Design a URL Shortener

Design a URL shortener that stores 10 billion short URLs, serves 100K redirects/sec globally with sub-5ms p99 server-side on cache hit, supports custom aliases and expiration, and runs click analytics without touching redirect latency. The hard parts: generating unique IDs without cross-region coordination, absorbing viral hot keys, and keeping the cache hit rate honest when traffic drifts away from a Zipf distribution.