**: \(O(1)\).
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-**Search**: \(O(1)\)for exact matches, \(O(\log n)\) with binary search for others.
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-**Insertion**: \(O(1)\)for the end, but usually \(O(n)\) to maintain order.
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-**Deletion**: \(O(1)\)for the end, but usually \(O(n)\) to maintain order.
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-**Append**: \(O(1)\)if appending a larger value, but can spike to \(O(n)\) if resizing or inserting in order.
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-**Access**: $O(1)$.
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-**Search**: $O(1)$ for exact matches, $O(\log n)$ with binary search for others.
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-**Insertion**: $O(1)$ for the end, but usually $O(n)$ to maintain order.
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-**Deletion**: $O(1)$ for the end, but usually $O(n)$ to maintain order.
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-**Append**: $O(1)$ if appending a larger value, but can spike to $O(n)$ if resizing or inserting in order.
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---
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@@ -182,15 +182,15 @@ This ordering provides various benefits, such as **optimized search operations**
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-**Adaptive Sizing**: Dynamic arrays adjust their size based on the number of elements, unlike fixed-size arrays.
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-**Contiguous Memory**: Dynamic arrays, like basic arrays, keep elements in adjacent memory locations for efficient indexed access.
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-**Amortized Appending**: Append operations are typically constant time. However, occasional resizing might take longer, but averaged over multiple operations, it's still \(O(1)\) amortized.
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-**Amortized Appending**: Append operations are typically constant time. However, occasional resizing might take longer, but averaged over multiple operations, it's still $O(1)$ amortized.
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### Time Complexity of Basic Operations
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-**Access**: \(O(1)\)
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-**Search**: \(O(1)\)for exact matches, \(O(n)\) linearly for others
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-**Insertion**: \(O(1)\)amortized, \(O(n)\) during resizing
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-**Deletion**: \(O(1)\)amortized, \(O(n)\) during shifting or resizing
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-**Append**: \(O(1)\)amortized, \(O(n)\) during resizing
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-**Access**: $O(1)$
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-**Search**: $O(1)$ for exact matches, $O(n)$ linearly for others
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-**Insertion**: $O(1)$ amortized, $O(n)$ during resizing
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-**Deletion**: $O(1)$ amortized, $O(n)$ during shifting or resizing
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-**Append**: $O(1)$ amortized, $O(n)$ during resizing
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