From c9dbef501c079cde50f55cb25bee4a5db31c22b2 Mon Sep 17 00:00:00 2001
From: Samarth Kolarkar <samarthkolarkar7777@gmail.com>
Date: Sat, 18 Oct 2025 16:58:55 +0530
Subject: [PATCH 1/2] =?UTF-8?q?Add=20-Dinic=E2=80=99s=20Algorithm?=
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---
 graph_algorithms/dinic_max_flow.r | 388 ++++++++++++++++++++++++++++++
 1 file changed, 388 insertions(+)
 create mode 100644 graph_algorithms/dinic_max_flow.r

diff --git a/graph_algorithms/dinic_max_flow.r b/graph_algorithms/dinic_max_flow.r
new file mode 100644
index 00000000..6a17e87c
--- /dev/null
+++ b/graph_algorithms/dinic_max_flow.r
@@ -0,0 +1,388 @@
+# Dinic's Algorithm for Maximum Flow
+#
+# Dinic's algorithm finds the maximum flow in a flow network from source to sink.
+# It uses level graphs and blocking flows to achieve better time complexity than
+# Ford-Fulkerson. The algorithm repeatedly finds shortest augmenting paths using BFS
+# and then uses DFS to send flow along multiple paths in each iteration.
+#
+# Time Complexity: O(V^2 * E) general case, O(E * sqrt(V)) for unit capacity networks
+# Space Complexity: O(V + E) for adjacency list and level array
+#
+# Applications:
+# - Network routing and traffic optimization
+# - Bipartite matching problems
+# - Image segmentation
+# - Airline scheduling
+# - Project selection and resource allocation
+
+#' Create an empty flow network
+#' @param n: Number of vertices
+#' @return: Flow network structure with adjacency list and capacity matrix
+create_flow_network <- function(n) {
+  list(
+    n = n,
+    graph = vector("list", n),  # Adjacency list
+    capacity = matrix(0, nrow = n, ncol = n),  # Capacity matrix
+    flow = matrix(0, nrow = n, ncol = n)  # Flow matrix
+  )
+}
+
+#' Add edge to flow network
+#' @param network: Flow network structure
+#' @param from: Source vertex (1-indexed)
+#' @param to: Destination vertex (1-indexed)
+#' @param capacity: Edge capacity
+#' @return: Updated network
+add_edge <- function(network, from, to, capacity) {
+  # Add forward edge
+  network$graph[[from]] <- c(network$graph[[from]], to)
+  network$capacity[from, to] <- network$capacity[from, to] + capacity
+  
+  # Add reverse edge for residual graph (with 0 capacity initially)
+  if (!(from %in% network$graph[[to]])) {
+    network$graph[[to]] <- c(network$graph[[to]], from)
+  }
+  
+  return(network)
+}
+
+#' Build level graph using BFS
+#' @param network: Flow network
+#' @param source: Source vertex
+#' @param sink: Sink vertex
+#' @return: Level array (-1 if vertex not reachable)
+bfs_level_graph <- function(network, source, sink) {
+  n <- network$n
+  level <- rep(-1, n)
+  level[source] <- 0
+  
+  queue <- c(source)
+  
+  while (length(queue) > 0) {
+    u <- queue[1]
+    queue <- queue[-1]
+    
+    for (v in network$graph[[u]]) {
+      # Check if edge has residual capacity and v is not visited
+      residual_capacity <- network$capacity[u, v] - network$flow[u, v]
+      
+      if (level[v] == -1 && residual_capacity > 0) {
+        level[v] <- level[u] + 1
+        queue <- c(queue, v)
+      }
+    }
+  }
+  
+  return(level)
+}
+
+#' Send flow using DFS (blocking flow)
+#' @param network: Flow network
+#' @param u: Current vertex
+#' @param sink: Sink vertex
+#' @param level: Level array from BFS
+#' @param flow: Flow to push
+#' @param start: Array tracking next edge to try from each vertex
+#' @return: Flow pushed
+dfs_send_flow <- function(network, u, sink, level, flow, start) {
+  # Reached sink
+  if (u == sink) {
+    return(flow)
+  }
+  
+  # Try all edges from current vertex
+  while (start[u] <= length(network$graph[[u]])) {
+    v <- network$graph[[u]][start[u]]
+    residual_capacity <- network$capacity[u, v] - network$flow[u, v]
+    
+    # Check if this edge can be used (level increases and has capacity)
+    if (level[v] == level[u] + 1 && residual_capacity > 0) {
+      # Recursively send flow
+      pushed_flow <- dfs_send_flow(
+        network, 
+        v, 
+        sink, 
+        level, 
+        min(flow, residual_capacity),
+        start
+      )
+      
+      if (pushed_flow > 0) {
+        # Update flow
+        network$flow[u, v] <<- network$flow[u, v] + pushed_flow
+        network$flow[v, u] <<- network$flow[v, u] - pushed_flow
+        return(pushed_flow)
+      }
+    }
+    
+    start[u] <- start[u] + 1
+  }
+  
+  return(0)
+}
+
+#' Dinic's algorithm to find maximum flow
