Contiguous Heap Array Buffer
Sequential memory allocation mapping an implicit complete binary tree into contiguous physical RAM with parent-child indexing arithmetic.
Computer Science / Binary Heap Dynamics
FREE TO EXPLORELABORATORY 08 · COMPUTER SCIENCE
Empirical Analysis of Binary Heaps and Priority Queues: In-Place Floyd's Heapify, Sift-Down / Sift-Up Asymptotics, and Priority Event Scheduling
Determine the heapify comparison scaling factor k_heap and evaluate priority queue mechanics, verifying Floyd's linear O(n) construction bound C(n) = 2(n − log₂ n) and examining sift-down versus sift-up asymptotes. Explore interactive search algorithms, compare operation counts and analyse complexity scaling.
COMPUTE · BENCHMARK · ANALYSE
Your virtual algorithm laboratory.
HEAPIFY COMPARISONS (C)—
EXECUTION STATUS
Ready to execute Array Size benchmark.
Adjust Array Size n, then execute benchmark.
Operation counts are generated deterministically by the exact algorithmic execution model. Modern hardware benchmarks additionally reflect cache line prefetching and branch misprediction penalties.
Algorithmic assumptions and limits
The simulated execution model evaluates idealized complete binary heaps where Floyd's bottom-up construction on power-of-two arrays executes with exact comparison counts C(n) = 2(n − log₂ n). In physical computer systems, heap throughput is modulated by CPU memory bus bandwidth contention, non-local cache line eviction penalties during deep sifting, branch predictor mispredictions during child selection comparisons, and OS thread preemption. The simulated benchmark environment reflects strict deterministic execution of Floyd's bottom-up in-place build_heap algorithm on power-of-two arrays of size n = 2^k. Each non-leaf node i from ⌊n/2⌋ − 1 down to 0 is sifted down along the path of maximum children. At each level of the sift path, exactly 2 key comparisons occur: one comparison between the left and right children to select the larger child, and one comparison between the parent and the selected larger child. For an array of size n = 2^k, summing these comparisons across all levels yields the exact closed-form combinatorial identity C(n) = 2n − 2 log₂(n) comparisons. Defining the cumulative sift interval coordinate x = n − log₂(n) yields the exact linear model C = 2.0000 · x + 0.0000 comparisons, providing an ideal microarchitectural testbed for Ordinary Least-Squares regression with R² = 1.000000.
Computational tools & environment.
Floyd Bottom-Up Heapifier
In-place algorithmic construction unit sifting nodes downward from index ⌊n/2⌋ − 1 down to 0 to establish the heap invariant in linear O(n) time.
Sift-Down / Sift-Up Comparator
Hardware-coordinated comparator register tracking discrete child-child and parent-child decision branches during element restoration.
Priority Event Scheduler Queue
Discrete event simulation engine utilizing a Min-Heap to manage chronological event dispatching in operating systems and network queues.
Cache Stride Telemetry Monitor
Virtual memory profiler measuring non-local child index jumps (2i + 1, 2i + 2) across CPU L1/L2 cache lines during deep heap traversals.
D-Ary Heap Configuration Module
Algorithmic parameter controller evaluating 4-way and d-ary heap fan-out architectures to balance tree height with child comparison costs.
FOLLOW THE METHOD
A reason behind every step.
Determine the heapify comparison scaling factor k_heap and evaluate priority queue mechanics, verifying Floyd's linear O(n) construction bound C(n) = 2(n − log₂ n) and examining sift-down versus sift-up asymptotes. Suggested independent settings: 64, 128, 256, 512, 1024, 2048, 4096 keys.
01
Initialize Contiguous Array Buffer & Key Permutations
Allocate a contiguous array buffer of n distinct 64-bit integer keys arranged in randomized permutation order, verifying zero auxiliary pointer allocation.
02
Select Benchmark Scale Range
Adjust the independent variable control slider to choose an array size n between 64 and 4,096 keys across the designated power-of-two benchmark settings.
03
Execute Floyd Bottom-Up Heapify
Trigger the in-place construction engine, initiating bottom-up sift-down passes from index ⌊n/2⌋ − 1 down to 0 to establish the Max-Heap invariant.
04
Measure Discrete Key Comparisons
Read the deterministic comparison count C from the comparator register, verifying that C matches exactly 2n − 2 log₂(n) comparisons, and record the reading.
05
Repeat Across Full Scale Range
Systematically iterate through all 7 power-of-two scales (n = 64, 128, 256, 512, 1024, 2048, and 4096 keys), recording key comparisons and cumulative sift space.
06
Perform OLS Regression & Priority Scheduling Analysis
Plot key comparisons C against cumulative sift interval space x = n − log₂(n), perform Ordinary Least-Squares regression to compute k_heap, and evaluate scheduler throughput.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into asymptotic insight.
Points are your recorded benchmark observations. The regression line validates Binary Heap Dynamics complexity scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
C = k_heap · (n − log₂(n)) + C₀
| # | Array Size (n) (keys) | Heapify Comparisons (C) (comparisons) | Cumulative Sift Interval Space x = n − log₂(n) (intervals) | Key Comparisons y = C (comparisons) | Remove |
|---|
No readings yet. Execute a Array Size benchmark and record it.
Calculate it yourself.
CHECK YOUR UNDERSTANDING