[Coding] Maximum Flow with Lower Bounds (Advanced Network Flow with Capacity Ranges) — Java DSA Pr

════════════════════════════════════════════════════════════ 📂 CATEGORY: Coding Interview 🎯 TOPIC: Maximum Flow with Lower Bounds (Advanced Network Flow with Capacity Ranges) ⚡ FOCUS: Optimal Time & Space Complexity | Clean Java 21 LTS Implementation 📚 PLAYLIST: Coding Interview Preparation ════════════════════════════════════════════════════════════ FAANG Technical Interview Preparation: Maximum Flow with Lower Bounds (Advanced Network Flow with Capacity Ranges) Target Complexity & SLA: Time: O(V·E) | Space: O(V+E) The algorithm first converts each lower‑bound edge (u→v, L, C) into a standard capacity edge of (C‑L) and records node imbalances caused by the mandatory L units of flow. By adding a super source and super sink that satisfy these imbalances, we reduce the feasibility problem to a single max‑flow computation. Dinic's algorithm, which runs in O(V·E) for integral capacities, provides the required performance. If the super source can push its total demand to the super sink, a feasible circulation exists; otherwise the instance is infeasible. After establishing feasibility, we keep the residual network, discard the auxiliary nodes, and run another max‑flow from the original source to sink. The resulting flow, combined with the previously subtracted lower bounds, yields the maximum flow respecting all lower bounds. This two‑phase approach cleanly separates feasibility from optimization, guaranteeing optimality while meeting the target time and space complexities. Chapters & Timeline: 00:00 Overview 00:38 Core Algorithm Implementation 01:24 Edge Cases & Invariants 02:26 Dry-Run & Unit Verification 03:18 Complexity & Interviewer Scoring Rubric Target Playlist: Coding Interview Preparation #CodingInterview #JavaAlgorithms #LeetCode #DataStructures #FAANG