+#' @param network: Flow network structure
+#' @param source: Source vertex (1-indexed)
+#' @param sink: Sink vertex (1-indexed)
+#' @return: List with max_flow value, flow matrix, and flow decomposition
+dinic_max_flow <- function(network, source, sink) {
+  if (source < 1 || source > network$n || sink < 1 || sink > network$n) {
+    stop("Error: Source and sink must be between 1 and n")
+  }
+  
+  if (source == sink) {
+    stop("Error: Source and sink must be different")
+  }
+  
+  max_flow <- 0
+  iterations <- 0
+  
+  # Repeat until no augmenting path exists
+  repeat {
+    iterations <- iterations + 1
+    
+    # Build level graph using BFS
+    level <- bfs_level_graph(network, source, sink)
+    
+    # If sink is not reachable, no more augmenting paths
+    if (level[sink] == -1) {
+      break
+    }
+    
+    # Send flow using DFS until blocking flow is achieved
+    start <- rep(1, network$n)  # Track next edge to try from each vertex
+    
+    repeat {
+      flow <- dfs_send_flow(network, source, sink, level, Inf, start)
+      
+      if (flow == 0) {
+        break
+      }
+      
+      max_flow <- max_flow + flow
+    }
+  }
+  
+  # Find minimum cut
+  level_final <- bfs_level_graph(network, source, sink)
+  source_side <- which(level_final != -1)
+  sink_side <- which(level_final == -1)
+  
+  # Find edges in the minimum cut
+  min_cut_edges <- list()
+  for (u in source_side) {
+    for (v in network$graph[[u]]) {
+      if (v %in% sink_side && network$capacity[u, v] > 0) {
+        min_cut_edges <- c(min_cut_edges, list(list(from = u, to = v, capacity = network$capacity[u, v])))
+      }
+    }
+  }
+  
+  return(list(
+    max_flow = max_flow,
+    flow_matrix = network$flow,
+    iterations = iterations,
+    min_cut_edges = min_cut_edges,
+    source_partition = source_side,
+    sink_partition = sink_side
+  ))
+}
+
+#' Print maximum flow results
+#' @param result: Result from dinic_max_flow
+#' @param network: Original network (optional, for displaying edges)
+print_max_flow <- function(result, network = NULL) {
+  cat("Maximum Flow Result:\n")
+  cat(strrep("=", 60), "\n\n")
+  cat(sprintf("Maximum Flow: %g\n", result$max_flow))
+  cat(sprintf("Iterations: %d\n\n", result$iterations))
+  
+  cat("Minimum Cut Edges:\n")
+  cat(strrep("-", 60), "\n")
+  if (length(result$min_cut_edges) > 0) {
+    cat(sprintf("%-15s %-15s %-15s\n", "From", "To", "Capacity"))
+    cat(strrep("-", 60), "\n")
+    total_cut_capacity <- 0
+    for (edge in result$min_cut_edges) {
+      cat(sprintf("%-15d %-15d %-15g\n", edge$from, edge$to, edge$capacity))
+      total_cut_capacity <- total_cut_capacity + edge$capacity
+    }
+    cat(strrep("-", 60), "\n")
+    cat(sprintf("Total Cut Capacity: %g\n", total_cut_capacity))
+  } else {
+    cat("No cut edges found\n")
+  }
+  
+  cat(sprintf("\nSource partition: {%s}\n", paste(result$source_partition, collapse = ", ")))
+  cat(sprintf("Sink partition: {%s}\n", paste(result$sink_partition, collapse = ", ")))
+  cat("\n")
+}
+
+#' Print flow on edges
+#' @param network: Flow network with computed flows
+print_flow_edges <- function(network) {
+  cat("Flow on Edges:\n")
+  cat(strrep("-", 60), "\n")
+  cat(sprintf("%-10s %-10s %-12s %-12s\n", "From", "To", "Flow", "Capacity"))
+  cat(strrep("-", 60), "\n")
+  
+  for (u in 1:network$n) {
+    for (v in network$graph[[u]]) {
+      if (network$capacity[u, v] > 0 && network$flow[u, v] > 0) {
+        cat(sprintf("%-10d %-10d %-12g %-12g\n", 
+                    u, v, network$flow[u, v], network$capacity[u, v]))
+      }
+    }
+  }
+  cat(strrep("-", 60), "\n\n")
+}
+
+# ========== Example 1: Basic 6-Vertex Network ==========
+
+cat("========== Example 1: Basic 6-Vertex Flow Network ==========\n\n")
+
+# Create network with 6 vertices
+network1 <- create_flow_network(6)
+
+# Add edges (from, to, capacity)
+network1 <- add_edge(network1, 1, 2, 16)
+network1 <- add_edge(network1, 1, 3, 13)
+network1 <- add_edge(network1, 2, 3, 10)
+network1 <- add_edge(network1, 2, 4, 12)
+network1 <- add_edge(network1, 3, 2, 4)
+network1 <- add_edge(network1, 3, 5, 14)
+network1 <- add_edge(network1, 4, 3, 9)
+network1 <- add_edge(network1, 4, 6, 20)
+network1 <- add_edge(network1, 5, 4, 7)
+network1 <- add_edge(network1, 5, 6, 4)
+
+cat("Network edges:\n")
+cat("1 -> 2 (16), 1 -> 3 (13)\n")
+cat("2 -> 3 (10), 2 -> 4 (12)\n")
+cat("3 -> 2 (4), 3 -> 5 (14)\n")
+cat("4 -> 3 (9), 4 -> 6 (20)\n")
+cat("5 -> 4 (7), 5 -> 6 (4)\n\n")
