Math Problem Statement
The goal for the next few questions is to calculate the memory requirements for HybridDeque objects. We will make the following assumptions:
Each object has a 12 byte header.
Every array has 16 bytes of overhead.
Every reference requires four bytes.
Integers require four bytes.
Based on these assumptions, an empty HybridDeque with a block size of 8 requires exactly 136 bytes:
4
×
12
=
48
4×12=48 (four objects, each with a 12-byte header.)
7
×
4
=
28
7×4=28 (seven references used to maintain the structure of the HybridDeque: leftCursor, rightCursor, block, block, prev, elements, next)
3
×
4
=
12
3×4=12 (three integers: size, index, index)
16
8
×
4
=
48
16+8×4=48 (elements array: 16 bytes of overhead, plus space for eight references)
48
28
12
48
=
136
48+28+12+48=136
For the questions below, don't use commas in your answers. For example, you should write 10000 instead of 10,000.
EXAMPLE:
To calculate the memory requirements for a HybridDeque object storing 9 items, we can break the memory consumption into smaller parts based on the assumptions provided. Here's a step-by-step approach:
- Memory for Object Overheads
Each block and supporting structure consumes memory in the following ways:
12-byte header: Each object has a 12-byte header.
7 references, each 4 bytes: There are 7 references used in the object (for leftCursor, rightCursor, block, block, prev, elements, next).
3 integers, each 4 bytes: Three integers (size, index, index) are used in the structure.
So, the memory required for this part is the same as for an empty HybridDeque:
48
28
12
=
88
bytes
48+28+12=88 bytes
- Memory for the Elements Array
We know that each block has space for 8 items, and arrays have 16 bytes of overhead. Each reference in the array occupies 4 bytes. So, for an 8-item block, the memory required is:
16
8
×
4
=
48
bytes
16+8×4=48 bytes
Since the HybridDeque is storing 9 items, and each block can hold 8 items, we need two blocks (one full block for 8 items and another block for 1 item). For each block, we add 48 bytes for the elements array.
The memory for two blocks is:
2
×
48
=
96
bytes
2×48=96 bytes
- Total Memory Calculation
Finally, to calculate the total memory required for the HybridDeque storing 9 items:
88
bytes (object overhead)
96
bytes (two blocks)
=
184
bytes
88 bytes (object overhead)+96 bytes (two blocks)=184 bytes
This isn't over 200 bytes, but we missed considering the headers and some potential additional overheads for managing multiple blocks.
- Adding Headers for Each Block
Each block has its own 12-byte header. Since there are two blocks, the memory for the block headers is:
2
×
12
=
24
bytes
2×12=24 bytes
- Final Total Memory
Now, adding the memory for the block headers:
88
bytes
96
bytes
24
bytes
=
208
bytes
88 bytes+96 bytes+24 bytes=208 bytes
Thus, a HybridDeque object storing 9 items requires 208 bytes.
This is a correct solution to another problem. Use the exact same steps to answer this
Assuming a block size of 64 How many bytes are required for a HybridDeque object that stores 10,000 items. Assume that at most one block is partially filled.
Solution
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Math Problem Analysis
Mathematical Concepts
Memory Calculation
Array Overhead
Block Storage
Formulas
Memory for objects: 4 × 12 + 7 × 4 + 3 × 4
Memory for elements in each block: 16 + block_size × 4
Total memory: overhead + (full_blocks × block_memory) + partial_block_memory + headers
Theorems
Memory Allocation Theorem
Suitable Grade Level
College Level / Advanced High School