Analyze and optimize a computationally intensive algorithm (such as a graph processing task, numerical simulation, or data transformation process). Provide: 1) Detailed time and space complexity analysis of the original implementation, 2) Identification of specific bottlenecks through profiling analysis, 3) At least three optimization approaches with theoretical justification, 4) Pseudocode or implementation of your optimized solution, 5) Expected performance improvements with mathematical justification, 6) Discussion of trade-offs introduced by optimizations, and 7) Cache-aware considerations for modern hardware architectures.