+
+# Find maximum flow from vertex 1 to vertex 6
+result1 <- dinic_max_flow(network1, source = 1, sink = 6)
+print_max_flow(result1, network1)
+print_flow_edges(network1)
+
+# ========== Example 2: Simple Network ==========
+
+cat("========== Example 2: Simple 4-Vertex Network ==========\n\n")
+
+network2 <- create_flow_network(4)
+network2 <- add_edge(network2, 1, 2, 10)
+network2 <- add_edge(network2, 1, 3, 10)
+network2 <- add_edge(network2, 2, 3, 2)
+network2 <- add_edge(network2, 2, 4, 4)
+network2 <- add_edge(network2, 3, 4, 9)
+
+cat("Network edges:\n")
+cat("1 -> 2 (10), 1 -> 3 (10)\n")
+cat("2 -> 3 (2), 2 -> 4 (4)\n")
+cat("3 -> 4 (9)\n\n")
+
+result2 <- dinic_max_flow(network2, source = 1, sink = 4)
+print_max_flow(result2, network2)
+print_flow_edges(network2)
+
+# ========== Example 3: Bipartite Matching ==========
+
+cat("========== Example 3: Bipartite Matching (5 workers, 4 jobs) ==========\n\n")
+
+# Create bipartite graph: source(1) -> workers(2-6) -> jobs(7-10) -> sink(11)
+network3 <- create_flow_network(11)
+
+# Source to workers (capacity 1 each)
+for (i in 2:6) {
+  network3 <- add_edge(network3, 1, i, 1)
+}
+
+# Workers to jobs (based on which jobs they can do)
+network3 <- add_edge(network3, 2, 7, 1)   # Worker 1 can do Job 1
+network3 <- add_edge(network3, 2, 8, 1)   # Worker 1 can do Job 2
+network3 <- add_edge(network3, 3, 7, 1)   # Worker 2 can do Job 1
+network3 <- add_edge(network3, 3, 9, 1)   # Worker 2 can do Job 3
+network3 <- add_edge(network3, 4, 8, 1)   # Worker 3 can do Job 2
+network3 <- add_edge(network3, 4, 10, 1)  # Worker 3 can do Job 4
+network3 <- add_edge(network3, 5, 9, 1)   # Worker 4 can do Job 3
+network3 <- add_edge(network3, 6, 10, 1)  # Worker 5 can do Job 4
+
+# Jobs to sink (capacity 1 each)
+for (i in 7:10) {
+  network3 <- add_edge(network3, i, 11, 1)
+}
+
+cat("Bipartite matching problem:\n")
+cat("Workers: 1-5, Jobs: 1-4\n")
+cat("Maximum number of worker-job assignments:\n\n")
+
+result3 <- dinic_max_flow(network3, source = 1, sink = 11)
+cat(sprintf("Maximum Matching: %d\n\n", result3$max_flow))
+
+# Show assignments
+cat("Worker-Job Assignments:\n")
+for (worker in 2:6) {
+  for (job in 7:10) {
+    if (network3$flow[worker, job] > 0) {
+      cat(sprintf("Worker %d -> Job %d\n", worker - 1, job - 6))
+    }
+  }
+}
+cat("\n")
+
+# ========== Example 4: Multi-source Multi-sink ==========
+
+cat("========== Example 4: Multi-Source Multi-Sink Problem ==========\n\n")
+
+# Convert to single source/sink by adding super source and super sink
+network4 <- create_flow_network(8)
+
+# Super source (1) to sources (2, 3)
+network4 <- add_edge(network4, 1, 2, 10)
+network4 <- add_edge(network4, 1, 3, 15)
+
+# Internal edges
+network4 <- add_edge(network4, 2, 4, 8)
+network4 <- add_edge(network4, 2, 5, 6)
+network4 <- add_edge(network4, 3, 4, 5)
+network4 <- add_edge(network4, 3, 5, 12)
+network4 <- add_edge(network4, 4, 6, 10)
+network4 <- add_edge(network4, 5, 7, 9)
+
+# Sinks (6, 7) to super sink (8)
+network4 <- add_edge(network4, 6, 8, 15)
+network4 <- add_edge(network4, 7, 8, 12)
+
+cat("Multi-source multi-sink converted to single source/sink\n")
+cat("Sources: 2, 3 | Sinks: 6, 7\n\n")
+
+result4 <- dinic_max_flow(network4, source = 1, sink = 8)
+print_max_flow(result4, network4)
+
+# ========== Algorithm Properties ==========
+
+cat("========== Algorithm Properties ==========\n\n")
+cat("1. Dinic's algorithm guarantees finding maximum flow\n")
+cat("2. Works on directed graphs with non-negative capacities\n")
+cat("3. Faster than Ford-Fulkerson for dense graphs\n")
+cat("4. Maximum flow equals minimum cut capacity (Max-Flow Min-Cut Theorem)\n")
+cat("5. Each iteration increases distance to sink for at least one vertex\n")
+cat("6. Number of iterations is O(V) for unit capacity networks\n\n")
+
+cat("========== Comparison with Other Max Flow Algorithms ==========\n\n")
+cat("Ford-Fulkerson:\n")
+cat("  Time: O(E * max_flow) - can be slow for large flows\n")
+cat("  Method: Find any augmenting path\n\n")
+cat("Edmonds-Karp:\n")
+cat("  Time: O(V * E^2) - specific Ford-Fulkerson with BFS\n")
+cat("  Method: Shortest augmenting path\n\n")
+cat("Dinic's Algorithm:\n")
+cat("  Time: O(V^2 * E), O(E * sqrt(V)) for unit capacities\n")
+cat("  Method: Blocking flows in level graphs\n\n")
+cat("Push-Relabel:\n")
+cat("  Time: O(V^2 * E) or O(V^3) depending on implementation\n")
+cat("  Method: Preflow-push operations\n")

