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Efficient Graph Traversal and Shortest Path Algorithms: BFS, Dijkstra, Prim Implementation

This document covers the implementation of key algorithms in graph theory including Breadth-First Search (BFS), Dijkstra's algorithm, and Prim's algorithm. It provides a structured approach to initializing and traversing graphs, highlighting the use of color coding to track vertex states (WHITE, GRAY, BLACK) and managing distances and parents for shortest paths. Code snippets illustrate the algorithms, emphasizing their utility for solving connectivity and shortest path problems in weighted and unweighted graphs.

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Efficient Graph Traversal and Shortest Path Algorithms: BFS, Dijkstra, Prim Implementation

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  1. Initialize def BFS(G, source):for v in G: color[v] <- WHITEQ <- MAKE-QUEUE()ADD(Q, source) color[source] <- GRAYwhile Q not empty:v <- REMOVE(Q) color[v] <- BLACKfor u adjacent to v:ifcolor[u] == WHITEcolor[u] = GRAYADD(Q, u) Relax

  2. Initialize def DIJKSTRA(G, source):for v in G: color[v] <- WHITE dist[v] <- INF parent[v] <- NIL dist[source] <- 0Q <- MAKE-PRIORITY-QUEUE()ADD(Q, source) color[source] <- GRAYwhile Q not empty:v <- REMOVE(Q) color[v] <- BLACKfor u adjacent to v:if dist[u] > dist[v]+weight(u,v) dist[u] = dist[v]+weight(u,v) parent[u] <- vif color[u] == WHITE: color[u] <- GRAYADD(Q, u) Relax

  3. Initialize def PRIM(G, source):for v in G: color[v] <- WHITE dist[v] <- INF parent[v] <- NIL dist[source] <- 0Q <- MAKE-PRIORITY-QUEUE()ADD(Q, source) color[source] <- GRAYwhile Q not empty:v <- REMOVE(Q) color[v] <- BLACKfor u adjacent to v:if color[u] != BLACK and dist[u] > weight(u,v): dist[u] = weight(u,v) parent[u] <- vif color[u] == WHITE: color[u] <- GRAYADD(Q, u) Relax

  4. SIMPLIFICATION (Dijkstra)

  5. Initialize def DIJKSTRA(G, source):for v in G: color[v] <- WHITE dist[v] <- INF parent[v] <- NIL dist[source] <- 0Q <- MAKE-PRIORITY-QUEUE()ADD(Q, source) color[source] <- GRAYwhile Q not empty:v <- REMOVE(Q) color[v] <- BLACKfor u adjacent to v:if dist[u] > dist[v]+weight(u,v) dist[u] = dist[v]+weight(u,v) parent[u] <- vif color[u] == WHITE: color[u] <- GRAYADD(Q, u) Relax

  6. Initialize def DIJKSTRA(G, source):for v in G: color[v] <- WHITE dist[v] <- INF parent[v] <- NIL dist[source] <- 0Q <- MAKE-PRIORITY-QUEUE()ADD(Q, source) ADD(Q, all v in G) color[source] <- GRAYwhile Q not empty:v <- REMOVE(Q)color[v] <- BLACKfor u adjacent to v:if dist[u] > dist[v]+weight(u,v) dist[u] = dist[v]+weight(u,v) parent[u] <- vif color[u] == WHITE:color[u] <- GRAYADD(Q, u) Relax

  7. Initialize def DIJKSTRA(G, source):for v in G: color[v] <- WHITE dist[v] <- INF parent[v] <- NIL dist[source] <- 0Q <- MAKE-PRIORITY-QUEUE()ADD(Q, all v in G) while Q not empty:v <- REMOVE(Q)for u adjacent to v:if dist[u] > dist[v]+weight(u,v) dist[u] = dist[v]+weight(u,v) parent[u] <- v Relax

  8. SIMPLIFICATION (PRIM)

  9. Initialize def PRIM(G, source):for v in G: color[v] <- WHITE dist[v] <- INF parent[v] <- NIL dist[source] <- 0Q <- MAKE-PRIORITY-QUEUE()ADD(Q, source) color[source] <- GRAYwhile Q not empty:v <- REMOVE(Q) color[v] <- BLACKfor u adjacent to v:if color[u] != BLACK and dist[u] > weight(u,v): dist[u] = weight(u,v) parent[u] <- vif color[u] == WHITE: color[u] <- GRAYADD(Q, u) Relax

  10. Initialize def PRIM(G, source):for v in G: color[v] <- WHITE dist[v] <- INF parent[v] <- NIL dist[source] <- 0Q <- MAKE-PRIORITY-QUEUE()ADD(Q, source) ADD(Q, all v in G) color[source] <- GRAYwhile Q not empty:v <- REMOVE(Q) color[v] <- BLACKfor u adjacent to v:if color[u] != BLACK and dist[u] > weight(u,v): dist[u] = weight(u,v) parent[u] <- vif color[u] == WHITE:color[u] <- GRAYADD(Q, u) Relax

  11. Initialize def PRIM(G, source):for v in G: color[v] <- WHITE dist[v] <- INF parent[v] <- NIL dist[source] <- 0Q <- MAKE-PRIORITY-QUEUE()ADD(Q, all v in G) while Q not empty:v <- REMOVE(Q) color[v] <- BLACKfor u adjacent to v:if color[u] != BLACK and dist[u] > weight(u,v): dist[u] = weight(u,v) parent[u] <- v Relax

  12. BFS utilizing dist[], parent[] fields dist[] keeps track of the level parent[] keep track of how vertex was reached

  13. Initialize def DIJKSTRA(G, source):for v in G: color[v] <- WHITE dist[v] <- INF parent[v] <- NIL dist[source] <- 0Q <- MAKE-QUEUE()ADD(Q, source) color[source] <- GRAYwhile Q not empty:v <- REMOVE(Q) color[v] <- BLACKfor u adjacent to v:ifcolor[u] == WHITE: dist[u] = dist[v]+1 parent[u] <- v color[u] <- GRAYADD(Q, u) Relax

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