From cefb14f38ad95fab588e30412e6893b373e06a2d Mon Sep 17 00:00:00 2001
From: Samarth Kolarkar <samarthkolarkar7777@gmail.com>
Date: Sun, 19 Oct 2025 12:40:14 +0530
Subject: [PATCH 2/2]  Improvements in Dinics algo

---
 graph_algorithms/dinic_max_flow.r | 62 ++++++++++++++++++-------------
 1 file changed, 36 insertions(+), 26 deletions(-)

diff --git a/graph_algorithms/dinic_max_flow.r b/graph_algorithms/dinic_max_flow.r
index 6a17e87c..8e3cfe51 100644
--- a/graph_algorithms/dinic_max_flow.r
+++ b/graph_algorithms/dinic_max_flow.r
@@ -15,16 +15,20 @@
 # - Airline scheduling
 # - Project selection and resource allocation
 
+# Formatting constant
+LINE_WIDTH <- 60
+
 #' Create an empty flow network
 #' @param n: Number of vertices
 #' @return: Flow network structure with adjacency list and capacity matrix
 create_flow_network <- function(n) {
-  list(
-    n = n,
-    graph = vector("list", n),  # Adjacency list
-    capacity = matrix(0, nrow = n, ncol = n),  # Capacity matrix
-    flow = matrix(0, nrow = n, ncol = n)  # Flow matrix
-  )
+  # Use environment to allow in-place (by-reference) mutation without superassignment
+  env <- new.env(parent = emptyenv())
+  env$n <- n
+  env$graph <- vector("list", n)           # Adjacency list
+  env$capacity <- matrix(0, nrow = n, ncol = n)  # Capacity matrix
+  env$flow <- matrix(0, nrow = n, ncol = n)      # Flow matrix
+  return(env)
 }
 
 #' Add edge to flow network
@@ -55,24 +59,30 @@ bfs_level_graph <- function(network, source, sink) {
   n <- network$n
   level <- rep(-1, n)
   level[source] <- 0
-  
-  queue <- c(source)
-  
-  while (length(queue) > 0) {
-    u <- queue[1]
-    queue <- queue[-1]
-    
+
+  # O(1) queue using head/tail indices; BFS enqueues each vertex at most once
+  queue <- integer(n)
+  head <- 1
+  tail <- 1
+  queue[tail] <- source
+  tail <- tail + 1
+
+  while (head < tail) {
+    u <- queue[head]
+    head <- head + 1
+
     for (v in network$graph[[u]]) {
       # Check if edge has residual capacity and v is not visited
       residual_capacity <- network$capacity[u, v] - network$flow[u, v]
-      
+
       if (level[v] == -1 && residual_capacity > 0) {
         level[v] <- level[u] + 1
-        queue <- c(queue, v)
+        queue[tail] <- v
+        tail <- tail + 1
       }
     }
   }
-  
+
   return(level)
 }
 
@@ -108,9 +118,9 @@ dfs_send_flow <- function(network, u, sink, level, flow, start) {
       )
       
       if (pushed_flow > 0) {
-        # Update flow
-        network$flow[u, v] <<- network$flow[u, v] + pushed_flow
-        network$flow[v, u] <<- network$flow[v, u] - pushed_flow
+        # Update flow (environment enables in-place mutation)
+        network$flow[u, v] <- network$flow[u, v] + pushed_flow
+        network$flow[v, u] <- network$flow[v, u] - pushed_flow
         return(pushed_flow)
       }
     }
@@ -194,21 +204,21 @@ dinic_max_flow <- function(network, source, sink) {
 #' @param network: Original network (optional, for displaying edges)
 print_max_flow <- function(result, network = NULL) {
   cat("Maximum Flow Result:\n")
-  cat(strrep("=", 60), "\n\n")
+  cat(strrep("=", LINE_WIDTH), "\n\n")
   cat(sprintf("Maximum Flow: %g\n", result$max_flow))
   cat(sprintf("Iterations: %d\n\n", result$iterations))
   
   cat("Minimum Cut Edges:\n")
-  cat(strrep("-", 60), "\n")
+  cat(strrep("-", LINE_WIDTH), "\n")
   if (length(result$min_cut_edges) > 0) {
     cat(sprintf("%-15s %-15s %-15s\n", "From", "To", "Capacity"))
-    cat(strrep("-", 60), "\n")
+    cat(strrep("-", LINE_WIDTH), "\n")
     total_cut_capacity <- 0
     for (edge in result$min_cut_edges) {
       cat(sprintf("%-15d %-15d %-15g\n", edge$from, edge$to, edge$capacity))
       total_cut_capacity <- total_cut_capacity + edge$capacity
     }
-    cat(strrep("-", 60), "\n")
+    cat(strrep("-", LINE_WIDTH), "\n")
     cat(sprintf("Total Cut Capacity: %g\n", total_cut_capacity))
   } else {
     cat("No cut edges found\n")
@@ -223,9 +233,9 @@ print_max_flow <- function(result, network = NULL) {
 #' @param network: Flow network with computed flows
 print_flow_edges <- function(network) {
   cat("Flow on Edges:\n")
-  cat(strrep("-", 60), "\n")
+  cat(strrep("-", LINE_WIDTH), "\n")
   cat(sprintf("%-10s %-10s %-12s %-12s\n", "From", "To", "Flow", "Capacity"))
-  cat(strrep("-", 60), "\n")
+  cat(strrep("-", LINE_WIDTH), "\n")
   
   for (u in 1:network$n) {
     for (v in network$graph[[u]]) {
@@ -235,7 +245,7 @@ print_flow_edges <- function(network) {
       }
     }
   }
-  cat(strrep("-", 60), "\n\n")
+  cat(strrep("-", LINE_WIDTH), "\n\n")
 }
 
 # ========== Example 1: Basic 6-Vertex Network ==========